Acoustics, Mechanics, Mathematical Flnalqsis

Acoustics, Mechanics, and the Related Topics of Mathematical flnalqsis This page intentionally left blank Proceedings of the International Conference to Celebrate Robert P. Gilbert's 70th Birthday

Acoustics, Mechanics, and the Related Topics of Mathema tical Analqsis

CAES du CNRS, Frejus, France 18 - 22 June 2002

Editor Armand Wirgin Laboratoire de Mecanique et dAcoustique Marseille, France

orld Scientific Jersey London Singapore Hong Kong Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202,1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WCZH 9HE

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ACOUSTICS, MECHANCIS, AND THE. RELATED TOPICS OF MATHEMATICAL ANALYSIS Proceedings of the International Conference to Celebrate Robert P. Gilbert’s 70th Birthday Copyright 0 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, eZectronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-264-X

This book is printed on acid-free paper. Printed in Singapore by Mainland Press PREFACE

The international conference Acoustics, Mechanics, and the Related Topics of Mathematical Analysis (AMRTMA) was held on 18-22 June, 2002 at the Villa Clythia, CAES du CNRS, in Frbjus, France. This interdis- ciplinary meeting was in some respects a smaller-format copy of previous ISAAC conferences. AMRTMA was sponsored jointly by the Rbgion PACA (Provence Alpes CBte d’Azur), the CNRS (Centre National de la Recherche Scientifique) and ISAAC (International Society for Analysis, its Applica- tions and Computation).

The objective of AMRTMA was two-fold. First, and above all, it was the occasion for friends and admirers of Robert P. Gilbert to celebrate his 70th birthday in a convivial atmosphere. Second, the conference was to

Figure 1. Bob Gilbert on the boat to St. Tkopez during the AMRTMA conference provide a means for bringing together scientists from many different fields to discuss the intricacies and usefulness of applied mathematics, notably in the realm of mechanics.

A group of over 60 persons, composed of scholars, scientists, engineers (some with their Spouses/companions and children) from the USA, Rance, Ger- many, Belgium, Italy, Greece, Austria, England, Sweden, Russia, Ukraine, Algeria, Tunisia and China, participated in this event. The quality of the (all half-hour) scientific communications (most of which appear in extenso

V vi in this book) was high and the banquet in honor of R. Gilbert a memorable birthday celebration. As the participants were together (all lodged in the Villa Clythia, a vacation and conference facility of the CNRS) most of the time during the five days of the meeting, they had many occasions (notably during the meals, cocktails, and boat trip to St. Tropez) to socialize as well as exchange ideas and impressions.

I would like to express my gratitude to the Centre National de la Recherche Scientifique, G. Berger, Directrice and E. Brun, Administrateur DkleguB, to the Conseil Wgional Provence-Alpes-C8te d’Azur, M. Hayot, Vice- PrBsident, to the International Society for Analysis, its Applications and Computation, H. Begehr, President of the Board, R.P. Gilbert, Member of the Board, and to the Laboratoire de MBcanique et d’Acoustique, M. Ritous, Directeur for their financial support and/or encouragement.

The Scientific Committee members: A. Ben-Israel , Rutgers Univ., USA, A.-S. Bonnet Ben-Dhia, ENSTA, France, A. Bourgeat , Univ. St. Eti- enne, fiance, J. Buchanan, US Naval Acad., USA, C. Depollier, LAUM, fiance, R.P. Gilbert, Univ. Delaware, USA, G. Hile, Univ. Hawaii, USA, A. Jeffrey, Univ. Newcastle/Tyne, U.K., G. Maugin, LMM/CNRS/UPMC, fiance, F. Nicolosi, Univ. Catania, Italy, A. Panchenko, Penn State Univ., USA, A. Wirgin, LMA/CNRS, France, and Y. Xu, Univ. Tennessee, USA did a great job in selecting the communications and reviewing the papers for the proceedings book.

The Organization Committee comprised of: G. Hile, Univ. Hawdi, USA, C. Tsogka, LMA/CNRS, France and A. Wirgin, LMA/CNRS, France acted efficiently in setting up the conference.

I. Czyz, DBlCgation RBgionale du CNRS, Marseille, France provided precious help on budgetary and financial issues, M. Morano, LMA/CNRS acted efficiently for the secretarial tasks, and E. Ogam, LMA/CNRS helped solve many problems connected with audiovisual presentations and informatics.

My old friend R. Tanteri from Toulouse, France, made the web page that helped draw attention to the conference and A. DBlinibre LMA/CNRS, France installed this page on the LMA web site. The task of designing the logo of the conference was submitted to P. de Sentenac, Paris, France and to V. Quesnel, Rouen, fiance (whose design was finally retained). vii

The success of the conference was largely due to the efforts of the kind and efficient personnel of the Villa Clythia: its Director, M. Chevalier, its Assistant Director, E. Porcu, and all the other members of the staff who made our stay an enjoyable experience.

I wholeheartedly thank these many persons for their precious help.

Next to last, but not least, I thank my wife Nicole and my daughter Zo6 for their encouragements and efficient help at the reception desk of the conference.

As concerns this book, I acknowledge the helpful aid and initiatives of Daniel Cartin and Chelsea Chin from World Scientific.

Armand Wirgin This page intentionally left blank Contents

Preface V A. Wirgin

Paen R.P. Gilbert 1 R. Carroll

About Bob Gilbert 4 A. Wirgin 8 Representation formulas in Clifford analysis H. Begehr

Imaging methods in random media 14 J. Berryman, L. Borcea, G. Papanicolaou and C. Tsogka

Resonances of an elastic plate in a duct, in the presence of a uniform 21 flow AS. Bonnet-Ben Dhia and J.-F. Mercier

First order asymptotic modeling of a nuclear waste repository 28 A. Bourgeat, I. Boursier, 0. Gipoulowc and E. Marusic-Paloka

Exact axisymmetric solution for temperature-dependent compressible 34 Navier-Stokes Equations P. Broadridge and T.M. Barrett

Recovery of the poroelastic parameters of cancellous bone using low 41 frequency acoustic interrogation J.L. Buchanan, R.P. Gilbert and K. Khashanah

Mathematical model of the interaction problem between 48 electromagnetic field and elastic body F. Cakoni and G.C. Hsiao

Bore evolution in inhomogeneous channels 55 J. G. Caputo and Y.A. Stepanyants

An inverse spectral problem for a Schrodinger operator with an 64 unbounded potential L. Cardoulis, M. Cristofol and P. Gaitan

ix X

Trapping regions for discontinuously coupled dynamic systems 71 S. Carl and J. W. Jerome

Differential calculi 78 R. Carroll

Reconstruction problem for a periodic boundary between two media 85 J. Chandezon, A. Ye. Poyedinchuk and N.P. Yashina.

A note on generalized Cesho operators 92 D.C. Chang, R. Gilbert and G. Wang

On the boundedness of functions from an anisotropic weighted space 100 satisfying some integral inequalities P. Cianci

Homogenizing a flow of an incompressible inviscid fluid through 108 an elastic porous media T. Clopeau and A. Mickelic

Approximation of a dynamic unilateral contact problem for a cracked 116 viscoelastic body M Cocou and G. Scarella

Principles of signal based ray tracing for 2D and 3D complex tectonics 123 P. Cristini and E. De Bazelaire

Regularity up to the boundary for a class of solutions of a functional- 130 differential system S. D 'Asero

On the Hardy spaces of harmonic and monogenic functions in the unit 137 ball of R~+~ R. Delanghe

Time domain wave equations for lossy media obeying a frequency 143 power law: application to the porous materials Z.E.A. Fellah, S. Berger, W. Lauriks and C. Depollier xi

A model for porous ductile viscoplastic solids including void shape 150 effects L. Flandi and J. B. Leblond

Acoustic wave propagation in a composite of two different poroelastic 157 materials with a very rough periodic interface: a homogenization approach R. Gilbert and MJ. Ou

Effective acoustic equations for a nonconsolidated medium with 164 microstructure R.P. Gilbert andA. Panchenko

A domain decomposition method for the Helmholtz equation in an 171 unbounded waveguide N. Gmati and N. Zrelli

Support function method for inverse obstacle scattering problems 178 S. Gutman andA.G. Ramm

Heat polynomial analogs 185 G.N. Hile and A. Stanoyevitch

Blow-up, shock formation, and acceleration waves in hyperelastic 192 media A. Jeffrey

Summability of solutions of Dirichlet problem 199 A. Kovalevsky and F. Nicolosi

On isophonic surfaces 207 R. Magnanini

A survey of pointwise interpolation inequalities for integer and 212 fractional derivatives V. Maz )a and T. Shaposhnikova

Non-uniqueness in connection with methods for the reconstruction of 222 the shape of cylindrical bodies from acoustic scattering data E. Ogam, T. Scotti and A. Wirgin xii

Dispersion identification using the Fourier analysis of resonances in 229 elastic and viscoelastic rods R. Othman, G. Gary, R. Blanc, M.N. Bussac and P. Collet

Application of the likelihood method to the analysis of waves in elastic 236 and viscoelastic rods R. Othman, G. Gary, M.N. Bussac and P. CoIlet

On the controlled evolution of level sets and like methods: the shape 243 and contrast reconstruction C. Ramananjaona, M. Lambert, D. Lesselier and J-P. Zol&o

Recent progress in the theoretical and numerical modeling of thin-layer 25 1 flow L. Schwartz

Seismic response in a city 258 C. Tsogka and A. Wirgin

Transmission of ultrasonic waves in cancellous bone and evaluation of 265 osteoporosis Y. xu

Hadamard singular integral equations for the Stokes problem and 272 Hermite wavelets L. Zhu and W. Lin

List of Communications 28 1

Author Index 285 PAEN R.P. GILBERT

ROBERT CARROLL University of Illinois, Urbana, IL 61801 USA Email: rcarrollOrnath.uiuc. edu

This is the talk given during the banquet dinner in honor of R.P. Gilbert on the occasion of his 70th birthday.

1. Propaganda I welcome this occasion to say a few words about Bob Gilbert. This is not just some garden variety guy so extensive comments are necessary in order to appreciate his magnitude "cornme une sorte de gros legume mais avec des qualit& humaines admirables". He refers to himself as an applied ana- lyst and I think this should be amplified to say that he has a deep physical intuition coupled with enormous mathematical insight, enabling him to formulate and solve many problems of genuine interest in a real technological world. It is a gift. His work is based on function theoretic methods applied to PDE and includes applications to inverse problems, plasto-elasticity, homogenization, hemivariational inequalities, flow of viscous fluids, ocean acoustics and search procedures, etc. Specific items are:

0 (A) 17 monographs, texts, and proceedings alone or with coau- thors, Begehr, Ben Israel, Buchanan, Colton, Howard, Kajiwara, Koepf, R. Newton, Panagiotopoulos, Pardalos, Weinacht, Wirgin, and Xu. 0 Over 260 papers, some joint, in many prestigious journals or proceedings, in mathematics and applied mathematics 0 Founding editor of journals Applicable Analysis and Complex Varz- ables; founding president of ISAAC, and on the editorial board of a number of journals and book series 0 (1) Two times Preis Trager of Alexander von Humboldt Stiftung (2) Fellow Deutsche Forschungsgemeinschaft (3) British Science Council Research Award at Oxford (4) Visiting appointments at Freie Univ. Berlin, Oxford, Univ. GIasgow, Univ. Dortmund, Hahn-Meitner Institut Berlin, Tech.

1 2

Univ. Danmark, Univ. Karlsruhe, Guangzhou Univ., China, Natl. Auton. Univ. Mexico City, Univ. Jean Monnet, St. Etienne, France 0 18 doctoral students at Maryland, Indiana, Delaware, and Firenze 0 Organizer of many important international conferences 0 Funded since 1962 by various agencies in USA and abroad.

2. Personal

0 (B)I've known Bob since 1970 and in my opinion he is one of the most important applied mathematicians of his generation. He is able to develop deep results using a rich combination of hard analysis, function theoretic methods, and functional analysis, com- plemented by a skillful use of computers. He is perhaps the world's primary authority in certain areas such as function theoretic methods in elliptic PDE and on scattering and detection problems in shallow oceans. In his capacity as editor he has contibuted greatly to the dissemination of important material in pure and applied mathematics and helped numerous scientists to find publication for their work. 0 (C)I quote here (more or less) from a book review I made for the Bulletin AMS of Vol. 1 of the book Trunsformations, Runs- mutations, and Kernel Functions by H. Begehr and R. Gilbert, Longman, 1992. "One of the Leitmotive for the application of function theoretic ideas in elliptic PDE is, of course, the possibility of exploiting various kernel functions and integral formulas that arise naturally from ellipticity... The function theoretic methods described in this book usually retain the feeling of classical PDE (i.e., calculus), while the functional analytic approach often looks more like operator theory... One of the nice features of the book is frequently to combine function theoretic methods and functional analytic techniques in various places to make the best use of both worlds and exhibit how a productive interaction can be achieved... This is one of the best examples I have ever seen of such interaction in solving problems in PDE... The book illustrates very well what rigorous applied mathematics can accomplish.. ." 0 (D) My own relations with Bob were perhaps indirectly influ- enced by his stay at Univ. Maryland and his interaction there with Alexander Weinstein, who was my thesis advisor. In any event I became an associate editor of AA in 1970 and found in Bob a sym- pathetic spirit. He was courageous enough to have his own ideas 3

and good enough to irritate or threaten various inflated egos in the community. This is not always a profitable nor easy role to play and there are penalties attached of which one need not speak. I found some of Bob’s work to be very interesting (as indicated above) and especially the nature of the now labeled Bergman-Gilbert operator as a transmutation (on which I wrote a paper). Peter Lax once re- marked that applied mathematics is sort of like mathematics only much harder. It has always appeared to me to be very difficult to fit the round pegs of symmetry in mathematical physics into the square holes of technology; hence I am frequently amazed when any success at all is achieved in applied mathematics and Bob’s achievements in this direction are extremely impressive. In addition to Bob’s professional expertise he is an enjoyable and passionate social being with good humor and good taste in literature and music. He also is an accomplished cook (French, German, and Chinese - not at the same time) and sails on the ocean (not immediately after eating I hope); perhaps this gives him insight into ocean structure and motivates the search for underwater objects. His daughter Jennifer seems to share his adventur- ous spirit and they have taken up ocean kayaking - which sounds almost as scary as playing violin-cello duets (which they also do). As to humor I remember once at a conference he and Jerry Goldstein spent the better part of an hour regaling each other with choice jokes (some quite good but I have unfortunately forgotten all of them). He has traveled and read extensively and plays the cello (as indicated above). He also was a very serious amateur ballet photographer and comes from a line of graphic artists on his mother’s side. Generally he makes things happen; this draws some people to him and perhaps scares others away. I believe he has accomplished more in generating and promoting mathematics, applied mathematics, and edu- cation than all but a few contemporaries. Thus he is a mover and shaker of sorts and in fact he has been responsible for generating an enormous amount of activity in applied mathematics at an international level. For this many of us are indebted to him and one hopes that the legacy will be perpetuated via further activities of ISAAC for example.

Therefore I join with others here in wishing Bob, his charming wife Nancy, and his lovely daughter Jennifer, salud, dinero, y amor, y el tiempo para gustarlos. ABOUT BOB GILBERT

ARMAND WIRGIN Laboratoire de Me’canique et d’Acoustique, UPR 7051 du CNRS, 31 Chemin Joseph Aiguier, 13402 Marseille cedes 20, fiance E-mail: ws’rginOlma.cnrs-mrs.fr

A portrait is sketched of Bob Gilbert on the occasion of his 70th birthday

1. Prelude Analysis is a branch of both mathematics and psychology. I am not an expert in either of these fields, but was happy to take the initiative of cel- ebrating the 70th birthday of the mathematician Robert Gilbert together with the participants of a meeting entitled Acoustics, Mechanics, and the Related Topics of Mathematical Analysis (AMRTMA for short) and will, in the following lines, attempt to sketch a psychological portrait of the man Bob Gilbert.

The AMRTMA birthday party took place in Europe which is where his (and my) parents were born. fiance (i.e., the country in which AMRTMA was held) plays a special role in Bob’s fantasies because of the singer Patricia Kass, and more tritely, on account of its food and wine. Bob also occasion- ally works with some frenchmen (Bourgeat) or foreigners (both naturalized and unnaturalized) who live in France (Mikelic, Panasenko, Wirgin). Once in a while, Bob flirts with the idea of moving to France beforelafter his retirement, but the prospect of making his enemies in the USA happy has deterred him from this project.

2. Childhood Bob and I were brought up in New York City, he in Brooklyn and I in Manhattan. In the nineteen fourties and fifties, youngsters in NYC whose ambitions were other than becoming football or baseball stars, gangsters or politicians, often went to one of three high schools: Bronx Science, Stuyvesant and Brooklyn Tech. Bob went to Brooklyn Tech and I myself

4 5 to Stuyvesant. Both of us found this to be a stimulating experience. Our college training also was similar in that we both were not majors in mathematics, but Bob (not I) later switched to mathematics. Having both been inhabitants of the NYC melting pot, we swam, although never together to my knowledge, in the waters around Long Island to cool off during those memorable summers prior to the era of air conditioning.

3. The role of the sea Once we met, which occurred in 1996, we started swimming together in the sea (Mediterranean) off Marseille and Montpellier and in the Atlantic ocean around Lewes (Delaware, where Bob has a summer home). Some short-lived sailing outings completed our common experiences in the sea environment, except one which seems worthy mentioning because of its professional significance (gathering data from an array of sensors (such as the eyes and ears), essential to solving inverse problems). Bob and I both noted that most, if not all, men in the USA wear boxer bathing suits, whereas many men in France wear a tighter variety of swimsuits. Although we don’t claim any correlation, we also observed that many French women wear only one half of two-piece bathing suits whereas most American women wear the totality of one-piece swimsuits. Processing this data, so as to reveal the motivations of these sets of individuals, has turned out to be a task beyond our analytic capabilities. But, with the complicity of Jim Buchanan and Steve Xu, we have undertaken the inversion of other pieces of sea-related data, such as would be recorded by floating or immersed hydrophones ”lis- tening” for swimmers (fishes or humans in any kind of bathing apparel), mines, boats and submarines, or remote sensing the physical properties of the water column.

4. From seawater to sand My sensorial and professional experiences with Bob were essentially limited to seawater, but Bob was interested, even before our first encounter, in sand, especially the water-saturated variety which can be found in the highest-lying layers of the seabed. Together with Buchanan, Xu, Lin, and Hackl, he studied how sound propagates in and above the seabed overlain by either a shallow or deep sea. Later, due to a certain uneasiness about the way Biot had derived his famous equations, Bob sought, together with Bourgeat, Mikelic and Panasenko, to put these equations on a firmer mathematical foundation, appealing as it were, to a device well-known to him in connection with the making of mayonnaise and ailloli: homogenization. 6

5. From sand to bones One day, during the second part of Bob’s career, for some mysterious reason the leaders behind the so-called Iron Curtain called it quits, notably as concerns waging the cold war against such countries as the USA, Great Britain, and fiance, whose navies were actively engaged until then in countering the ”threat to world peace”. For this reason, military, and in particular Navy, research funding in the ”western world” was cut drastically, and interest in ocean acoustics (and sea sediments) waned considerably. Bob, Jim, Steve and myself decided that bones could be an interesting alternative to sand as a poroelastic object of interest for revealing its constitutive properties by means of sound. Bob, Jim and Steve are now actively engaged in this field and hoping to give some clues to the medicd community as to how to obtain an early diagnosis of osteporosis. Bob even thinks that the homogenization techniques he brings to bear on this problem will be useful in characterizing anomalies in other types of tissue.

6. From bones to snow Bob’s young neighbor in Lewes, who also is one of the sailing partners of Bob’s daughter, is interested in avalanches and one day spoke to Bob of the related problems of how to characterize the snow cover to prevent such ”natural” events and find buried victims in the fluffy stuff. Snow, like underwater sand sediments and bones, is a more-or-less fluid saturated porous medium with a solid frame, so that it should be possible to analyze it by the same mathematical tools as sand and bones, notably as concerns the way it affects sound propagation as a function of its physical state. Bob is, of course, actively engaged in solving this problem.

7. Back to childhood Kids in the USA (such as Bob and myself when we were young) with euro- pean parents, had the reputation of being troublemakers, perhaps because their parents had been the cause or victims of world-war 1 and/or world- war 2 and/or other gang wars. This has produced at least two classes of grown-ups: those who want to be mainstream and those who want to live up to their reputation. I believe Bob is predominantly of the second variety (this is undoubtedly a source of our friendship). Being a troublemaker often is thoroughly unproductive. Bob’s itinerary provides a striking counter example. He is often unsatisfied, a man with great curiosity and drive, perpetually in search of tough problems to cope with. He has an aver- sion for (other-than-his-own) authority, especially of the type that rests on 7 incompetence, and is outspoken about this. He likes dogs (especially his own), underdogs, Leicas and ginger (in his culinary creations). In short, Bob Gilbert is one of those complex personalities we need more of to make life (and mathematics) enjoyable. REPRESENTATION FORMULAS IN CLIFFORD ANALYSIS

H. BEGEHR Beie Uniuersitat Berlin, I. Mathematisches Institut, Arnamallee 3, 14195 Berlin E-mai1:begehrOmath.f-berlin.de

Integral respresentation formulas of Cauchy Pompeiu type in Clifford analysis were developed in l. Higher order representations can be found from first order ones through iteration. The formulas in which are related to powen of the Laplacian are valid for any power only in the case of odd dimensional spaces. For even dimensions these formulas hold only for small powers not exceeding half of the dimension. Here the missing formulas are presented.

1. Introduction Introducing a multiplication in the Eucledian space Rn with orthonormal basis {e, : 0 5 Y 5 n - 1) via eo = 1 , epe, = -e,e, , ePe, = -1 , for 1 5 p < v 5 n - 1 leads to the Clifford algebra G over C with the basic elements

e@=l,eA=e,, ...e,, forA = {a1,02,..., ~b}, 15 al< ... < ~lc

dz = = 2 -n , 1z12=1z12 = 22 , Izla=lzIQ 8 = Q IzI~-~z ,

8 9

a(P + zk)= (Tk + zk)a= 2kZk-' , a(Z IZ I-") = (F Izl-")a = 0 . The last relation shows that F I z I-" is a fundamental solution to the Dirac operator leading to a Cauchy type kernel.

2. Cauchy Pompeiu representation formulas From the Gaufi theorem first order Cauchy Pompeiu representations follow, see '. Combining them leads to higher order representation formulas on one hand for powers of the Dirac and for powers of its conjugate operator on the other hand for powers of the Laplace operator. While the formulas for the powers of the Dirac operator and those for its conjugate holds without restrictions this is not true in general for the Laplace operator. In it is proved that for w E C2k((o;@In) rlC2"' (a;@n) where D c R" is a regular domain

u=l This representation obviously holds for any k if n is odd. But for even n it is valid only for 0 5 2k < n . Applying formula (1) for m = 1 in the case n = 2m to Am-lw and observing

log I? - z 12 22m-1 (-l)m-l(m - 1)!2

the formula 10

w(2) =

follows. Treating Amw as Am-'w before and applying induction leads to the general formula. Theorem 1 For 0 5 k and a regular domain D c R2m any w E ~2(m+"((~;G~) n ~2(m+k)-l (D,Gm) -. can be represented bgl 11

The basic formula for proving (3) is

where the two terms 4 of boundary integrals form a harmonic function in < , i.e. 4(z7<)d& = o . Theorem 2 For 1 5 m the weak singular kernel functions

I 12(k-m)+l 7n=2m-1, lsk, 2"-'(k - l)!II;=1(2~ - 2m + 1)~2m-1

7n=2m,l

,n=2m7k=m, (4)

I n=2m7 m+l

Theorem 3 For f E Ll(D;@,)and regular domain D C Rn

Tk,nf(z):= JKk,n(C-z)f(C)dU(<) (5) D provides a particular solution to Akw = f in D satisfying Akw = 0 in W\D. As can be shown

so that

and, see 6, AkTk,,f = f in D . Remark The boundary integrals in (1)-(3) form proper polyanalytic functions of respective orders as can be seen from the properties of the kernel functions (4). These representations are in general not appropriate to some related boundary value problems. Only if the solution to such a problem does exist it can be represented by (1)-(3) where the boundary values of the lower order terms are inserted in the boundary integrals. However, the representation

= Pk,n -k Tk,nf with polyharmonic @,-valued pk,n of order k in Rn can be used to transform differential equations for &-valued functions in Rn with leading term Ah into a singular integral equation. For this purpose the properties of the weakly singular integral operators have to be investigated, in particular those of all their k-th order derivatives. For the case n = 2 which is excluded in the above considerations the respective representation formulas are developed in 2. 13

References 1. H. Begehr: Iterated integral operators in Clifford analysis. ZAA 18 (1999), 361-377. 2. H. Begehr, G. N. Hile: A hierarchy of integral operators. Rocky Mount. J. Math. 27 (1997), 669-706. 3. F. Brackx, R. Delanghe, F. Sommen: Clifford analysis. Pitman, London, 1982. 4. R. P. Gilbert, J. L. Buchanan: First order elliptic systems: A function theoretic approach. Acad. Press, New York, 1983. 5. K. Giirlebeck, W. SproBig: Quaternionic and Clifford calculus for physicists and engeneers. John Wiley, Chichester, 1997. 6. G. H.Hile: Hypercomplex function theory applied to partial differential equations. Ph. D. thesis, Indiana Univ., Bloomington, Indiana, 1972. 7. V. V. Kravchenko, M. V. Shapiro: Integral representations for spatial models of mathematics1 physics. Addison Wesley Longman, Harlow, 1996. 8. E. Obolashvili: Partial differential equations in Clifford analysis. Addison Wesley Longman, Harlow, 1998. 9. Z.-Y. Xu: A function theory for the operator D - A. Complex Variables, Theory Appl. 16 (1991), 27-42. IMAGING METHODS IN RANDOM MEDIA

JAMES G. BERRYMAN University of Californaa, Lawrence Livermore National Laboratory, P.0. Box 808 L-200,Livermore, CA 94551-9900, USA, E-mail: berrymanl OIlnl.gov

LILIANA BORCEA Computational and Applied Mathematics, Rice University, Houston, TX ‘7’7005-1892, USA, E-mail: borceaOcaam.rice.edu

GEORGE C. PAPANICOLAOU Department of Mathematics, Stanford University, Stanford, CA 94305, USA, E-mail: papanicoOmath. stanford.edu

CHRYSOULA TSOGKA CNRS/LMA, 31 Chemin Joseph Aiguier, 13402 Marseille Cedex 20, FRANCE, E-mail: tsogtaOlma.cnrs-mrs.fr

Detection and localization of targets embedded in random media is performed by analysing array data of the scattered field. As the random medium is a source of scattered energy we assume that the targets are more reflective than the background fluctuations so that a clear distinction can be made between targets and background scatterers. We show that the key to successful imaging is finding statistically stable functionals which provide estimates of scatterer locations.

1. Introduction Imaging in ultrasonics is closely related to recent studies of time-reversal acoustics8>6. The work of Prada and Fink and Prada et al. on the D.O.R.T. method (diagonalization of the time reversal operator) has clari- fied the connection between scatterers and the eigenfunctions of the time- reversal operator. After decomposing the array response matrix using eigenfunctions, we can either use the eigenfunctions to refocus acoustic energy back onto the scatterers, or we can use them to form an image of the scatterers’ spatial distribution. Both of these applications are relatively straightforward when the background medium is homogeneous ’. But, if the background medium is heterogeneous, several difficulties arise. Our

14 15 focus in this paper concentrates on the difficulties introduced by spatial heterogeneities of the acoustic medium, and on what can be done with acoustic array data to achieve reliable images of scatterers in such media. There have been many methods of estimating target location using acoustic array data. One of the most popular is matched-field (MF) processing 33435. We will be discussing here necessary modifications of the MF method, since the randomness we consider has a different character than that usually envisioned in traditional analyses of acoustic array data, because it comes from multipathing that is generated by the random medium. In the first section, we briefly present the problem to be studied. Section 3 focuses on the standard matched-field functional in frequency domain. This method does not provide statistically stable results and, therefore, is not useful for imaging in random media with strong multipathing. Section 4 then shows how the same objective functional may be transformed into the time domain in order to produce statistically stable and, therefore, useful images that localize the target cross-range in a satisfactory manner. Section 5 then goes further to show how range information may be obtained from the time-domain arrival data after careful processing and subsequent averaging of multiple copies of the pertinent singular vectors contained in the multistatic array data. Our conclusions are summarized in Section 6.

2. Problem Statement and Notation A linear array composed by N transducers, located at xp,for p = 1,.. . ,N, probes an unknown acoustic medium containing M (M 5 N) small scatterers by emitting pulses and recording the back-scattered echos. We call the resulting data set response (or transfer) matrix P(t)= (PPP(t)),where p and q range over all the array elements. Our goal is to detect and localize the M targets in the random medium. We assume here that the response matrix in the frequency domain p(w) is symmetric (but not Hermitian), and this is consistent with our simulations. All of our analysis nevertheless carries over to the non-symmetric case. We introduce the singular value decomposition (SVD) of the response matrix,

F(w) = fi(U>C(W)P(W), (1) where the singular vectors eT(w),T = 1, . . . ,N of p(w) are the columns of fi(w) and the singular values cp.(w) of p(w) form the diagonal matrix C(w). We denote by &(y",w) the vector observed at the array for a source located at ys in a deterministic medium (ie., a homogeneous medium with the averaged velocity of the random medium ca). In our simulations, ~0 16 is constant but it could vary in space, assuming prior knowledge of the environment. Our simulations assume that X 5 C << a << L, where X is the central wavelength, C is the correlation length of the inhomogeneity, a is the array aperture, and L is the distance to the targets from the array. This is the regime where multipathing, or multiple scattering, is significant even when the standard deviation of sound speed fluctuations is only a few percent. All the formulas are presented in their general form in terms of Green's functions. Thus, these formulas are valid either in 2D or in 3D. Due to the high cost of numerically simulating wave propagation in a random medium, with significant multipathing, we have only done 2D simulations up to now. We solve the wave equation in 2D with a numerical method based on the discretization of the mixed velocity-pressure formulation for acoustics. For the spatial discretization we use a new finite element method 2, which is compatible with mass-lumping techniques (ie., it leads to explicit time discretization schemes) and for the time discretization we use a centered second order finite difference scheme. In the numerical simulations, we have statistically homogeneous Gaussian random velocity fields generated using a random Fourier series, with constant mean = 1.5km/s, and Gaussian correlation function having correlation length C = 0.3mm and standard deviation ranging from 1% to 5%. The probing pulse is a Rcker with central frequency v = 3MHz, the carrier wavelength is X = 0.5mm and the aperture of the array is a = 2.5mm. The targets, which are soft scatterers, are modeled by small squares (the size of a small one is X/30 x X/30, while the size of a larger one is X/15 x X/15).

3. Matched field - central frequency We present here a well-known frequency-domain imaging method, the matched-field processing 31475. We compute GEL (y") and display the objective functional RMF(Y")(see 2) for a discrete set of points y" in the target domain. Examples of MF processing are displayed in Fig. 1.

When the background medium is homogeneous, the significant singular vectors are linear combinations of &(y$,u),where y$ are the target locations and the MF functional has local maxima at the target locations. But in the case of random media, as the random fluctuations in the velocity increase, there are false peaks and the functional may not peak at the targets 17

9 = 0%. M.F. * 0% s rn 2.5% M.F. m 4.38%

4 4

2 2

0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 s I4.134%~ M.F. = 8.38% s m 6.96%, M.F.I 12.05%

-0 2 4 6 0 10 12 wO 2 4 6 8 10 12

Figure 1. The Matched Field central frequency estimate (2) in random media. The exact location of the targets is denoted by a star. The standard deviation s and maximum fluctuations (M.F.)are indicated on the top of each view. The horizontal axis is the range in rnm and the vertical axis is the cross-range in mrn. at all. When the realization of the random medium is changed, the images change also, which is what we call “lack of statistical stability”.

4. Matched field in time We transform here MF into the time-domain to take advantage of the statistical stability that can be gained this way. We compute

Since the factor multiplying e-wt in the integrand is real and non-negative, this integral takes its maximum value for t = 0. We thus display the following objective functional for points y8 in the target domain,

Examples of MF processing in the time domain are displayed in Fig. 2. Cross-range results are now dramatically improved. Range information is still not to be found here, but the statistical stability of the “comet tails” is now easily observed. The images shown are for specific realizations, but the results do not change significantly when the underlying realization of the random medium is changed. This fact has been repeatedly shown in our simulations, and is the main characteristic of statistically stable methods. 18

s I095, M F. m 0% s I 253%, NF - 4.38%

4 4

2 2

0 0 2 4 6 8 10 12 '0 2 4 6 8 10 12 s 4 e.3m = sa,M.F. . S I 6 96%. M.F I 12.05%

4 4

2 2

0 '0 '0 2 4 6 8 10 12 0 2 4 6 81012

Figure 2. The Matched Field time estimate (4) of the location of two targets in random media. The exact location of the target is denoted by the green star. The standard deviation s and maximum fluctuations (M.F.) are indicated on the top of each view. The horizontal axis is the range in mm and the vertical axis is the cross-range in mm.

5. Time domain processing and range estimation methods To complete the localization of the targets, we also need an estimate of range. Good range estimates can be obtained in near field either from amplitude move-out information or from arrival time information. In the far field, only the arrival time information is useful, and we will concentrate on arrival times in the present analysis.

5.1. Matched field in time combined with times from avemged singular vectors We would like to use the singular vectors 6j (w) to estimate the travel times from target j to the array. Note though that the singular vectors 6j(w) which are normalized (ll6j(w)))= 1) carry an arbitrary, frequency dependent, phase. Because of this Uj(t) look incoherent in the time domain. We can, however, calculate N, coherent in time, versions of singular vectors by projecting the columns of the response matrix onto them

6y(w)= [6j(W)w)(W)] Q(w), p = 1,.. . ,N, j = 1,. . . , M. (5)

Here e(P) is the pth column of the response matrix p(w).Clearly 6y)(w) are singular vectors of p(w) and carry the phase of its pth column. We use these various versions of the singular vectors to estimate T;), for j = 1,..., M, and p = 1,..., N, the travel times from target j to the array 19 element p as the minimizers of

The ATSV (Arrival Times from averaged Singular Vectors) functional is

We combine MFT with ATSV to obtain

where

Examples of MFT - ATSV estimates are displayed in Fig. 3. This method is statistically stable and gives good estimates of the target locations.

s-O%,M.F -0% s = 2 53% M.F 4.38%

s = 4 84%, M.F.= 8.38% s = 6 36%. M.F = 12.05%

-0 2 4 6 8 10 12 "0 2 4 6 8 10 12

Figure 3. Combined MFT and ATSV estimation (6) of two targets location in random media. The exact location of the targets is denoted by the green star. The standard deviation 8 and maximum fluctuations (M.F.) are indicated on the top of each view. The horizontal axis is the range in mm and the vertical axis is the cross-range in mm. 20

6. Conclusions For imaging in randomly inhomogeneous media, the foregoing results lead us to the following conclusions: Single frequency methods are not statistically stable, and therefore cannot be used without modification in the presence of significant spatial heterogeneities in the acoustic wave speed. In contrast, time domain methods are statistically stable for any objective functional having the characteristic that the random Green’s functions appear in pairs of gg* This has been shown here to be true for Matched Field, and is expected to be true more generally’. Statistical stability is a necessary, but not a sufficient, condition for optimal imaging in random media, so satisfaction of this criterion is not enough in itself. To locate the targets in random media, we need either multiple views (using multiple arrays) so we can triangulate, or we need to extract a direct measure of range from the data. In the examples chosen here, we concentrated on arrival time and this information was obtained by combining MF with ATSV (arrival times from averaged singular vectors). Another alternative is to use Synthetic Aperture Imaging (SAI)as range estimator’.

References 1. L. Borcea, G. C. Papanicolmu, C. Tsogka, and J. G. Berryman, “Imaging and time reversal in random media,” submitted in Inverse Problems. 2. E. Bbcache, P. Joly, and C. Tsogka, “An analysis of new mixed finite elements for the approximation of wave propagation problems,” SIA M J. Numer. Anal. 37, 1053-1084 (2000). 3. H. P. Bucker, “Use of calculated sound field and matched-field detection to locate sound sources in shallow water,” J. Acoust. SOC.Am. 59, 368-373 (1976). 4. A. B. Baggeroer, W. A. Kuperman, and H. Schmidt, “Matched field processing: Source localization in correlated noise as an optimum parameter estimation problem,” J. Acoust. SOC.Am. 83, 571-587 (1988). 5. F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics (AIP Press, New York, 1994). 6. C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151-163 (1994). 7. C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: Detection and selective focusing on two scatterers,” J. Acoust. SOC.Am. 99, 2067-2076 (1996). 8. M. Fink, D. Cassereau, A. Decode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas,and F. Wu, “Time-reversed acoustics,” Rep. Prog. Phgls. 63, 1933- 1995 (2000). 9. T. D. Mast, A. I. Nachman, R. C. Waag, “Focusing and imaging using eigenfunctions of the scattering operator,” J. Acoust. SOC.Am. 102, 715-725 (1997). RESONANCES OF AN ELASTIC PLATE IN A DUCT, IN THE PRESENCE OF A UNIFORM FLOW

A.S. BONNET-BEN DHIA AND J.-F. MERCIER Laboratoire de Simulation et de Mode'lisation des phehom2nes de Propagation, ENSTA, URA 853 du CNRS, Paris, fiance E-mail: bonnetOensta.fr and jmercierOensta.fr

It is well known that the presence of a rigide plate, in the center of a bidimensional duct and parallel to the boundary walls, can give raise to fluid vibrations trapped around the plate. The theoretical study of these resonant modes is based on the spectral theory for a self-adjoint operator and is lead thanks to the Min-Max principle. We propose to extend the method to the cases of an elastic plate and/or a uniform flow. We show that in these different cases, trapped modes may still exist and we study their dependance versus the caracteristics of both the plate and the fluid, and versus the Mach number. Finally numerical results are obtained by a mixed method spectral/finite elements.

1. Introduction We are interested in the resonances of a plate placed in a duct, and we study the influence of both the presence of a flow in the duct and the elastic deformations of the plate. The study of resonances of coupled systems fluid/plate is useful for practical applications since resonances correspond to large amplitude of vibration, which may damage the structures. In particular, hydrodynamic resonances, analogous to acoustic resonances have a great practical interest. They have been studied in the case of surface waves trapped by a cylinder placed vertically in a fluid layer. This kind of configuration is involved when studying the stability of floating bridges, floating airports and offshore plateforms. Many studies of resonances have been lead in the case of a rigid plate and in absence of flow. Numerically l, and theoretically 3, it has been shown that when the plate is in the center of the duct, resonant modes can exist and are more numerous when the plate is longer. The vibrations of the fluid are then antisymmetric with respect to the plate, and a,re trapped by the plate. The influence of a flow and of plate deformations has already

21 22 been studied, but when considering an incompressible fluid and taking into account only the two first modes of the plate vibrations 5. We have extended these studies to the case of a compressible fluid, in motion and with an elastic plate. Our study is motivated by experimental results (UME, ENSTA) which have shown that an instability occurs in the coupled system fluid/plate above a critical flow velocity. The leading edge of the plate is fixed and the trailing edge is free, as in the experimental study.

2. Equations of the problem We consider a two-dimensional infinite duct D of height h and of boundary dD. The elastic plate is defined at rest by r = { (2,y); y = 4 and 0 < x < L}. The duct without the plate is noted fl (see Fig. 1). We restricted

g=h A’ a0 wave guide

Figure 1. Geometry of the problem ourselves to the case of a plate placed in the center of the duct and to antisymmetric vibrations of the fluid with respect to the plate.

2.1. Fluid The velocity of the uniform flow is noted U. In time harmonic regime of frequency w, the linearized Euler’s equations lead to the ”convected” Helmholtz equation for the velocity potential 4 :

U W M = - < 1 is the Mach number, c the sound velocity and k = - the C C acoustic wavenumber, which will be called frequency in the sequel. The acoustic pressure p is linked to the velocity potential through the relation p+p Gxu--iw ) +=o, 23 where p is the fluid density.

2.2. Elastic plate The vertical displacement of the plate ~(x,t) = w(x)e-iwtfollows the equation of elastic deformations of the plates a4Q a2v B -+ Ms - = -[PI, (3) ax4 at2 where B is the bending stiffness, Ms the specific mass of the plate , and B b] designs the pressure jump through the plate. Introducing p4 = -Msc2 ' <=-Ms and q5 = c'p, Eq. 3 thanks to Eq. 2 becomes P

5 (p4% - k2) w = (M& - ik) ['p] (4)

2.3. Boundaw conditions The slip-condition on the boundary of the duct and the continuity of the normal displacement at the fluid/plate interface, which differs from the a'p continuity of the normal velocity if M # 0, lead to - = 0 on dD and aY 9 = (M& - ik) w on r. For the plate, we have on the leading edge 8Y w(0) = 0 = w'(0) and on the trailing edge w"(L) = 0 = w"'(L). We seek trapped resonant modes, and therefore the condition imposed.

3. Characterization of the resonance frequencies Equations (1) and (4) as well as the boundary conditions are written in a variational form. In this aim w_e define the bilinear _form = a [kM; ('p, w),($7 V)l strict ourselves to seek velocity potential belonging to V = {'p E H1(fl)/'p(x,h - y) = -'p(x, y)}. Moreover W = {w E H2(I')/w(0)= 0 = w'(0)) is introduced. Looking for resonance frequencies is equivalent to solve the eigenvalue problem: 24

Find k E R / 3(cp,w) E V x W,(p,w)# (0,O) such that V($, w) E V x W, (5) aP, M;(cp, w),($,.)I = k2 (J* cp4 + 5 Jr wq

nnry Remembering that the modes of the duct read cp; = eib,fxcos - 6, sim- h ple calculations show that the criticd frequency of the first antisymmetric duct mode is k, = Ed=, and that for k < k, all the antisymmetric h modes of the duct are evanescent. We will restrict ourselves to determine resonances below the critical frequency: therefore since any fluid vibration produced by the plate can be decomposed on the duct modes, resonant modes will be trapped by the plate. If we note XF(k), n = 1,2, . , N(k)the eigenvdues of a(k,M), ordered by increasing values, determine resonance frequencies is equivalent to find the roots kfaM of Xr(k) = k2, with k E [O,k,[.

4. Theoretical tool The general method to study the resonances will be presented when applied to the determination of the first resonance frequency. To study the spectrum of a(k,M), we use the Min-Max principle. Starting from the a[k,M; (cp, (9, the first Min-Max Rayleigh ratio R(k,M; cp, w) = w>? w>l, Ja Id2+ 5 sr lWl2 formula defines the function ry(k)= inf R(k,M;cp,w). The Min- (V,W)€VXW Max principle indicates that if ry(k)< k:, then a(k,M) has at least one eigenvalue ~f"(k) equal to rp ( k). The interest of this principle is to predict the existence of an eigenvalue without calculating it explicitely: in the sequel we will seek test fields (91,wl) such that R(k,M; cpl, wl)is a constant independant of k and stricly lower than Ic; . Then the Min-Max principle proves that Xf" (k) exists. Since it is also proved to be continuous on [0,kc[, at least one resonance frequency exists . The test fields giving the best existence results are the fields close to the real resonant modes. More generally, the Min-Max principle gives access to the eigenvalues of a(k,M)larger than Xp, and the procedure presented in the previous paragraph can be generalized to determine several resonance frequencies. We have studied two configurations with a uniform flow: a rigid plate and an elastic plate. 25

4.1. Rigid plate in presence of a flow The case of a rigid plate is interesting since we are able to find both a lower and an upper bound to the number of resonances. Resonances in a moving fluid can be determined by studing the resonances in a fluid at rest.

4.1.1. Study of a fluid at rest ?r In a fluid at rest, the critical frequency becomes k, = - and the Rayleigh h ratio simplifies in Rrigid(k;‘p) = ” We choose a test field localized S’ Id2 h around the plate: for y > -, ‘pl(z,y)= sin (:z) if 0 < z < L, ‘p1 = 0 2 ?r2 out side of z €10, L[. Then we obtain &igid(k; cpl) = . Therefore, if (-)L h < L, at least one resonance frequency exists. More generally we get that if we note Nf the integer such that Nfh < L < (Nf + l)h, Nf or Nf + 1 resonance frequencies exist. Numerically and only for short plates (L 5 6h), it has been proved that exactly Nf + 1 resonances exist I. Results for longer plates have been obtained theoreti- L cally, and the number of resonant modes varies like - when L + 00 4, in h agreement with our results,

4.1.2. Fluid in motion When the fluid is in motion, there exists a change of variables and unknown which allows us to study the resonances in a fluid at rest. Then from previous results is deduced that if we note Nf the largest integer such that Nfhd- < L, Nf or Nf + 1 resonance frequencies exists (and thus more resonance frequencies when the fluid velocity is increased).

4.2. Elastic plate in the presence of a flow When the plate is elastic and the fluid in motion, we consider a trial field taking into account only the plate behaviour: ‘p = 0 and w = w,, the nth vibration mode of the plate without fluid. The eigenfrequencies for a plate surrounded with vacuum are kP, = (pF)2,- where a, is the nth positive root of cosancoshan = -1. We can then find a lower bound to N(L,M), the number of resonance frequencies for an elastic plate placed in a uniform flow. In this aim, we introduce Np the number of eigenfrequencies of the plate without fluid located below k,: it is defined as the largest integer for 26

which the inequality ~CYN~ < L holds true. Then the Min-Max principle indicates that both existence criterions of resonances presented above apply together: we get that N(L,M) 2 max(Nf,Np) (contrary to the case of a rigid plate, we can not access to any upper bound for N(L,M)). Remark that since k, decreases when M increases, Np decreases when the fluid is accelerated, contrary to Nf.

5. Numerical study In order to illustrate our theoretical results, we have determined numerically the resonance frequencies for an elastic plate in presence of a flow. The numerical method consists in submiting the plate to an incident wave associated to a variable frequency k, and to measure the diffracted field by the plate. A bounded domain RR is built around the plate, and for a fixed energy of the incident wave (lRIqinc12 = 1) , the plate energy E(k) = J, lwI2 is measured. The numerical method couples finite elements introduced in the bounded domain to modal decompositions of the velocity potential outside RR '. For a plate of length L = 10, placed in a duct of height h = 4 and in presence of a flow of Mach number M = 0.4, the critical frequency is k, = 0,72. To p = 1 and E = 100 (light fluid) correspond Nf = 2 and Np = 3. Therefore at least 3 resonance frequencies exist. The curve log E versus k presents six picks corresponding to the resonance frequencies (Fig. 2). The fouth resonant mode is represented on Figs. 3 and 4.

References 1. D. V. Evans et C. M. Linton, Tkapped modes in open channels, J. Fluid Mech. 225 (1991), 153-175. 2. D. V. Evans, C. M. Linton et F. Ursell, napped mode frequencies embedded in the continuous spectrum, Q. J. Me&. Appl. Math. 46(2) (1993), 253-274. 3. D. V. Evans, M. Levitin et D. Vassilev, Existence theorems for trapped modes, J. Fluid Mech. 261, 21-31 (1994). 4. N. S. A. Khallaf, L. Parnovski et D. Vassilev, Trapped modes in a waveguide with a long obstacle, J. Fluid Mech. 403 (2000), 251-261. 5. Y. Auregan et N. Meslier, Mode'lisation des apndes obstructives du sommeil, C. R. Acad. Sci. Paris, serie I1 316 (1993), 1529-1534. 6. M. Bruneau, Manuel d'acoustique fondamentale, Hermes (1998). 7. Daniel MARTIN, Code dldments finis MELINA, http://www.maths. univ-rennesi.fr/"dmartin/melina/www/homepage.html 27

0' I 1 0 0.2 0.4 0.6 k

Figure 2. Logarithm of the plate energy versus the frequency of an incident wave

-10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 3. Real part of the velocity potential of the fourth resonant mode

-6M) 0' 2 4 6 8 10 X

Figure 4. Plate shape associated to the fourth resonance FIRST ORDER ASYMPTOTIC MODELING OF A NUCLEAR WASTE REPOSITORY

A. BOURGEAT AND I. BOURSIER MCS-ISTIL, Universite' Lyonl, Bit. ISTIL, 43 Bd. du 11 novernbre, 69622 Villeurbanne Cedex, fiance

0. GIPOULOUX MCS-Faculte' de sciences, Universite' de St-Etienne, 23 Rue Dr.Pau1 Michelon, St-Etienne Cedex 2, fiance Laboratoire de Me'canique et d'Acoustique, UPR 7051, 31 Chernan Joseph Aiguier, 13402 Marseille cedex 20, fiance

E. MARUSIC-PALOKA Department of Mathematics, University of Zagreb, BajeniEka 30, 10000 Zagreb, Croatia

1. Introduction The goal of this paper is to give a mathematical model describing the global behavior of a nuclear waste disposal process in order to answer the upscal- ing problem in the benchmark COUPLEX section 3 '. The disposal site can be described as an array made of high number of containers inside a low permeability layer( e.g. clay) included between two bigger layers with higher permeability (eg. limestone). In porous media, the pollutant is transported both by convection produced by the water flowing slowly through the rocks and by diffusion coming from the dilution in the water. Herein, for simplicity, the repository consists of a set of units lying on a hypersurface C and we represent the leaking of a disposal unit by a localized density source inside the domain or by a hole in the domain with a given flux on its boundary. Since there is a large number of units, each of them with a small size compared to the layers size, a direct numerical simulations of the full field, based on a microscopic model taking into account all the details, is unrealistic. The ratio between the width of a single unit 1 and the layer length L,is of small order, and can be considered as a small parameter, E, in the microscopic renormalized model. Now, in this

28 29 renormalized model, the units, have a height of order aP;P 2 1 and are imbedded in a layer of thickness a; the model behavior, as E tends to 0, by means of the homogenization method, gives a first order asymptotic model which could be used as a global repository model for numerical simulations. For this, we use herein similar methods to those applied for modelling the flow through a sieve, like in 3, or '.

2. Setting the problem 2.1. Renomalization The equation describing the transport of the concentration of a radioactive element after COUPLEX is: aP Rw-at - V . (AVp) + (V .V)p+ XRwp = 0 in the domain, where R is the latency retardation factor, w the porosity, V the Darcy's velocity and X = with 7 the half life time of the ra- dioactivity element. Due to the dimensions of the storage we may consider, without lost of generality, only a 20 vertical section of the storage. We take as characteristic length L the length of the impervious layer, and for characteristic time the diffusion characteristic time T, = in this same layer. With this time and space rescaling, 5 = $, t = &, the thickness of the impervious layer and of the units are of order, respectively a and 8;the rescaled releasing time, t, = t' is then very small compared to the total time T = TIT, and to the leaking flux during this short releasing period is after renormalization, CP = CP' x L.

2.2. The Geometry For the purpose of this paper, assuming the above renormalization, we will simplify the geometry by assuming that in R =]- 1, 1[2 all the thin units have the same shape. Then the median plane C =] - 1/2,1/2[ x (0). More precisely, we define a normalized unit M, = Mx] - ao-l, af'-' [; M =] - m, m [ , 1/2 > m > 0 . By periodic translation of M,, we define BE= UaEJ(E)EM; , where M," = a+M, and J(E)= {a E Z ; aM,"nC # O}. We also assume that the small parameter E << 1, is such that a-1 E N. We define the all units boundary Fa = aB, and I?: = r,x]O,T[.The a-th unit boundary is denoted = aM,". Finally, we denote RT = Rx]O,T[, 0: = RE x]O,T[where RE = R\BE is the porous media around all the units. 30

2.3. The Equations Now X = > 0, with r being the normalized half life of the radioactive element, and let us assume the initial concentration of the radioactive material in the soil cpo € H1(R,). The diffusion tensor A € LW(R;Rzx2)is a positive definite matrix function, and since the layers of soil involved in our model have different properties, we write for instance

Now we write the diffusion matrix in the form AE(x2)= A(?). With the convection velocity v E C([O,T];H1(Rx a)'),the situation is similar and the dependence on y2 is like in the diffusion matrix:

For simplicity, we assume moreover that the last component v2 does not depend on y2. Next, we suppose that div,v = 0, in order to have the divergence free convection velocity. Finally we pose vE(x,t) = v(x, 2,t). At last, we define the porosity of the medium as

and we put

WE(52) = W(Z2/&) . We have chosen to represent the units' leaking by assuming the porous domain Re to have many holes corresponding to the units BEand by giving the flux behavior n . u through these holes. An other option would be to assume the units to be part of the porous domain R and to consider any unit as a source term fa with support on the unit volume EM:; then we would have the obvious relation

Finally we assume 9 E G( [0,TI), the function describing the time behaviour of a unit's flux through the porous media, and we suppose, for simplicity, that iP(0) = 0 and has a compact support [0,tm] C [0,TI. Now, after the above adimensionalization, the process is governed by the following convection-diffusion type equation:

WE " - div (A"Vq,) (v' . V)cp, Xw" cpE = 0 in 0: Eat + + (2) 31

cpE(0,Z) = cpo(z) z E R& (3) n -0 = n - (A'V(P, - V' cp,) = @(t)on I?: (4) with the boundary condition on the exterior boundary S = dR; S = S1 US2:

(P, = 0 on 4, (5) n ' (A"Vq, - vEqE)= 0 on S2 . (6)

3. A priori estimates The starting point of any asymptotic analysis is a sharp a priori estimate. In our case the only difficulty comes from the boundary integral over I?& which needs to be estimated. The main result of this section is:

Prop 3.1. Let cp, be a unique solution of (2)-(6); then there exists a constant C > 0 independent of e such that

The proof relies on the following auxiliary result:

Lemma 1. There exists a constant C > 0, independent from e, such that

v$ E H1(%) 7 l$lL1(T,) ICbbIHI(S2,) - (9)

Proof of prop 3.1. To prove (8) and (7) we use (P, as test function in (2)-(6)and we obtain

4. Weak convergence

The solution (PE is defined on the variable domain OF, but to use the weak convergence methods, we need to extend cpE to the whole domain OT and preserving the a priori estimates (7) and ((8). Unfortunately the standard results do not apply, unless /3 = 1 and we have to adapt the ideas from 32

to the situation resulting from the third scale EP in order to obtain the following lemma.

Lemma 2. For any q!J E H1(R,) there exists an edension 6 E H1(R) such that

In the sequel we will assume that cpa is extended using lemma 2 and we will denote that extension, for simplicity, by the same symbol. Then, due to the proposition 3.1, we can conclude that there exists some cp E L2(0,T;H1(R)) n LCO(O,T;L2(R)) such that (up to a subsequence)

(PE cp weak* in ~~(0,T;L~(R)) (12) 09, 2 Vcp weakly in L2(0,T; L2/lP(R)) for b < 2 (13) weak* in L2(0,T; M(S2)) for = 2; (14) where M(R) is the space of measures on R. The main goal of this section is to identify the limit cp .

Theorem 1. The limit function cp defined in (12-14)is the unique solution of the problem u2--V.(A2Vcp)+(v2.V)cp+Xu2cp=OinfiTacp at (15) cpb, 0) = cpo(z> 2 E fi (16) cp=O on Sl (17) n . (A2Vcp - v2 cp) = 0 on S2 (18) [cp] = 0 , [e2 - (A2Vcp- v2 cp)] = -2 IMJ on C , (19) where fi = s~\c,fiT = fi x 10, T[,[w](zl) = ~(21,o+) --w(z1, 0-) , denotes the jump over C and IMI denotes the external area of M.

Proof. Let $ E C"([O, TI;Cr(S2)) be such that $( . , T)= 0 . Using $ as test function in (2)-(6) we get 33

The passage to the limit for the first four integrals is straightforward like for instance in:

where obviously I: + JOT A2Vp V$ , and I: + 0, due to the Lebesgue theorem, since /(A1-A2)Vp, V$~LI(RT)5 C and (A1-A2)Vp, V$ + 0 (a.e.) in RT. For the last integral in (20), we have

Remark 1. In fact we did not use the periodicity of the units distribution and the same proof holds in case of each unit being randomly placed in a mesh of an e-net. The units do not even need to have the same shape as long as their thickness is small enough (< E). In a general case where the flux depends also on the space = @(z,t)and the units have different shapes M,(z), then the right hand side of (19) will be replaced by lim IM&(z>l@(z’,t). E-bO

References 1. Bourgeat A., Gipouloux O., MaruSiC-Paloka E., Mathematical modelling and numerical simulation of non-Newtonian flow through a thin filter, SIAM J.Appl.Math, 62, (2001), 597-626. 2. Cioranescu D.,Saint Jean Paulin J., Homogenization in open sets with holes, J.Math.Anal.Appl., 71 (1979), 590-607. 3. Conca C., Etude d’un fluide traversant une paroi perforhe, I, 11, J.Math.Pures et Appl., 66 (1987), 1-69; Mascarenhas M.L., PoliSevski D., The warping, the torsion and the Neumann problems in a quasi-periodically perforated domain, RAIRO Math.Mod.Num.Anal., 28 (1994), 37-57. 4. Sanchez-Palencia, E., Boundary value problems in domains containing perforated wds, Se‘mdnaire Colldge de fiance, Research Notes an Mathematics No 70, Pitman, London. 5. https://mcs.univ-lyonl.fr/MOMAS EXACT AXISYMMETRIC SOLUTION FOR TEMPERATURE-DEPENDENT COMPRESSIBLE NAVIER-STOKES EQUATIONS

P. BROADBRIDGE Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA E-mail: [email protected]

T. M.BARRETT School of Electrical, Computer and Telecommunicataons Engineering, University of Wollongong, Wollongong NS W 2522, Australia E-mail: [email protected]

For steady axisymmetric flow of a compressible Boltzmann gas with temperature- dependent viscosity and conductivity, the Navier-Stokes system for momentum, density and temperature is explicitly solvable. The temperature is positive only in the exterior of a thin cylinder whose size is microscopic compared to the dimensions of a vortex. At the center of this thin core, the non-negative temperature has an accumulation point of zeros. In the central thin core of a stable vortex, more general transport mechanisms must be in effect.

1. Introduction Nonlinear partial differential equations, representing physical conservation laws, have long been a source of inspiration for developments in mathematical analysis. In particular, the Navier-Stokes equations provide us with many challenging problems related to existence and number of solutions, stability, bifurcations, dimensionality of asymptotic solution manifolds, and numerical and analytic solution methods, e.g. 1>2. Although the Newtonian (Navier-Poisson) fluid assumes a simple linear relationship between stress and rate of strain, the resulting equations of motion predict an array of complicated dynamical behavior that is still not well understood. Indeed this must be the case if the equations are to

34 35 faithfully represent the chaotic unpredictable behavior observed experimentally in common fluids. For compressible fluids with temperature-dependent properties, we are dealing with a system of nonlinear equations with five dependent variables, conventionally taken to be mass density, temperature and three components of momentum density, all of which are functions of space and time. Any non-trivial exact solutions of this system would necessarily represent smooth flows that could be stable mean flows only at low Reynolds numbers and low temperature gradients. Nevertheless, they would provide valuable test beds for approximate solution methods and for stability analysis. More subtly they test the physical modelling assumptions by showing whether or not it is possible to duplicate experimentally observed behavior. In this short article we focus on the direct construction of a vortex solution or more generally a Couette flow in a compressible viscous fluid with temperature-dependent properties. Remarkably, we are able to construct a full explicit solution for the case of an ideal Boltzmann hard-sphere gas, even when we incorporate the predictions of kinetic theory that has both viscosity and thermal conductivity proportional to square root of temperature. Although it is not used nearly as often as the incompressible inviscid vortex solution, the compressible inviscid constant-temperature vortex solution has been known for a long time3. This and our Boltzmann gas vortex, have interesting physical properties that cannot be fully elaborated here because of space limitations.

2. Steady Compressible Temperature-Dependent Navier-Stokes Equations For background on dynamical equations of compressible fluids we refer to the text by Thompson4. Assuming the Einstein summation convention in n spatial dimensions, the steady compressible Navier-Stokes system is:

(momentum) pujt~ju~= -8iP + ajcij i = 1,2,.. . ,n. (1) (mass) &(pi)= o (2) (heat) &[K4T]+ = 0. (3) Pressure P(p,T) depends on density p(r, t) and temperature T(r,t). Ther- mal conductivity K(T)depends on temperature. r is the viscous dissipation,

r = aikD? = Tr(GD) 36 where Dt is the rate-of-strain tensor, 1 D! = -(diuk &ui), 2 + and sik is the stress tensor, 6ik - -Psik + Ci, with Xik = 2p(T)[Dik- 4dikDf]+pv(T)6ikDf. The shear viscosity p and bulk viscosity pv are assumed to depend on temperature. Now we restrict our attention to two-dimensional axisymmetric flow around origin 0. From (2),the total rate of mass flowing across any curve OP must be constant. This quantity is a stream function $(r). In conventional notation rt = (2, y) E (s1,s2)ds1and ut (u,v) E (u1,u2),

For axisymmetric flows, Y X p(r)u = --$'(r) and p(r)v = -$'(r). r r (4) For this special case of axisymmetric flow, Vpis radial and u is circumferential. Expanding (2),we then deduce

v.u=o. (5) This means that we could define the usual volume-based stream function. However we prefer to use the mass-based stream function since it exists also for non-axisymmetric steady flows. Importantly, (5) means that any terms in the stress tensor involving D: may be ignored. In particular, the bulk viscosity plays no part in steady axisymmetric dynamics. Direct calculation now gives 2D"djp = -p'(T)T'(r)X'(r)y for i = 1 = +p'(T)T'(r)X'(r)s for i = 2 (6) and I' = pr2[X'(r)I2 (7)

By taking radial and circumferential components of the vector momentum equations, we have 37

(centripetal acceleration is driven by pressure gradient) and

The heat equation reduces to

The mass continuity equation (2) is automatically satisfied by (4). We are now left with the system (8), (9), (10) for the variables $,p and T.

3. Full Axisymmetric Solution for Boltzmann Gas By defining A = kg ($), we find that (9) integrates twice to

f = al (T(rl))rL3drl + u2 (ul,a2 constant). (11) Pr 0 Hence the velocity components are

and the vorticity vector is

Hence, the parameter 132 is merely the angular velocity magnitude of a rigid rotation. Since this is scarcely relevant to a fluid, hereafter we assume a2 = 0. Now the heat equation (10) reduces to

This can be solved for several choices of p(T)and K(T).In particular, for an ideal gas, hard-sphere kinetic theory predicts

/.L = cgdT (15) 38 and

K = up (16) where c5 = %$* and u = x-te N 0.31. Here, k is Boltzmann's constant, m is molecular mass and 6 is molecular diameter. For this case of particular physical significance, (14) is equivalent to r2R"(r)+ 37-77,' [pu3 2 -1 c5r2 -2 + I]R = o + (17) where R = $. The solution of (17) is well known (e.g. see 4.6 of Bell5). Hence the general solution is

T = [b3 JO(h/r) + b4~o(di/r>l+(b3,b4 constant) (18) where b = $u:u3cg, and JOand y0 are Bessel functions of order zero, of the first and second kinds. If temperature is not to grow without bound as r + 00, we must 2 have 13.4 = 0. Temperature then approaches bi as r + 00. Hence, T = T,Jo$(&/r) where T, = bj. However, we now have the surprising result that temperature is absolute zero and is not differentiable at an infinite number of locations r = fi/Xj within the region

r < r min = -Ji; A1 where XI x 2.4 is the lowest zero of the function Jo. We conclude that the Navier-Stokes system is not a good model of a fluid within distance rmin of the center of a vortex. For r > rmin,T is positive increasing and convergent. For a small macroscopic vortex in a Helium-like gas with velocity = 0.1 ms-l at r = 0.01 m, m = 7 x kg and 6 = 5 x 10-lOm, (19) gives rmin = 40 micron. Since this is smaller than the size of objects used to stir vortices, r < rmin is outside of the experimental domain. However for a vortex of tornado-like dimensions, with IuI = 100 kmh-' at r = 2 km, rmin is of the order of 1 km. This is small on meteorological scales. The failure of the model to predict non-vanishing temperatures for r < rmin might be rectified by incorporating time-dependent fluctuations with turbulent transport of heat. Now the remaining equation to solve is (8) for p(r), 39

by (11). For an ideal gas with P = c4pT and q = 5, (20) integrates directly to give

(TO arbitrary constant). For the case of the Boltzmann hard-sphere gas, we may substitute p = cgT* and T = T,J;(&/r) directly in (21). This completes the full analytic solution. The solution (21) predicts p(r) decreases monotonically to zero as r + 00. However, the exponential factor in (21) may be extremely close to one until T is extremely large. This means that pressure may be close to a uniform value until r is extremely large. This will be further elaborated in the next section.

4. Ideal Gas with Constant Transport Coefficients In the solution for the Boltzmann gas, p and K approach p(T,) and K(T,) as T + 00. By way of comparison we solve the above equations for the case of a compressible ideal gas with p and K constant, independent of T. With a2 = 0, (12) gives the velocity field of the point vortex of potential flow theory. The general solution to (14) is

1 a:p3 T = b3 + bqlog~- -- 4 Kr2 *

Assuming T is bounded as T + 00, we take b4 = 0 and b3 = T,. Then T is positive for

For a small macroscopic vortex with alp = 10-3m2s-1 and T, = 300K this is of the order of 50 micron whereas for a tornado-scale vortex, it is seven orders of magnitude larger. 40

Finally, equation (21) gives

-1 -a1/(4c463)

with a5 constant. Hence, p + 0 as r + 00 and p + 00 as r + rmin. The pressure is given by

For large r, P N P0r-P where p = a1/(2c4b3) and POis an arbitrary constant. The exponent p may in practice, be very small. For the small macroscopic vortex considered earlier, is approximately 4 x This means that P is imperceptibly close to PO out to the edge of the universe. For the tornado-scale vortex considered earlier, p is 7 orders of magnitude larger, implying that P decreases rapidly beyond the central core. In summary, the solution for the small vortex is very close to the standard vortex of potential flow but the large vortex solution, without external heat sources and without vertical motion, is like nothing observed.

Acknowledgments The authors are grateful to Elizabeth Mansfield and Peter Clarkson who collaborated with us on a symmetry analysis of the Navier-Stokes equations.

References 1. R. Salvi (ed.), Nauier-Stokes equations: theory and numerical methods, Marcel Dekker, Inc., New York (2002). 2. V. K. Andrew, 0. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov, Ap- plications of group-theoretical methods in hydrodynamics, Kluwer, Dordrecht (1998). 3. R. Von Mises, Mathematical Theory of Compressible Fluid Flow, Academic Press, New York (1958). 4. P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York (1972). 5. W.W. Bell, Special hnctions for Scientists and Engineers, Van Nostrand Reinhold, London (1968). RECOVERY OF THE POROELASTIC PARAMETERS OF CANCELLOUS BONE USING LOW FREQUENCY ACOUSTIC INTERROGATION

JAMES L. BUCHANAN Mathematics Department, United States Naval Academy, Annapolis, MD 21402, USA

ROBERT P. GILBERT Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA

KHALDOUN KHASHANAH Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA

We examine the conditions under which the poroelastic parameters of cancellous bone-like material can be recovered by acoustic interrogation of a slab specimen.

1. Introduction Cancellous bone is a porous type of bone found in the spine and at all articulating joints. It is a porous cellular solid, consisting of an irregular three-dimensional array of bony rods and plates, called trabeculae. Bone marrow fills the spaces of the pores. The study of cancellous bone is important, among other reasons, because the thinning of the trabeculae is a cause of osteoporosis. Hence determination of the porosity (pore space volume/total volume) of cancellous bone by acoustic interrogation would be useful in diagnosing osteoporosis. In view of the structure of cancellous bone Biot’s model of a poroelastic medium as an elastic frame with a connected pore space filled with fluid may be applicable. Williams 4, McKelvie and Palmer 3, and Hosokawa and Otani have compared the predictions of the Biot model for cancellous bone with experimental measurements. In order to do this the parameters upon which the Biot model depends had to be determined. Some of the parameters were measured, but others were estimates taken from the literature or based

41 42

Symbol I Parameter Symbol I Parameter Symbol I Parameter Aggregate Frame Fluid P Porosity Pr Density Pf Density of material

~ k Permeability K, Material bulk Kj Bulk modulus modulus a Pore size Kb Complex bulk 7 Viscosity parameter modulus a Structure /I Complex shear I I constant I I modulus I I I

Table 2. Estimated values of some Biot parameters at different porosities taken from McKelvie and Palmer or Hosokawa and Otani. The second set of values for the permeability (Column 3) were calculated from the indicated value of the pore size parameter using the Kozeny-Carmen equation.

0.75 7 x 10-9 2.40 x 10-8 8.00 x 10-4 2.69 x 109 1.10 x 109 0.81 2 x 5.83 x 1.20 x 1.80 x log 7.38 x lo8 0.83 3 x 7.56 x 1.35 x 1.55 x lo9 6.27 x lo8 0.95 5 x loF7 2.30 x 2.20 x 2.57 x lo8 1.05 x lo8 upon empirical formulas. Thus there is both expense and uncertainty in the process of ascertaining these parameters. The question we address is whether these parameters can be recovered by acoustic interrogation from an experiment such as that described in Hosokawa and Otani.

2. The parameters of the Biot model Table 1 gives the parameters of a poroelastic medium upon which the Biot model depends. Numerical experimentation indicated that of the thirteen real Biot parameters five, porosity, permeability, pore size, and the real parts of the bulk and shear frame moduli seemed to influence the acoustic field significantly when varied over their expected ranges. The estimates for theses five parameters at five different porosities are given in Table 2. In obtaining the values of those parameters that were not directly measured we followed the estimation procedures used by McKelvie and Palmer and Hosokawa and Otani: The real parts of Kb and p were calculated using the formulas of Williams 43 used by Hosokawa and Otani. Here Vf = 1- B is the bone volume fraction. Theoretically n = 1 for waves travelling in the trabecular direction and is between 2 and 3 for transverse waves, however there is enough randomness in the trabecular direction in bone that investigators have empirically adjusted the exponent to agree with experiment. Williams arrived at a value of n = 1.23 for his samples taken from bovine tibia. Hosokawa and Otani found that n = 1.46 agreed well with their data from experiments on bone specimens from bovine femora. For the estimates of the moduli in Table 2 we used the exponent of Hosokawa and Otani and also their values E = 2.2 x lolo,v = 0.32 for the Young’s modulus and Poisson ratio of solid bone. The pore size parameter was estimated by McKelvie and Palmer using electron microscopy and by Hosokawa and Otani using x-ray examination. Figure l(Top) indicates that pore size is approximately a linear function of porosity. Permeability is a difficult parameter to estimate. Figure l(Bottom) shows the values for the five specimens of McKelvie and Palmer and Hosokawa and Otani. The estimates indicate that permeability is approximately a log-linear function of porosity. However McKelvie and Palmer characterize their values without elaboration as ”estimates” and Hosokawa and Otani state that their estimates are based upon those of McKelvie and Palmer. Hence the apparent log-linear relation should be regarded circum- spectly. Indeed according the Kozeny-Carmen formula

where K M 5 is an empirical constant, the relation is not log-linear. Figure l(Bottom) shows that if pore size is indeed a linear function of porosity, then permeability, as predicted by (l), will deviate significantly from log- linearity.

3. A simulated experiment We simulated an experiment similar to that described in Hosokawa and Otani in which a small rectangular piece of cancellous bone is placed in a tank of water. The equations of the Biot model cannot be solved analytically for this configuration and so the resulting pressure fields were approximated using the finite element method. Discussion of the implementation of the finite element method for this problem can be found in 44

0 0.7 0.75 0.8 0.85 0.0 0.05 J1

10P10.7 J 0.75 0.8 0.65 0.0 0.86 PdV

Figure 1. Top: Estimated values of pore size (*) for five bone specimens. Dashed line: regression line. Bottom: Estimated values of permeability (*) for five bone specimens. Dashed line: regression line. Solid line: Value of premeability predicted by the Kozeny- Carmen equation assuming a linear relation between pore size and porosity.

Buchanan et al.I. We denote the values of the acoustic field generated by the point source at points (xi, yj), i = 1,. . . , L,j = 1,.. . ,M located in the tank by PG . A set of Biot parameters which produces trial values Pij is compared to the "measured" values using the objective function

Thus the problem is to minimize the objective function. The minimization was carried out using the Nelder-Mead simplex algorithm. Typically inverse problems such as this contain numerous local minima and hence the success of such a procedure depends upon the quality of the initial guesses for the parmeters that are sent to the minimization algorithm. Lacking experimental data, the measured values PG had to be calculated using the finite element method for the Biot model, just as the trial values Pij were. This ignores errors due to measurement, modelling and discretization that would be present were experimental data used and may lead to unrealistic success in the parameter recovery process. To ameliorate this problem a finer finite element mesh was used for the simulated data fields than for the trial fields. 45

4. An algorithm for recovering the Biot parameters A simple approach to the parameter recovery problem in which we at- tempted to recover the parameters of the specimens of porosities p = 0.72,0.75,0.83 and 0.95 given in Table 2 using the parameters for p = 0.81 as the initial guesses in the simplex algorithm did not prove consistently successful' and so we devised a more elaborate method of generating initial guesses. We first formulated the problem as a univariate minimization problem for the single parameter porosity. This is feasible because of the various formulas relating other Biot parameters to porosity discussed above. Since these formulas were used in creating the target Biot parameters that we were trying to recover, variations based on the uncertainties mentioned above were incorporated where possible in order to avoid unrealistically good outcomes. For a given value of porosity p: The moduli ReKb(p) and Rep(p) were determined from the formulas (2). Since the values for the target specimens of Table 2 were calculated using Hosokawa and Otani's value n = 1.46 we used Williams value n = 1.23 in calculating the trial data in the minimization process. Pore size a(@)was determined from the regression line for the data in Table 2. Since this parameter was measured independently by McK- elvie and Palmer and Hosokawa and Otani for their bone specimens and the results indicate an approximately linear relation between pore size and porosity (Figure l(Top)), this is justifiable. Permeability k(p) is determined from the regression line shown in Figure l(Bottom). Since, as indicated earlier, there is some uncertainty about the log-linear relationship indicated by the data in Table 2, we also tested the algorithm when the permeability of the target specimen is calculated from the Kozeny-Carmen equation (1) using the pore size parameter in Table 2.

5. Results In our simulation the tank size was approximately 0.18x0.12m. Following Hosokawa and Otani the size of the bone specimen was taken to be one centimeter thick and three centimeters high. It was assumed that measurements were available at 10 points in the middle 80% of the tank along two vertical lines, one midway between the source and the bone specimen, the other midway between the bone and the right edge of the tank. Since we had doubts about the estimates for permeability in McKelvie and Palmer and Hosokawa and Otani, and since the algorithm described above presumes a log-linear relation between porosity and permeability, we also tested it when the values of permeability (given in Column 3 of Table 46

Table 3. Percentage errors in determining five Bilot parameters. The for permeability were the first set in Table 2 (Column 2). Frequency p k a Re Kb Re P 6000 Target 0.72 5.00 x 0.000471 3.18 x log 1.30 x log Result 1 2.1% 6.2% 23.8% 0.3% 16.1% Result 2 1.9% 2.0% 24.3% 7.1% 6.6% 6500 Target 0.75 7.00 x 0.000800 2.67 x lo9 1.10 x lo9 Result 1 2.3% 17.4% 8.8% 3.4% 4.5% Result 2 2.3% 44.3% 67.5% 3.4% 4.5% 6500 Target 0.83 3.00 x loF8 0.00135 1.53 x lo9 6.27 x lo8 Result 1 2.2% 56.3% 58.5% 29.7% 27.7% Result 2 2.8% 42.3% 17.0% 45.7% 43.8% 6500 Target 0.95 5.00 x 0.0022 2.57 x lo8 1.05 x lo8 Result 1 1.9% 45.3% 11.4% 165.8% 172.4% Result2 1.8% 2.0% 65.0% 94.2% 99.0%

2) were those predicted by the Kozeny-Carmen equation (1). Tables 3 and 4 give the outcomes of the simulation. In these tables the percentage error for an approximation x to x* was calculated by . The univariate minimization procedure describedII above was used to generate initial estimates for the five Biot parameters for which values were sought. The frequency used was the one that produced the best agreement between trial and "measured" data in the frequency range 5.5-10 kHz. The simplex method was initialized with these values. The estimates for porosity and the other parameters that the univariate minimization procedure generated were good when the target permeabilities were the first set (Column 2) in Table 2, but sometimes were poor when the second (Kozeny-Carmen) set of permeabilities (Column 3) was used. The simplex method was always fairly successful in determining the porosity, but was sometimes less successful in determining the other parameters. The result of sending the estimates arising from the univariate minimization procedure is referred to as "Result 1" in the Tables. Since the application of the simplex method usually produced values for porosity and the two moduli that were improvements over those generated by the univariate minimization process, we decided to see whether these improvements could be used to refine the results. A new initial guess for the simplex method was constructed using the porosity, bulk and shear moduli determined by the first application, but with the pore size and permeability calculated from the value of porosity found by the first application using regression lines for the permeability 47

Table 4. Percentage errors in determining five Biot parameters. The values for permeab ty were the second set in Table 2 (Column 3). Frequency Pk a Re Kb Re P 6500 Target 0.72 7.99 x 0.000471 3.18 x log 1.30 x lo9 Result 1 0.1% 12.5% 90.7% 15.6% 23.1% Result 2 0.1% 12.5% 91.5% 16.1% 23.1% 6500 Target 0.75 2.40 X lo-' 0.000800 2.67 X lo9 1.10 X log Result 1 2.5% 47.9% 60.0% 76.8% 47.7% Result 2 2.5% 10.0% 2.4% 86.7% 32.7% 6500 Target 0.83 7.56 x lo-' 0.00135 1.53 x log 6.27 x lo8 Result 1 1.9% 131.5% 152.6% 64.5% 61.6% Result 2 1.4% 20.4% 47.9% 92.5% 59.1% 6500 Target 0.95 5.00 x 0.0022 2.57 x lo8 1.05 x 10' Result 1 2.8% 187.8% 23.6% 206% 197.5% Result 2 2.8% 254.8% 61.4% 244.5% 234.4% data (Column 2) and the pore size data. The outcome of this second application of the simplex method is labelled "Result 2" in the Tables. The second application did in some cases improve the results when the second (Kozeny-Carmen) set of permeabilities in Table 2 were used for the target specimens (Table 4). In assessing the results it should be noted that even when the correct parameters for the target specimen were sent to the simplex method relative errors on the order of 50% occurred for some of the parameters'. On the whole we deem the algorithm described successful at recovering the parameters for the specimens with porosities ,8 = 0.72,0.75 and 0.83. This was not the case for ,8 = 0.95. It is possible that the four parameters other than porosity simply do not affect the acoustic field enough in this extreme case for their values to be recoverable.

References 1. J. L. Buchanan, R. P. Gilbert, and K. Khashanah. Determination of the parameters of cancellous bone using low frequency acoustic measurements. 2002. Preprint . 2. A. Hosokawa and T. Otani. Ultrasonic wave propagation in bovine cancellous bone. J. Acouist. SOC.Am., 101:558-562, 1997. 3. M.L. McKelvie and S. B. Palmer. The interaction of ultrasound with cancel- low bone. Phys. Med. Biol., 10:1331-1340, 1991. 4. J. L. Williams. Prediction of some experimental results by biot's theory. J. Acoust. SOC.Am., 91:1106-1112, 1992. MATHEMATICAL MODEL OF THE INTERACTION PROBLEM BETWEEN ELECTROMAGNETIC FIELD AND ELASTIC BODY

F. CAKONI AND G. C. HSIAO Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA E-mail: cokoniOrnath.udeE.edu, hsiaoOmath.udel.edu

In this paper we model the scattering of a time harmonic electromagnetic wave by an elastic body. Assuming that the interaction between electroamgnetic field and the elastic body occura on the surface of the scatterer we obtain a suitable transmission problem. Equivalent boundary-field equation formulations and a weak variational formulation in the appropriate Sobolev spaces are presented. Unique- nes results are also established.

1. The Mathematical Model Let Ri E R3 be a bounded, simply connected domain with smooth boundary r. We denote by Re the exterior domain R3 \ ni and nz the outer normal vector at x E r. Assume that 52i represents a homogeneous isotropic elastic body immersed in the medium fle where an electromagnetic field is considered. Furthermore we assume that, due to the physical properties of the elastic body on the surface r, the electromagnetic field produces a tangential stress on r which initiates elastic deformations of the elastic body and does not considerably penetrate inside the elastic body. We consider that the system "electromagnetic field - elastic body" is closed related to exterior forces, heat sources and any exterior energy transformation. Hence the electromagnetic energy crossing the surface r of the elastic body, which is not zero because of the surface coupling, is totally transformed into elastic energy. The problem is to determine the electric and magnetic scattered fields in the exterior medium 52, and the elastic displacement field in the body Ri. In Oe, the electric field E and magnetic field H satisfy time dependent

48 49

Maxwell’s equations

V.E=O, V.H=O where the electric permittivity ee, the magnetic permeability pe and the electric conductivity u, are given constants. On the other hand, in Ri the elastic displacement field u satisfies the source free elastodynamic equation d2U P~AU (Xi pi)V(V * U) - pi- --0 + + Ot where Xi, pi are Lamb constants and pi > 0 is the density 12. In it is shown that the electric and magnetic energy in a given closed volume R with smooth boundary dR satisfy the relation

The left-hand side of (1) is the rate at which the electric and magnetic energy is decreasing. In particular the second term on the right-hand side gives the rate at which energy is being converted into heat in this volume. Under the considered physical assumptions, any loss of energy which is not accounted for by heat must be occurring by a flow through OR. Accordingly we conclude that the rate of the flow of electromagnetic energy across the surface I? of the elastic body is given by Jr E x H . n ds. Next, the total energy of the elastic body in the absence of the exterior forces is given by

where the symmetric bilinear form &(u,v) := X(V . u)(V * V) + p C:,,=,% (2+ 2)corresponds to the strain potential energy. By applying the divergence theorem we obtain that the rate of the flow of the elastic energy across the closed surface I’ oriented to outside of the elastic body is given by d --Energ = l(piutt - A*u) . utdx + dt where T := 2pii . Vu(r) + XiiV - u(r) + pii x (V x u(r)) denotes the elastic surface traction operator. Thus, the basic law of energy conservation, applied to the closed system “electromagnetic field-elastic body “ determines that the interaction on the boundary l? must be such that the following relation holds -I E x H-rids = I Tu.Ut d8. (4) 50

Since (4) must be true for every subset of the closed surface I?, we can rewrite the interaction condition point-wise as follows H(x,t) x E(x,t) .n, = Tu(x,t).ut(x,t) for x E I', and t > 0. (5) Now, let us be restricted to the time harmonic electromagnetic field with frequency w, namely ~(x,t)= (ee + +)-1/2 E(x)eciUtand H(x,t) = pe1/2H(x)e-iwt,where E(x) and H(x) satisfy

V x E -ikH = 0, V x H+ikE = 0 (6) and the wave number k is given by k2 = (ee + *)pew2 with the sign of k chosen such that 8(k) 2 0. Consequently, the elastic displacement is also a time-harmonic field with the same frequency w and after factoring out e-iwt it satisfies

piAu + (Xi + pi)V(V * U) + piw2u = 0 (7) In this case the boundary interaction condition (5) takes the following form i -H(x) x E(x) . n, = Tu(x) . u(x) for x E r k (8) There are infinite many decomposition of the above condition. In particular Voigt's model (see e.g. 13) states that the stress tensor is proportional to the magnetic field. This justifies the following decomposition i -n, x H(x) = Tu(x) and nz x E(x) = n, x u, for x E r (9) k Note that, since Tu is assumed to be tangential on the boundary, simple calculations show that Tu . u = iH x E . n, whence (9) implies (8).

The classical interaction problem between the electromagnetic field and the elastic body, which we refer to as (CIP), can now be formulated as follows: Given smooth incident electromagnetic field Einc, Hinc that is an entire solution to (6) in R3,find E, H E C1(s2,) n C(s2, U I?) and u E C2((ni)n C1(0, u r) satisfying V x E - ikH = 0, V x H + ikE = 0 in fie, (10) A*u + piw2u = 0 in fIi (11) i -ii x (H + Hint) = Tu, ii x (E + Einc)= n x u on r, (12) k and the Silver-Muller radiation condition X E(x) x - + H(x) = o (13) 1x1 51 uniformly in all directions x/lxl. Here and in the sequel, for a vector valued function u, we use the notation u E X, where X is a function space, to mean that each component of u belongs to this space.

2. Uniqueness Let (E,H,u) be a solution of (CIP) corresponding to EiVac= Hinc = 0. We denote by RR := Ren {xE It3;1x1 < R} the annual region with dRR = I? U SR. We assume that the radius R is sufficiently large so that 52i is completely contained in SR. Applying the divergence theorem for E, E, H, H and using the transmission conditions we obtain

(lkI21Hl2- k21EI2)dx = ik ii x E . H ds + k2

We consider the following cases: 1) k and w2 are real. By taking the imaginary part of (14), using Betti’s formula we have O=&(Jsa iixE.Hds) =k,(L’Ih-iids), since

is real. From the Silver-Muller radiation condition and (15) (see 2, it follows

whence from Rellich’s lemma H = E G 0 in fie. Thus u satisfies A*u+piw2u=0 in Ri, (17) Tu=O and iixu=O onr, which not necessarily implies that u is zero in Ri. We denote by C(w) the set of pathological frequency w for which (17) has nontrivial solution. It is not our intention to discuss here these eigenfrequencies that, if they exist, depend on the geometry and the physical properties of the elastic body. 2) S(k)is strictly posative. In this case, it is shown in that E and H decay exponentially at infinity and 92 (ssRn x E . H ds1 + 0 as R + 00. Letting R + 00 in (14), we obtain

9 [Le(EIH12 - k(E12)dx - k (&(u, ii) - P~w~(u(~)dx 52

Obviously, if S(kw2) 5 0, (18) implies that E E 0 and H = 0 in fie, and u 3 0 in Ri. So we have proved the following uniqueness result Theorem 2.1. Assume that either k, w are real and w @ L(w) or S(k) > 0 and S(kw2)5 0. Then (CIP) has at most one solution.

3. Nonlocal and Variational Formulations In this section we give equivalent formulations for the problem (CIP) in order to provide a mathematical framework for proving the existence of the solution as well as approximating it by boundary element method or finite element method in bounded domains. It is known, from the fluid-solid interaction, there are two different kind of such approaches 59891133. One, which we call the boundary integral formulation, transforms the field interaction problem in Ri as well as in 52, into a system of integral equations on the boundary r. The other one, the so- called non-local boundary condition formulation, combines field equations in the bounded domain and appropriate boundary integral equations on r. The later leads to a variational formulation in the appropriate Sobolev spaces. Letting L;oc(52e),H1(52i) and Hi(r) denote the usual Sobolev spaces we define (see for details)

Hlo,(curl, 0,) := {U E ~12,,(0,) : v x u E L;oc(Oe)} HL~(r) := { u E ~-4(r), divr u E ~-4(r)}

Hc;kl(r) := {u E H-h(r), curlr u E x-t(r)}, Moreover we define the following boundary integral operators Ma : = 2nx x V, x s, a(y)@(x,y)ds, Na : = 2nx x V, x V, x I n, x a(y)@(x,y)ds, and

where +(x,y) is the fundamental solution of the Helmholtz equation and G(x,y) is the elastic fundamental tensor (mapping properties of these operators can be found in 1,5,618110). Now we are ready to give several equivalent 53 formulations of (CIP). They are obtained by using the Stratton-Chu formula, Betti's formula and integration by parts. Boundary integral formulation: Given an incident electromagnetic field EinC, HinC E H loC (curl,fle), find n x H E Hd

A*u + piw2u = 0 in Ri (21) i -n x (H + Hint) = Tu on r k (22) where n x H satisfies i -N(nk x n x H) + (I - M)(n x U) 5 (I - M)(n x Einc) (23) or i i (I - M)(n x H) - -N(n x ii x U) = --N(fi x ii x Einc). k k (24) As a consequence of the non-local boundary formulation, we may consider the following. Variational formulation: Given Einc,Hinc E HtOc(curl, a,), determine 1 (u,n x H) E H1(!Ii) x HA: (r)satisfying

d(u,n x H; v, T) = F(Einc,Hint; v, T) for any (v,~)E H1(fli) x Hci!l(I'), where i d(u,iixH;v,T):=u(u,v)-w~~~[u,v]--(~xH,?) k + (N(nx n x H),T)+ ((I - M)(n x u),T) ik i F(Einc,HinC;v,~):= -(nxHinC,?)+((I-M)(n~ Einc),?), k where (-,-) and (.,.) denote the duality pairing on HGt"(I') x Hcit{2(I') and the L2 duality pairing respectively, and a(.,.) is the sequilinear form 54

corresponding to the Lam6 operator (the expression can be found in 59s). We remark that for the existence result, we need a kind of Girding’s inequality for the corresponding bilinear form A(.,.). We hope to pursuit this in a separated communication.

Acknowledgments This research is supported in part by a grant from the Air Force Office of Scientific Research. The authors are indebted to Professor R.C. MacCamy for his valuable comments on the early version of this paper.

References 1. M.Cessenat, Mathematical Methods in Electromagnetism, Linear Theory and Applications, World Scientific Publishing Co. Pte. Ltd, (1996). 2. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering The- ory, Springer Verlag, 1992. 3. R.P. Gilbert and Z. Lin, Acoustic fields in a shallow, stratified ocean with a por-elastic seebed, Z. Angew. Math. Mech., (9) 77 677-688, (1997). 4. G.C. Hsiao, The coupling of boundary element and finite element methods, Z. Angew. Math. Mech., (6) 70 T493-T503, (1990). 5. G.C. Hsiao, On boundary-field equation methods for fluid-structure interactions, Problems and Methods in Methematical Physics, Teubner- Texte zur Methematik, 79-88, 1994. 6. G.C. Hsiao and R.E. Kleinman, Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics, IEEE Trans. Antennas, Propagat., (3), 45, 316-329, (1997). 7. G.C. Hsiao, Mathematical foundations for the boundary-field equation methods in acoustic and electromagnetic scattering, Analytical and Computational Methods in Scattering and Applied Mathematics, (Santosa, Stakgold eds.), Chapman Research Notes Maths., 417, 149-163, (2000). 8. G.C. Hsiao, R.E. Kleinman and G.F. Roach, Weak Solutions of fluid-solid interaction problem, Mathematische Nachrichten, (1) 218, 139-163, (2000). 9. D.S. Jones, Acoustic and Electromagnetic Waves Oxford University Press, New York, 1986. 10. A.de La Bourdonnaye, Some formulations coupling finite element and integral equation method for Helmholtz equation and electromagnetism, Numer. Math., 69, 257-268, (1995). 11. C.J. Luke and P.A. Martin, Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle, SIAM J.App1. Math., 55, 904-922, (1995). 12. V. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.U. Burchuladze Three- dimensional Problems of the Mathematical Theory of Elasticity and Thermoe- lasticity, North-Holland, New York, 1979. 13. G.A. Maugin, Continuum Mechanics of electromagnetic Solids, North- Holland, Amsterdam, 1988. BORE EVOLUTION IN INHOMOGENEOUS CHANNELS

J.-G. CAPUTO Laboratoire de Mathe'matdques, INSA de Rouen, B.P. 8, 76131 Mont-Saint-Aignan cedex and Laboratoire de Physique The'orique et Modelisation, Universite' de Cergy-Pontoise and C.N.R.S., fiance E-mail: caputoOinsa-rouen.fr

Y. A. STEPANYANTS Environment at ANSTO, PMB 1, Menai (Sydney), 2234, NSW, Australia, E-mail: [email protected]. au

The propagation of nonlinear surface waves in channels of arbitrary but smoothly varying cross-section has been studied theoretically and numerically. The mathematical model describing the evolution of these waves is a generalized Korteweg-de- Vries equation written in signaling coordinates with an additional term due to the spatial inhomogeneity of the channel. Specifically, we consider rectangular channels of variable depth or width and study the breaking of Riemann waves and the disintegration of a hydraulic jump into solitons. Simple rules are obtained which could be useful to understand qualitatively the propagation of bores in various river configurations.

1. Introduction The problem of shallow water propagation in rivers or channels has been a matter of interest for at least the two last centuries and remarkable progress has been achieved. Despite this, some questions remain unresolved concerning the influence of inhomogeneities in the river bed. These could be described as a damping in the case of small-scale inhomogeneities or need to be incorporated in the equation for large-scale inhomogeneities. Here we will study the latter in the very simple case of a rectangular river bed of variable depth or width. Such a simple approach was also chosen by Whitham and allowed him to obtain some very useful estimates for the description of a bore. Here we follow the same direction and analyze within the framework of Korteweg de Vries (KdV) equation the influence of the channel geometry on Riemann-wave breaking and the decomposition of a hydraulic jump into solitons. We ignore dissipation and show that a de-

55 56 crease of the depth of the river gives a much shorter breaking distance and decomposition distance than a decrease of the width. Other effects that we do not discuss here are the amplification of the wave amplitude and the de- lay in propagation. We present all these effects together with an extensive numerical study in an full article to appear2.

2. The model To describe the evolution of the water elevation q in a rectangular channel of depth h and width 1 we can use the derivation of the KdV equation for a rectilinear channel of constant cross-section done by Das3. In the dimensional variables this reads 677 1 arl - + - [l - alq]- - a2- a3v = 0, at co ax ax3 where the damping term has been omitted. The coefficients are the velocity of long linear waves, co = m, the nonlinearity a1 = 3/(2h~)and the dispersion a2 = h2/(64).These coefficients are identical to the ones for an infinite width channel1. We now extend this to a curvilinear coordinate frame (see Fig. 1) so that the undisturbed free surface is taken as the z = 0 plane with the axis z directed vertically and intersecting the lower part of the river bed. The axis x is directed along the river and may smoothly bend in space (it is the so-called ray coordinate - the distance along the ray from some fixed point). The axis y is directed across the river perpendicular to both the x and z axes. The x, y and z axes form an orthogonal coordinate frame at each point in space. To describe the variation of the channel’s cross-section with distance, one needs to consider all the coefficients of the KdV equation (1) as z-dependent (because the channel’s depth is variable, in general) and add also an extra term (q/2A)(dA/dx),where A = QZ. This term comes from the law of conservation of wave energy flux across the channel’s cross- section area, Q = pg~(x)l(x)q~/2,where Z(z) is channel’s local width. In a second step we transform (1) into a Cauchy boundary value problem by going to signaling coordinates because the aim is to predict the spatial evolution along the channel of a perturbation that is known as a function of time at some fixed point, at the boundary of the domain of interest4. Note that the constant coefficient KdV equation has been elegantly solved on the quarter plane x < 0, t > 0 by Marchant and Smyth5 by matching particular solutions of the KdV equation on the infinite line. However this approach cannot be used in our case because of the inhomogeneity. Instead, we note 57

<+ __-- Y

Figure 1. Sketch of a channel bend and a curvilinear coordinate frame. that the derivation of the KdV equation assumes that in the zero-order approximation, wave processes obey the simple wave equation -+q)-=oa77 877 dt ax (2) and all other effects (nonlinearity, dispersion, dissipation, etc.) appear as higher order terms. This implies that the following relation between derivatives -d = ---ld ax co at' (3) which follows from (2), can be used for the higher-order approximations. Hence, one can replace the spatial derivatives by time derivatives in the nonlinear and dispersive terms. As a result, Eq. (1) takes the following form (c. f. ') a77 1 all a377 1 dA(z) - + -[l - a(x)Q]77- - P(z),. + --77 = 0. (4) ax co(x) at at 2A(x) dx

2.1. Conservation laws Equation (4) is not completely integrable, although it possesses at least two integrals of motion. The first one corresponds to the conservation of momentum in a homogeneous medium 58 where the integral is taken over a period of the wave for periodic perturbations or on the whole line for solitary waves. The second conserved quantity corresponds to the energy in a homogeneous medium: A s q2 dt = const. (6) From these conservation laws, one can estimate how the amplitude of a given initial perturbation varies in space. In particular a linear dispersion- less perturbation has a duration T which is independent of its amplitude A. In this case both these integrals give the same law of wave amplitude variation with distance if the parameter of the channel cross-section A(x) is known

A(~)- A(~:>-I/~+ A(~) h-1/41-1/2 (7) In the nonlinear case the situation turns to be a bit more complex because the wave duration and the amplitude are usually linked. For a KdV soliton of the form

q(z,t)= Asech2 [$ (t - [K)] V(x')

there are relationships between the amplitude and duration (see, e. g., 6y7!1), T = J-, and between the amplitude and wave velocity, I/ = ca/(l- aA/3) M co(l+ aA/3). Substituting these expressions into (6),one obtains instead of (7)

, -- A(z)A2(z>T(x)/ sech4(8)do = A(x)A~/~(~) = const. (9) -a For channels of variable depth and constant width this implies a nonlinear analog of Green's law A(x) - h-'(s) (see where generalized Green's laws were obtained both for solitary and periodic perturbations). When the width of the river varies and the depth is constant, Eq. (9) gives ~(z)- 1-2/3(4.

2.2. Simplification of the main equation (4) First of all, let us exclude the inhomogeneous term from (4). This can be done via the change of dependent variable: - u(x,t) = s(z)q(x, t), where s(x) = {w.*(XI 59

The resultant equation takes the form:

dU 1 dU 63u - - p(z)- = 0, dz + [qq-P(.)"] dt dt3 where p(z) = a(z)/(~(z)s(z)).In the remainder of the paper we will assume that the boundary condition is u(z = 0,t) = U9(t/T),where a is a given function of unit amplitude.

3. Breaking of Riemann waves We first consider very long and smooth perturbations for which dispersive effects are insignificant. This means that the corresponding dispersive term (w p) in Eq. (11) can be dropped to yield

where V(u,z)= q,(z)[l - a(z)u/s(z)]-'.The solution can be obtained in implicit form as

According to the theory of nonlinear waves1, the evolution of a smooth "initial" perturbation described by (13) leads in general to the development of a discontinuity in the wave profile. The distance at which the discontinuity appears, xb,is calculated as the minimum root of the transcendental equation (cf. ')

After trivial manipulations this condition can be presented as:

For a detailed analysis of this formula one can specify the character of the river inhomogeneity and wave profile at a given boundary. In the case of a channel of linearly decreasing depth h = ho(1- ~z)one obtains 60 where

Taking for example: ho = 10 m, g = 9.8 m s-' leads to Q = 0.15 m-l, P = 0.017 s3 m-l. -4ssuming a sinusoidal initial perturbation @(t)= Asin(wt), A = 1 m, w = 0.5 s-l, one obtains X: = 132 m, for K = 0, and xb = 60 m for K = lo-' m-l. In the case of a linearly decreasing width 1 = lo(1 - KZ) the breaking distance is

xb=- 1- l--KXbO MXb" 1--nx; . K"(3'] (:> Notice that the coefficient 1/4 is 3.5 times smaller in the latter case than in the former one, so that the influence of a decrease of the width on the distance of breaking is much weaker.

4. Bore disintegration into KdV solitons The wave breaking described in the previous section occurs only in the limiting cases when dispersion and dissipation can be neglected. In general, these effects are small but present and can prevent breaking. In particular, third order dispersion leads to the formation of a train of KdV solitons from a step-wise initial condition l. This disintegration of a bore into KdV solitons has important practical consequences since these have an amplitude that is double of the initial condition l. These precursor waves can be seen in many bore observationsg, for example, in the Seine riverlo and can be very dangerous for the navigation. This also comes from the fact that these KdV solitons have a velocity that is proportional to their amplitude, so that the larger solitons ride ahead of the smaller ones. First, for simplicity, consider the homogeneous case (s G l), when (4) has constant coefficients. In this case, rescaling variables

leads to the following KdV equation in dimensionless form av -- v- av - -asv = 0. at a7 a+ The boundary condition for the function v is just the unit Heaviside function in this case: u(0,~)= H(7). 61

One can then define a criterion for the emergence of a first soliton if the height of the first trough is less than 10% of the amplitude of the first pulse. This distance was estimated numerically for (19) by one of the authors1’ and found to be 6, M 45. Going back to the dimensional units one obtains X, = &8Jm= 10h5/2U-3/2, so that for U = 1 m and h = 10 m we get X, = 3160 m, while if U = 2 m, then X, = 1120 m. A smaller river depth h = 5 m yields a significantly smaller X, = 560 m for U = 1 m. In the inhomogeneous case, it is not possible to use the estimates given above since the coefficients of the equation are x dependent. One can however reduce the KdV equation (11) to its simplest form via the transformations

This results in av av a3v - - v- - B([)- = 0, at 67 ar3 where

Assuming a channel of linearly decreasing depth h(z) = ho(1- nz),one obtains

where ( = (Up(0))31/”/(n~).If the channel’s width is linearly decreasing, the dispersion coefficient varies as

Fig. 2 shows 3d and B, (lines marked dl and wl) as a function of the variable E = r/(. Notice how Bd decreases much faster than B, and also that B, reaches zero at a finite 5. This unwanted feature is eliminated if we assume an exponential taper of the river mouth, a very common hypothesis for many rivers1/”like the river Seine13. We present in Fig. 2 3d and B, for an exponential decrease of the river depth and width (lines marked d2 and w2). The fact that Bd decreases much faster than B, indicates that X, will be much smaller when the river depth is decreased than when the river width is decreased. This is confirmed by the numerical solution of (4) 62

Figure 2. Evolution of Bd and B, as a function of the normalized variable 2.

2.5

1300 1320 1340 1360 z

Figure 3. Time dependence of V(T) for < = (2/3)<. The parameters are ho = 10m,U = lm and IE =

v((, T) displayed in Fig. 3 for E = (2/3)( and for the homogeneous, variable depth and variable width cases. The homogeneous case is shown as a solid line, the variable depth case as long dashes, and the variable width case as short dashes. One easily sees that a soliton has completely emerged from the variable depth solution as opposed to the other two cases. Notice also the narrowness and amplitude (> 2) of this leading soliton due to the small value of the dispersion coef- 63 ficient. This amplification of the wave is even stronger in the laboratory coordinates because of the impact of the factor ~(2)~.

5. Acknowledgments The authors thank E. N. Pelinovsky for useful advice and many discussions. Y. S. is also grateful for the invitation from the INSA de Rouen, France where this work was initiated, and appreciative of the hospitality during his visit in April 2001.

References 1. G. M. B. Whitham, "Linear and Nonlinear waves", Wiley (1974). 2. J.G. Caputo and Y.Stepanyants, Nonlinear processes in geophysics, submitted (2002). 3. K. P. Das, Phys. Fluids, 28, n. 3, 770-775 (1985). 4. A. R. Osborne, Chaos, solitons and fractals, 5, 2623-2637 (1995). 5. T. R. Marchant and N. F. Smyth, I.M.A. J. of Applied Mathematics, 47,247 (1991). 6. M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Tkansform, SIAM, Philadelphia (1981). 7. V. I. Karpman, Nonlinear Waves in Dispersive Media, MOSCOW,Nauka (1973) (in Russian). (Engl. transl.: Pergamon Press, Oxford (1975).) 8. L. A. Ostrovsky and E. N Pelinovsky, Sow. Phys. Izvestia. Atmospheric and Oceanic Physics, 6, n. 9, 552-555 (1970); Sou. Phys. Izvestia. Atmospheric and Oceanic Physics, 11, n. 1, 68-74(1975). 9. See the very well documented site by Hubert Chanson http://uuu.uq.edu.au/'e2hchans/mascaret.html 10. See for example the sites http://molay.chez.tiscali.fr/mascaret.htm and http://vvv.univ-lehavre.fr/cybernat/pages/mascaret.htm 11. E. Pelinovsky and Y.Stepanyants, Izv. VUZov, Radiojizika, 24,n. 7,908-911 (1981) (in Russian). 12. N. C. Mazumber and S. Bose, Journal of Waterway, Port, Coastal and Ocean Engineering, 121,n. 3, 167-169 (1995). 13. J. F. Le Floch, Propagation de la mare'e dans l'estuaire de la Seine et en Seine-maritime, These de Doctorat, Centre de Recherches et d'etudes oceanographiques, Paris (1960). AN INVERSE SPECTRAL PROBLEM FOR A SCHRODINGER OPERATOR WITH AN UNBOUNDED POTENTIAL

LAURE CARDOULIS Universite' de Toulouse 1 Place Anatole fiance, 31 000 Toulouse, E-mail: cardouliOmath.univ-tlsel

MICHEL CRISTOFOL AND PATRICIA GAITAN Universite' de Prowence, CMI,UMR CNRS 6632, 39, rue Joliot Curie, 13453 Marseille Cedex 13, fiance, E-mail: cristoQcmi.univ-rnrs.fr, gaatanOcmi.uns'v-rnr-s.fr

In this paper, we prove an uniqueness theorem for the potential V(z) for the following Schrijdinger operator H = -A + q(lz1) + V(z) in R2,where q(lz1) is an increasing radial known potential satisfying limlzl-t+oo q(Iz1) = +ca and V(z) is a bounded potential.

1. Introduction Recently, a lot of papers have dealt with inverse problems for Schrodinger

Operators in the whole or half space (see for example 394,5,9). In these papers, some decreasing hypothesis for the potential at infinity are assumed. Our aim here is to study an inverse problem for a potential which tends to infinity at infinity. To our knowledge, such inverse problem has never be investigated. Let Q be a potential such that:

Q E LtoC(R2), Q(z) 1 Q where Q is a constant, lim &(z) = +cm. Ixl++oo We recall the well-known result that the operator H = -A + Q, considered as an operator in L2(R2), is with compact inverse. Its spectrum is discrete and composed with eigenvalues: A1 5 A2 5 ... 5 A, +n++oo +m. hrther- more, the first eigenvalue XI is simple and associated with an eigenfunction b1 > 0. (see 1,2i8). In the present paper, one considers the following problem (-A + q(Iz1) + V(z))u= Xu in R2 (1)

64 65 where X E R*+, q is an increasing radial known potential, q(Ix1) E C2(R) : limlzl++ooq(lz1) = +m and V is a bounded potential. If 1x1 = T, then

where J(q1/2 - %q-'/2) denote a primitive of (q1l2 - %q-'j2) and u is a solution of (1.1). Let the following items: (Hl) q(Ix1) + V(x)2 a where a is a constant (H2) VN E N*,3CNE R*+, IV(T,d)I 5 CN(l+r-)-Ne-.fq"2 033) limr-b+Oo $+) = 0

(H4) and $!$) are q.zql:2(T), $@) Jq)(I (T (H5) *, *and $# E L1 [I,+4 (H6) q-1/4e-.f (91J2-*9-1J2) E L2[1, +m[ (H7) s,'" 0 (e-2S (91/2-49-1'2)) = 0 (e-2S (91J2-X 2 q -1/2 1)

x -1/2 (H8) ~-'/~e=f(q'/~-Tq) f! L2[1, +oo[ For example, q(lx1) = 1xI2 satisfies each previous hypotheses. Our aim is to prove the following theorem. Theorem: Let V(x) and W(x)be two bounded potentials with compact support both satisfying (Hl) and (H2). Denote (&(V))l (resp. (&(W))E)the eigenwal- ues of the operator -A+q(lzl)+V(x) (resp. -A+q(lxl)+W(x)). Assume also (H3) to (H8).

We follow a method used by H.Isozaki for the anharmonic operator in R2. In the first step, we study the asymptotic behaviour of the solutions of a second order differential equation which derives from equation (1). Then we prove that under hypotheses (Hl) to (H8), the constant Cx,,,(V)exists. In the second step, we prove that the first eigenfunctions associated to V and W are the same and since the first one is strictly positive we are able to come to the above conclusion.

2. Asymptotic behaviour of a solution Here we study the asymptotic behaviour of the solutions of the following equation (-A + q(lx1) + V(x))u= Xu in R2, where q satisfies q(lz1) E C2(R),lim~,~,+,~(I~~)= +GO and V is a bounded potential. Denote 66

1x1 = r. We use polar coordinates to define up(.) = r1l2s,"" u(r, 8)e-ipedB. We obtain -$(r) + (y+ q(r) - X)up(r) = fp(r), where fp(r>= -r1/2 s,"" v(r,e)u(r, 6)e-ipede. Theorem 2.1. Let q and V be potentials defined as before satisfying (Hl) to (H8) , X and a two positive reals. Then the equation a -u"(r) + (7+ q(r) - X)u(r) = 0 r (2) has a system of fundamental solutions with the asymptotic behawiour:

-- Furthermore, the Wronskian GI;~ - = 1.

Proof : first steD We set y = (:'), then using (2) we have :

We denote

If we combine the two relations (5) and (6), we obtain :

where P' is the derivative of the matrix P with respect to the r variable. Next, we compute for each term of the matrix PI a limited development; this allows us to write (7) in the form

Remark : We denote R(q) a 2 x 2-matrix where all the coefficients are in the form 0 ( T2q1;2(T)7 *, $#) = O(.2g'/2(p))1 +O(*) +O($#) * second stel, 67

We look for P2 such that z' = A(q)z+ R(q)z with A(q) a diagonal matrix. Then (8) can be written :

10 with = A0 ( 0 -1 ) and A2 = (-'-1 -1-'>. The derivative of the equation (9) and the equation (10) allows us to write

z' = ($/2 - - Q' 2q1,2x )A0 + -(P2Ao - AoP2 + A2) z + R(q)z. [ 4q 1 We denote A(q) = (q1/2- &)A0 + $(P2A0 - AoP~+ A2). Then, if we set P2 = ( 1y2 -:I2), we have :

z' = A(q)z R(q)z, A(q) = + -qv + third step We look for the asymptotic behaviour of z. We search a diagonal matrix E(q) such that E'(q) = A(q)E(q).For this we introduce the new variable w defined by z = E(q)v,with 0 E(q) = 1(q-1/4e-.f2 0 E2

Then we have v' = E-l(q)R(q)E(q)v=

We define the application T and the subset F as follows :

v1 = - J," KlV(t)dt w2 = (2 + Jr.',K2v(t)dt

-2~(~1/2-a and 7 = {(wI, w2);vl = o(e P'), Iv2(z)l < +m). Then for a suitable choice of TO and (2 # 0, the application T : F -+ F is a contraction and therefore we can find an unique w solution of (11). So we construct z then 5 and y. From the asymptotic behaviour of z, which is given by the terms of the matrix E(q),we obtain the asymptotic behaviour of u in the form u - Elvl - E2v2. Since v E 3, &v1 = o(1) and v tends to 52 # 0 when r tends to infinity, we deduce that E2 is an asymptotic behaviour of a solution of (2). Then we prove that El is another asymptotic behaviour of a solution of (2) using a classical change of function. Then we 68 deduce (3) and(4) and that the wronskian is equal to 1. This concludes the proof. Applying Theorem 2.1 and using up (above defined) we obtain the following lemma :

Lemma 2.1. Let u be in L2(R2) such that (H - X)u = 0 for X a positive real with

where q is an increasing radial potential which tends to infinity at infinity and q and V satisfy the hypotheses (Hl) to (H8). Then the foIlowing limit exists

3. Unicity Theorem We can now state the unicity theorem. We consider the operators - A +q(Izl)+V(z)and -A+q(la:l)+W(z) where q(Iz1) is an increasing radial potential verifying the hypotheses (H3) to (H8), V(z)and W(z)with compact support.

Theorem 3.1. Let V and W two bounded potentials with compact support. If

A1 (V)= A1 (W) Vl E W,Vp E N, and then V = W. ICX,,P(V) = CXi,P(W) Proof: Let XI < X2 < . . . the eigenvalues of the operators - A +q(lzl) + V(z) and -A+q(lzl)+W(z). Let (PI, cpz, cp3,. .. the normalized eigenfunctions associated to V,and let $1, $2, $3,. . . the normalized eigenfunctions associated to W. We assume that suppV c {z; 1x1 < R} and suppW C {z;1x1 < R},R is a fixed real. We are going to prove that cp1(z) = $1 (z),for all 2. first step We prove that V1, cpl(z) = $l(z), if 1.1 > R. Recall that if 1x1 > R then (- A +q.(l~l>)(Pl= Xlcpl and (- A +4l4))$l = h+1- We decompose cp1 and $1 in the trigonometric functions basis {eWike}kand we prove that all the coefficients are equal. For this, we state 69

kz-l Ifr > R then bl,k(r) satisfies -b”l,k(T)+(~+q(T)--l)bl,k(T) = 0. This equation has the following fundamental system of solutions (cf theorem 2.1)

But, bl,k(~)E L2 * c2 = 0, so we can write that bl,k(T) = clul(r). In the same way, we prove that if 6l,k are the functions associated to $JL(T) then ?)L,~(T)= 611u1(~). By using these hypotheses, Xl(V) = Xl(W) and CX,,~(V)= CX,,~(W),we have bl,k(T) = ?)l,k(T), ‘dl and Vk. This concludes the first step. second step It is necessary to prove that ‘pl and $1 are equal everywhere. We consider in the distribution’s sense

K(z,Y) =cVl(!/){$l(z)-Vl(z)) =c$Jl(z){ql(Y) -$Jl(Y)) 1 1 Note that K is an ultrahyperbolic operator and K(z,y) = 0 if 1x1 > R and x # y. We can add that Vz K(z,y) := (- A +q(lzl) + v(~))~K(s,y) = 0 if 1x1 > R and Iyl < R.

This series converges because we can write

~t(z,y)= Ce+’ ~lrl(z)cpr(y)- Ce-tAt$~l(z)$~l(y). 121 I2 1

Since e-txl = C+?n-0 n! (-Xl).l, using (12), we obtain

+m tn = -i(-Vz)n K(z,y) = 0 if 1x1 > R and IyI < R. n. n=O 70

Now, multiplying Ft(z,y)by etXl, we obtain, etXIFt(s, y) = o = +l(x){cpl(y)-+l(y)}+X e-t(xt-xl) $l(Z> {cpl (Y)-$1 (Y)1. 1 >2 We prove that the limit, when t tends to infinity, of the second term of the previous sum is equal to zero, then we have $l(z){cpl(y) - $l(y)} = 0 for 1x1 > R and IyI < R. So, since $I(z)has no zero, (see '), cpl(y) - $l(y) = 0 for lyl < R. Now, we can write, [- A +q(lzl) + V(x)](cp1(x)) = [- A +q(IzI) + W(x)](cp1(z)), and, since (cpl(s)) has no zero, we can conclude that V(z) = W(z)for all x E R2. We can note that such potential like q(Ix1) = lxln, (n 2 2) works fine if we replace the hypothesis (H2) by

Moreover, since the exponential function is increasing too rapidly ((H4) is not checked) and since the logarithmic function is not increasing sufficiently ((H5)is not checked), so these two potentials do not satisfy our hypothesis.

References 1. S. AGMON,Bounds on Exponential Decay of Eigenfunctions of Schrodinger Operators, Schrodinger Operators, (Como, 1984), Springer, Berlin, 1985, pp 1-38. 2. D.E. EDMUNDSAND W.D. EVANS,Spectral Theory and Differential Opera- tors, Oxford Mathematical Monograph, Clarendon Press Oxford University Press, New York, 1978. 3. G. ESKINAND J. RALSTON,Inverse coefficient problems an perturbed half space, Inverse Problems 15, 1999 vol 3, p.683-699. 4. J.C. GUILLOTAND J. RALSTON,Inverse ScatterPng at Fixed Energy for Lay- ered Media , J. Math. Pures Appl. vol 1, 1999 p 27-48 5. H. ISOZAKI,Inverse scattering theory for wave equations in stratified media, J. Diff. Eq., vol 138, 1 (1997) p.19-54 . 6. H. ISOZAKI,Cours sur les ProblBmes Inverses, Juin 1991, Universitk de Provence, Marseille, France. 7. H.P. MCKEANAND E. TRUBOWITZ,The Spectral Class of the Quantum- Mechanical Harmonic Oscillator, Commun. Math. Phys., p.471-495, 1982. 8. REEDAND SIMON,Method of Modern Mathematical Physics, vol. 4, Analysis of Operators, Academic Press, New York, (1978). 9. R. WEDER,Multidimensional inverse problems in perturbed stratified media, J. Diff. Eq., vol 152, 1 (1999) p.191-239. TRAPPING REGIONS FOR DISCONTINUOUSLY COUPLED DYNAMIC SYSTEMS

S. CARL Fachbereich Mathematik und Informatik, Institut fir Analysis Martin-Luther- Universitlt Halle- Wittenberg 0-06099 Halle, Germany E-mail: carlOmathematik. uni-halle. de

J.W. JEROME Department of Mathematacs, Northwestern University Evanston, IL 60208-2730, USA E-mail: jwjOmath.northwestern. edu

1. Introduction Let R C RN,N 2 1 be a bounded domain with C1-boundary do, Q = R x (0,T) and r = dR x (O,T),with T > 0. We consider the following initial-boundary value problem (IBVP for short): k = 1,2

&L k uk = 0 on R x {O}, -+gk(ul,u2) = 0 on r, avk where ak E Loo(Q) with ak (5,t) 2 pk > 0 in Q, and d/dVk denotes the outward conormal derivative at I' related to the corresponding elliptic operator. It should be noted that the method we are going to develop is applicable to more complicated systems, in which the elliptic operators may be the sum of a monotone divergence type operator and lower order convection terms, the vector fields f and g may depend, in addition, on the space-time variables (x,t)and the initial condition may be nonhomogeneous, i.e., of the form Uk(2,O) = '$k(Z) with '$k E L2(R). Even mixed Dirichlet-Robin type boundary conditions can be treated. Only for the sake of simplifying our presentation and in order to emphasize the main idea we consider here problem (l),(2) as a model problem. Existence results for discontinuously coupled elliptic systems have been obtained by the authors in 235

71 72

The novelty of the IBVP (l),(2) is that the vector fields f and g may be discontinuous in all their arguments. In order to formulate the conditions imposed on the vector fields we introduce the following terminology.

Definition 1.1. A vector field h = (hl,h2) : R2 -+ Rz is said to be of competitive tgpe if the component functions hl(sl,s2) and h2(~1,s2) are both separately increasing in s1, SZ. The argument sk of hk (sl,s2) is called the principd argument. A vector field h = (hl, hz) : R2 + R2 is said to be of cooperative type if hk (s1,SZ) is increasing in its principal argument and decreasing in its nonprincipal argument.

From Definition 1.1 it follows that IBVP (l),(2) with f and g of cooperative type can be transformed into a system of competitive type by the simple transformation (w1, w2) := (u1, -212). Thus cooperative and competitive systems are qualitatively equivalent. Throughout the rest of this paper we assume the following hypotheses on the vector fields f and g:

(Hl) The component functions fk : R2 -+ R and gk : R2 + JR are Baire- measurable and satisfy a growth condition of the form

Ifk(Sl,S2)1 Ic(l+ Is11 + Iszl), v (SlrS2) E R2, 19k(~l,S2)15 c(l+ Is11 + 14, v (S1,SZ) E R2, where c is some positive generic constant. (H2) The vector fields f and g are assumed to be of competitive type.

2. Notations and Preliminaries Let W'>2(R) denote the usual Sobolev space of square integrable functions and let (W'3z((n))*denote its dual space. Then by identifying L2(R) with its dual space, W'J(52) c L2(R) c (W'>2(R))*forms an evolution triple with all the embeddings being continuous, dense and compact, cf. 6. We let V = L2(0,T;W1i2(R)),denote its dual space by V* = L2(0,T; (W'>2((n))*), and define a function space W by dW W={w€VI -eV*),at where the derivative dldt is understood in the sense of vector-valued distributions. The space W endowed with the norm llwllw = llwllv + Ilaw/atllv* is a separable and reflexive Banach space. We introduce the natural partial ordering in L2(Q) by u 5 w if and only if w - u belongs to the cone 73

L$(Q) of all nonnegative elements of L2(Q). This induces a corresponding partial ordering also in the subspaces V and W of L2(Q),and if 5 w then [u,w] := {w I u 5 o 5 w} denotes the order interval formed by u and w. Further, if (B,<)is any ordered Banach space, then we furnish the Cartesian product B x B with the componentwise partid ordering, i.e., z = (z1,z2) 5 (y1,y2) = y iff za 5 ya, k = 1,2. Thus the order interval [z,y] c B x B corresponds to the rectangle R = [XI,y1] x [z2,2/2] c B x B. In what follows we will make use of the following Cartesian products: X := V x V, Y := L2(Q)x L2(Q),and Z := L2(r)x L2(r).We denote by (., -) the duality pairing between V* and V, and by y : V + L2(r)the trace operator which is linear and continuous, and if considered as a mapping y : W + L2(r)it is even compact. In order to apply functional analytic methods to the IBVP (l),(2) we introduce operators Ak generated by the elliptic operators -V . [akVw],and Fa and Ga related to the vector fields f and g, respectively, as follows: Let k = 1,2, and cp E V

One easily verifies that Ak : V + V* is linear and monotone. By (Hl) the operators Fk : X + V*and GI,oy : X + V*are well defined and bounded, but not necessarily continuous. The time derivative d/dt : V + V*is given bY

with < ., . > denoting the duality pairing between (W'%2(Q))*and W1i2(Q), and we denote its restriction to the subspace of functions having homogeneous initial data by L, i.e., L := d/dt and its domain D(L) of definition is given by

D(L) = {u E W I u(x,O)= 0 in Q}.

The linear operator L : D(L) C V + V* can be shown to be closed, densely defined and maximal monotone, e.g., cf. 6. With u = (u1,u2), Au = (Aiui,Azuz),YU = (y~1,~2),F(u) = (Fi(u),F2(~)),G(u) = (Gl(u),Gz(u)),and Lu = (Lul,Lu2) the weak formulation of a solution 74 of system (l), (2) reads as follows: Find u E [D(L)I2c X such that the following vector equation holds:

Lu + Au + F(u)+ G 0 ~(u)= 0 in X*. (3) In view of the discontinuous behaviour of the operators F and G this notion is, however, too restrictive for establishing a solution theory. One can construct simple examples of systems satisfying the given hypotheses without having any solutions in the sense of (3). Therefore, to establish a consistent solution theory for the discontinuous system (l),(2) we extend the notion of its solution by introducing multivalued vector fields a = (al,a2) and P = (PI,P2) associated with f and 9, respectively, as follows:

Ql(Sl,S2) = [fl(Sl-7S2),fl(Sl+, s2)1, (4) Q2(S17S2) = [f2(Sl,S2-),f2(sl,s2+)1, where f1 (slf,82) and fz(s1,saf) denote the corresponding one-sided lim- its. Thus ak : R2 + 2’ \ 8 is the maximal monotone graph of fk with respect to the principal argument Sk. Correspondingly, Pk : R2 + 2’ \ 8 is the maximal monotone graph of gk with respect to its principal argument. In what follows we consider instead of the IBVP (l),(2) the following multivalued version of it: awe -at + Akuk + ak(u1,~2)3 0 in Q,

Next we develop the concept of a trapping region for systems which is an appropriate extension of the notion of super- and subsolutions in the scalar case. To this end let R = [g, fi] be the rectangle formed by the ordered pair -u = (g1,g2)and ii = (Gl,ii2), where 2, fi E W x W.

Definition 2.1. The vector field &/at + Au + F(u)+ G o ~(u)is called a generalized outward pointing vector on the boundary dR of the rectangle R if the following inequalities hold for all cp E V n L:(Q):

(8~1/at + AIU~+ F1 (21 v) + GI 0 Y(U~7 v), ‘P) 5 0, v 21 E [IL~,fiiz]; (a22/dt+A2g2++2(21,g2)+G20~(~,212),(~)5 0, Vv E [gi,fiil]; (a.iil/at+Alfii,+Fl(fil,v)+Glo~(iil,v),c~) 20, VvE [~2,G2];

(afi2/at+ A2fi2 + F2(v,%) + GZ0 ~(v,&), ~p)L. 0, v E [21,fill. Using the notion of the generalized outward pointing vector we define the trapping region. 75

Definition 2.2. Let u, ti E W x W satisfy 5 a, and u(z,O) 5 0 5 O(z, 0). Then R = [a,ti] is called a trapping region for the system (l), (2) if &/at + Au + F(u)+ G o y(u) is a generalized outward pointing vector on dR. Our main goal is to show that each trapping region for the system (l), (2) contains a solution of its multivalued version (5), (6) in the following sense. Definition 2.3. The vector u E D(L) x D(L) C X is a solution of the IBVP (5), (6) if there is an E E Y and an q E 2 such that for k = 1,2 the following holds: 6) 5kb,t) E %(%(z,t),u2(z,t>),for a-e. (z,t)E &, (ii) vk(4E Pk(yul(z,t),~~12(z,t)),for a.e. (z,t)E r, (iii) (Luk + Akuk + C$k + y*vk,cp) = 0, V cp E V, where y* : L2(r) + V* denotes the adjoint operator to the trace operator y with

For the analysis of the multivalued system (5), (6) it will be convenient to use its equivalent formulation in terms of a discontinuously coupled system of evolution variational inequalities of the form: Find Uk E D(L) such that for all cp E V we have

(Lui + Aiui, 50 - .I) + J1 (~p,u2) - Ji(u1, '112) +a1 0 y(cp, 212) - 91 0 y(w,u2) 1 0, ( 7) (Lu2 + -42~2,~- ~2) + Jz(u1,~) - J2(~1,~2)

+92 0 Y(% ,cp) - 92 0 Y(W, u2) 1 0, (8) where the functionals Jk and 9k are defined by 76

By hypotheses (Hl) and (H2) the functionals JI,: Y + R and : 2 W are well defined, convex and locally Lipschitz continuous with respect to their principal argument. Since X is dense in Y and y(X) is dense in Z we obtain by applying Theorem 2.2 and Theorem 2.3 by Chang in along with the chain rule (see, e.g., 7, p.4031) as well as the sum rule for subgradients (see, e.g., 7, Theorem 47.B]) that systems (5), (6) and (7), (8) are equivalent.

3. Main Result The proof of our main result is based on existence and comparison results for evolution variational inequalities obtained in and the following fixed point theorem in ordered normed spaces, (see l, Proposition 1.1.1). Lemma 3.1. Let b, 74 be a nonempty order interval in an ordered normed space (N,r), and let P : [g,C] + [14, C] be an increasing mapping, i.e., v 5 w implies Pv 5 Pw. If monotone sequences of P([g,ii])converge weakly or strongly in N, then P has the least fixed point u* and the greatest jixed point u* in [g,a]. Our main result is given by the next theorem.

Theorem 3.1. Let R = b, C] be a trapping region in the sense of Definition 2.2. Then the mudtivalued system (5), (6) and its equivalent system of evolution variational inequalities (r), (8) possesses solutions within R.

Proof. For convenience we recall the system (7), (8) of evolution variational inequalities: Find uk E D(L) such that for all cp E V we have

(Lui + Aiui, 9 - ui) + Ji (9, UP) - Ji (211,212) +a1 0 Y(cp, u2) - $1 0 y(u1, u2) 2 0, (9) (Lu2 + A2~2,9 - u2) + J2(uit9) - J2(~1,~2)

+a2 0 y(u1, 'p) - a2 0 y(u11.2) L 0. (10)

Let g = [g1,u2]and ti = [ii1,C2]. Then we define first a mapping 7 as follows: [gl,GI] 3 v1 I+ 7wl = z, where z is a solution of the evolution variational inequality (10) with u1 := v1 fixed. By applying the property of the trapping region a thorough analysis shows that for any 211E [gl,611 the functions fi2 and g2 are super- and subsolutions for the corresponding equation related to (lo), which ensures the existence of a unique solution z = 7vl of (10) satisfying z E [u2,fi2],~f.~. Moreover, by means of the monotonicity assumptions on f and g it can be shown that 7 : [gl,011 C 77

V + [u2,fiz] is decreasing, cf. ’. Now, by means of the evolution variational inequality (9) we define a mapping S on the range of 7 in the following way: If z = Tvl, then z ++ Sz := u1, where u1 is the uniquely defined solution of the evolution variational inequality (9) with ug = z fixed, which in view of the property of the trapping region satisfies u1 E [gl,fil].Moreover, in an analogous way as for 7 one can show that S : 7([g1,fi1])+ [gl,fil] is also decreasing, see ’. Hence, it follows that the composed operator P = S 0 7 : [gl,7311 -+ bl,fill is an increasing operator from the interval [gl,fil]C V to itself. To apply the abstract fixed point result given in Lemma 3.1 we need to show that any monotone sequence of the image P([gl, GI]) converges weakly or strongly in V, a proof of which can be found in ’. Thus Lemma 3.1 ensures the existence of extremd fixed points of P in [gl,fill. Finally, let u1 be any fixed point of P,i.e., u1 = Pu1 = S(7ul). Then if u2 := 7u1, it follows that u = (ul, ug) is a solution of the system (9),(10) within R, which completes the proof. 0

Remarks. A thorough analysis including existence and comparison results for evolution variational inequalities that appears in the proof of Theorem 3.1 has been given by the authors in ’. In applications a trapping region for (l), (2) can be found very often in form of a rectangle R = [a,b] in Rg , where a = (al,a2) 5 0 and b = (bl, b2) 2 0 are constant vectors such that f and g are outward pointing vectors on dR.

References 1. S. Carl and S.Heikkila, Nonlinear Differential Equations an Ordered Spaces, Chapman & Hall/CRC, London, 2000. 2. S. Carl and J.W. Jerome, Trapping region for discontinuous quasilinear elliptic systems of mixed monotone type, Nonlinear Analysis, (to appear). 3. S. Carl, S.Heikkila and J.W. Jerome, Trapping regions for discontinuously coupled systems of evolution variational inequalities and application, (in preparation). 4. K.C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102- 129. 5. J.W. Jerome, A trapping principle for discontinuous elliptic systems of mixed monotone type, J. Math. Anal. Appl. 262 (2001), 700-721. 6. E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. II A/B: Monotone Operators, Springer-Verlag, New York, 1990. 7. E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. 111: Vari- ational Methods, Springer-Verlag, New York, 1985. DIFFERENTIAL CALCULI

ROBERT CARROLL University of Illinois, Urbana, IL 61801 Email: rcarrollQrnath.uiuc. edu

We discuss here various aspects of "quantum" calculus.

1. Hopf algebras and differential calculi This is more than just a study of q-special functions and involves an attempt to build quantum mechanical (QM) ideas into the calculations from the beginning (and eventually also relativistic ideas). One recalls the ideal of q-calculus with e-g- (All Dqf(z) = [f(az)-f(z)ll"a-l)f(z)l; Dq(fg) = f(a.)D,g(z) + g(z)D,f(z); Dqz" = [nIz"-l; 1.1 = [a" - ll/[a - 11;

So where is the quantum? One needs quantum groups to do this properly; quantum groups are quasitriangular Hopf algebras which means essentially Hopf algebras plus braiding and leads e.g. to quantum invariants of knots (see 3317,19). Historically one goes back to quantum inverse scattering (QIS) B la the Faddeev school (we refer to " for an exposure via classical inverse scattering). Alternatively one can start with WZW models, conformal field theory (CFT), and the Knizhnik-Zamolodchikov (KZ) equation (cf. 'J9). First for QIS consider a vertex (statistical mechanical) model based on an M x N lattice (with periodic boundary conditions); adjacent bond states are i,j,t, t with (Boltzman) weights Ri; and indices 1 + n (cf. 7317). One considers R E End(V @ V) with V - C". The partition function (generating functional of correlation functions) is the sum over all bond states of products of weights and the model is physical if R satisfies certain unitarity, symmetry, and positivity conditions. It is exactly solvable if R = R(A) (parameter A) satisfies the quantum Yang-Baxter equation (QYBE) (A2) R12 (X)R13(A')R23 (A") = R23 (A")R13 ( A')R12 (A) (A" + A = A' sometimes - the subscripts refer to different positions in

78 79 tensor product action). Given (A2) one constructs transfer matrices and the partition function can be expressed in a simple manner via the transfer matrices from parameters of the model. Thus (A3) T(X)$i:::t: = R(A)$;llR(X)R;$;. . . R(X)$"-lkN where T(X)$E End(VN),and when the QYBE holds, (A4) R (X)E,T (X')jmT (A''); = T (A") kT (A') 2 R(X)r.

Definition 1.1. An algebra A over a field k is denoted by (A,., +, k) with (A,., +) a ring and a compatible multiplication of k on A, so (A,+, k) is a vector space and X(ab) = (Xa)b = a(Xb). In terms of commutative diagrams one has (A5) . (id 8 -) = -(. 8 id) with -(q 8 id) = id = -(id8 q) where qu : k + A is a linear map qu(X) = Xu with qu(l) = a (so q N qla). A coalgebra (C, +,A, E, k) over k is a vector space (C,+, 1) and a coproduct A : C + C 8 C which is coassociative and for which there exists a linear counit map E : C + k such that (A6) (A 8 id) o A = (id 8 A) o A; c = (E 8 id) o A(c) = (id @ E) o A(c)

One writes here (A7) A(c) = C c1 8 c2 N c1 8 c2 (Sweedler notation) and leading to (A8) c18c218 c22 = c11 8 c12 8 c2 = CI 8 c2 @ c3. One can think here of a "sharing", e.g. A(c) = C c1@Q with c1 A c2 = 0 and c1 V c2 = c. There are adjoint maps to A and E defined via (A9) . : C* 8 C* + C* and q : k + C* where (A10) (4. +)(c) = 4(cl)$(c2); q(X)(c) = XE(C) for 4, + E C* and c E C. The unit in C* is Ice = E and (C*,-,q, k) is an algebra. Note also (All) A(c 8 d) = c18dl 8 c2 8 d2 and a coalgebra map respects coalgebra structure via (f @ f)o A = A o f and E o f = E.

Definition 1.2. A bialgebra (H,+, ., q,A, E, k) is a vector space (H,+, k) which is both an algebra and a coalgebra in a compatible way. Thus e.g. (A12) A(hg) = A(h)A(g); A(1) = 181; E(hg) = E(h)E(g); ~(1)= 1 while . : H 8 H + H and q : k 4 H are coalgebra maps where H @ H has the tensor product coalgebra structure (~(1)= 1 is automatic). A Hopf algebra (H,+, ., q,A, E, S, k) is a bialgebra with a linear antipode S : H + H such that (+) (S 8 id) o A = -(id 8 S) o A = q o E. S2 = 1 is not required nor is S assumed to have an inverse but S is unique and satisfies A13) S(hg)= S(g)S(h), S(l)= 1, (S 8 S) o A(h) = T o A o S(h),and ESh = Eh where ~(a8b)= b@u. Note also that the axioms imply (A14) €(a)= C alS(az) = c S(a1)az; a = C al~(a2)= c ~(al)a2.For a coalgebra A and an algebra B there is a natural convolution product on L(A,B) = k-linear maps A + B, namely (me - multiplication in B) (A15) (f * g)(a) = (me o (f @ g)AA)(a) = c f(al>g(a2) and L(A,B) becomes an algebra with unit 0 €A. 80

Definition 1.3. We follow 15717 and recall that a Hopf algebra H is cocommutative if TOA= A (where ~(a@b)= b@a) and this can be weakened by considering a Hopf algebra that is only cocommutative up to conjugation by an element R E H @ H, and obeys other suitable properties. A quasitriangular bialgebra or Hopf algebra is a pair (H,R) (R - R-matrix) where H is a bialgebra or Hopf algebra with R E H @ H invertible and satisfying (A16)(A @ id)R = 72137223; (id@ A)R = R13R12; T 0 Ah = R(Ah)R-l. Writing R = C R1@ R2 the notation involves (A17)Rij = 1 @ . -.@ R' @. +.&I R2IB. . . @ 1 so Rij involves R action in the ith and jth positions. rn

It can be easily shown that for a quasitriangular bialgebra (A18)(c @ id)R = (id &I c)R = 1 while for a Hopf algebra one has in addition (A19)(S @ id)R = R-l and (id &I S)R-l = R and hence (S @ S)R= R. F'urther one has the QYBE (A20)R12R13R23 = R23R13R12. Now for quantum groups and the R matrix we follow 23 (cf. also 15!17118). As a 2-dimensional model consider matrices with ab = qba, ac = qca, and

T = (: :) ; ad = da + Xbc, bc = cb, bd = qdb, cd = qdc (1.2) where q E C, q # 0, and X = q - q-l. Note det,T = ad - qbc is central (i.e. commutes with a, b, c, d). One considers the free associative algebra generated by l,a,b,c,d modulo the ideal of relations above and in this algebra Q formal power series are also allowed; if det,T = 1 then T E SLq(2).

Consider SL,(2) with relations (cf. 15318,23) x 12x = qx 21x ; 61x1 = l+q2X%l+(q2-1)5262; 61x2 = qx261; 62x1 = qx162;

11- 21 62x2 = 1 + q2x262; 6162 = q-ld261; dx x - x dxl; dx1x2= qx2dx1+ (q2 - l)X1dX2;dx2x1 = qX1dX2; dx2x2 = q2s2dx2(1.3)

Here x1 and x2 generate a quantum plane V N Ci or H = SL,(2) comod- ule where SL,(2) is described via the R matrix and relations above (with det,T = 1); we will use H and Q = t)(SL,(2)) interchangeably at times (Q - coordinate Hopf algebra generated by 1, a, b, c, d modulo relations and det,T = 1). One goes on to define actions, coactions, antipodes, units, etc. and for C; one has FODC (first order differential calculi) Fk with e.g. r+ described via

xi . d~j= qdxj . xi + (a2 - l)d~i. ~j (i < j); xi * d~i= q2dxi . xi; (1.4) 81 xj.dxi = qdxi-xj (i < j); dxi Adxj = -q-'dxj Adxi (i < j); dxi Adxi = 0 and (4) ~i~j= QSjXi (i < j); &tlj = q-'aj& (i < j); 8iXj = qXj& (i # j) with &xi - q2x& = 1 + (q2 - 1)&i xjaj and r- arises by replacing q by qP1 and i < j by j < i in the formulas above. Note for q # 1, I?+ and r- are not isomorphic and for q = 1 they both give the ordinary differential calculus on the correponding polynomial algebra C[xl,. . . , x,]. From these formulas one derives by induction the expressions for the actions of 8, and 8, on general elements of D(Ci) and for polynomials g and h, e.g. (A211 r+ : &(g(Y)h(x))= g(qy)Dqz(h)(x);%(g(y)h(x)) = Dq2(g)(y)h(z).Similarly for polynomials f, g (A22) &(fg) = Dq2(fg)= D,.(fg) == g(x)W(x)+ f(q2z)aZg(4.

2. Formulation for integrable systems (A) The proper setting for differential calculi seems to be via quantum groups (cf. Woronowicz 24) but we do not use this here. (B) Rather we go to a long series of papers by Dimakis, Miiller- Hoissen, et al. (cf. 1~3~8~g~10~11).Take A an associative algebra with 1. A differential calculus (DC) is of the form (R(A),d)with R = @TOk where Ro = A, Rk - A bimodules, d : Rk + Rk+', d2 = 0, d(ww') = (dw)w' + (-l)kw(dw') for w E Rk,and di2k-1 generates flk as an A bimodule. Here (R'(A),d) is a FODC and one takes m or p. as multiplication m : A @ A + A : f @ h + fh; this is a bimodule homomorphism (but not generally an algebra homomorphism). Let d = ker(p.) - A2 and define (A23) 2 : A + d1 : f + 1 @ f - f @ 1. Then (dl(A),d) is the universal FODC on A and with fip = fil @A ... @A d1 (p times) d(A) = @rfiP (graded associative algebra) is universal DC on A. 2 is extended via d(f0 @ . . . @ fp) = EE" (-1)qfo @ . . . @ fq-l @ 1 @ * * . @ fp and C linearity (note e.g. d(fo @ fi) = 1 ~3fo @ fi - fo @ 1 @ fi + fo @ fl @ 1). One can produce a number of interesting insights into integrable models and their deformations. Example 2.1. Consider

0 (A) dtdt = dtdx + dxdt = dxdz = 0 0 (B) d(fg)= (df)g+ (-l)deg(f)fdg and d2 = 0 0 (C)[dt, t] = [dx,t] = [dt,x] = 0; [dz,X] = qdt Then e.g. d(zz) = (dx)x + z(dx) = xdx + (xdx + qdt) = 2xdx + qdt with d(x2x)= (ds2)x+ z2(dx)= (2xdx + qdt)x + x2dx = 3x2dx + 3xqdt, etc. leading to (A24) d(x") = nxn-'dx + $n(n- l)qxnP2dtand dzf = 82 f dx + q fxdt; df = fxdx+ (ft + !jq fxx)dt Take a connection A = wdt + udx with curvature (field strength) F = dA + A2;then dA = dwdt + dudx = w,dxdt + (ut + (1/2)qu,,)dtds = (ut - wx + (q/2)uX,)dtdx while A2 = (wdt + udx)(wdt+ uds) = quu,dtdx. Hence (A25) F = (ut - w, + $uzx + quux)dtdx = 0 + ut - w, + !fuxx + quu, = 0. If e.g. w, = 0 one has Burger’s equation. rn Example 2.2. Consider

(A) [dt,t] = [dx,t] = [dt,~]= [dy,t] = [dt,y] = [dy,y] = 0; [dz,z]= 2bdy; [ds,y]= [dy,z]= 3adt (B) dtdt = dtds + dxdt = dydy = dxdz = dydt + dtdy = dyds + dxdy = 0

This yields then (A26) df (x,y,t) = f,dx + (f, + bfxx)dy + (ft + 3afx, + abfxx,)dt. Now set A = vdx + wdt + udy and write out dA + A’; thus (A27) dtf = fdt; dyf = fdy + 3afxdt; dxf = fdx + 2bfZdy+ 3a(f, + bf,,)dt. and F = 0 requires u, = v, + bv,, + 2bvvx; w, = 3avxY+ abv,,, + 3awX+ 3av(v, + bvxx);(2.1)

wy+ bw,, = ut + 3auxy+ abu,,, + 3auux - w[2bw, - 3a(u, + bu,,)] Now take e.g. (*) w, = (3a/2b)uY + (3a/2)uX, in the third equation to decouple (using compatibility) and one obtains (A28) a, (ut - ~u,,,ab + 3auux) = 8, ($uy) For suitable choices of a,b this is the famous KP equation. Further if there is no y dependence of u,v, w then formally a KdV equation arises for suitable a, b by rewriting. rn Similar calculations can be made involving differential calculi for quantum planes (cf. 6). One can also generate many features of integrable systems using bidifferential calculi (with d and S) and/or bicomplexes. We recall now the f- Moyal (star) product (6 - h/2) (A29) f *g = f exp [$(%,SP- apSx] g (A301 f * &>= ezP [ew( pk4j f (x+ t>dX+ 5)1,=,=,. For a - quantum plane TY - 9YX (~31)f * = Q(1/2)(-z’a,,~a~+xa,~’aYl)f (x,y)g(x’, Y‘)ld,y‘+z,y. There are now legitimate inquiries about deeper connections between quantum mechanics (QM) and integrable systems (aside from questions concerning quantum integrable systems (QIS) via R matrices, etc.). One knows that e.g. KP can be considered as a quantization of dKP via deformation quantization with star products and further q-KP can be also considered as a QM 83 theory (cf. 3). For example in l3 it is observed that there seems to be no new physical idea connected with quantizing a simple Hamiltonian via deformed commutation relations when this would be equivalent to a complicated Hamiltonian quantized canonically. However new phenomena can arise in the realm of time evolution. In particular the reduction of time dependent Schrodinger equations to eigenvalue (EV) problems via exp( -iEt/h) can be extended to more complicated time evolution in a natural manner leading to integrable systems. Thus e.g. an EV equation of Schrodinger type (A32) - diqh(x,t)+ u(x,t)qh(z,t)= Xqh(z, t) with X independent of t is compatible with a time evolution qht = Bqh provided L = 3: + u and B satisfy a Lax pair condition (A33) dtL + [L,B] = 0 This opens up the idea of treating e.g. KdV in a QM spirit (for B = -48: + 3ud, + 38,~).In connection with KP and Moyal we recall from l4 (cf. also 2y16s21i22) that if L = d + xy u,(x, t)d-, be the Lax operator for KP then one can apply the geometrical framework to obtain a Moyal KP hierarchy KP,, based on deformation of dKP, which is equivalent to the Sat0 hierarchy based on PSDO. Similar considerations apply to Toda and dToda, KdV and dKdV, etc. It seems from this that if one starts with dKP as a basic Hamiltonian system with Hamiltonians 23, and standard P brackets then KP, can be considered as a quantization of dKP in some sense with quantum integrals of motion B,(n) which for K, = 1/2 say is equivalent to KP (cf. 14). In any case one can formulate the KP hierarchy as a quantization of dKP under the Moyal bracket. The actual correspondence is not important here and one could simply define KP as (KP), for K, = 1/2 and express it through phase space (X,P)Moyal brackets. In fact there exist similar correspon- dences for q-KP and dKP under suitable q-Moyal type brackets and this is pursued in more detail in along with variations involving qKP and q- pseudodifferential operators (qPSDO), the latter suggesting a sort of qQM.

Remark In 'i3 we also discuss at some length the Seiberg-Witten (SW) map of 2o between ordinary and noncommutative (NC) gauge theories based on a Moyal type product (cf. for zero curvature representations).

References 1. H. Baehr, A. Dimakis, and F. Miiller-Hoissen, Jour. Phys. A, 28 (1995), 3197-3222 2. R. Carroll, Quantum theory, deformation, and integrability, North-Holland, 2000 3. R. Carroll, Calculus revisited, Kluwer, in preparation 4. R. Carroll, Jour. Nonlin. Sci., 4 (1994), 519-544 5. R. Carroll and Y. Kodama, Jour. Phys. A, 28 (1995), 6373-6378 84

6. R. Carroll, Differential calculi for quantum planes, in preparation 7. V. Chari and A. Pressley, A guide to quantum groups, Cambridge Univ. Press, 1004 8. A. Dimakis and F. Miiller-Hoissen, Jour. Phys. A, 29 (1996), 5007-5018; math-ph 9809023, 9908016; nlin.SI 0008016, 0008029; hep-th 0007015; Jour. Phys. A, 27 (1994), 3159-3178; Jour. Math. Phys., 35 (1994), 6703-6735; hep- th 9401150; Phys. Lett. B, 295 (1992), 242-248; Inter. Jour. Mod. Phys. A, Supp. 3 (1992), 474-; Lett. Math. Phys., 28 (1993), 123-127; hep-th9401151, 9408114; q-alg 9707016; physics 9712004; hep-th 9608009; math-ph 9908015; gr-qc 9808023, 9908022; physics 9712002; nlin.SI 0006029, 0104071; hep- th 0006005; math-ph 9809023; math-ph 9908016; hep-th 0007015, 0007074, 0007160; Jour. Phys. A, 29 (1996), 5007- 5018; Lett. Math. Phys., 39 (1997), 69-79; Jour. Phys. A, 25 (1992), 5625-; Inter. Jour. Mod. Phys. A, Supp. 3A, 1993, pp. 474-477. Note that a reference such as hep-th 9608009 refers to an entry in the well-known (in the theoretical physics and mathematics communities) arXiv electronic bulletin board. To access this reference, go to (e.g., the french mirror site of arXiv) website http://fr.arXiv.org, then to the hep-th section, and finally enter the paper number 9608009. 9. A. Dimakis, F. Muller-Hoissen, and F. Vanderseypen, hep-th 9408114 10. A. Dimakis, F. Muller-Hoissen, and T. Striker, Jour. Phys. A, 26 (1993), 1927-1949; Phys. Lett. B, 300 (1993), 141-144; quant-ph 9509014 11. A. Dimakis and J. Madore, Jour. Math. Phys., 37 (1996), 4647-4661 12. L. Faddeev and L. Takhtajan, Hamiltonian methods in the theory of solitons, Springer, 1987 13. D. Fairlie, Differential geometric methods in theoretical physics, World Sci- entific, 1993, pp. 196-200 14. J. Gawrylczyk, Jour. Phys. A, 28 (1995), 593-605 15. A. Klimyk and K. Schmudgen, Quantum groups and their representations, Springer, 1997 16. B. Kupershmidt, Lett. Math. Phys., 20 (1990), 19-31 17. S. Majid, Foundations of quantum group theory, Cambridge Univ. Press, 1995 18. Yu. Manin, Quantum groups and noncommutative geometry, CRM, MontrBal, 1988 19. T. Ohtsuki, Quantum invariants, World Scientific, 2002 20. N. Seiberg and E. Witten, hep-th 9908142 21. I. Strachan, Jour. Phys. A, 28 (1995), 1967-1975 22. K. Takasaki and T. Takebe, Inter. Jour. Mod. Phys. A, Supp. 1992, pp. 889-922; Rev. Math. Phys., 7 (1995), 743-808 23. J. Wess, math-ph 9910013, Springer Lect. Notes Physics, 493, 1999, pp. 311-382 24. S. Woronowicz, Comm. Math. Phys., 111 (1987), 613-665; 122 (1989), 125- 170; Lett. Math. Phys., 23 (1991), 251-263 RECONSTRUCTION PROBLEM FOR A PERIODIC BOUNDARY BETWEEN TWO MEDIA

J. CHANDEZON Universitb Blaise Pascal, LASMEA, 24 Avenue des Landais, 63167 Clermont-Ferrand, France

A.Ye. POYEDINCHUK, N.P. YASHINA Institute of Radiophysics and Electronics of National Academy of Sciences of Ukraine, 12 Ak. Proskury St., Kharkov, 61085 Ukraine E-mail: nataliya@lin. com.ua

The reconstruction problem for a periodic (with arbitrary profile within a period) boundary between two homogeneous media is considered. This class of inverse problems is of interest for remote sensing and monitoring of the earth’s surface. Here we apply analytical numerical methods (C-method, completed by Tikhonov regularization) for solving the reconstruction problem. A scheme for numerical tests of the algorithm and criteria for reconstruction accuracy are proposed and verified. Results of extensive numerical experiments are presented.

1. Introduction

Non-destructive control and remote sensing of land, agricultural fields and soil, and of their inner geometrical, structural and physico-chemical properties, is a problem of great importance nowadays’. Existing methods and approaches for wave diMaction by a local or periodic inhomogeneity can be separated conditionally as follows (see references in24): 1. approaches, using the Green’s formulae technique, which lead to volume and boundary integral equations of the first and the second kind, including methods of analytical regularization; 2. the method of partial domains, including semi-inversion procedures; 3. the methods of perturbation theory (small inhomogeneities, small slopes of the boundary surface, etc.); 4. the “differential” methods; 5. the methods of boundary “straightening”, which are a rather efficient specialization of “differential” methods; 6. the incomplete Galerkin method.; 7. methods, based on field representations by means of Rayleigh harmonics (theory of series, involving non-orthogonal functions, in general);.

85 86

8. asymptotic methods.

The theoretical background for remote sensing problems requires not only the efficient resolution of the direct diffraction problem, but also a solution of the corresponding inverse problem of the reconstruction of the interface boundary between two media. The latter is an ill-posed problem. This is why its solution requires the construction of a relevant regularizing algorithm. Fundamental results in the area of inverse problems are connected with the names of Lavrent’yev and Tikhonov’. Tikhonov’s regularizing method reduces, as is well-known, to the minimization of a regularizing functional based on the resolution of a series of corresponding direct problems. The specificity of the problem of interface boundary reconstruction, which is considered below, requires a corresponding definition of the regularizing hctional. The robust and clear implementation method that is presented herein for solving the problem of acoustic wave scattering by rather arbitrarily-shaped surfaces contains certain modifications of one known in optics as the C method4*.

2. Direct problem

We consider two-dimensional diMaction problems for plane pressure waves striking an arbitrarily-profiled boundary between two media with sound velocities c, , c2 and densities pl and p2.The boundary curve between the two media is described by the function z = a(y) with period d and maximal deviation from the y axis equal to h. The incident pressure wave propagates in the first medium with angle of incidence cp . The time factor is chosen to be e-jU , The excitation field has the form pi = eik(Ysin(qbcos(q)), with k, = @Ic1, The diMaction field must meet the following requirements: i) the acoustic wave equations; ii) the radiation conditions at infinity; iii) the transmission boundary conditions, iv) the conditions of continuity of pressure and normal-to-the boundary components of velocity; v) the quasi periodic Floquet conditions); vi) the condition of energy boundedness in any finite domain. It can be proved, (see, for example 2, that these conditions guarantee the uniqueness of the direct diffraction problem solution. For convenience we introduce the following variables: Z = Kz, = @, i?= 2x/d . Then the acoustic equations for the pressure and velocity fields acquire the forms 87

with n = 1 and n = 2 referring to the first and second media, P, denoting the pressure field, and (V_.Vm) the velocity fields in the different media. The equation defining the boundary between the media now takes the form Z = 4u(J), A, = 2xh/d, where a(J) is a periodic fimction with period 2n such that OIu(J)Il. For the sake of simplicity, we consider the case q=O. All derivations for the case cp # 0 can be obtained in the similar way. Notations such as b(J) means derivation with respect to the argument. Following the conventional C method, we introduce the new variables v = 7 , u = Z - %&), which transform (1) into the form

wherein K,, = m/cn G,, = P, , G,, = -iwp,lrV,. The equation describing the boundary becomes u = 0 and transforms the boundary conditions into

Further operations are connected with: i) the transformation of (2) and (3) into an infinite system of linear algebraic equations with respect to the coefficients of the expansions of functions Gl , Gl, over the system of eigenfunctions of relevant spectral problems of the C method (see '), and ii) the application of a regularizing procedure'. This leads to the equation 88

Here F'is the conjugate of the operator F defined via GI, and GI, and ad is a regularizing parameter of the direct problem. The form of the operators in (4) are presented in '. Thus the original acoustic problem results into (4), which can be efficiently solved numerically by a truncation method.

3. Inverse problem

The input data for the inverse problem, formulated as a minimization problem, is the set of complex amplitudes R = (R, (A)r=-Nof reflected propagating pressure waves, A being the wavelength in the upper medium. We suppose that this data is known in a certain range [A,, 43. The period of the boundary, sound velocities and densities of the media are also known. The problem is to determine the boundary between the two media. Let a = (urn):=--be the Fourier coefficients of this boundary function. The solution of operator equation (4) gives the mapping that associates the set a = (an):=-_with the set of complex amplitudes R = (Rn(A)r=_N. Thus, the non-linear operator

F(aJ)=R(A), h.4.-21 (5) is defined on a certain set of vectors DF cI, . The mathematical statement of the inverse problem is: find the solution of (5) such that the residual F(a,A)- R(A) is minimized in a relevant metric. Having found the Fourier coefficientsa = (a,);=, from (9,we can obtain the function describing the boundary between the two media. This can be done by means of a stable procedure which is simply the summation of a Fourier series with approximate (in 1, space metric) coefficients5. Consider the functional in domain DF of the operator F :

where a > 0 is the regularization parameter, R = 1 (in general, R 2 1 is a parameter of the functional), Am E [A,,&], ab)=$,a,$"y. Vector a, =(~,,iwhich "=-Q provides the hctional(6) with a minimum, is considered to be a solution of (5).

4. Numerical experiments 89

The search for vector a, is organized by means of a regularized quasi Newton method with step adjustment, using only first derivatives. The minimum residual method is applied for the choice of the regularizing parameter a . Here we present several numerical illustrations for test problems., We simulated the input data R~ (Am) = (R; (A, )r=-.,,,m = 1,2.. .P by relying on the solutions to (5) for

4Y) 1"""I

Figure 1. Reconstruction of boundary shape for the profile given by fhction a,(y) (Fig.a) a,(y) (Fig.b), for P=6, c,lc,=2.25, p, =p,, 0.5.SddlL<3.5, 2hld=0.4. two different types of boundary profiles: a, (,,)= h[ 0.5 + -2(: -- 1 1 1 -- :)I , a2G) = h [0.375+ 0.25~h(2a- + 0.125Co {:)I - which are periodically continued from interval [O,d] onto the interval (- w,+m). Parameters d and h meet the restriction 2nh/dI1. The wavelength of the incident wave varies within the range 0.5Id/il13.5, the relation of sound velocities in media being c2/cI =2.25. Functions a,(y), i = 1,2, were chosen so as to belong to essentially different classes. In particular, a,(y) can be expressed as a finite series of its Fourier coefficients, whereas a,(y) requires an infinite number of tenns due to the fact that the Fourier coefficients have only an algebraic type of decay. The results of numerical tests are presented in Fig. 1. The solid lines correspond to the exact functions a,(y), i = 1,2 . The lines depicted as crosses are the graphs of the functions af((y),i=1,2 that have been defined via input data Re(Am)= (Ri (Am)x=-Naccording to the above-described algorithm. As they almost coincide with graphical accuracy, the deviations 10h-' (a,(y)- up (y]), i = 1,2 are 90

presented in the same figures as dotted lines. It worth emphasizing that the maximum absolute value of deviation essentially decreases with increasing P . It is well known, that one of most complicated problems in solving unstable reconstruction problems is that of matching the regularizing parameter a with the given level of errors in the input data R:" . A tradeoff can be obtained be means of the residual method'. We demonstrate this statement for the test problem of shape reconstruction of the surface, described by the flmction u2(y) . Based on the solution to the direct problem, we have calculated the input data Rk = R:(An)(l + y.Rand) for various levels of relative error y. The error Rand has been simulated by the normal distribution random number generator. One of the numerical examples is presented in Fig. 2. Panel a) depicts the characteristic behavior of the relative error of profile reconstruction estimated according to the

point point b

a a

Fig.2 (a) relative error; (b) reconstruction with non-optimal choice of regularization parameter; (c) reconstruction with optimal choice of regularization parameter.

n - 1,2.. .N; N I1 0 . Here uz are the exact values of the Fourier coefficients of the function u2(y).Function &(a)has a pronounced minimum which manifests itself for all considered levels of error y. In Fig. 2 (b) and (c) are presented the results of the boundary shape reconstruction for various 6,a. Relyng on our numerical experiment, we can conclude that the reconstruction can be performed with best accuracy for that value of a , which provides a minimum to the fknction 91

6(a).The residual method can give rise to a reliable determination of the optimal value of a according to the relation sop, = sup(a : Acl,(y)ly). Here (I

is the relative residual of input data R& and RZ,, which are the results of the solution of the direct problem, calculated for function a:(y) found fiom minimization of (8) for a given y . From the results of the numerical experiments, we see that Am(a) depends monotonically on a , and, thus aopris unique for each level of input data error. The suggested algorithm, which reconstructs the shape of the periodic boundary between two media relying on information pertaining to diffraction harmonics that are known within a certain interval of wavelengths, requires a starting approximation for the Fourier coefficients of function a(y) describing the boundary. Such an approximation can be constructed by generalizing results described in ’,

References

1. Van Oevelen P., and D. Hoekman , Radar Backscatter Inversion Technique for Estimation of Surface Soil Moisture: EFEDA-Spain and HAPEX - Sahel case Studies, IEEE Trans.Geosci.Remote Sens., GE-37, 1999, 1 13-124. 2. Petit R., ed., Electromagnetic theory of gratings. (Springer, Heidelberg, 1980). 3. Shestopalov V. P., Yu. A. Tuchkin, A, Ye. Poyedinchuk and Yu. K. Sirenko. Novel Methods of Solving of Direct and Inverse Problems of Dfjaction Theory. Analytical Regularization of Boundary Value Problems in Electromagnetics. (Kharkov, Osnova, 1997). (in Russian). 4. Li L., J. Chandezon, G. Granet, and J. -P. Plumey, A rigorous and efficient grating analysis method made easy for optical engineers, Pure Appl. Opt., 21, 1999, 121- 136. 5. Tikhonov A. N. and V. Ya. Arsenin. Metho& for the solution of ill posed problems, (Mir, Moscow, 1986). (in Russian). 6. Chandezon J., A. Ye. Poyedinchuk, and N. P. Yashina, Dfjaction of electromagnetic waves by periodic boundary between two media: C method and Tikhonov ’s regularization, in 2001 URSI International Symposium on Electromagnetic the0ry. Victoria, May 13-17,2001,24 1-242. 7. Hadson V. and J.Pim. Application of Functional Analyses for the Operator Theory. (Mir, Moscow, 1987). (in Russian). A NOTE ON GENERALIZED CESARO OPERATORS

DER-CHEN CHANG Department of Mathematics, Georgetown University Washington D. C., 20057, USA E-mail: [email protected]

ROBERT GILBERT Department of Mathematics, University of Delaware Newark, Delaware, 1971 6, USA E-mail: gilbertOmath.udel.edu

GANG WANG Department of Aerospace Engineeeng, University of Maryland College Park, Maryland, 20742, USA E-mail: gwangOeng.umd.edu

In this note we show that the adjoint operator of the generalized Ceshro operator is bounded on the weighted Bergman spaces dP(dVa) if and only if a + 2 < p. We also study the link of this operator with the Libera transform on the polydisk D, in Cn.

1. Introduction and preliminaries Let D, be the unit polydisc in the n-dimensional complex space C" and dV(z) = rnny==, rjdrjddj the normalized Lebesgue area measure on D,. Let X((D,) be the space of all analytic functions in D,. Recall that for p > 0 and d = (al,.. .lan)with aj > -1 for j = 1,.. . ln, the weighted Bergman space dp(dV5) consists of the functions f E Z(Dn)such that

92 93 where

and r . eie = (rleiel,.. . ,meien). For a complex number y with Re (y) > -1 and a nonnegative integer k, denote A; the kth coefficient in the expression 1 03 = A:xk. (1 - .)7+1 k=O

It is easy to see that A: = ~r+l)"'(r+k~.k! Let us first concentrate on the case n = 1. For an analytic function

@=O on D, the generalized Ceshro operator is defined by

These operators were first introduced in and have been studied extensively by many mathematicians (see e.g., 2,4,8J3J4. For y = 0, we obtain the classical CesBro operator Co = C. It is known that the operator C is bounded on the weighted Bergman spaces (see e.g., 1y335,12) as well as other function spaces (see e.g., 4~6y13915 1. For y E C with Re (y) > -1, let C; be the adjoint operator of C,, which is formally defined by

where f has the Taylor series (2). Unlike the operator C,, the adjoint operator C; is only formally defined. This definition can be formulated as an integral

where cpt(z) = 1 - t + tz for 0 < t < 1. The adjoint operator was first considered in in the case y = 0, in which the author proved that the operator C; is bounded on dp(dV5) if and only if Q + 2 < p. In this paper, we will generalize this result to the case when y # 0 with Re(?) > -1. 94

Let @ = (41,.. . ,4,) : D, + D, be an holomorphic map. Then it is well-known that the composition operator T+(f)= f o @ is bounded on dP(dV,) when n = 1 (see e.g., and the references therein). In fact, we have the following result for general n. Lemma 1. Let @ = (41,.. . , 4,) : D, + D, be analytic and noncon- stant and p > 0. Then the norm of the operator Ta (f)= f o @ satisfies

on weighted Bergman space dP(dVz).Here Ci = 1 if Cri 2 0 and Ci =

22, ..., z, we have

where Cl = 1 if 01 2 0 and Cl = (-\I"' if -1 < a1 < 0. Integrating (5) over D in z2, then applying Fubini's theorem and [g, Lemma 11 on the right hand side in the second coordinate, we obtain that for almost all z3,..., zn

Repeating this procedure we obtain the result. The following lemma is well known, see [7, p.651. Lemma 2. For each 1 < p < 03 there is a positive constant C = C(p) such that

J -??

The aim of this note is to generalize the result on C,* to the operators C;, when y # 0. Indeed, we have the following theorem. Theorem 1. The operator C; defind by (4) is bounded on the weighted Bergman spaces dP(dVor)af and only if a: + 2 < p. 95

2. Proof of the main result In this section we prove the main result in this note. Proof of Theorem 1. Without loss of generality, we may assume that -1 < y < 00. Case 1. a + 2 < p. If a + 2 < p, then p > 1, because a > -1. By Minkowski integral inequality (see e.g., [I1, p. 2711) and Lemma 1, with = qt, we obtain:

Since the last integral converges for a + 2 < p, we obtain the result in this case. Case 2. a + 2 > p. Let fl(z) = 1/(1 - z). Using polar coordinates and applying Lemma 2 we obtain

for a + 2 > p. Hence fl E dP(dVa) for a + 2 > p. On the other hand, for all -1 < y < 00, C; have no sense on the function f1 and consequently C;fl is not in dP(dV,) in this case. Indeed, it is well known that (see e.g., [16, p. 771)

from which easily derive the following sequence

does not converge. 96

Case 9. a + 2 = p. Note that p > 1 in this case. Let 1 -1 f2(z) = - (1 log L) . 1-z z 1-2

The only singularity of f2 is at z = 1. Using polar coordinates centered at z = 1, we see that the integral

s, If2(Z)lP(l - Iw-2~v(4 is equiconvergent with J:'2 p-l(log $)-Pdp < oo since p > 1. Hence f2 E dP (dVp-2). On the other hand, from the following equality

we obtain that for dl y E (-l,oo), C; have no sense on the function f2 and consequently C; f2 is not in dp(dV,) for a + 2 = p. The proof of the theorem is therefore complete. For a fixed zo E D, and f E %(Dn)we define the linear operator Azo (f) by

This operator is one of the most natural averaging operators on %(D,). For n = 1 and zo = 0, it is called the Libera transform which has many applications in geometric function theory. We shall allow that zo E D, restricting the operator's domain. As we have seen in (4),the formal adjoint operator C; of the generalized CesAro operator C, is closely related to the Libera transform. Now let us move our discussion to the generalized Libera transform on D,. We have the following theorem. Theorem 2. The generalized Libera operator Azo is bounded on the space dp(dV5) if and only if cyj + 2 < p for all j = 1, ..., n. Proof. Case 1. aj + 2 < p. In this case, one has p > 1 since aj > -1. By Minkowski integral inequality and Lemma 1, with cp = cpt = (tlzl+(l-tl)z;, . . . ,t,z,+(l-tn)z;), 0 < tj 5 1, j = 1,.. . ,72, we obtain 97

Since the last integrals converge for aj + 2 < p, j = 1, ..., n, we obtain the result in this case. Case 2. aj + 2 > p for some j E (1, ..., n} and zy E dD1. Let fl(zj) = l/(zy-zj). Then using polar coordinates and [7,p.84, Lemma 31, we obtain

< c J, (1 - Tj)aj+l--P drj < 00, for aj + 2 > p. Hence f1 E AP(dV3) if aj + 2 > p for some j E (1, ..., n). On the other hand h,o have no sense on the function f1 and consequently h,o(fl) is not in dP(dV3) in this case. Case 3. aj + 2 = p for some j E (1, ..., n}. Note that p > 1 it this case. Let

f2(zj) = -(Log- >-'. 1 - zj zj 1 - Zj

The only singularity of f2 is at zj = 1. Using polar coordinates centered at zj = 1, we see that the integral

is equiconvergent to f2p-l(ln $)-pdp < co, since p > 1. Hence f2 E dp(dVz), where d = (a1,..., aj-1,p - 2, aj+l,..., a,). On the other hand, it is easy to see that Ael have no sense on the function f2 and consequently her(f2)is not in dP(dV3) for aj + 2 = p. Here e' = (zy, ..., Z~O-~,1, ..., z",. The proof of Theorem 2 is therefore complete. Remark The techniques in this paper can be generalized to obtain similar results on Hardy space. Let Hp(D,) be the space of all holomorphic func- 98 tions defined on the unit polydisk such that supOlr

Form this inequality, it is not so hard to obtain the gereralized Libera operator if bounded on HP(Dn) if 1 < p < 00. We will discuss this in a forthcoming paper.

3. Acknowledgments The authors would like to thank the organizing committee, especially Pro- fessor Armand Wirgin, for organizing the conference. They would also like to thank the warm hospitality during their visit to France. This research project is partially supported by a William Fulbright Research Grant and the FY96 Defense University Research Instrumentation Program (DURIP) Contract No. DAAH-0496-10301.

References 1. A. Aleman and A. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997) pp. 337-356. 2. K.F. Andersen, Cesko averaging operators on Hardy spaces, Proc. Royal SOC. Edinburgh 126A (1996) pp. 617-624. 3. G. Benke and D.C. Chang, A note on weighted Bergman spaces and the Cesho oprator, Nagoya Math. J. 159 (2000) pp. 25-43. 4. D.C. Chang, R. Gilbert and J. Tie, Bergman projection and weighted holomorphic functions, to appear in Reproducing kernel, Hilbert spaces, positivity, function theory, system theory and related topics, ed. by D. Alpay (Birkhaiiser- Verlag, Berlin, 2002). 5. D.C. Chang and S. SteviC, The generalized Cesho operator on the unit polydisk to appear in Taiwanese Math. J., (2003). 6. N. Danikas and A. Siskakis, The CesZlro operator on bounded analytic functions, Analysis 13 (1993) pp. 195-199. 7. P. Duren, Theory of Hp spaces (Academic Press, New York, 1970). 8. A. Siskakis, Composition semigroups and the Cesko operator on Hp(D),J. London Math. SOC.36 (1987) pp. 153-164. 9. A. Siskakis, Semigroups of composition operators in Bergman spaces, Bull. Austral. Math. SOC.35 (1987) pp. 397-406. 10. A. Siskakis, On the Bergman space norm of the Cesiro operator, Arch. Math. 67 (1996) pp. 312-318. 11. E.M. Stein, Singular Integrals and Differentiability of Functions (Princeton University Press, Princeton, New Jeresy, 1970). 12. K. Stempak, CesZlro averaging operators, Proc. Royal SOC.Edinburgh 124A (1994) pp. 121-126. 99

13. S. SteviE, Composition operators on the generalized Bergman space, to appear in J. Indian Math. SOC.69 (2002). 14. S. SteviC, Ceslrro averaging operators, to appear in Math. Nachr. (2002). 15. J. Xiao, Cesko-type operators on Hardy, BMOA and Bloch spaces, Arch. Math. 68 (1997) pp. 398-406. 16. A. Zygmund, Trigonometric Series I, (Cambridge University Press, Chicago, 1968). ON THE BOUNDEDNESS OF FUNCTIONS FROM AN ANISOTROPIC WEIGHTED SPACE SATISFYING SOME INTEGRAL INEQUALITIES

PAOLO CIANCI Department of Mathematics, University of Catania, Viale A. Doria 6, 90125 - Catania, Italy E-mail: cianciOdmi.unict.it

In this paper we give a result which is a basic part of Moser’s method to obtain boundedness of solutions for degenerate non linear partial differential equations in anisotropic case.

1. Introduction In this paper we give a result which is a basic part of Moser’s method to obtain boundedness of solutions for degenerate non linear partial differential equations in anisotropic case. The application of the given result to degenerate anisotropic higher order equations one can consider as the main reason of this obtaining. We note in this, connection that direct use of generalization of corresponding results of the previous paper and of earlier paper ( see for istance 294,7,8) where isotropic case was studied, for establishing of boundedness of solutions in complete anisotropic case makes difficulties. Due to this fact some details of result given below have difference as compared with isotropic case. We note also that some results on boundedness of solutions for anisotropic second order equation with use of other method, was obtained in ’.

2. Main Result Let 52 be a bounded open set of Rn, n 2 2. Let, for every multiindexes a, IayI = 1, qa, be a number such that 1 < qa < n. We set

q = {qa : la1 = 1).

100 101

Let, for every multiindexes a, la1 = 1, v, be a positive function in R such that (6)* E L;*,(R).

Hypotheses of this kind have also been considered before in 4,536. We set

v = {v, : la1 = 1).

By W1rq(v,R) we denote the space of all functions u E Lq- (R) such that their derivatives ( in the sense of distributions) D"u are functions which have the following property

v,(D~upE L1(R). W1?Q(v,R) is a Banach space with the norm

By Wilq(v,52) we denote the closure in W1iq(v, R) of the set C,'jO(R).

Hypothesis 2.1. There exists ij > q+ and E > 0 such that for every u E wyv,R),

In the further consideration we shall use the sequence of function h, : R + R which was considered in 7. Recall these functions have the following properties:

h8(v) = 9 if 101 Is (1) h,(rl) = (s + 1)swrl if lrll 2 s + 1 (2)

0 I I c1 VQ E R (3) IrlIhXrl) I2~llhs(rl)1 vq E R (4) ~1711I 21~1 vq E R (5) 102

IIuI14 5 mo. Let, for eoery s E N and r > 0,

then

vrai maxR 1.1 _< Mo ( 7) where the positive constant MO depends only on 72, Q-, Q+, 4, e, c1, mo,m1, m2, tllt2, l~l~ll~1ll~~~ll~2ll~*. Proof: We shall denote by ci, i = 2,3,. . . , positive constants which depend only on the mentioned parameters. Define

It is clear that r > 1. We denote

717 =- ,g=-. @ 7-1 T'Q+ By (8) we have 8 > 1. Let us fix s E N. For every r > 0 we define

T,(r) = 1 + 2-74 Iuy+"[l+ h341TT1dX s, (9) Now let us fix r > &(QI - q+r') and define 1 1 p = :[rrl - -(@ - q+rl)] Q 2 It is clear that p > 0 and

PQ+ < From (10) it follows that 1 rr' = p4 + s(@- q+rl) and r1 Q+ PQ+ = -J - $Q+ - 8) 103

Using (9) and (12) we obtain

T,(r) = 1-I- 2-74 1u1"""1+ h;(u)y'dx + Ll + 2-T'r Iup+'"l + h3u)]3(6-4+T')[l + h3U)]-P"ds 5 4,U1>1l -< w + 2--Pg+g--9+7' s, Iu)g[l+ h3u)lP"x.

Hence setting

v = [l + h3u)]-Pu we obtain

Since u E Wi"((~,fl)we have v E W,'>q((v,fl)and for every multiindex 0,101 = 1, DQv = {[I+ h:(u)]P+ 2p[1+ h:(~)]-P-~h,(u)hl,(~)~)D~u,a.e.inR. Hence due to (4),

5 (1 + 4clp)1~%I[1+ h:(u)]p a.e. in 0 then

By (6) we have

This inequality and (16) imply that 104

Let us estimate the integral in the right-hand side of (18). Evidently,

s,(@lIul + @2)[1+ h;(u)]PqQdzI

= I1 + 12, where

Taking into account that > 1, q+rl-qa = 1 and using Holder's +q+r + 4+7' inequality we get I1 I

Taking into account that Qa < -, q- qa-1 - q--1 the integral 12 we estimate as follows:

From (19)-(21) it follows that

This inequality and (18) imply that 105

Since v E W:lq(v,a), by Hypothesis 1 we have

then

[T(PQ+)le* This inequality, (11),(13) and (15) imply that

We set 1 1 Q+ d = -(@27' - q+T'), di = s(q+ - 7) and define for every j = 0,1, . . . ,

6j - 1 rj =6jd+- 6di 6-1 Since 6 > 1, for every j E N we have rj > d, then by (23) T,(rj) I c~(1+ rj)(mz+1)i[T(5- d1)le, by definition of rj, we have ?$ - dl = rj-1, therefore

T(rj)5 c7(1 + rj)(m2+1)i[~(rj - 1 )Ie. Hence, noting that rj 5 c&, we get T(rj)_< (24) Using iteration process we obtain,Vj E N,

Note now that

then by (25) we have

T(T~)I & v j E N. (27) 106

Due to definition of T,(r) and (27) we get

Since h,(u) 4 u as s + 00, by Fatou's Lemma we obtain

Hence it follows that

and so the lemma is proved.

References 1. L. Boccardo, T. Gallouet, P. Marcellini, Anisotropic equations in L', Differ- ential Integral Equations 9 (1996), n. 1, 209-212. 2. S. Bonafede, S. D'Asero , Holder continuity of solutions for a class of nonlinear elliptic variation inequalities of high order, Nonlinear Anal. 44 (2001), n. 5, Ser A: Theory Methods, 657-667. 3. P. Cianci, Boundedness of solutions of Dirichlet problem for a class of nonlinear elliptic equations with weights. (to appear). 4. S. D'Asero, Integral estimates for the gradient of solutions of local nonlinear variational inequalities with degeneration, Nonlinear Stud. 5 (1998), n. 1, 95- 113. 5. F. Guglielmino, F. Nicolosi, Existence theorems for boundary value problems associated with quasilinear elliptic equations, (Italian) Ricerche Mat. 37 (1988) n. 1, 157-176. 6. F. Guglielmino, F. Nicolosi, W-solutions of boundary value problems for degenerate elliptic operators (Italian) Ricerche Mat 36 (1987) suppl. 59-72. 107

7. A. Kovalevsky, F. Nicolosi, Boundedness of solutions of variational inequalities with nonlinear degenerated elliptic operators of high order, Appl. Anal. 65 (1997) n. 3-4, 225-249. 8. F. Nicolosi, I.V. Skrypnik, Nirenberg-Gagliardo interpolation inequality and regularity of solutions of nonlinear higher order equations, Topol. Methods Nonlinear Anal 7 (1996) n. 2, 327-347. HOMOGENIZING A FLOW OF AN INCOMPRESSIBLE INVISCID FLUID THROUGH AN ELASTIC POROUS MEDIA

T. CLOPEAU AND A. MIKELIC UFR Mathimatiques, Universite' Claude Bernard Lyon 1, B6t. 101 43, bd. du 11 novembre 1918, 69622 Villeurbanne Cedex, fiance E-mail: androOrnaply.une'v-lyonl.fr

We give a derivation of the Biot's law describing vibrations of an inviscid fluid in an elastic porous medium, using homogenization. We obtain not only the Biot's equations but also the expressions for the coefficients .

1. Introduction Modeling of elastic porous materials is a classical issue, undertaken in the extensive works on the general three-dimensional continuum theory by M. Biot in fifties. Biot's theory is used to predict the propagation of elasto- acoustic waves through porous media. The Biot's effective equations take into account the displacement of both the fluid in the pores and the solid skeleton and the coupling between these. We study here the case of a porous medium saturated by an inviscid fluid. Materials of this type are actually used in aerospace applications for reducing the noise transmission. The characteristic length of the non- homogeneities was t? and in Biot introduced the averaged displacements ii and d of the solid and fluid phase, respectively. They correspond to averages over volumes proportional to l3of the microscopic displacements. Then by generalizing the classical procedure from the linear theory of elasticity, he introduced the generalized stress-strain relations through the elastic potential energy 24= AlD(ii) : D(G) + A2(div ii)' + B(div d)2+ C(div ii)(div d), where D is the symmetrized gradient. Next, in Biot obtained dynamical relations in the absence of dissipation. It was argued that compressible fluids behave as incompressible ones if the wavelength of the elastic waves was much bigger than the pore size and that the microscopic flow pattern of the fluid, relative to the solid, depended only on the direction of the flow

108 109 and not on its magnitude. Then the kinetic energy E, of the statistically isotropic system is

Through invariance considerations, it was found that ~11= (1 - 9)~s+ pa, pi2 = -Pa and ~22= 9Pf + Pa, where cp is the porosity, ps and pf are the mass densities for the solid and the fluid, respectively, and pa is an additional mass due to the fluid. After considering the Lagrangian L = E, - Ep,the following system of equations is obtained a2ii a23 p11- + Pl2F= at2 C A1 div(D(d)) A2V div u'+ -V div d + 2 (1) 8% a2d -c p12- + p22- = BV div U + -V div ii . at2 at2 2 The morei realistic dissipative case is modeled by adding the friction effects due to the relative motion between the fluid and the solid, but we don't discuss it in this short expository paper. The coefficients in (1) are not given and their determination is in general not clear. In this direction we refer to the paper by Biot and Willis. Furthermore, it is not clear that coefficients in (1) are scalars. All those open questions give motivation to undertake a derivation of the equations (1) starting from the first principles, i.e. from the coupled system containing the linear elasticity system for the solid skeleton and the linearized Euler equations for the inviscid fluid. The homogenization limit, when the pore size E tends to zero should give an upscaled model. The fluid could be supposed incompressible, the compressible case being simply the penalization of the incompressible case. We are going to present an outline of the rigorous derivation from 7. In order to fix the idea, we study a porous medium obtained by a periodic arrangement of the pores, with connected fluid part. The formal description as in ?. Now we see that 52 =]0,L[3is covered with a regular mesh of size E, each cell being a cube y," = E(Y+i), with 1 5 i 5 N(E)= (n(~-~[l+O(l)]. We define the solid matrix 52; = UkETEys", , I?" = afl;, and the pores 52; = 52 \ 52;. Obviously, 852; = 852 U re. The domains 52; and 52; represent, respectively, the solid and fluid parts of a porous medium 52. We suppose small deformations and the displacements are described by linearized equations in both media. More precisely, in 52; we have the linearized momentum equation for the time derivative of the fluid displacement ue, in 110

Eulerian variables. il: is the reference configuration of a deformable elastic body and the equation of linear elasticity for the displacement w" are in Lagrangean variables. In both domains all quadratic terms are neglected. At the linearized interface between two media we have continuity of the normal displacements. Let F E Cm([0,T];L2(il)3)and curl F E C""([0,T];L2(R)3). Then we consider the system for {u",wE,p"} L - periodic in x a2W" Ps- - div(r(w")) = Fp, and ~(w")= AD(w') in R:x]O,T[, (2) dt2 d2U& dU" + Vp" = Fpf and div - = 0 in 0; x]O, T[, PS dt2 dt (3) (up- w"). v = 0 and (pel + a(w")) v = 0 on I', x]O, T[ (4) The above system has a unique variational solution in Cw([O,TI, V) V = {cp E L2((n)3; cp E H1(R:)3, div cp = 0 in 07, div cp E L2(R) and cp is L-periodic}. After taking u" E H3(0,T;L2(REf)3)n H2(0,T;Hb,,(REf)3) , w" E H3(0, T;L2(Rz)3) and p" E H1(0, T;L2(ilEf)) as the test function in (2)- (4) we obtain

(5) The linearized incompressible Euler system doesn't involve derivatives of the velocity field with respect to x and an H1-estimate for the velocity doesn't follow directly. One way to proceed is to use the H(div;REf) estimate in the fluid part. Nevertheless, after taking the curl of the linearized Euler system, we get

div 24" = 0 md IIcurl u"IJLm(0,TiL2(n;)3)- is zero. By assumptions, Oil> is a connected 2D manifold. Each such manifold is homeomorphic to a sphere with "handles" and the first Betti number or the genus of ail; is the number of handles. For a simply connected domain, the first Betti number is zero. Our domain is multiply connected and the estimate requires some effort. Proposition 1.1. (see ) By supposing that aYf E C1il we get 111

We note that for isolated fluid parts the velocity gradient is uniformly bounded, leading to different results. For more details we refer to . The extension of the pressure field pE to 0; x]O,T[ is given by substracting its mean. Then SjEdivcp = s p&Evcp, Vcp E H;,, (R)3, and n "7 1117" llH1(o,T$;(")) + WEllH1 (O,T;H,A (")a) <- c. (8) In order to prove the main convergence results of this paper we use the notion of two-scale convergence (see Allaire '). The main interest of this convergence lies in the compactness properties. Then we have Theorem 1.1. (see ) There exist subsequences such that, Vt E [O,T], xn;u" + xn: wE+ uo(x,t) + xy, (y)v(x, y, t) in the 8-scale sense, (9) xn:Vw" + ~~,(y)[V,u~(x,t)+ Vyul(x,y,t)] in the 2-scale sense, (10) xn; EVU~+ xyf (y)V,v(x, y, t) in the 2-scale sense, (11) @' + @O(x,y, t) in the 2-scale sense, (12) with w E H3(0,T; L2(R; H(div;Y))3) fl H1(O,T; L2(R; Hk,, (Yf))3), v = 0 on Y,, div $$ = 0 in R x Yfx]O,T[ and div, Jy, $$ dy E H1(lO,T[, L2(W E H3(0,T;L2(R; Hker (Ys)/R)3)7 uo E ~3(0,~;~;,,(52)~)and ~(x,y,t) = XY,(y)p(x,t) + ~(t),p E H1(]O,T[; L2(R)), B(t)= -#Jn p(z,t) dx. It should be noted that the above convergence result relies on the connectivity of the solid part. The connectivity of the fluid part is not required. Using standard procedures from the theory of 2-scale convergence it is straightforward to pass to the limit in the system (2) - (4). For details we refer to . We obtain the following two-scale system for the effective displacement uo,the effective relative pore displacement w between phases, the correction Dy(ul),the effective pressure p and the second pressure 7r a2 pf-(uo+v)+V,p+V,7r=Fpf inRx Yfx]O,T[, (13) at2 -divy{A(D,(uo) + Dy(ul))}= 0 in R x Y,x]O,T[, (14)

- div, [[A(Dx(uo) + Dy(ul))dy1 = Fp in Rx]O,T[, (15) 112

A(Dz(uo)+ D,(u'))v = -pv on R x (aY,\aY)x]O,T[, (17)

w . v = 0 on R x (aYf\aY) x]O,T[, (18) where P = IYfIpf + (YSl~,. Theorem 1.2. (see Let {uo,p,ul,w}be given by (9) - (12) . Then they define the unique variational solution for (13) - (18) .

Hence the two-scale system (13) - (18) is the complete upscaled problem. Nevertheless, it is too complicated for numerical calculations and it is necessary to eliminate the fast scale y. It is eliminated using the ansatz

and 1-periodic wo with zero mean by

-div, {AD,(wo)} = 0 in Y,, AD,(wo)v = --v on aY,\aY,

The auxiliary functions define the following effective coefficients :

BH = J AD,(wo)dy,

div,wij (y) dy = - AD, (wo) : D, (wij) dy = Bz Y.J Y. /div,wO dy = - s AD,(wo) : D,(wo) < 0.

The tensors AH and BH defined by (22) and (23), respectively, are positive definite and symmetric. Ansatz for w and T is more complicated . Here we write IJ as 113

where is the unique 1-periodic solution with zero mean of

The effective coefficient is now

The matrix Aij, given by (29), is positive definite and symmetric. For a viscid porous medium the accelerated law (26) was established in ’. Therefore, we have the following system for {uo,p} L-periodic in x : (PI - pfA) - div, {AHD,(uO)}+ (lYfl1- BH - A)V,p(x,t) = (PI- pfA)F(x,t) in 0x10, T[ (30) 1 - d2p 1div,wO dy - - div (AV,p(x, t))+ at2 Pf Y* 62UO div { (JYflI- A - I?*)-} = - div (AF) in 0x]O,T[ (31) at2 We refer to for the proof that {uo,p}is a unique solution for the system (30) - (31). The new system doesn’t contain the fast scale and it is much easier to study. It remains to compare it with the Biot’s system (B).First, we introduce the effective displacement of the solid part as ii = uo and the effective displacement of the fluid part as = uo+ J v dy. Then, after averaging Yf the equation (26), we get

i.e., even an incompressible fluid gave rise to a compressible effective 2- phase medium, as noticed by M. A. Biot. This formula coincides with the formula proposed by Biot for a general anisotropic material. Let us now obtain the system (B). After some transformations we have 114

BHV div(BHZ) - lyfl BHV div fi. (33) - J div,wO - J div,wO Y8 Y,

If the system (33) - (34) is compared with the original Biot's system (l),then we see that they have the same structure. Equation (33) corresponds to the first equation in (B)and (34) to the second. The difference is that we got matrices for pij. p22 = (YfI2pfA-', ~12= pflYfl(I - 1yrlA-l) and p11 = p,lY,II + IYflpf(IYflA-' - 1). The added mass is pa = pflYfl(lYflA-' - I). The matricial coefficient IYflA-' correspond to the tortuosity factor. Also in the compressibility terms V div, we have BHG and not Z. Nevertheless, the four matricial coefficients corresponding to the effective stress-strain relations in the Biot's theory are reduced to only two: AH and BH.To the contrary, the study of the purely elastic and dilatation waves for our system is much more complicated. The advantage of our approach is that we are not only able to justify the Biot's model without ad hoc assumptions, but also we are able to calculate the coefficients from the first principles. References 1. G. Allaire. Homogenization and two-scale convergence, SIAM J. Math. Anal. 23.6 (1992), 1482-1518. 2. M. A. Biot, Theory of Elasticity and Consolidation for a Porous Anisotropic Solid, J. Appl. Phys., 26, 182-185 (1955). 3. M.A. Biot. Theory of propagation of elastic waves in a fluid-saturated po~ous solid. I. Lower frequency range, and II. Higher frequency range, J. Acoust SOC.Am. 28(2) (1956), 168-178 and 179-191. 4. M. A. Biot and D. G. Willis, The Elastic Coefficients of the Theory of Con- solidation, J. Appl. Mech., 24, 594-601 (1957). 5. M.A. Biot. Generalized theory of acoustic propagation in porous dissipative media, Jour. Acoustic SOC. Amer. 34 (1962), 1254-1264. 6. T. Clopeau, J.L. Ferrin, R.P. Gilbert and A. MikeliC, Homogenizing the acoustic properties of the seabed: Part 11, Mathematical and Computer Mod- elling , 33(2001), p. 821 - 841. 115

7. J.L. Ferrin , A. Mikelib , Homogenizing the Acoustic Properties of a Porous Matrix Containing an Incompressible Inviscid Fluid, to appear in Mathemat- ical Methods in Applied Sciences 2002. 8. R.P. Gilbert and A. Mikelib. Homogenizing the acoustic properties of the seabed: Part I, Nonlinear Analysis, 40 (2000), 185-212. 9. A. MikeliC, L. Paoli. Homogenization of the inviscid incompressible fiuid fiow trough a 2D porous medium, Proceedings of the AMS, vol. 17 (1999), 2019- 2028. APPROXIMATION OF A DYNAMIC UNILATERAL CONTACT PROBLEM FOR A CRACKED VISCOELASTIC BODY

M. COCOU132 Laboratoire de Me‘canique et d’Acoustique - C.N.R.S., 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, fiance,

Universite‘ de Provence, Marseille, fiance, E-mail: [email protected]

G. SCARELLAlS3 INRIA Rocquencourt, Domaine de Voluceau - B.P. 105, 78153 Le Chesnay, fiance, E-mail: [email protected]

We investigate a dynamic contact problem for a cracked viscoelastic body, when Signorini’s conditions between the two faces of the crack are taken into account. Firstly, we present the classical and variational formulations of the problem. In order to solve the variational problem, we then consider an auxiliary penalized problem with a normal compliance law. We obtain estimates which enable us, by using compactness results, to select a sequence of penalized solutions converging to a solution of the unilateral contact problem. Finally, the convergence of a numerical method based on an internal approximation in space and an implicit time discretization is proved.

1. Introduction The aim of this paper is to study the existence and the approximation of a solution to a dynamic contact problem for a cracked viscoelastic body, when unilateral contact (or Signorini’s) conditions between the two faces of the crack are taken into account. A few results concerning dynamic unilateral contact problems exist in the literature. An existence and uniqueness result for the wave equation with unilateral boundary conditions in a half-space was proved by G. Lebeau and M. Schatzman and an existence result for the wave equation with unilateral boundary conditions in a smooth bounded domain was

116 117 proved by J.U. Kim 4. Dynamic contact problems with normal compliance law were analyzed by J.A.C. Martins and J.T. Oden and a dynamic unilateral contact problem for a viscoelastic body without friction was studied by J. Jaruiek '. More recently, M. Cocou and J.M. Ricaud, E. Pratt and J.M. Ricaud 2,7 have obtained abstract existence results with applications to dynamic contact problems for viscoelastic bodies.

2. Formulations of the problem and existence results 2.1. Classical formulation We consider a cracked viscoelastic body of the Kelvin-Voigt type which initially occupies a bounded domain 0. The crack, denoted by rc, may include an initial gap g. We suppose that R can be split into two open subsets fl+ and 0-, such that R = R+ u R- u rV,rv = r: U I?;, where rv is a part of the boundary of aR+ and bR-, as it can be seen in figure 1 and mes (r:) > 0, mes (r;) > 0. We assume that rCcan be parametrized by a subset E which will enable us to express unilateral contact conditions on the faces of the crack.

" ...... '...... ) ......

/ +I-- r"

Figure 1. A representation of R in 2D.

The strong formulation of the dynamic unilateral contact problem for a cracked viscoelastic body is the following. 118

Problem PO:Find u = u(t,s)such that

' attu - diva(u) =f in ]O,T[x R, (u) = dE(u)+ BE(&u) in 10, T[x 0, u = 0 in ]O,T[x rU, a(u)n= F in ]O,T[x FF, (1) [UN] 5 9, (T;(u) = a$(u) 5 o in ]0,T[x2, ai(u)([uN] - g)= 0, a~(u)= a,'(.) = o in ]o,T[xE, u(0)= uo, atu(0) = u1 in R, where d, l? satisfy classical symmetry and ellipticity conditions, UN,a~, UT, (TT are respectively the normal and tangential components of the displacement and of the stress vector and [u~]is the relative normal displacement on rC.

2.2. Variational formulations We introduce the following functional framework: H = [L2(0)]" Hs = [HS(0)ld,Vs E IR, d v = [H:~(a)] = {u E [H~(0)l ; u = 0 a.e. on rU}, K = {u E V;[VN] 5 g a.e. in E}, a:VxV+lR, b:VxV+lR,

dE(v) : E(w)~x,b(u,w) =

L : V + R,(L,w) =

We assume that uo E K,ul E V, g E HiJ2(Z),f E W1>"(O,T;H), and F E W1@(O,T; [L2(F~)ld). A variational formulation, equivalent to problem Po, is as follows: Problem PI:Find u E W1>'(O,T;V)n C1([0,T],H-1/2), u(0)= uo,dtu(O)= u1 in 0, such that u(t)E K for all t and

(2)

We shall study an auxiliary variational formulation, based on the decomposition of R, which takes into account more regular domains and which 119 will allow us to prove the existence of a variational solution to problem PI. We consider the functional framework:

H, = [L~(W)I~,v, = {v E [H~(RQ)I~;w = o a.e. on ra},U o =+,-, I2 = H+ x H-, C = {s = (w+,w-) E V+ x v-;w+= w- a.e. on I?,,,}, B = (8 E P; [8~]_< g},I;Ts = [H"(R+)ld x [HS(R2-)ld,~E R.

We shall use the following notations for 6 (and similarly for 6):

&(a,$)= U+(U+,W+) +u-(u-,v-) d&(u+): ~(w+)dz + l-dc(u-) : ~(w-)dz, V G, 8 E V. = l+ Problem Pz: Find G = (u+,u-) E W132(0,5"; P) n C'([O, TI,&-'I2), G(t) E K for all t and

(W(T),qq- G(T))&-1/2,$1/2 - (Gl,S(O) - Go)

2.3. Existence results We will prove the following existence result.

Theorem 2.1. Under the above assumptions, there mists a solution to problem PI,

To obtain this result, we consider an approximate penalized problem. We first derive the existence of a unique solution to this problem. Then, we establish estimates on the solution. We consider a penalized problem with the solution G, = (u:, u;), verifying similar equations and conditions as PO in 0, rcr,rF. The boundary conditions on the crack faces are the following (normal compliance law): 1 o$(G,) = o&(G,) = -;( [G,N]-g)+, ~T+(fi~)= o+(G,) = o on ]0,5"[xE.(4) 120

Problem P2,,: Find 4, E W1~O0(O,T; P) n W2ioo(0,T; ?') such that (atta&,G)+ s(a&,s)+ i@&&,G) + Q&(G&,G)= (L,G), (5)

Using standard existence results we can prove the following theorem.

Theorem 2.2. There exists a unique solution to problem P2.&.

Proposition 2.1. The following estimates hold, with a constant M independent of c:

Thanks to (7), there exists 4 = (u+,u-)and a subsequence, still denoted by (a,), such that: u,"- u" weak * in LO"(0,T; Va), at.," 2 dtu" in L2(0,T;V,),weak * in Lw(O,T;H,), (8)

attu; 2 dttu" in L~(o,T;[H-'(W)]')), Q = +, -.

Using a compactness result of J. Simon (8, Corollary 4), we have atu: + atu" in L2(0,T;Ha), at&, + at4 in L2(0,T; a), atu: + atu" in G([o,T],[H-'/'(w)]~), u; + uo in c([o,T],[H'~~(w)]'). Hence

(atu; (T), w" (T)- u; (T))H- 1 / 2,Hl /* + (atu" (T),w" (T)- u" (T))H- *,2, Hl /* . We may also use convexity and lower semi-continuity arguments for lT12 and LT&. By a diagonal process, we obtain, for all t E [O,T]:

GE(t)2 &(t)in V+ x V-.

Therefore, by using trace properties, we have, VtE [O,T], 4&(t)- 4(t)in P, so that u+ = u- a.e. on rV,and 4 E K by (7)s. 121

Now the previous convergence properties allow to pass to the limit in the penalized problem. Thanks to a monotonicity argument, the limit is solution to the unilateral problem..

3. Semidiscrete and fully discrete approximations for the penalized problem 3.1. A regulas-ity result for the penalized problem Assume the following compatibility condition between initial velocity and initial displacement:

Then, we can prove the following regularity result.

Proposition 3.1. Under the compatibility condition (9), we have at&& E Lye,T; H).

3.2. Enor estimates for the semidiscrete problem Let (Vh) be an internal approximation of V. Problem P:.& : Find u," E W1im(O,T;Vj)n W2>"(O,T;V,)such that

Proposition 3.2. Under the compatibility condition (9) there exists M independent of h (it may depend on E) such that:

Proposition 3.3. The following semidiscrete estimates hold with a positive constant C independent of h: 122

3.3. filly discrete approximations and stability estimates iT Problem P,h>i: Find uh E Vh, 2 i N - 1, ti = - such that I I N (~X,w~h)+a(uh,wh)+b(di,wh)+~.,(uk,wh)=(Li,wh),Vwh EV~,(12) T where the following notations are used for i 2 2, At = - . N'

Proposition 3.4. Under a compatibility condition on the discrete initial At data and the stability condition - 77, where q > 0, there exists a h I positive constant c, independent of h and At, such that: Ilrill, I c, Ildj9lV 5 c, IIMV I c, Il.hNII, I c and IIuiIIv I C,IISXItv 5 C,IIrXIIH I c, 1 I i I N - 1.

3.4. Convergence results We can define internal approximations u,, d,, Gn on ]O,T[such that u, - u," and d, - at.," in L2(0,T; Vh), Gn - u," in W2,2(0,T; Vh). Using (ll),one can prove the convergence of fully discrete solutions to the continuous solution of the penalized problem. For numerical aspects, a work is in progress to use the same method as E. Bbcache, P. Joly and C. Tsogka in * to solve the elastodynamic problem with unilateral contact conditions on the faces of the crack.

Acknowledgments The support of this work by E.D.F. - Recherche Developpement, SINETICS, Clamart, fiance, is gratefully acknowledged.

References 1. E. BBcache, P. Joly and C. Tsogka, J. of Comp. Acous. 9, 1175 (2001). 2. M. Cocou and J.M. Ricaud, Int. J. Engrg. Sci. 38, 1535 (2000). 3. J. JaruSek, Boll. Unione Mat. Ital. 7 (9-A), 581 (1995). 4. J.U. Kim, Commun. in Partial Differential Equations 14, 1011 (1989). 5. G. Lebeau and M. Schatzman, J.Diff.Eqs. 53, 309 (1984). 6. J.A.C. Martins and J.T. Oden, Nonlinear Analysis, Theory Meth. Applic. 11, 407 (1987). 7. J.M. Ricaud and E. Pratt, Math. Meth. Appl. Sci. 24, 491 (2001). 8. J. Simon, Ann. Mat. Pura Applic. 146, 65 (1987). PRINCIPLES OF SIGNAL BASED RAY TRACING FOR 2D AND 3D COMPLEX TECTONICS

P. CRISTINI Universite‘ de Pau et des Pays de I’Adour CNRS/UMR5831 Imagerie Ge’ophysique PO Box 1155 64013 Pau Cedex, FRANCE Email: pad.cristiniOuniu-pau. fr

E. DE BAZELAIRE TotalFinaElf CSTJF avefiue Larribau 64000 Pau, FRANCE Email: Eric.De-Bazelaire@totalfinaelf. corn

A new method that allows the fast computation of the traveltimes between a source and an array of receivers for a two-dimensional velocity model is presented. This method computes all rays without any omission at a much lower cost in computing time than classical methods. In the first part of this paper, we give its basic principles, and then discuss its extension to 3D models in a second part.

1. Introduction Ray tracing is commonly used for the calculation of traveltimes and amplitudes of seismic events. Many algorithms exist5 which all have their advantages and drawbacks. Nevertheless they all provide poor results when the geometry of the interfaces is complex. Some rays are not found and it makes the interpretation of complex tectonics dangerously complicated. Furthermore, even if ray methods are much faster than exact solutions they are still not fast enough for 3D inversion. A very fast and comprehensive ray tracing algorithm is then a crucial requirement. In this paper we present a method which meets these objectives.

2. Principles of the method in 2D The starting point of the method goes back to the observation that the tool used to probe the earth interior is not perfect. It is a broadband seismic wave which has its own resolving power. As a consequence, uncertainties are

123 124 inherent in the process of determining the times of arrival. It is not possible and necessary to determine the exact traveltime of a ray with such a signal. If B is the bandwidth of the seismic source signal then the resolution in time associated with such a signal is At = 1/B. This property is analoguous to the Heisenberg principle since it indicates that it is not possible to be accurate both in the time and frequency domain. An error in the evaluation of the time of arrival is acceptable if it is within this limit. Furthermore, if the velocity model is known, an error in time can be transformed into an error in space. The question arising after these observations is what can we do with such a possibility ? Our answer to this question is a two-step process. The first step consists in modifying the model to get a simpler one by using the space error. Interfaces are interpolated with an error control with simple functions (circles with continuous first order derivative) as shown in figure 1. The time error is then used in the second step to construct beams which have a traveltime difference between their extremal rays less than a fixed value.

Approxi mat ion n

Figure 1. Circle arcs approximation of an interface

The difficult problem of finding the rays is then reduced to a set of elementary problems (reflection or refraction of a beam on a circle arc) which can be solved analytically. The calculation of the width of the beam is done by minimizing the traveltime difference (Fermat principle). The nullification of first order terms gives the rays (Snell’s law). The nullification of the second order gives the caustic i.e. the location of the centers of curvatures of the outgoing wavefront. The third order which cannot be nullify then gives the width of the beam as indicated in figure 2. More details on the procedure can be found in reference2. As a result, the caustic is sampled and the pair point source and interface is replaced by 125

Source Ai Ai

, Caustic

Refracted wavefront sample \

Figure 2. Caustic sampling several point sources which are the result of this sampling all of them lying in the second medium. The wavefront is then approximated by circle arcs. This procedure is repeated for all the new sources and the next interface. An example of a beam propagating together with all the sources generated by the crossing of different interfaces is indicated in figure 3. It must be noticed that one can choose the interfaces on which the reflections occur. Separation of contribution from each interface is then easily done. The method is found to be at least 10 times faster than classical methods. Figure 4 shows an example where only the reflections from the fifth interface are considered. The cputime needed to compute these ray paths is negligible.

3. From2Dto3D The principles established previously can be extended to 3D models. They are basically identicals but the objects on which we will apply them are very different. This is particularly true for the reflection-refraction caustics which become two-sheeted surfaces. Recalling the principles of the method in 2D, two types of problems are to be solved : interpolation of interfaces and caustic sampling. Interfaces will be approximated by elliptic and hyperbolic paraboloids in order to be able to handle surfaces having curvatures of the same or opposite sign. The caustic sampling or wavefront decomposition will be done with torus elements. The caustic of a torus is a circle and a straight line orthogonal to the plane of the circle passing 126

Figure 3. A beam travelling

Figure 4. Example of ray paths over salt dome

through its center. As the caustic is sampled with points in 2D, in 3D we will use curves for the sampling of the two sheets. The choice of a torus led to approximate on one sheet with a circle arc and with a straight line segment on the other sheet. There is no difficulty to get the segment and a natural choice for the circle arc is to take it as a part of the osculating circle. We will show in this section how to construct this circle. 127

Let x(u, v) be the principal patch for a surface (wavefront) : ds2 = Edu2 + Gdv2 The two sheets of the caustic surface are : X1,z = x + p1,2N where N is the normal to the surface and p1,z the radii of principal curvature and we have6 :

ds; = dp: + G (1 - fi)2dv2 and ds; = E (1 - E)2du2 + dp; (1) P2

As shown by these equations, the parametric lines : p1 = py are isochrons and v = vo axe geodesics on the sheets. A point moving on a line of principal curvature of the surface is moving on a geodesics on the corresponding sheet and on an isochron on the other sheet. The center of the osculating circle for an isochron is given by :

mi= -p2 k, Tu + -'M (2) P1 - P2 P2 - PI where ni is the normal to the isochron, Xu a tangent vector,k, the geodesic curvature of the corresponding line of curvature and IG the curvature of the isochron. Consequently we will need to know on the outgoing wavefront the geodesic curvatures of the lines of principal curvature which are given by : 1 dkl 1 dk2 k,, = -- kg2 = -- (3) kl - k2 ds, kl - k2 dsc where the derivatives are taken along the two lines of curvature and k1,2 are the principal curvatures.

3.1. Outgoing wavefront identification As, in the 2D case, we will have to go to the third order minimization of the traveltime as the geodesic curvatures are proportional to the derivative of the principal curvatures. Several methods ll3 were developped for the calculation of the curvatures (second order) of the outgoing wavefront but none of them can be extended to third order in a straightforward way. Fortunately, another method initiated by Stavroudis can be extended more easily. It is based on the properties that through every point of a surface passes a geodesic in every direction. As indicated by Snell's law, the normal to the outghoing wavefront N' belongs to the plane of incidence defined by the normal to the incoming wavefront N and the normal to the interface iV :

N' = ~1 N + yiV, = - ~COS~ (4) 128 where p is the ratio of the two velocities, i is the angle of incidence and r is the angle of reflection-refraction. Stavroudis' method, instead of working with lines of principal curvature, uses orthogonal geodesics, one belonging to the plane of incidence as indicated in figure 5 and makes extensive use of directional derivatives of Eq. (4).

Figure 5. Geometry of incidence

The connections between the lines of principal curvatures and these geodesics are given by the following system of equations :

k, = kl cos2 8 + k2 sin2 6 kl = k, cos2 6 + k, sin2 8 - T sin 26 k, = kl sin2 8 + k2 cos2 6 k2 = k, sin2 8 + k, cos2 6 + T sin 28 (5) ~=+sin26(k2-kl) tan26 = 2 b-k, where k, is the curvature of the geodesic perpendicular to the plane of incidence, k, is the curvature of the geodesic in the plane of incidence, T the torsion associated to these geodesics and finally 8 the angle between principal directions and a geodesic. Reflection-refraction laws for the geodesics become : - Ic:, = p k, + y k, cosrT'=p cosiT+yT kk cos2 r = k,p cos2 i + y %, (6) (P0)y= sin r (T - 7') (&V)y = sin r (k; cos T - k, cos i) (N0)y= sin T (k, sin i - kh sin r) The last three equations are evaluation of the directional derivatives in the direction of vectors P,& and R. They were not given by Stavroudis4 129 because there are of no use for obtaining the curvatures but if the derivatives of the curvatures are to be obtained these equations are absolutely necessary. As an example, we give the calculation of the derivative of the first equation of the reflection-refraction laws given previously. It provides an expression for the derivative of the curvature in the direction of the tangent to the geodesic orthogonal to the plane of incidence. dkb - dk, dzp -- p-+y-+E,sinr(~-~') ds, ds, ds, The geodesic curvatures can then be calculated as well as the the third order equations of the minimization of traveltime leading to the caustic sampling.

4. Conclusions We presented the basic principles for the extension to 3D subsurface models of a new ray tracing algorithm. Its main drawback is its limitation to blocky models but the extension to media with a constant gradient of velocity is possible. Further improvements will be reported in the future.

Acknowledgements We thank Pierre Thore and Wasiu Makinde for their help.

References 1. Hubral P and Krey T Interval velocities from seismic reflection time measurements SEG (1980) 2. De Bazelaire and Derain J.F. SEG Abstract (1988) 3. Stavroudis O.N. The optics of ray wavefronts and caustics Academic Press 1972 4. Stavroudis O.N. J. Opt. SOC.Am. 66 (1976) 5. Cerveny V. Seismic ray theory Cambridge University Press (2001) 6. Eisenhaxt L. P A treatise on the differential geometry of curves and surfaces Dover (1960) ( reprint from 1909 edition ) REGULARITY UP TO THE BOUNDARY FOR A CLASS OF SOLUTIONS OF A FUNCTIONAL-DIFFERENTIAL SYSTEM.

S. D’ASERO Dipartamento di Matematica ed Inforrnatica, Universitd di Catania Viale A. Doria, 6 - 95125 Catania, Italia E-mail: daseroQdmi.unict.it

We investigate regularity properties for a class of solutions of a functional- differential system. We obtain, by using a modified Moser’s method and estimating the oscillation of solutions near the boundary of R, Holder’s regularity of the solutions up to the boundary.

In this paper we investigate regularity properties for solutions of a functional-differential system. In particular we consider the following problem: 0 14 Find a pair (u,Q) E w2,p(Q,v,p)x Lq(Q) such that:

where:

is the differential operator of the fourth order:

2 du = C (-l)laIDa~a(z,u,D1u, D2u) + ~(5,U, Qu) (2) la(=l that will be defined more precisely later. 0 14 Here W2,,(Q1v, p) is a Banach space connected with weighted functions v, p of Guglielmino-Nicolosi kind; we have used the following notation: Dku(z)= {Dau(z): la1 = k}. b(z,q, 0, ua(z, [), u(z,q, <) are CarathBodory functions satisfying some conditions.

130 131

Problems of the kind (1) arise in homogenization problems, in particular, under study of asymptotic behavior of solutions of elliptic boundary value problems in domains with complicated structure (for non-degenerate case corresponding second order equations see, for instance 6, 7).

We shall suppose that Rn (n 3 2) is n-dimensional Euclidean space. Let R be an open bounded subset in R". Let p 2 2, q be two real numbers such that n > q > 2p. Hypotheses concerning weights

Let Y be a measurable positive function in R such that: -- v E L:,,(52), v A E L:,,(R)

Under this hypothesis W1>q(R,Y), the functional space of all functions u E Lq(R) having weak derivatives Dau, la1 = 1, with the property Y+D% E LQ(R),is a Banach space with the norm:

0 I>Q The closure of Cr(R) in W1"(R,v) will be denoted by w (0,~)

Let p be a measurable positive function in 52 such that: -- P E L:Om, P pL1 E L:Om

We shall denote by Wi:;(52, v,p) the functional space of all functions u E W1gq(R,v) having weak derivatives D"u, la1 = 2, with the property p$D"u E Lp(R), la1 = 2. Wi;i(52, v,p) is a Banach space with the norm:

The closure of Cr(R) in Wi;l(R,v,p) will be denoted by

Previous hypotheses are general, so we need further assumptions concerning weight Y, in order to guarantee some embedding. 132

There exists t > such that uPt E L1(Q) we set !l= n(l;:;-qt From the above hypothesis, it follows that:

0 Id and there exists E > 0 such that for every u EW (0,u) the following inequality:

holds.

We set:

a There exists t* > $& such that: u* E Lq*(R) and:

u, fi = (u*)q* E Lt'(0)

For every y E Rn, p > 0 we denote:

%, P) = {. E It" : Iz - YI < P) a There exists F > 0 such that for every y E 0 e p > 0 with B(y, p) c Q, the following inequality:

holds. Hypothesis on the boundary of S2 133

This hypotheses means that R belongs to the class S (cfr. ').

Define:

h R = {x E Rn : dist(x,R) < p*}. There exist a positive function P : fi + R and ? > 0 such that: (1) G(x)= v(x) in R; (2) 6 E L@), P E Lt'(E), (3) for every y E dR and p €]O,p*],

Hypothesis concerning coefficients of A

We denote: R2>nthe space of all sets t = {ta : 1.1 _< 2} of real numbers.

0 Let, for every multi-indm a, la1 I 2 a,(x,t) : R x Rni2 + R, Icy1 = 1,2 be Carathe'odory functions. Moreover there exist c1, c2 > 0, such that for almost every x E R and every t E W2 the following inequalities hold:

Let a(x,q, c) : R x R x R + R be a Carathe'odory function and we suppose that there exists c3, such that the following inequalities:

b(x,%C>l I C3{bdq-' + ]

Hypothesis concerning b(z,77, S) Let b : R x W x R + R be a Caruthbodory function. We suppose that there exist c4, c5, q > 0 and ,L? ~]0,1[such that the following inequalities:

Ib(z,77, 01 Ic4 (I77Iq-l + Iclq-') + c5

(bb77, 4-1 - b(z,77, r,)(C - C') 1 CSlC - C'lq

(bb777 0- b(z,77'7 0)(77 - 77') I0

Ibk7 7770 - b(Y, 77'7 511 I c4(1 + 14-1 + 1771 + 177'1)q-2 x x (177 - 77'1 + 15 - YIP) hold for almost every x, y E R and every 77, q', <, E R. In order to define the operator A, we need the following:

Proposition 0.1. For every u E Lq(R), there exists GU E Lq(R) such that: b(z,u(z),+,(x))= 0 a.e. in 52 This proposition derives directly from the theory of monotone operators. 0 1,q 0 14 Let A : W2,p(R,v,p) + (Wz,p(R,u,p))* be the operator such that for

(Au,v) = / { ,kacu(z,u, D1u, D2u)Daw + a(z,u, &)v}dz 0 a)=l where q!.~~is the function that comes from Proposition 0.1 and u E 0 1,q WZ,p(fl,~,Cl)- Remark 0.1. We observe that the operator A is bounded, hemicontinous, monotone and coercive. 0 1A Definition 0.1. We say that the pair (u,+) E Wz,p(R,v,p)x Lq(R) is a solution of the system (1) if the following identities hold: b(z,u(z), $(x)) = 0 a.e. in R (14) 135

From the theory of monotone operators (cfr. 12, 13), system (1) has a 0 14 solution of kind (u,&) E w2,,(R, v,p) x LQ(0) Now we can formulate our main result: Theorem 0.1. We assume that all previous hypotheses are satisfied. If 0 14 (u,&) E W2,+,(R, "3)x Lq(S2) is a solution of the system (l),then there exist a functaon G : R + R such that G(z) = u(z) a.e. in R and a function 6%: n + R such that &(z) = qU(z)a.e. in R and for every z, y E n: lG(z) - G(Y)l 5 CIS - !/I0 (16) I&(z> -&(Y)I I CIz - Y1° where positive constants C and a depend only on known values and on 11.1 11.1 IL+(R) Note on the proof. Let (u,$J,J be a solution of our Problem. The proof of Theorem is performed in two steps: at the beginning, by using a modified Moser's method, we prove the Holder continuity up to the boundary of the solution u(z) of du = 0 estimating the oscillation of u(z) near the boundary of S2 and using an interior regularity of this solution obtained in and then we prove the same property for +%(z). Examples An example illustrating the fulfillment of the assumptions on weight is given by: let R = B(0,l), m = 3, p >_ 2, 3p < q < n. We fix a number r such that: -4~ - 4 rq' qt-n' 136

An example of function that satisfies all hypotheses on b(s,q, C) is given by the following:

References 1. S. D’Asero, Boundedness and regularity for a class of solutions of a functional- differential system, (to appear Nonlinear Analysis) 2. P. Drabek, F. Nicolosi, Existence of bounded solutions for some degenerated quasilinear elliptic equations, Ann. Mat. Pura Appl., vol 165 (1993), 217-238. 3. F. Guglielmino, F. Nicolosi, Sulle W-soluzioni dei problemi al contorno, Ric. Mat., vol 36 (1987), 59-72. 4. F. Guglielmino, F. Nicolosi, Teoremi di esistenza per i problemi a1 contorno relativi alle equazioni ellittiche quasilineari, Ric. Mat., vol 37, (1988), 157- 176. 5. D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. 6. E.Ya., Khruslov, Homogenization models of diffution in fractured cellular media, Dokl. Acad. Nauk USSR, vol 309 n.2(1989), 332-335. 7. A.A. Kovalevsky, Homogenization of Neumann problems for nonlinear elliptic equations in domains with accumulators, Ukr. Math. J., vol 47 n. 2, (1995), 227-249. 8. A.A. Kovalevsky, F. Nicolosi, Boudedness of solutions of degenerate nonlinear elliptic variational inequalities, Nonlin. Anal. 35 n.8 Ser B (1999), 987-999. 9. A.A. Kovalevsky, F. Nicolosi, On Holder continuity of solutions of equations and variational inequalities with degenerate nonlinear elliptic high order operators, Atti de12 Simposio Internazionale ”Problemi attuali dell’Analisi e della Fisica Matematica” dedicato alla memoria di Gaetano Fichera, Taormina, 15-17 ottobre 1998. 10. A.A. Kovalevsky, F. Nicolosi, On regularity up to the boundary of solutions to degenerate nonlinear elliptic high’order equations, Nonlin. Anal. 40 (2000) 365-379. 11. O.A. Ladyzhenskaya, N.N. Uralt’seva, Linear and quasilinear elliptic equations, Moskow Nauka 1967. 12. J.L. Lions, Quelques methodes de rksolution des problbmes aux limites non linhaires, Dunod et Gauthier-Villars, Paris 1969. 13. M.K.V. Murthy, G. Stampacchia, Boundary value problems for some dege- narate elliptic operators, Ann. Mat. Pura Appl. (4), vol. 80 (1968),1-122. 14. F. Nicolosi, Soluzioni deboli per operatori parabolici che possono degenerare, Ann. Mat. (4) vo1.125 (1980), 135-155. 15. F. Nicolosi, I.V. Skrypnik, Niremberg Gagliardo interpolation inequality and regularity of solutions of nonlinear higher order equations,Topol. Method Nonlin. Anal., 7 n. 2 (1996), 327-347. 16. I.V. Skrypnik, Nonlinear elliptic equations of high order, Naukova Dumka, Kiev, 1973 ON THE HARDY SPACES OF HARMONIC AND MONOGENIC FUNCTIONS IN THE UNIT BALL OF RM+l

R. DELANGHE Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Gent, Belgium E-mail: Richard. [email protected]. be

DEDICATED TO ROBERT P. GILBERT ON THE OCCASION OF HIS 70TH BIRTHDAY Using the decomposition of the Poisson kernel in terms of the Szego kernel, a decomposition is obtained of the Hardy space of harmonic functions on the unit ball of Rrn+l.

1. Introduction Let R c U2 be a bounded and simply connected domain with C,-boundary I' = an and let C be the Cauchy transform on L2(r) with adjoint C*. If P : L2(I') + H2(I')is the so-called Szego projection, H2(I') being the Hardy space on I', then in lo Kerzman and Stein proved the following relation between C and IP : C = P(1+ A) where A = C - C* is the so-called Kerzman-Stein operator on C. The Kerzman-Stein operator A is a smoothing operator mapping Lz(C) into C,(W In ', Bell showed how the above Kerzman-Stein formula may be applied to solve the classical Dirichlet problem : Au=Q in R { u1~= f on I', with f E C,(r) being given. As a basic result he obtained that u=h+R where h and H both are holomorphic in 0 and moreover Cm(n).

137 138

2. The Cauchy transform in Euclidean space The aim of this section is to show how Clifford analysis offers a natural framework for generalizing the Kerzman-Stein formula to Euclidean space. Let &,m+l be the universal Clifford algebra constructed over the vector space Rm+l equipped with a quadratic form of signature (0,m+ 1) and let e = (eo, el,. . . ,em) be an orthonormal basis of Ro>m+l. Then noncommutative multiplication in b,m+lis governed by : e:=-1, i=O,1, ...,m and

ei ej + ej ei = 0, i # j.

Any element a E &,,+I may be written as

a=x aAeA, A where eA = ei,ei, . . . ei,,, 0 5 il < a2 < . . . < ih 5 m. Putting e,#, = 1, the identity element of &,m+l, we thus have that R is imbedded in &,m+l. For a E RJ,~+~,we call a4 = [a10 the real part of a. The conjugation a + E -is the anti-involution on &,,+I defined by Ej = -ej, j = 0,1,. . . ,m, and ab = $2 for a, b E &,m+l. m An element x = (xo,x1, . . . , 2,) E Rm+l is identified with x = C xj ej E j=O lP%m+l.Notice that in &,m+l, x2 = -Ix12 and xZ = Zx = )xI2. Now define the Dirac operator 8, in Rm+l by m a, = C ejaxj. j=O It is a strongly elliptic first order differential operator satisfying

8,"= -Ax, (2.1) A, being the Laplacian in Itm+'. Its fundamental solution is given by

where Am+l is the area of the unit sphere S" in Rm+l. In what follows, all functions considered are &,m+l-valued. A function f(x) = C fA(X)eA defined on a subset D of Rm+' is said to belong to some A classical class of functions on D if each of its components f~ belongs to that 139 class. If R is open in Rm+' and f E C,(R), then f is said to be (left) monogenic in R if axf = 0 in R. Notice that in view of (2.1), monogenic functions are &,m+l-valued harmonic functions in R. Important examples of monogenic functions are induced by the so-called Cauchy transform C. To that end, assume that R is a bounded smooth domain in Rm+' with C,-boundary C = 130.For any two elements f,g E Lz(C), define the inner product (f,g)E by

JE Furthermore, define the Cauchy kernel Cz(y) in Rm+' by

CX(Y) = 4y)E(Y - where v(y) is the outward pointing unit normal at y E C. Then for each f E L2(C),its Cauchy transform Cf is given by Cf(Z) = (Cz,f)E

Clearly Cf is monogenic in Rm+l \C.

3. The Kerzman-Stein formula in Euclidean space We consider Rz(R) (resp. H2(R)), the Hardy space of &,,+l-valued harmonic (resp. monogenic) functions on R having non-tangential LZ(C) boundary values and also H2(C),the Hardy space on C consisting of the boundary values of elements in H2(R). Then we have the following results

(see e.g. 415) : (i) H2(C)is a closed subspace of Lz(C) and Lz(C) admits the orthogonal decomposition LZ(C) = H2(C) @ vH2(C) (3.1) (ii) C maps L2(C) onto H2(R),and for each f E Lz(C),the non-tangential boundary value of Cf is given by the Plemelj-Sokhotzki formula : 1 Cf = z(f + Hf). Here we have introduced, for a.e. x E C, 140 the so-called Hilbert transform H. (iii) C may be extended to a bounded linear operator on L2(C);moreover, it is a skew projection operator, mapping L2(C) onto H2(C). (iv) H2(C) is a (right) &,,+l-Hilbert module with reproducing kernel, the Szego kernel S,(y), i.e. for f E L2(C) and 5 E 51,

(3-2)

Moreover, the associated Szego integral operator S with kernel S,(y) maps L2(C) onto H2(R). (v) Denoting by P the orthogonal projection of L2(C) onto H2(C)we have the Kerzman-Stein formula : P(1 +A) = c. The operator A = C - C* is a smoothing operator; it is also given by 1 A = -(H - 2 vHv). Furthermore P maps C,(C) into itself.

REMARKS (1) Hilbert modules with reproducing kernel over a (finite dimensional) H*- algebra were introduced independently in 619. Examples of such modules, 0 in particular the Bergman and Hardy spaces for the unit ball B(1)in Rm+l, were given in 7.

(2) The properties (i) - (v) listed above are generalizations to Euclidean space of similar results in the complex plane (see 2,10).

4. The case of the unit ball 0 In this section it is shown how X2((B(1))may be decomposed in terms of H2(h(l)),h(1) being the unit ball in Rm+l. Using the Szego kernel for the unit ball which is known to be (see 3, : 1 1+wx S,(w) = - 5 E h(l),bJE S", Am+l 11 + WZ[~+' ' 0 we obtain that the Poisson kernel P,(w) for B(1), given by 141 may be decomposed as follows :

P, (w) = sx(w) + sx(w) (4.1) where S,(w) = w Sx(w)Z. Now at each w E S", v(w) = w, and so we get in view of (3.1) that any g E L2(Sm) may be orthogonally decomposed as g = Pg +wP(L29). Hence, in view of (3.2) and (4.1) we have, for all g E Lz(S"),

(Px,g)s, = (SZ,9)S" + (Sx,g)sm or P=S+S where

Sg(w) = x S(P(Gg))(x), x E i(1). As P maps Lz(Sm)onto X'(h(1)) and S maps H2(Sm)onto Hz(i(l)), we obtain the following theorem.

0 Theorem 4.1. The space X'(B(1)) admits the decompositicin

X"i(1)) = HZ(i(1)) a3 5 HZ(i(l)),

0 i.e. given U E Xz(B(l)), there exists a pair of monogenicfunctions FI,FZ E H'(h(1)) such that

U(x) = Fl(X) + x FZ(2). (4.2) REMARKS

0 (1) If U,harmonic in B(l), belongs to Cm(l?(l)), then F1, FZ are monogenic 0 in B(1)and belong to Cm(l?(l)). (2) Let k E N be fixed and denote by H(k)(resp. M(k))the set of homogeneous harmonic (resp. monogenic) polynomials of degree k in Itm+'. If in Theorem 4.1., U is an element of H(k),say U = Sk, then the decomposition (4.2) reads

sk = Pk + X 9-1 1 42 where Pr, E M(k)and 9-1 E M(k - 1). (3) Theorem 4.1 may be considered as a generalization of Theorem 7.2 to Euclidean space.

0 (4) It is easily checked that on an open ball B(a,R) in Rm+', Sx(y) = CX(Y>.

PROBLEM : If for an open bounded smooth and simply connected domain 0 in Rm+l, Sx(y) = Cx(y),does this imply that 0 is an open ball ?

If the answer is affirmative, a generalization would thus be obtained of a result proved in the complex plane by N. Kerzman and E.M. Stein (see 10).

References 1. S. Bell, Solving the Dirichlet problem in the plane by means of the Cauchy integral, Indiana Math. J. 39 (1990), 1355-1371. 2. S. Bell, The Cauchy transform, potential theory, and conformal mapping (CRC Press, Boca Raton, 1992). 3. F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis (Pitman, London, 1982). 4. D. Calderbank, Clifford analysis for Dirac operators on manifolds with boundary (Max-Planck-Institut fur Mathematik, Bonn, 1996). 5. J. Cnops, An introduction to Dirac operators on manifolds (Birkhauser, Basel, 2002). 6. R. Delanghe, On Hilbert modules with reproducing kernel, In : Funtion Theo- retic Methods for Partial Differential Equations (Eds. V.E. Meister, N. Weck and W.L. Wendland), Lecture Notes in Mathematics 561 (Springer, Berlin, 1976), 158-170. 7. R. DeIaaghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, Proc. London Math. SOC.37 (1978), 545-576. 8. J. Gilbert and M. Murray, Clifford algebras and Dirac operators in harmonic analysis (Cambridge Univ. Press, Cambridge, 1991). 9. R.P. Gilbert and G. Hile, Hilbert function modules with reproducing kernel, Nonlinear Analysis 1 (1977), 135-150. 10. N. Kerzman and E.M. Stein, The Cauchy kernel, the Szego kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 85-93. TIME DOMAIN WAVE EQUATIONS FOR LOSSY MEDIA OBEYING A FREQUENCY POWER LAW: APPLICATION TO THE POROUS MATERIALS.

Z.E.A. FELLAH, S. BERGER AND W. LAURIKS Laboratorium voor Akoestiek en Thermische Fysica, Katholieke Universiteit Leuven, Belgium E-mail: Zine. [email protected]. be

C. DEPOLLIER Laboratoire d'Acoustique de 1 'Universite' du Maine, Avenue Olivier Messaien, 72085 Le Mans cedex 9, fiance

The acoustic attenuation in porous media having rigid frame is described by a fractional power law of frequency. In order to write the propagation equation in time domain, the concept of fractional derivatives is used given a good correspondence between theory and experience. A generalization for lossy media having a power law dependance of frequency of the attenuation a = aoIwIY is treated. Classical lossy time domain wave equations exists only for restricted cases where y = 0 or y = 2. For the frequently occurring practical situation in which attenuation is much smaller than the wave number, a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence. In order to obtain the corresponding time domain lossy wave equation, time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions.

1. Porous material having a rigid frame In the acoustics of porous materials, one distinguishes two situations according to whether the frame is moving or not. In the first case, the dynamics of the waves due to the coupling between the solid skeleton and the fluid is well described by the Biot theory In air-saturated porous media, the structure is generally motionless and the waves propagate only in fluid. This case is described by the model of equivalent fluid which is a particular case of the Biot model. In the model of equivalent fluid, the density and the bulk modulus are "renormalized" by the fluid-structure interactions. A prediction of the acoustic comportment of the porous material requires the determination of the dynamic tortuosity E(W) and the dynamic com-

143 pressibility p(w). These functions depend to the physical characteristics of the fluid in the pore space of the medium and are independent of the dynamic characteristics of the structure. The basic equations of the model of equivalent fluid are:

dVi --Awl dP = -v.v. PfE(W)Z = -vip, Ka dt In these relations, v and p are the particle velocity and the acoustic pressure, pf is the fluid density, K, = yP0 is the compressibility modulus of the fluid. The first equation is the Euler equation, and the second one is a constitutive equation obtained from the equation of mass conservation associated with the behavior (or adiabatic) equation. E(W) and p(~)are the dynamic tortuosity of the medium and the dynamic compressibility of the air included in the porous material. These two factors are complex functions which heavily depend on the frequency f = w/2n. Their theoretical expressions are given by Johnson et a1 ', Allard and Lafarge 4:

where i2 = -1, y represents the adiabatic constant, P, the Prandtl number, the tortuosity, ko the static permeability, k; the thermal permeability, A and A' the viscous and thermal characteristic lengths, r,~the viscosity of fluid. The functions E(W) and p(w) express the viscous and thermal exchanges between the air and the structure which are responsible of the sound damping in acoustics materials. These exchanges are due on the hand to the fluid-structure relative motion and the other hand to the air compression- dilatations produced by the wave motion. The parts of the fluid affected by these exchanges can be estimated by the ratio of a microscopic characteristic length of the media, as for example the sizes of the pores, to the viscous and thermal skin depth thickness 6 = (277/wp0)~/~and 6' = (2v/~poPr)~/~.

1 .l. High-frequency approximation When the frequency increases, the skin thickness becomes narrower and the viscous effects are concentrated in a small volume near the frame since 6/r << 1. In this case the compression / dilatation cycle is much faster than 145 the heat transfer between the air and the structure. The high-frequency approximation of the responses factors E(W) and p(w) when w + 00 are given by the relations

.(W) = Ew (1 + ;(&) '") , (4)

The attenuation Q(W) is given by

In the time domain the expressions of the dynamic tortuosity and the dynamic compressibility are given by Fellah and Depollier 5:

(7)

where * denotes the time convolution and 6(t) is the Dirac function. In this model the time convolution oft-'/' with a function is interpreted as a semi derivative operator following the definition of the fractional derivative of order v given in Samko and coll 6,

where r(x)is the gamma function. A fractional derivative no longer represents the local variations of the function but on the contrary, it acts as a convolution integral operator. When the wave propagates along the coordinate axis 02, the propagation equation in time domain is given by:

where the coefficients A, B and C are constants respectively given by:

The first one is related to the velocity c = 1/,/= of the wave in the air included in the porous material. The other coefficients are essentially 146 dependent on the characteristic lengths A and A' and express the viscous and thermd interactions between the fluid and the structure. Fig.1 shows the comparison between experimental transmitted signal, and simulated transmitted signal using Eq. 10 and boundary condition at the interfaces of a slab of porous plastic foam (thickness 2 cm, porosity: 0.95, a, = 1.07, A = 200pm and A' = 600pm)

Time (s) x 10"

Figure 1. experimental (solid line) and simulated transmitted signals (dashed line)

2. Generalization to lossy media obeying a frequency power law In a variety of media (e.g., porous media, liquids and tissue) over a finite bandwith, the attenuation of acoustic waves appears to be adequately modelled by a power-law dependence on frequency 71819310

where w is angular frequency, and a0 has units of Np/m and the loss is exp(-az). The absolute sign is a consequence of the real even properties of absorption as a function of frequency, y is a real positive finite number. 147

For most materials, the power law exponent y has value from 0 to 2. It is possible to write a general dispersion relation for the propagation of ultrasonic waves in a wide variety of media as: k2(w)= (w/c)2 + i2(w/c)(ao I w I”, (13) where k(w) = w/c + p’(w) + ia(w),p‘(w) is the extra term of dispersion, a(~)is the attenuation and c is the high frequency limit of the velocity of the medium. This relation is valid if (a(w)/$) << 1, this inequality defines a finite frequency range in which the general lossy wave equation developed is valid. This approximation is used widely for the linear case and nonlinear cases such as the Burger’s equations and KZK equations ll. In the case of porous materials, the above approximation is well verified when 6/r << 1, for high frequency range. In the time domain, all coefficients of the differential terms of the waves equations are real constants. This characteristic insure real results for real excitation signals. Even when a consistent, valid, frequency domain plane wave solution approach is used through either a complex compressibility or complex elastic constant, it cannot apply directly to the solution of pulsed case, as also pointed out by Nachman 12. A problem arises when we attempt to transform the general frequency domain lossy wave equation above back to the space and time domain 5, under conventional Fourier transform, the inversion is defined only when y is an even integer n:

p,, - l/c2ptt - (-l)7422ao/cptn+1 = 0. (14) For other powers of y, the classical Fourier transform derivative relations fail to recover differential terms with real constants. It is the case for the propagation in porous material at high frequency range in which the introduction of the fractional derivative definition is needed to write the wave equation in time domain. The approach taken by here is to apply generalized functions and theirs transforms to the problem, if y is an odd integer, it is helpful to rewrite Eq. 13 in an equivalent form

kz((w)= (w/c)~+ iw(2ao/c)sgn(w)w~ (15) for this odd integer case, a useful generalized function pair can be recast in the Fourier transform convention used for time (angular) frequency

FT-~[u%~II(~)]= y!/[.(it)g++’] (16) where sgn(w) is the sign function, this pair permits to write equivalent time domain lossy wave equation

pzz - l/c2ptt + (2/nc)ao(y + l)!(-l)(’+1)/2/p * l/t(”’) = 0. (17) 148

Another set of cases of interest are those in which y is a noninteger, an appropriate generalized function transform pair from Lighthill 13is given by:

FT;~(Iw 1,) = r(y + 1) COS[(Y + 1)~/2]/(~I t lY+l), (18) this non integer case include the case of fractional derivative defined previously in Eq. 9. The general propagation equation in time domain is that given by P,, - (i/c2)ptt+ (2/7~)~r~r(~+ 2) COS[(Y + I)T/~]P* I/ I t pis)=0~9) In summary these wave equation can be expressed in the compact form:

P,, - 1/C2Ptt - (2/c)d/dt[L,,,,t *PI = 0. (20) where L,,,,t is a time domain convolution loss operator that is a function of time t, loss Q, and y, and it differs for y as an even or odd integer or as a noninteger.

2.1. Causality Causal means that an effect cannot precede its cause. For a time waveform initiated at t = 0, its spectral characteristics must meet certain requirements so that complete time cancellation occurs for t < 0. To take advantage of these Hilbert transform relations, we use the definition of the propagation factor,

Y(W) = jWW) = -4W) + APO + P‘(W)l (21) where k(w) is the complex wave number and p’ is the extra dispersion term needed for causal propagation, PO = W/C, c = cg/& in high frequency range. It has been recognized that the causal Hilbert transform relationships have more general applicability and that specifically they also relate the real and imaginary parts of complex propagation constant 879J10.Both .(w) and P‘(w) are also related through their Hilbert transform’s in the present sign convention,

P’(4 = [-l/(nw)l* [-.(W)l, -4W) = [l/(.rrW)l* P’W. (22) By defining

La,y,t = FT,-l[-a(41, Lp,y,t = F53P’(w>l. it is easy to write the above Hilbert transform in time domain:

Lp,,,t = -i sgn(t)La,,,t La,y,t = i sgn(t)Lo,,,t (23) 149

Equations (23), are the time causal relations. These relations can be shown to be the Fourier transforms equivalent of the Hilbert transforms. Because generalized function, time domain operators satisfying Eq. (23) have no restriction on the value of y (assumed to be finite, real), they have more general validity ( for y > 1) than the Kramers-Kroning relations expressed in the frequency domain. While the frequencies domain Kramers-Kronig relations require knowledge of either Q or p’ at all frequencies, the convolution of a time domain causal operator, each one is naturally limited by a finite length of an imput pressure of total propagation operator. The temporal propagation operator L,,y,t = FTT1[y(w)] is given by

L,,y,t = La,y,t + iLp,y,t = [1+ sgn(t)lLa,,,t = 2H(t)Lx,,,t, (24) here H(t)is the step function [12] defined as : H(t) = 0, t < 0, H(t) = 1/2, t = 0, H(t) = 1, t > 0. (25) The final causal lossy wave equation in time domain is given by

Pzz - 1/C2Ptt - (4/c)Wt[H(t)La,,,t*PI = 0 (26) Because of the step function, the propagation time operator L,,,,t is causal. The attraction of a time domain based approach is that analysis is naturally bounded by the finite duration of ultrasonic pressures and is consequently the most appropriate approach for the transient signal.

References 1. M. A. Biot, J. Acoust. SOC.Am. 28, 168 (1956). M. A. Biot, J. Acoust. SOC.Am. 28, 179 (1956). 2. D.L. Johnson, J. Koplik, R. Dashen, J. Fluid. Mech. 176,379 (1987). 3. J.F. Allard, Chapman and Hall. London, (1993). 4. D. Lafarge, P. Lemarnier, J. F. Allard and V. Tarnow, J. Acoust. SOC.Am. 102, 1995 (1996). 5. Z.E.A Fellah and C. Depollier, J. Acoust. Sac. Am. 107,683 (2000). 6. S. G. Samko, A. A. Kilbas and 0. I. Marichev, Gordon and Breach Science, Amsterdam(l993). 7. P. He , IEEE Trans. Ultrason. Ferroelectr. Reg. Control. 45, 114-125 (1998). 8. N. Akashi, J. Kushibiki and F. Dunn, J. Acoust. Soc. Am. 102, 3774-3778 (1997). 9. K. R. Waters, M. S. Hughes, J. Mobley, G. H. Brandenburger and J. G. Miller, J. Acoust. SOC.Am, 108,556-563 (2000). T. 10. L. Szabo, J. Acoust. SOC.Am, 97,14-24 (1995). 11. B. K. Novikov, 0. V. Rudenko and V. I. Timoshenko, AIP, New York, English translation by R. T. Beyer (1987). 12. A. I. Nachman, J. F. Smith and R. C. Waag, J. Acoust. SOC.Am, 88 (1990). 13. M. J. Lighthill, (Cambridge University), Chap.3 (1962). A MODEL FOR POROUS DUCTILE VISCOPLASTIC SOLIDS INCLUDING VOID SHAPE EFFECTS

L. FLANDI AND J.B. LEBLOND Laboratoire de Mode‘lisation en Me‘canique, Universite‘ Pierre et Marie Curie (Paris VI), 8 rue du Capitaine Scott, 75015 Paris, France E-mail: jlandiQlmm.jussieu.fr, EeblondOlrnm.jussieu.fr

The aim of this paper is to propose a new model for porous viscoplastic solids incorporating void shape effects. The sound matrix is assumed to obey a simple Norton law with exponent n. Several previous well-accepted models pertaining to various special cases are used as references. These models include: (i) that of Gologanu, Leblond and Devaux, for arbitrary spheroidal voids but an (ideal-) plastic matrix only (n = m); (ii) that of Leblond, Perrin and Suquet, for spherical or cylindrical voids only, but arbitrary n; (iii) that of Ponte-Castaneda and Zaid- man, for spheroidal voids but a linearly viscous matrix only (n = 1). Use is also made of the nonlinear Hashin-Shtrikman bound for voids of arbitrary shape and arbitrary n, but low triaxialities. The approach used basically consists of looking for a suitable heuristic expression for the ‘Lgaugefunction” of the voided material, which is required to reduce to those corresponding to the reference models in the relevant special cases. The validity of the model is messed through comparison between the analytical, approximate “gauge surface” proposed and that determined by considering a spheroidal RVE and performing some numerical minimization of the viscoplastic potential over a large number of trial velocity fields.

1. Introduction An impressive number of models, which cannot all be cited here, have been proposed for the overall behavior of porous viscoplastic materials, in order to predict ductile rupture of metals at high temperatures. In almost all models, the behavior of the sound matrix is assumed to be governed by a simple Norton law with exponent n, connecting the local strain rate d and the local stress u.This constitutive law includes linearly viscous and (ideal-) plastic materials as special cases, for n = 1 and n = 00 respectively. The problem is to define the resulting relation, which must necessarily involve the void volume fraction, between the overall strain rate D and the overall stress E. Only 2 models, however, try to account for the influence of void shape.

150 151

The first one is due to Ponte-Castaneda and Zaidman’ (PCZ). It relies on the use of the nonlinear Hashin-Shtrikman (HS) bound (based on consideration of some “linear comparison material”) for the overall viscoplastic potential, adopted as an approximate value instead of a limiting one. The PCZ model is very comprehensive, especially in later versions also incorporating the influence of the spatial distribution of voids. It nevertheless suffers from the fact that the HS nonlinear bound, although fully rigorous as a bound, is known to provide a poor estimate of the overall behavior for high values of the triaxiality (ratio of the overall mean stress over the overall equivalent stress), when the Norton exponent n is itself high. The PCZ model can therefore be considered as a good reference model only for n = 1 (linearly viscous material) or low triaxialities. The second model incorporating void shape effects is due to Garajeu, Michel and Suquet2 (GMS). Unlike the PCZ model, the GMS model does not rely on the use of a linear comparison material but on direct homogenization of some viscoplastic voided RVE. It undoubtedly yields better predictions than the PCZ model, except for values of n close to 1. It nevertheless also suffers from several drawbacks. First, in the plastic cme (n = GO), the GMS model bears a close resemblance to the model of Golo- ganu, Leblond and Devaux3 (GLD), but unfortunately rather in its early version than its final one. The GLD model, which has gained relatively wide acceptance and applies to spheroidal voids but plastic matrices only, indeed exists in 2 versions, the second one3 being much better than the first one, as shown by Sovik4. Second, the GMS model considers only prolate cavities, no proposal being made for oblate ones. Finally, the expression proposed for the necessary evolution equation of the void shape is very inaccurate for deviatoric stress states, as is evident from reference3 in the plastic case. We look here for a model for porous viscoplastic materials incorporating void shape effects but free of the defects of the PCZ and GMS models. The approach used is the same as in the work of Leblond, Perrin and Suquet5 (LPS) devoted to viscoplastic matrices but spherical or cylindrical voids only; namely, instead of looking for an approximate analytic expression of the overall plastic potential, which is a very difficult task, one looks for an expression of the overall “gauge function” (as defined below). Several classical models pertaining to various special cases are used as references:

(1) the GLD model, applicable to plastic materials (n = GO) only but arbitrary void shapes; (2) the LPS model, applicable to viscoplastic materials (n arbitrary) 152

but spherical or cylindrical voids only; (3) the PCZ model, applicable to linearly viscous materials (n = 1) only but arbitrary void shapes.

The analytic expression of the gauge function looked for is required to match the expressions corresponding to these models in the relevant special cases. A further requirement is that it must match the value of the overall viscoplastic potential provided by the HS nonlinear bound for arbitrary n but low triaxialities, since it then provides a good estimate.

2. The GLD, LPS and PCZ Models We briefly present here the GLD, LPS and PCZ models used as references. The void volume fraction (porosity) is denoted f. The voids are assumed to be spheroidal, with axis of symmetry parallel to the direction z. The semi-axis of the voids along this direction is denoted a, and the common semi-axes along the perpendicular directions x and y, b. The void shape is thus characterized by the single parameter a/b; prolate voids have a/b > 1 and oblate ones a/b < 1. Only axisymmetric loadings with the same axis of symmetry as that of the voids are considered: the sole non-zero components of the overall stress tensor are thus C,, = C,, and E,,.

2.1. The GLD Model This model is applicable to voids of arbitrary shape (oblate as well as prolate), but the material is assumed to be (ideal-) plastic. The yield stress in simple tension is denoted 00. The expression of the overall yield function, which extends that of Gurson applicable to spherical voids only, reads:

-(g + q2- (9 + f)2 = 0,

Cd c,, - c,,, Ch 2a2E,, + (1 - 2a2)E2,

(Ed is the deviatoric part of E). In this expression G, 77, g, K and a2 are parameters depending on f and a/b, the detailed expressions of which are given in reference3. The quantity g is of purely geometric nature and plays the role of a %econd, fictitious porosity”; in the special case of penny- shaped cracks (a/b = 0) for instance, it is identical to the porosity which would result from spherical voids with radius equal to that (b) of the cracks. 153

2.2. The LPS Model This model applies to viscoplastic materials but spherical or cylindrical voids only. Instead of providing an approximate expression for the overall viscoplastic potential 9(X,f, n), it gives an approximation of the overall gauge surface, composed of those stress tensors S corresponding to some constant, specified value of 9. This value is chosen in such a way that in the plastic case (n = m), the gauge surface becomes identical (up to an unimportant factor) to the yield surface. The gauge surface is described by some equation F(S,f,n) = 0 where F is the gauge fisnction. Prescribing F(S,f,n) is sufficient to prescribe 9(X,f,n), that is to fully define the model. Indeed 9 is a positively homogeneous function of degree n + 1of Z. For any given X,there is a unique positive scalar A such that S ZE X/A lies on the gauge surface, i.e. such that F(S,f,n) = 0. The value of 9(X,f,n) is then deduced from that of 9(S,f, n) through multiplication by An+’. For spherical voids, the equation of the LPS gauge surface reads: n-1 F(S,f,n) Si + f n + 1H(Sm)

where sd E S,, -Sxxand S, 3 gtr S denote the deviatoric and mean parts of S. The function H is chosen in such a way that the model reproduces the exact solution of the problem of a hollow viscoplastic sphere loaded hydrostatically. A similar expression is proposed for cylindrical voids.

2.3. The PCZ Model As explained above, this model was developed for arbitrary values of alb and n, but in fact provides a good estimate in the whole stress space only for n = 1. The equation of the gauge surface proposed then reads:

(3) 9 +-sk+3(1-3ai)SdSm -1=o 4 1 where a1 and a; are coefficients depending on f and alb, the expressions of which are given in reference1. 154

3. The Model Proposed 3.1. Expression of the Gauge Function The gauge function proposed is deduced from the GLD yield function (1) in the same way as the LPS gauge function (2)from the Gurson yield function. Indeed the problem (extending an expression valid for n = co to arbitrary n) is the same, the only difference being the void shape (spherical for the Gurson and LPS models, spheroidal for the GLD model and here). We thus replace the “cosh” by i(H + Sh)and introduce a factor of in the term -(g + f)2 in the equation of the gauge surface:

U F (s, f,pn) C(Sd 7m2 (9 l)(g f) H(Sh) -- = + + + + + n + 1H(Sh) n-1 -(9 - -(g + n+l + fy = 0,

This expression can match (l),(2) and (3)in the relevant special cases provided that the coefficients are ascribed suitable values then. However, general expressions applicable for arbitrary values of f, u/b and n are needed.

3.2. Choice of Coeficients The parameter g being of purely geometric nature (see above), it is logical to ascribe it the same value as in the GLD model3, independent of n. The parameter a2 determines the weighting coefficients of the radial (Szz)and axial (Szz)stress components in the stress Sh, which is the essential parameter governing void growth. There is no clear reason why it should depend on the matrix rheology. This suggests to also take a2 as independent of n and given by the same value as in the GLD model3. Now for n = 00 or 1, enforcing coincidence of (4) and ((1)or (3)), one gets distinct values d”), dl)of IF.The following “interpolation” formula for arbitrary n is then proposed for this parameter:

Finally, the nonlinear HS bound provides a good estimate of that zone of the gauge surface corresponding to low triaxialities. Enforcing coincidence 155 and tangency of the HS and present gauge surfaces at some typical point in this zone, we get formulae for the remaining 2 parameters C and 17. The gauge function proposed then meets all desired requirements.

4. Comparison of Approximate and “Exact” Gauge Surfaces and Directions of Flow For all values of f, a/b and n, a numerical, supposedly exact gauge surface can be obtained by considering a suitable RVE and performing some minimization of the average value of the local viscoplastic potential over a large number of trial velocity fields. In practice, we adopt a spheroidal RVE confocal with the void, and use the family of velocity fields proposed by Lee and Mear6, especially adapted to the spheroidal geometry. Figure 1 shows the results obtained for a prolate void, typical values of n and a/b and several values of f.

n = 5, a/b = 5 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 4 5 6 S, Figure 1. Approximate and numerical gauge surfaces

The direction of the normal to the gauge surface governs, via some “nor- mality rule”, that of viscoplastic flow. Figure 2 allows for a comparison, in a typical case, of the approximate and numerical directions of this normal. The quantity on the horizontal axis is the angle tan-’ (Sm/Sd)(in degrees), characterizing position on the gauge surface, and that on the vertical one 156

the angle tan-' [Dm/(fDd)] (also in degrees) where Dd E D,, - D,, and D, z Str D, characterizing the direction of the normal.

- 140 2 120 Q 5 100 \ 80 Y2 *; 60 40 Y3 20 0 -20 0 20 40 60 80 100 120 140 160 180 tan-* (S,/Sd)

Figure 2. Directions of the normals to the approximate and numerical gauge surfaces

5. Perspectives This work will now be pursued in 2 directions. First, we shall look for a suitable evolution equation for the void shape, extending to arbitrary n that proposed in the GLD model for n = 00. Second, we shall compare model predictions with numerical (FE) simulations performed for more physically meaningful RVE (cylindrical instead of spheroidal).

References 1. P. Ponte-Castaneda and M. Zaidman, J. Mech. Phys. Solids 42, 1459 (1994). 2. M. Garajeu, J.C. Michel and P. Suquet, Comput. Methods Appl. Mech. Engrg. 183,223 (2000). 3. M. Gologanu, J.B. Leblond, G. Perrin and J. Devaux, in Continuum Microme- chanics, P. Suquet, ed., Springer, pp. 61-130 (1997). 4. O.P. Sovik, Ph.D. Thesis, Norwegian Univ. of Science and Technology (1996). 5. J.B. Leblond, G. Perrin and P. Suquet, Int. J. Plasticity 10, 213 (1994). 6. B. Lee and M.E. Mear, J. Mech. Phys. Solids 40, 1805 (1992). ACOUSTIC WAVE PROPAGATION IN A COMPOSITE OF TWO DIFFERENT POROELASTIC MATERIALS WITH A VERY ROUGH PERIODIC INTERFACE: A HOMOGENIZATION APPROACH

ROBERT GILBERT Department of Mathematics, University of Delaware Newark, Delaware, i97i6, USA E-mail: gilbertOmath.udel.edu

MIAO- JUNG OU Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA E-mail:miouOima.umn. edu

Homogenization is used to analyze the system of Biot-type partial differential equations in a domain of two different poroelastic materials with a very rough periodic interface. It is shown that by using homogenization, such a rough interface can be replaced by an equivalent layer within which a system of modified differential equations holds.The coefficients of this new system of equations are certain “effective” parameters. These coefficients are determined by the solutions of certain auxiliary problems which involve the detailed structure of the interface. In this paper, the auxiliary problems are derived and the homogenized system of equations is given.

1. Introduction Poroelasticity, the mechanics of porous elastic solids with fluid-filled pores, has received attention in the last few decades for its important role in oil recovery, the study of the triggering of earthquakes, liquid waste disposed by underground seepage into pores, and underwater acoustics involving propagation in the water-saturated, porous bottom of the ocean, etc. In this paper, the interface between the two poroelastic layers is assumed to be very rough with a periodic geometry, i.e. the ratio of the amplitude to the “wavelength” is large. Problems involving rough boundaries or interfaces can usually be analyzed using perturbation methods when the amplitude/wavelength ratio is small. For problems with a large ratio, other methods of analysis are required. For example, the homogenization method used by Kohler, Papanicolmu and Varadhan ’, or Nevard and Keller ’.

157 158

In their works, it is shown that for certain partial differential equations, the problem of the PDE in the region which contains the rough interface can be replaced by another problem of a homogenized PDE in an equivalent flat layer. The basic assumption underlying the method is that the scale length, or “wavelength” E of the roughness is small compared with all other relevant lengths, especially the roughness amplitude. We use a homogenization method, similar to the one mentioned above, to study the time-harmonic acoustic wave propagation in a composite of two different poroelastic materials. The auxiliary problems and the homogenized equations in the equivalent layer will be derived.

2. The Constitutive Equations

The poroelastic equations derived by Biot 3)4,5 have long been regarded as standard in solving problems in poroelasticity. However, the validity of these equations has recently been questioned. For this reason, various authors have used homogenization methods to derive the governing equations of linear poroelasticity by starting with the micro-structure of the pores. For example, Burridge and Keller 6, Gilbert and MikeliC 7, Auriault 8, and Auriault et al. 9. In these papers, the pores are assumed to have periodic structure, and the linearized equations of elasticity and the linearized Navier-Stokes equations are used to describe the behavior of the solid part and the fluid part, respectively. These works have shown that the newly derived equations coincide with Biot ’s equations when the dimensionless viscosity of the fluid is small. Furthermore, their works also enable us to perform a complete calculation of the effective parameters in Biot’s equations. Since the pore fluid in the seabed can be regarded as Newtonian and incompressible , we will adapt the equations given by Auriault et al. in g. For time harmonic motion u(x,t) = u(x)eiwt,p(x, t) = p(x)eiwtin an isotropic poroelastic medium, these equations can be reduced to lo

dzj(Cijhldzluh) + w2(pYijuj) = -&ijaxjp, 23 # h(z1,22), (1) 2 --Kijz (ax,dxjp-plw aziuj) - haxiui = 23 # h(z1,52)1 (2) W BP, 1 59 where

[F(x*)]:= lim lim F(x). Y - 3x+x* a- F(x) - y+ 3X--tx* a- In these equations, Z = G,Cijkl is the elasticity tensor of the skeleton, p the pressure( positive for compression), w the acoustic frequency, ui the displacement of the solid part, f the porosity, p1 the density of the pore fluid, ps the density of the skeleton and Kij is the generalized Darcy permeability tensor, which is introduced by the homogenization theory and is w-dependent. The effective parameters ii and can be computed. The interface between the two different poroelastic materials is described by 23 = h(xI,xz),where h is €-periodic in both the XI and x~ direction. The physical parameters are assumed to be constants above and below the interface and have a jump discontinuity across the interface. In what follows, we adopt the summation convention, Latin subscripts taking the values 1, 2, 3 and Greek subscripts taking the values 1, 2. In (1)-(6), 6 is

Figure 1. Schematic representation of the z1z2 profile of the periodicity cell Y. the Kronecker delta tensor and n is the unit normal vector of the interface 23 = h(x1,xz)pointing in the positive 53 direction. The interface conditions (3), (4), (5) and (6) represent the continuity of displacement, normal stress, pressure and the averaged relative fluid velocity across the interface, respectively.

3. Formal Expansions and the Homogenized Equations We introduce a new independent variable y, where ycu := -,xff a = 1, 2. € Using the fact that n is proportional to (€-'ay,h, e-'dynh, -l), (4) and (6) 160 become

(av,h) Cifiki&l~k - Ci3kr&,urc] = 0, (7) [E-' Kaj (azj~- plw2uj) (ag,h) - K3j (azj~- plw2uj)] = 0. (8)

Define new dependent variables wi and q such that

wi(X,~,t,e): =~i(~,t,€), a(x7 Y, t, 4 : = P(X, t, 4. Note that az,ui = (aza+ E-ldP,)Wi and dz,p = (az, + E-'du,)q. Now we assume that w and q have the following asymptotic expansions for E small: 2 w(x, y,t, €) = U(O)(X, y,t) + c u(k)(x,y,t)2 + 0(€3), (9) k=l 2 q(x,Y,t,E) = P(0)(x,Y,t) + CP(k)(X,Y,t)Ek+ 0(E3). (10) k= 1 We also assume that each term in the expansions is Y-periodic and that the asymptotic forms of the derivatives of w and q are given by term by term differentiation of (9) and (lo), respectively. First, We consider the equations corresponding to

a,, (cifikaausuf)) = 0, (11)

[Cipkaa,s"k(0)a,,h] = 0, (12)

By scalar multiplying (11) by u(o), then integrating the product over the periodicity cell Y ( see Figure 1 ) and applying the Y-periodic condition to u(O), we have

where

V,h := (a,, h, a,,h)T Applying the jump conditions on r,we obtain 161

By the symmetry of cijkl and dgpul0),we have

$,uy = 0. (15) Applying similar argument to the equations for p(O),we get

This implies

aarpp(0)= 0. (16) Next, we consider the O(E-')equations. Applying (15) and (16) to these equations, we get

~cYp~,u%pP(l)= 0, 23 # h(Yl, Y2), (17)

To solve (Pl), we introduce a new variable 4j(23, y) by writing p(') in the form

P(')(X,Y,~)= 4j(23,~)(d,,d0)(x,t) - prw2uy)(x,t)). (18)

Substituting (18) into (Pl), we can see that p(')(x, y,t) will solve (Pl) if $j (23,y) satisfies

K&gQq&j = 0, 23 # h(Yl,Y2), (19) [(Ka&/p4j + Kaj) (a,J)I = 0.

We also require 4j(x3, y) to be Y-periodic in the y variable, continuous in Y and to have zero average over Y. These conditions uniquely determine 4j- Similarly, we have the following system of equations for u(l):

(p2) : {a,, (cipkad,,ur)) = X, 23 # h(yl,y2), [ (cipkaagauf) + cipklas,ur)) (dgph)] = 0.

To solve (P2), we introduce a new variable Xkmn by writing uf) in the form

ur)(x,Y, t) = Xkmn(237 Y)&,ug)(x, t). Substituting this into (P2) gives

Cigk6 (dppaysxkmn) (0) - - 0, 23 # hbl,Y2), [(CiflkcYdysxkmn + Cipmn) ( agph)] (&,,d))= 0- 162

Therefore, xkmn (23,y)ax, ug) (x, t) solves (P2) if Xkmn satisfies

(AP2){ "z4kb (agpay6Xkmn) = 0, x3 # h(Y1, Y2), (20) [(CiflkbagsXkmn + Ciomn) ( agph)] = 0. We also require Xkmn to be continuous in Y,Y periodic in the y variable and to have zero average over Y. These conditions uniquely determine Xkmn. Finally, we consider the O(to)equations. We integrate these equations with respect to y over Y and divide it by the area of Y,which is denoted by A. We then apply the divergence theorem and the assumed Y-periodicity of each term in the formal expansions. Using the jump conditions on I?, we may convert the higher order "flux" terms into lower order terms. This gives us the following theorem.

Theorem 3.1. Let u(x,t, E) and p(x,t, E) satisfy equations (I) and (2) on both sides of the periodic surface 53 = h(x1,xZ) in a poroelastic composite with constant solid density ps, fluid density pl, Darcy permeability K, porosity f, effective parameters d and b on each side. Suppose u(x,t, E) and p(x,t, E) also satisfy the continuity conditions (3)-(6) across the interface x3 = h(xl,x2), which is between x3 = 0 and 53 = a. If u(x,t, E) and p(x,t, E) have the asymptotic forms (9) and (lo), respec- tiveZy. Then u(O)(x,t) and p(O)(x,t) satisfy the followang system of homogenized equations for 0 < x3 < a,

where 163

Here, $j und xkmn are solutions to the auxiliary problems (AP1) and (AP2), respectively.

References 1. W. Kohler and G.C. Papanicolmu and S. Varadhan. Boundary and interface problems in regions with very rough boundaries in Multiple Scattering and Waves in Random Media, P. Chow and W. Kohler and G.C. Papanicolmu, eds., pp. 165-197, North-Holland, Amsterdam, 1981 2. J. Nevard and J.B. Keller. Homogenization of Rough Boundaries and Inter- faces, SIAM J. Appl. Math., Vol. 57, No. 6, pp. 1660-1686, 1997 3. M.A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. J. Acoustical Society of America, Vol. 28, No. 1, pp. 168-178, 1956 4. M.A. Biot. Generalized theory of acoustic propagation in porous dissipative media. J. Acoustical Society of America, Vol. 34, pp. 1254-1264, 1962. 5. M.A. Biot. Mechanics of deformation and acoustic propagation in porous media.Journal of Applied Physics. Vol. 33, pp. 1482-1498, 1962 6. R. Burridge and J.B. Keller . Poroelasticity equations derived from microstructure. J. Acoustical Society of America, Vol. 70, No. 4, pp. 1140-1146, 1981 7. R.P. Gilbert and A. MikeliC. Homogenizing the acoustic properties of the seabed: Part I. Nonlinear Analysis, Theory, Methods and applications, Vol. 40, pp. 185-212, 2000 8. J.L. Auriault. Dynamic behavior of a porous medium saturated by a Newto- nian fluid. Int. J. Engng. Sci. Vol. 18, pp. 775-785, 1980 9. J.L. Auriault and L. Borne and R. Chambon. Dynamics of porous saturated media, checking of the generalized law of DarcyJ. Acoustical Society of Amer- ica, Vol. 77, No. 5, pp. 1641-1650, 1985 10. C. Boutin and G. Bonnet and P.Y. Bard. Green functions and associated sources in infinite and stratified poroelastic media. Geophysics J. R. Astr. SOC.Vol. 90, pp. 521-550, 1987 EFFECTIVE ACOUSTIC EQUATIONS FOR A NONCONSOLIDATED MEDIUM WITH MICROSTRUCTURE

ROBERT P. GILBERT University of Delaware, Newark, DE 19716 USA E-mail: gilbertOmath.udel.edu

ALEXANDER PANCHENKO Washington State University, Pullman, WA 99164 USA E-mail: panchenkoOwsu.edu

In this work we show how the heuristic development of the acoustic equations for a non-consolidated media can be derived using the methods of homogenization.

1. Introduction

In Buckingham’s model of an unconsolidated sediment 435,6, the dissipation of acoustic energy is caused primarily by the rubbing of grains against one another. Moreover, the thinness of the pore fluid sandwiched between the grains is thought to be responsible for strain hardening in the granular medium. When two grains begin to slide against one another first there is a period when the particles stick, or in Buckingham’s terminology their asperities are effectively pinned 6. On being triggered into slipping the contact will become tighter and the effective viscosity of the lubricating fluid film will increase giving rise to strain hardening. The methodology we shall assume is that of homogenization where there is assumed a small parameter 6 which describes the microstructure. Com- pared to the previous work in this field, our approach is most reminiscent of that in the paper of Burridge and Keller 7. A microscopic length scale 6 is introduced by assuming that material properties are of the form f(z,t), where y = is the so-called fast variable. To develop such a framework, we sought a model in which the length scale separation is maintained by imposing more flexible assumptions than the overly restrictive periodicity. We shall refer to a material as being admissible, if these fluctuations are comparable to 6 in size.

164 165

We assume several other rather general conditions, such as the density p of the fluid is bounded from above and below independent of c (no vacuum assumption), and the vibrations of the interface are insignificant even on the microscale.a

2. Microscopic Equations We consider our regime to be a fixed large cube U in three-dimensional space, and we assume that there are no voids. Then U can be written as a union of two non-intersecting domains: the solid domain Wcand the fluid domain V', where the subscript indicates dependence on the small parameter c. For a fixed cube C define C' to be the cube shrunk by a factor of E: C' = {x : 6-l~E C}.

Assumption 2.1. For any 0 < E 5 1, U can be broken up into disjoint cubes Ci with the properties

with GI, CZ andependent of e, k.

The volume fraction a: of the solEd phase in each cube satisfies

c, IaiI c4, (2) with C3, C4 independent of E, k and such that 0 < C3 < C4 < 1.

Next we consider the characteristic function 8(x,t, E) of the fluid domain Vc,which we take to be smoothed-out and differs from zero in the domain only slightly larger than V'. It is clear that Assumption 2.1 forces oscillations of 6 with a wavelength comparable to c. An important mathematical consequence of this assumption is that for every C1-function f on Ck such that f = 0 on W1n Ck, the Poincare inequality

holds with a constant L independent of the choice of Ck. aThe latter assumption is needed to avoid a possibility of friction between the different parts of the solid phase. 166

Assumption 2.2. Time-dependence of 8 does not influence the effective equations and thus can be ignored.

Assumption 2.3. There exists a function Q(x,y), where x ranges over U, and y belongs to R3,with the following properties. X e(Xc,E> = &(z, 5)' (4) for 0 < E 5 1, and partial derivatives of Q are bounded uniformly. We consider a composite medium that contains a porous elastic phase and a fluid phase. To this end we postulate the following microscopic equations. Solid Phase. The displacement in the solid, u, satisfies

po(z)d?u - div (A(z)e(u))= f, (5) where po is a known density, A is an elastic tensor, e(u)denotes the symmetric part of the deformation gradient, and f is the body force density. We assume that A is symmetric and its components are bounded from above and below.

Fluid Phase. The state variables here are density p, velocity v and pressure P. We use the system of equations of compressible barotropic fluid.

Mass balance: dip + div (pv) = 0. (6) Momentum balance: &(pv) + div (pv BV) - pAv - [divv + VP= f, (7) where (v @ v)ij = vivj. Equation of state: P=ap,6 (8) where a is a material constant and 6 > 1 . In Eq. (7) p and < are viscosity coefficients.

3. Homogenized Equations Let us define the displacement in V' by 167

Then the linearized equation for 0 is equivalent to

po@u - div (Afe(&u)+ apodiv uI) = f, (9) where I is the second order identity tensor, and f- Aijkl - 2Pdikhjl + (t- P)&jkl. In what follows we will use 8(x, :) to denote the characteristic function of V'. The elastic tensor of the solid will be denoted by A". Combining Eq. (9) with the elastic equations in the solid we obtain the system

p@u - divT = f (10) in U,where the microscopic stress tensor T is given by T = (1 - 8)A8e(u)+ 8Afe(atu)+ 8apo div uI. (11) In order to obtain the effective stress we multiply divergence of T by a certain parameter-dependent test function 6,integrate by parts, and then pass to the limit in the resulting integrals. This procedure is known as the method of oscillating test functions and is due to Tartar Is. The crucial ingredient here is the choice of @. We set X (Pt(X, t>= 4@, + €41(t, z, 5)' where the function d, is arbitrary, and 41 depends on 4, namely, T dl (4 z,Y) = NP4kY)e(4)pq + /- MPQ(t- 775, ?/)e(d)pq(T,z) fh.(12) t The the vectors NPQ and MPq are found from the local problems analogous to the cell problems of periodic homogenization. To describe the local problems we need some notation. Denote by cij the square matrix with the components ctj equal to one if k = i, 1 = j and zero otherwise. Then NPQ is required to satisfy

div,([(l - 8)Ag+ 8Af](e,(Np4)+ cpq))= 0. (13) After the NpQ are determined, we find the initial value M,PQ(9) by solving

div ,(8[Af (e(M,Pq) + epq)+ apodiv ,NPqI]) = 0. (14) Finally, the MPQ are required to satisfy the equation div, [(l - 8)Ase(Mpq)- 8Afe(dtMpq)+ @upodiv, MPQI]= 0. (15) To avoid componentwise notation in what follows, we define the fourth- order tensors K1,K2 and the second-order tensors K3,K4 as follows.

Kiljkl(Y1 = e,(Ni%, (16) 168

Ki = div,Mij. Next we observe that for a differentiable function $(x, y),

Using this we obtain - lTs, div T . 4'dxdt = lTs, Te(#) dx dt =

The last integral is bounded uniformly in E so that the corresponding term disappears in the limit E + 0. Integrating by parts in the remaining integral we obtain

lTiT.e(@)dxdt= -E iTiu.divu(B1-Bz)dxdt+ (18)

~T~~.divZ(B1- B2)dxdt +O(E),

where B1 = (1-@)AS(e(4)+e,(41)) +8apo(div4+div,41)l, and

~2 = eAr(e(at4)+ e,(atqM). Due to the choice of 41, the €-'-term in the above equation is zero. This can be verified by straightforward computation using the fact that NPq and MPq solve the local problems. To obtain the effective stress, we need to pass to the limit in the €'-term. Justification of the passage to the limit is given in g. Using the definition of 41 and the tensors Kj defined in Eq. (16) we write 169

where the tensors T: are defined by T; = (1 - 8)AS(I+ K1) - 8AfK2(0,.) + 8K31, (19) TG = 8Af (I + K1), (20) Ti = (1 - 8)A8K2- 6AfdtK2+ K41. (21) Now we find the effective tensors TI(x), Tz(x), T3(t1 x) that satisfy div (Tle(q5))= lim div, (Tfe($)), (22) €+O div (T2e(dtq5))= lim div, (T;e(dtq5)), (23) e+O div T3(t- r,x)e($)(T,x)dT= lim div, 7, -)d.r. lT €-to (T,’(t - .)e($)(r, (24) Returning to Eq. (17) we see that

Now an integration by parts yields

for any test function $(xlt). The expression in brackets defines the effective stress tensor that combines elastic and memory effects. The term TIe(u0) represents the elastic part of the overall stress, while the other two terms describe the viscoelastic part. Using Eq. (25) we can write down the effective equations as t podtuo - div (TIe(u0)+ TZe(&uo) + I T3(t - 7,.)e(uo)(~, -1d.r) = f@6) If we compare our equations with those of Buckingham, they are seen to be very similar. Of course, the resemblance is qualitative, since without additional assumptions it is difficult to derive estimates on the longtime 170 behavior of the effective tensor T3,which is a crucial feature of the Buck- ingham model. Current investigations are directed to this problem and will be reported on in a subsequent paper.

Acknowledgments The work of both authors was supported in part by the Office of Naval Research through grant N00014-001-0853.

References 1. A .Yu. Beliaev and S.M. Kozlov, Communications in Pure and Applied Math- ematics, v. 69, (1996), 1-34. 2. M. A. Biot, J. Acoust. SOC.Amer. 28 (1956), 168-178, and 179-191. 3. M. A. Biot, J. Applied Physics 33 (1962), 1482-1498. 4. M. J. Buckingham, J. Awust. SOC.Am. 102,(1997), 2579-2596. 5. M. J. Buckingham, J. Acoust. Soc. Am. 103,(1998), 288-299. 6. M. J. Buckingham, J. Acoust. SOC.Am. 108 (6), (2000), 2796-2815. 7. R. Burridge and J. B. Keller, J. Acoust. SOC.Amer. 70 (1981), 1140-1146. 8. R. P. Gilbert and A. Mikelic,Nonlinear Analysis 40 (2000) 185-212. 9. R. P. Gilbert and A. Panchenko, Zeitschrift fir Analysis und ihre Anwendun- gen 18(4) (1999) 977-1001. 10. D. Hoff, Arch. Rational Mech. Anal. 139 (1997), 303-354. 11. J. T. Jenkins and S. B. Savage, Journal of Fluid Mechanics, v. 130 (1983), 187-202. 12. V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Op- erators and Integral finctionals, Springer, Berlin (1994). 13. K. L. Kuttler and M. Shillor, Comm. Contemp. Math. 1 (1) (1999) 87-123. 14. P.-L. Lions and N. Masmoudi, J. Math. Pures Appl., 77 (1998), 585-627. 15. T. Levy, Fluids in porous media and suspensions in: Homogenization Tech- niques in Composite Media, Springer, Berlin (1987). 16. J. T. Oden and J. A. C. Martins, Computer Methods in Appl. Mech. and Engin. 52 (1985) 527-634. 17. E. Sanchez-Palencia, Non-Homogenious Media and Vibration Theory, Springer, Berlin (1980). 18. L. Tartar, Cows Peccot au Collkge de France, (1977), (Preprint). 19. L. Tartar, Appendix in Non-Homogenious Media and Vibration Theory, Springer, Berlin (1980). 20. W. P. Ziemer, Weakly Dafferentiable finctions, Springer-Verlag, New York (1989). A DOMAIN DECOMPOSITION METHOD FOR THE HELMHOLTZ EQUATION IN AN UNBOUNDED WAVEGUIDE

N. GMATI AND N. ZRELLI LAMSIN, Ecole Nationale des Inge'nieurs de Tunis, Tunisie BP 37 Le Belve'dire 1002 Tunis E-mail: nabil.gmatiOipein.rnu.tn, naouel.zrellaQenat.rnu.tn

The aim is to simulate an acoustic wave propagating through an unbounded waveguide. We propose an original numerical algorithm using the finite element method and which takes into account in a exact way the behavior of the solution at infinity. The method is based on a fixed point technique applied to the problem set in the truncated domain by imposing on a fictitious border a non-standard boundary condition by means of a Fourier expansion. It can be interpreted as a domain decompostion solver. The numerical results confirm the advantages of this method compared to a classic implementation.

1. Position of the diffraction problem Let 52 = 52, U !I*be the bidimensional infinite waveguide (Figurel), a generic point in 52 is designated by (sty).

i2- j ci 0 -a a

Figure 1. geometry of the diffraction problem

Let k be the wavenumber involved in the problem. We suppose that the infinite waveguide is submitted to an incident field uin, the governing equation for the diffracted field u is the Helmholtz equation in the unbounded domain 52, with Neumann boundary condition on I', and must be completed

171 172 by a radiation condition (Cutzach-Luneville'): Find u E iR$o,(R), u # 0 such that: Au+k2u = 0 in R

u satisfies a radiation condition awn In (l),f = -- arises from uin, and is assumed to be compactly sup- dn ported i.e. supp (f) c 52, for some a. In order to use a finite element method, we introduce in the next section a classical method to bound the domain for computations.

2. Localized finite element method Writing explicitly the Dirichlet-to-Neuman operator, which is made by a variables separation, we reduce the initial problem (P)to a bounded equivalent problem (Pa),with transparent boundary condition on the fictitious boundaries E*, defined by C* = {(z,y) E R2 with z = fa and 0 < y < b}. This method has already been proposed for several problems (Bonnet- Gmati2, Lenoir-Tounsig, MahB'O, Razafiarivelo12,Ferreira6). We introduce the following operator: Tf : Wb(C*) + IH- 4 (P)

where the constant of propagation k, related to the mth mode is given m2n2 1 kb by k, = (k2 - 7)Tfor m < 7,and k, = i(q- k2)i for m 2 -y. The orthonormal Hilbert basis of lL2(Cf) is defined by: cp,(~) = rmCOs(yy) such that: 7, = for m 2 1 and 70= i.An equivalent bounded problem to (P),with a6 transparent boundary condition can now be stated as follows: Find u E W1(aa), u # 0 such that: IAU + k2u = 0 in Ra dU - = f onr an dU - = T*(u)onC* dn 173

3. Fixed point method It is well known that it is not generally easy to handle the operators Th from a numerical standpoint, the difficulty stemming from their non-local character. This destroy the typical sparse "profile" of the finite element matrix. We propose using a fixed point method for (P,). To do that, the main point is to write, at each step of the iterative procedure, the boundary aU"+1 condition as follows : -= T*(u") on C*. The diffraction problem an obtained at each step is not well-posed for it countable set of values of the wave number, which correspond to the irregular frequencies of the domain 0,. This suggests to modify the boundary condition on C'. The problem to be solved at each step can be stated as follows:

=f on ~R,\c* 1%- ikun+l = Tfu" - iku" on Cf

The matrix of the resulting linear system remains imaginary but its sparsity is preserved. In the following, in order to make the convergence analysis of the proposed method, we suggest to give an interpretation of our method under the form of a subdomains iterative method.

4. Iterative domain decomposition method The idea consists in splitting the domain by the fictious boundary C* into three subdomains 52, and R* (Figure 1) and solve a sequence of problems on these subdomains. The boundary conditions are adjusted iteratively by some appropriate transmission conditions between adjacent subdomains. Refer to Collino-Ghanemi-Joly for problems in bounded domains, and Boubendir for unbounded domains. We now define the following iterative procedure using three sequences (u")"EN, (vg)nEw and (u14)nEN. We denote 174 by v2, the function solution of the following problem:

Find vz E QOc ($2') (A + k2)vz = 0 in $2'

and define u"+' which solves: Find un+' E W' (R,);u"+l # 0 such that:

Aun+' + k2un+' = 0 in $2, (P?+') dun+' on dS2a\C' dn =f &P+l av2 -- ikun+' = - - ikv" on C' an dn The definition of the operators T', and the interface conditions leads to n the following identities: % = T*(un),and vn * = u+. We can than I dn IC conclude that (P?") and (P,"+') are the same problem. Refer to Ben Bel- gacem, Fournik, Gmati and Jelassil (see also Jelassi8) for a detailed discussion on how to use domain decomposition techniques, and in particular the Schwarz method, to handle the boundary conditions at infinity. Also, one can find therein a complete convergence analysis for the Poisson problem on unbounded domains.

5. Convergence study in a simple case We are able to prove the convergence of the iterative procedure in the simple cme of a semi-infinite rectangular waveguide (Figure 2 ). We denote by en+' = (tinf1 - u) the (n + l)thstep error in 0, and ek+' its projection on (P, defined below. An explicit calculation shows that e$+'(a,y) = A,ek(a,y), for all m,n E N and y €10, h[,where A, = (k, - ik)(ekma+ e-kma) kh . We get IA,I < 1 if m < - and IA,I 1 (k, - ik)e"a - (k, + ik)e-"-" 7r kh 1 if m 2 -. This is enough to prove the convergence of the domain 7r decomposition for an incident propagative wave. 175

0 a

Figure 2. Rectangular waveguide

6. Numerical results:

6.1. The rectangular waveguide In this section, we perform numerical experiments, for a two-dimensional infinite rectangular waveguide fl = R x [0,1]. The comparison with the exact solution proves the convergence of the proposed method with respect to the mesh size and the fixed point iterations (Figure 3 ). For this purpose, we consider the computational domain denoted by fla= [0,1] x [0,1]. We keep the wavenumber k = 5 and consider four meshes of different sizes h. The computational duration on a Silicon Graphic computer is given in table 1 for first and second order finite elements (Pl,P2). We compare the accuracy and computational duration between the classical algorithm (LFEM) and our method, based on the subdomain iterations method (SIM). As expected both methods give the same accuracy, with smaller computational durations for the second method.

Table 1. Computational durations.

1/32 0.95 1/16 6.15 2.82 55.99 176

Figure 3. Relative error between iterative and exact solutions as a function of iteration order. K=5. Left: P1. Right: P2

6.2. The elliptic waveguide In this section, for an elliptic waveguide, the comparison between the classical methods and the proposed one shows that we again have the same accuracy, but the difference of computational durations is less than for the previous case. This is with smaller computational durations for the second method and is due to the fact that the number of degrees of freedom on the boundary C is small with regard to the total number. The full part of the finite element matrix is then much less important. We represent below the isovalues obtained by the proposed method.

23 2 h 1 0.5 0 10

Figure 4. Real part of the pressure field

Acknowledgments : The authors would like to thank Dr. Faker Ben Belgacem for his valuable help. All the Numerical results were obtained by the Melina Finite Element Code (Martin"). 177

References 1. F. Ben Belgacem, M. FourniB, N. Gmati and F. Jelassi, Schwarz Methods for Exterior Problems, In preparation. 2. A.S.Bonnet Ben Dhia and N.Gmati, Spectral approximation of a boundary condition for an eigenvalue problem, SIAM J. Nurner.Ana1. 32, 1995, 1263- 1279. 3. Y. Boubendir, Techniques de dBcomposition de domaine et mBthode d’gquations intBgrales, Th&sede Doctorat de E’Institut National des Sciences Applique‘es de Toulouse, 2002. 4. F. Collino, S.Ghanemi, and P.Joly, Domain Decomposition Method for Har- monic Wave Propagation: A General Presentation, Rapport de recherche INRIA, no. 3473, 1998. 5. P.-M. Cutzach and E. Luneville, Diffraction d’ondes acoustiques par un guide semi-infini, C.R. Acad. SciParis Se‘r.1 Math. 326 1998,1151-1154. 6. A. Ferreira, Etude numerique de quelques problsmes de diffractions d’oydes pax des rBseaux pQiodiques en dimension 2, Thbse de doctorat de I’Ecode polytechnique Paris, 1998. 7. C. Hazard and M. Lenoir, ModBlisation et rksolution des problbmes de diffraction, ENSTA Cours EL 340, DEA de MBcanique, Paris 6, 1995. 8. F. Jelassi, MBthode de Schwarz alternee pour le problbme de Poisson extbrieur, Me‘rnoire de DEA de I’Ecole Nataonade d’Inge‘nieurs de Tunas,2002. 9. M. Lenoir and A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem, SIAM J. Numer.Anal. 25, 1988, 729-752. 10. F. MahB, Etude mathkmatique et nurnbrique de la propagation d’ondes Bl6ctromagnktiques dans les microguides de l’optique intkgrbe, Rapport de recherche, Ecole nationale supe‘rieure de techniques awance‘es, 1993 . 11. D. Martin, Documentation MELINA, Rennes 1997. http://www.maths.univ-rennesl.fr/dmartin/melina/www/homepage.html. 12. J. Razafiarivelo, Optimisation de la forme de transition entre guides BlBctromagnktiques par une mBthode intkgrale d’61Bments finis. Th6se de doctorat de l’Uniwersit6 Pierre et Marie Curie, 1996. SUPPORT FUNCTION METHOD FOR INVERSE OBSTACLE SCATTERING PROBLEMS

SEMION GUTMAN Department of Mathematics University of Oklahoma Norman, OK 73019, USA E-mail: sgutmanOou.edu

ALEXANDER G. RAMM Department of Mathematics Kansas State University Manhattan, Kansas 66506-2602, USA

LMA/CNRS, 31 Chemin Joseph Aiguier Marseille 13402, cedex 20, fiance E-mail: rammOmath.ksu. edu

The knowledge of the Support Function (SF) of a smooth and strictly convex obstacle allows one to reconstruct the obstacle. The SF can be computed if the Scattering Amplitude is known for large values of the wave number k. For the values of k in the resonance region, the SF can only be found approximately. Nevertheless, the results of this paper show that such an SF can be used to find an approximate location of the obstacle. The method is inexpensive, and it does not require extensive data.

1. Introduction Let an obstacle be a bounded domain D c R2 with a Lipschitz boundary r. Fix a frequency k > 0 and denote the exterior domain by D' = R2 \ b. Let Q E S1, and the incident field be uo(x)= eikx'ff, x E D' . (1) The Direct Acoustic Obstacle Scattering Problem for the Dirichlet boundary conditions consists of finding the total field

U(X,k) = UO(X)+ v(x), x E D' (2) such that Au+k2u=0, XED', (3)

178 179

=o, x E r, (4) and the scattered field w(x) satisfies the Sommerfeld radiation condition

where the limit is attained uniformly for all the directions x/ 1x1 , x E R'. The scattered field w(x) has an asymptotic representation

where the uniquely defined function A(a', a) is called the Scattering Am- plitude of the Obstacle Scattering Problem, see, e.g. '. The Inverse Scattering Problem consists of finding the obstacle D from the Scattering Amplitude, or similarly observed data. In this note the Support Function Method (SFM) originally described in a 3-D setting in ' is used to approximately locate the obstacle D. The SFM is described in Section 2, and the numerical results are presented in Section 3.

2. The Support Function Method (SFM). The SFM was Originally developed in ', pp 94-99. It identifies smooth, strictly convex, obstacles from the knowledge of the Scattering Amplitude at high wave numbers k. In this paper we study the numerical performance of the SFM in the resonance region. One can restate the SFM in a 2-D setting as follows. Let D C R' be a smooth and strictly convex obstacle with the boundary r. Let ~(y)be the unique outward unit normal vector to I' at y E r. Fix an incident direction a E S1. Then the boundary I' can be decomposed into the following two parts:

I?+ = {y E r : ~(y).a < 0) , and r- = {y E r : ~(y). CY 2 0) , (7) which are, correspondingly, the illuminated and the shadowed parts of the boundary for the chosen incident direction a. Given a E S1, its specular point so(a) E I?+ is defined from the condition:

s,-,(cY) . a = min s . a (8) sEr+

Note that the equation of the tangent line to r+ at SO is

< 51,5' > . a = %(a)'a, (9) 180 and

u(so(a))= -a. ( 10) The Support function d(a) is defined by

d(a)= so(0) . a. (11) Thus ld(a)I is the distance from the origin to the unique tangent line to F+ perpendicular to the incident vector a. Since the obstacle D is assumed to be convex

D = naUESl{X E R2 : x . LY 2 d(a)}. ( 12) The boundary r of D is smooth, hence so is the Support Function. The knowledge of this function allows one to reconstruct the boundary I’ using the following procedure. Parametrize unit vectors 1 E S1 by I(t) = (cost, sint), 0 5 t < 27r and define

p(t) = d(l(t)), 0 5 t < 27r. (13) Equation (9) and the definition of the Support Function give

slcost+~sint=p(t). (14) Since I’ is the envelope of its tangent lines, its equation can be found from (14) and

-zl sin t + x2 cos t = p’(t) . (15) Therefore the parametric equations of the boundary r are

x1 (t)= p(t)cos t - p’ (t)sin t, x2 (t) = p(t)sin t + p’( t)cos t . (16) So, the question is how to construct the Support function d(l), 1 E S1 from the knowledge of the Scattering Amplitude. In 2-D the Scattering Amplitude is related to the total field u = uo + v by

In the Kirchhoff (high frequency) approximation one sets

on the illuminated part r+of the boundary r, and 181 on the shadowed part r-. Therefore, in this approximation,

Let L be the length of I?+, and y = y(<),0 5 C 5 L be its arc length parametrization. Then

Let 50 E [0, L] be such that SO = ~(50)is the specular point of the unit vector 1, where

Then .(SO) = -1, and d(1) = y(C0) . 1. Let cp(C) = (Q - a‘>. Y (5) . Then p(<) = 1. y(C) la - ~’1. Since v(s0) and y’(Q) are orthogonal, one has

cp‘(C0) = 1. y’(C0) (a- a‘\ = 0. Therefore, due to the strict convexity of D, is also the unique non- degenerate stationary point of v(<)on the interval [0,L], that is cp’(50) = 0, and cp”(C0) # 0. According to the Stationary Phase method

ask+oo. By the definition of the curvature IE(

-cp”(C0) = 1. ICP” (io) I Using (23)-(24), expression (21) becomes: 1 82

At the specular point one has 1. a' = -1 - a. By the definition a - a' = 110 - a'[.Hence l.(a-a'>= la - 0'1 and 21.a = la - a'l. These equalities and d(1) = y(50) * 1 give

Approximation

can be used for an approximate recovery of the curvature and the support function (modulo 27r/kla - a'l) of the obstacle. The uncertainty in the support function determination can be remedied by using different combinations of vectors a and a'.

3. Numerical results. In this numerical experiment the obstacle is the circle

D = {(XI, ~2)E R2 : (XI- 6)2 + (~2- 2)2 = 1). (28) It is reconstructed using the Support Function Method for two frequencies in the resonance region: k = 1.0, and k = 5.0. Table 1 shows how well the approximation (27) is satisfied for various pairs of vectors a and a' all representing the same vector 1 = (1.0,O.O) according to (22). The Table shows the ratios of the approximate Scattering Amplitude A, (a',a) defined as the right hand side of the equation (27) to the exact Scattering Amplitude A(a',a). Note, that for a sphere of radius a, centered at xo E R2,one has

where a' = x/ 1x1 = eie, and a = eib. Vectors a and a' are defined by their polar angles shown in Table 1. Table 1 shows that only vectors a close to the vector 1 are suitable for the Scattering Amplitude approximation. This shows the practical importance of the backscattering data. As mentioned at the end of Section 2, any single combination of vectors a and a' representing 1 is not sufficient to uniquely determine the Support Function d(1) from (27) because of the phase uncertainty. However, one can remedy this by using more than one pair of vectors a and a' as follows. Let 1 E S1 be fixed. Let R(1) = {a E s1 : la. 11 > l/JZ}. Table 1. Ratios of the approximate and the exact Scattering Am- plitudes A,(a’,a)/A(a’,a)for 1 = (1.0,O.O).

ff’ a k = 1.0 k = 5.0

x 0 0.88473 - 0.174872 0.98859 - 0.058462 23x124 x/24 0.88272 - 0.176962 0.98739 - 0.06006Z 22x124 2x124 0.87602 - 0.184222 0.98446 - 0.064593 21x124 3x124 0.86182 - 0.199272 0.97977 - 0.074322 20x124 4~124 0.83290 - 0.224112 0.96701 - 0.088732 19x124 5~124 0.77723 - 0.254102 0.95311 - 0.103212 18x124 6x124 0.68675 - 0.271302 0.92330 - 0.141952 17x124 7x124 0.57311 - 0.253602 0.86457 - 0.149592 16x124 8x124 0.46201 - 0.198942 0.81794 - 0.229002 15x124 9x124 0.36677 - 0.126002 0.61444 - 0.19014Z 14x124 10~124 0.28169 - 0.054492 0.57681 - 0.310752 13x124 11~124 0.19019 + 0.000752 0.14989 - 0.094792 12x124 12x124 0.00000 + 0.000002 0.00000 + 0.000002

Define Q : R + Rf by

where a’ = cr‘(a)is defined by 1 and a according to (ZZ), and the integration is done over a E R(1). If the approximation (27) were exact for any a E R(l), then the value of Q(d(1)) would be zero. This justifies the use of the minimizer to E R of the function Q(t) as an approximate value of the Support Function d(1). If the Support Function is known for sufficiently many directions 1 E S1, the obstacle can be localized using (12) or (16). The results of such a localization for k = 1.0 together with the original obstacle D is shown on Figure 1. For k = 5.0 the identified obstacle is not shown, since it is practically the same as D. The only a priori assumption on D was that it was located inside the circle of radius 20 with the center in the origin. The Support Function was computed for 16 uniformly distributed in S1 vectors 1.

4. Conclusions. The Support Function Method is an inexpensive algorithm for solution of Inverse Obstacle Scattering problems. The SFM can be used in conjunction 184

2 4 6 a

Figure 1. Identified (dotted line), and the original (solid line) obstacle D for k = 1.0.

with other methods (such as the Modified Rayleigh Conjecture (MRC) method, see ,4), which can provide a more precise identification, given an approximate location of the obstacle.

References 1. Gutman S., Ramm A.G. [2002] Numerical Implementation of the MRC Method for Obstacle Scattering Problems, J. Phys. A, to appear. 2. Ramm A.G. [1986] Scattering by Obstacles, D. Reidel Publishing, Dordrecht, Holland. 3. Ramm A.G. 119921 Multidimensional Inverse Scattering Problems, Long- man/Wiley, New York. 4. Ramm A.G. [2002] Modified Rayleigh conjecture and applications, J. Phys. A, 35, L357-361. HEAT POLYNOMIAL ANALOGS

G. N. HILE Department of Mathematics, University of Hawaii, Honolulu, HI 96822 E-mail: hileOhawaii.edu

ALEXANDER STANOYEVITCH Department of Mathematics, University of Guam, UOG Guam Station, Mangilao, GU 96923 E-mail: alexOmath. hawaii. edu

The classical heat polynomials are polynomial solutions of the heat equation. They are useful in a function theoretic treatment of the heat equation, serving as a basis for expansion and approximation of other solutions. We generalize these polynomials to more general evolution equations, of arbitrary order and with coefficients depending on the time variable. We have explicit formulas for these polynomials, obtained through expansion of a generating function. We also exhibit pointwise upper bounds on the polynomials, useful in the analysis of series expansions in terms of generalized heat polynomials.

1. Introduction The classical heat polynomials {pa} are polynomial solutions of the initial value problem for the heat equation

where z = (21, . . . ,2,) E IW", p = (PI,. . . ,Pn) represents a multi-index in Rn, and zp = zlP1. . .znPn. These polynomials appear in early work of Appell on the heat equation, and were later investigated in detail by

Rosenbloom and Widder 516,798. Several authors have generalized the heat polynomials to equations besides the heat equation, but these equations are rather specialized in scope and usually involve only one space variable. References to much of this work can be found in the papers of the present authors 213. We consider, more generally, polynomial solutions of the initial value problem

.cpp (z,t>= 0 , Po (z,O) = zB 1 (1)

185 186 where C is a linear differential operator of the form cu (2,t) = atu (2,t) - c a,(t)a,"u (2,t) . (2) a Here 2 E Rn, t E I& and the coefficients {a,}, indexed by multi-indices a in Rn and finite in number, are real valued continuous functions oft on an interval li containing the origin. It turns out that each pp is for fixed t a polynomial in z of degree 1/31, having the form

P&,t) = c cmzv7 yip with each coefficient c, a real valued function in C1 (I). Moreover, when the coefficients {a,} are constant and JC has no zero order term, each pp is a polynomial in both z and t. The present authors 2,3 have derived explicit algebraic formulas for these polynomials, described some of their properties, and established pointwise upper bounds on the polynomials.

2. The Polynomials With the operator C of (1) and (2) we associate real valued functions {b,(t)},Q(t,z),R(t,z), wheretEII,zERn,accordingto

R(t,z)= Q (s,z) ds = zb, (t) . I' a We introduce a generating function associated with C,

G (2,t, z) = ez'zeR(t9z), (3) where "-" is the ordinary dot product in Rn . We expand G (2,t, z) in powers of z, getting an expansion of the form

It is readily checked that, for each fixed z E Iwn, the function G(.,.,z) is a solution in En x II of CG = 0; we may use this fact to confirm that Cpp (2,t) = 0 for each p. Moreover, from the initial condition 187 it follows further from (4) that pp (z,0) = zp for each P. With some algebraic manipulations we can derive from (3) and (4) the explicit formula

The notation in (5) requires some explanation. First we number the

coefficients {a,} in (2) as {a,, a2,. +. ,UK}, and the corresponding multi- indices as {a',a2,. . . , Q"}, so that L can be represented as K Lu(2,t) = a,u(z,t) -Cak(t)a,"*U(z,t) . kl Then we introduce vector functions a and b in BK, a=(Ui,az,-..,a~), b= (bi,bz,*.-,b~), where t t bk (t)= 1 ak (s) ds , b (t)= 1a (s) ds , as well as a "vector of multi-indices", - a = (al,a2,...,aK).

In (5), y = (71,.. . ,yn) denotes a multi-index in B", u = (~1,.. . , UK) a multi-index in RK , and the notation E ' u refers to the "dot product" K

k=l LY Note that E.u is a multi-index in R". The summation in (5) is taken over all multi-indices y in Rn and u in IRK such that y + 72'. u = P. With the substitution y = P - E.u, we may write (5) in the alternative formulation

(For multi-indices a and p, Q 5 ,8 means that a( 5 Pi for each i.) Note that (5) shows that pp (2, t) is a polynomial in 5 for each fixed t. If the coefficients {a,} of (2) are constant, then formula (5) becomes

If furthermore L has no zero order term, then Q = (0,.. .O) does not occur in (2), and there are only a finite number of indices y and u such that 188 y + E.0 = P; in this case the summation in (6)has only a finite number of terms, and consequently pp (z, t) is a polynomial in both z and t. Differentiation of (3) and (4) leads to the formulas

We have also recursion formulas, for 1 5 i 5 n,

where ei denotes the i-th unit coordinate vector in R".

3. Homogeneous Parabolic Operators We specialize now to parabolic partial differential operators having constant coefficients, and involving space derivatives only of the highest order e. The general form is Lu(z,t) = ~(z,t)- C a,a$u(z, t) = ~(z,t)- Q(&)u(z,t) , lffl=e where Q is the polynomial

l+e The parabolicity condition requires that, for all z E Rn and some 6 > 0, Re &(iz) = Re C am (iz)" 5 -6 lzle . I+e A fundamental solution for L is given for z E R" and t > 0 as the function K(z,t):= (2*)-" ineiX.2 e t&(iZ) d z. It can be shown that each polynomial solution pp can be represented for t>Oas m(z,t) = 1 K(z - Y,t) YP dY. Wn Following the development of Rosenbloom and Widder for the heat equation, we introduce functions {qp}, indexed by multi-indices P in R", according to qp (z,t)= (-1)b,PK(z,t) . 189

It can be verified that each qo solves Lqo = 0, that K has the expansion

and that the families {pp} and {qp} obey for t > 0 the biorthogonality relation

We have also the "quasihomogeneity" conditions

pp (Ax, A%) = AIfllpp (2,t) , QP (Ax, A%) = A-"-IpIq, (2,t) . It turns out that the polynomials {hp},defined for x E R" by

h4 (5) = PP (2,-1) 7 are in some respects analogous to the well known Hermite polynomials.

4. Polynomial Bounds Returning back to the general operator (2), we assume the order of the operator is m, m 2 2, so that 101 5 m for each multi-index Q appearing in the summation. On the coefficients {a,} of (2) we assume the finite bounds

We let M denote the number of lower order multi-indices a appearing in (2) such that 0 < 1.1 < m. Then, expanding on the techniques of H. Kemnitz 4, we can derive for z E R" , t E I[, and for any 6 > 0 the bound

In the case that L is of order m and has no lower order terms, this bound simplifies to 190

In deriving these pointwise bounds from (5), it is necessary first to confirm new estimates involving the factorial function,

x!=r(x+i) , O~~

a! 5 \a\!= (a1 + a2 + * *. + an)! 5 nlaIa! (if ai 2 o far each i) ,

(Here 1x1 is the Euclidean norm and 1.1 the multi-index norm.)

5. Series Expansions Again for the general operator (2), we consider series expansions in the polynomial solutions {pp}, of the form

u(5,t)=qp4(Zit) (8) 4 with real coefficients {cp}. We define s, 0 < s 5 00, according to 1 (~p(~ll'Imell - = lim sup (9) S B+m IPI With use of our bounds on {pp}, it can be confirmed that the series (8) converges absolutely and uniformly on compact subsets of the strip s, = {(z,t): x E R", t E n, It1 < s} . (10) Similar convergence holds for differentiated versions of (8) involving no time derivative of order greater than one. Also, u, as prescribed by (8), solves in S, the equation Lu = 0, and for each P we have 13fu (0,O) = cp. Observe that (9) and (7) show that the width s of the strip of convergence for (8) depends only on the bounds on the coefficients of the highest order terms of the operator L; the lower order coefficients play no role. The preceding result on series allows us, under certain conditions on the initial data function f, to obtain a series solution of the Cauchy problem LU(2,t) = 0 , u(x,O)= f (x) . (11) We say that the function f has growth {p, r},where 0 < p, r < 00, provided that f has a power series expansion

f (z)= Ed0zp = 9p! z4 ' 4 191 with

It can be shown that (13) is equivalent to

and that these conditions imply that the series (12) converges to an entire function f in Cn satisfying the growth condition

asz + 00, foralZT'> T . (15)

Conversely, if f is an entire function in Cn with growth condition (15) and the expansions (12), then (13) and (14) must hold. If we assume that the initial data function f of (11) has growth {m/ (m- 1) , T}, where m is the order of ,C and 0 < T < 00, then the series (8) solves the Cauchy problem (11) in the strip (lo), provided that

References 1. P. Appell, Sur l'e'quation d2z/ax2 - az/ay = 0 et la the'orie de la chaleur, J. Math. Pures Appl. 8 (1892), 187-216.. 2. G. N. Hile and A. Stanoyevitch, Heat polynomial analogs for higher order evolution equations, Electronic Journal of Differential Equations 2001 (2001), NO. 28, 1-19. 3. G. N. Hile and A. Stanoyevitch, Expansions of solutions of higher order evolution equations in series of generalized heat polynomials, Electronic Journal of Differential Equations 2002 (2002), No. 64, 1-25. 4. H. Kemnitz, Polynomial ezpansions for solutions of Dgu(x,t) = Dtu(x,t), t = 2,3,4,.. ., SIAM J. Math. Anal. 13 (1982), 640-650. 5. P. C. Rosenbloom and D. V. Widder, Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. SOC.92 (1959), 220-266. 6. D. V. Widder, Series expansions of solutions of the heat equation in n dimensions, Ann. Mat. Pura Appl. 55 (1961), 389-410. 7. D. V. Widder, Expansions in series of homogeneous temperature functions of the first and second kinds, Duke Math. J. 36 (1969), 495-510. 8. D. V. Widder, The Heat Equation, Academic Press, New York, San Francisco, London, 1975. BLOW-UP, SHOCK FORMATION, AND ACCELERATION WAVES IN HYPERELASTIC MEDIA

A. JEFFREY Department of Engineering Mathematics, Universiiy of Newcastle upon Tyne, England E-mail: [email protected]

This work starts by reviewing some fhdamental results concerning the development of singularities in solutions of nonlinear hyperbolic equations and systems. First the propagation of Lipschitz discontinuities and their blow-up into a shock wave is considered and then, in the case of a system, the blow-up of the solution itself is demonstrated. An application of acceleration wave propagation in an isotropic incompressible hyperelastic medium is then mentioned where two types of wave can propagate, one to the left and the other to the right, though only one of these can ever lead to shock formation. Wave propagation in a layered medium is also mentioned.

1 Singularity Formation in Nonlinear Hyperbolic Equations and Systems

The formation of singularities in solutions of nonlinear hyperbolic equations is a well known phenomenon, with the most commonly occurring type of singularity being the development of a shock wave. However other types of singularity can also arise, ranging from the propagation of acceleration waves in solids (Lipschitz discontinuities) , to shock wave formation, and even to the blow-up of the solution itself. The non-dissipative Burgers' equation u, + a(u)u, = 0 for u(x, t) subject to the initial condition u(x, 0) = Ax), where a(u) and the initial data Ax) may even be C functions has the implicit solution u =Ax - a(u)t).This implies the possibility of non-uniqueness due to the intersection of characteristics leading to the evolution of a shock. It is important to notice that the equation can be written in the consewation form (divergence form)

where A(u) = 5 a(u)uxdx, in which case shock solutions are possible satisfylng the generalized Rankine Hugoniot jump condition %[[u]]= [[A(u)]], where [[a]]is the jump in a across a shock moving with speed % , see for example''2. It follows that u, = f '(A) /[I +a'( f (A))f '(A)t],so u, becomes infinite on a Ca characteristic with equation x = A + aV(;l))t at some time t 2 0 if

192 7 93

a’(f(A))f’(A) c 0 for all A, but it will remain finite if dflA))f’(A) 2 0 for all 2. So defining a critical time tc as

t, = [sup(-u’(f(A))f’(A)] a 2 0 ,

a necessary condition for u, to remain finite at time t is 0 I t < z I tc. This strip in the (x, t)-plane defines the lifespan of a differentiable solution of the equation and, if blow-up of u, occurs, this depends only on a(u) andfix) and not on any smoothness properties of the initial data, even though as here this may be a Cmhction . Consider a function Ax) with compact support, with fix) = 0 for I x 1 > xo, then it is seen that u(x, t) also has compact support in x for all t, with the result that u(x, t) = 0 for x < - xo + a(0)t or x > xo + a(0)t. However the condition that the solution remains finite is simply that aV(A)) is a non-decreasing function of A, because then the characteristics along which initial values of the solution are propagated will be divergent, but as a(f(A))= a(0) for [ill >xo this implies that aMA)) = a(0) for all A. If the equation is genuinely nonlinear in the sense of Lax (see’*3) then a ’(0)# 0, and the condition a(@)) = a(0) can only hold iff= 0, showing that then u(x, t) = 0. Thus in a genuinely nonlinear equation all non-trivial solutions with compact support must blow-up in a finite time tc. Conversely, if d(A)= 0 for some A(u(x, 0)),the solution cannot blow-up along that Ca characteristic. When d(A)= 0 for all A(u(x, 0)) the solution is said to be exceptional with respect to the Cn characteristic field, and it then follows that such a solution must remainJinite for all time t > 0. Similar results can be established for nonlinear second order hyperbolic equations, see for example ‘. To close this section we give an example of a system where the blow-up is more dramatic, because it is the solution u(x, t) that becomes unbounded after a finite time, and not just its spatial derivative. Consider the system where -cosh(2u,) 0 -sinh(2u,) coshu, 0 sinhu, , sinh(2u,) 0 cosh(2u,) 1 subject to initial data with compact support uI(x, 0)= x/ah, uz(x, 0)= 0, u3(x,0) = -x/a h for -h < x < h and U = 0 for 1x1 > h. This system has the exact solution 194

1 X 1 X u1 =- +--1, u2=In1---, u ----- 1, I-tlah ah 1 :hi 3-l-tlah ah from which it is seen that the solution U itself blows-up when t = a h. Notice that in this case the eigenvalues of A(U) defining the three families of characteristics are A1 = -1, & = 0 and A3 = 1, so the system is completely exceptional, because for a system the exceptional condition can be shown to be V,XA(U) ) = 0 (see [l]]),where Vudenotes the gradient operator with respect to U . In this case the blow-up of the solution is not related to intersection of characteristics because each family of characteristics comprises a family of parallel straight lines, with each family diverging as t increases. This example does not contradict the earlier findings about the blow-up of only a spatial derivative in Burgers’ equation, ore indeed of a hyperbolic system, because unlike Burgers’ equation the previous example is not a system of conservation laws. This demonstrates that when a system is not a conservation system, even if the solution is completely exceptional, meaning it is exceptional with respect to each family of characteristics, it is still possible for the solution itself to blow-up after a finite time. In this case, unlike systems of hyperbolic conservation laws where the solution can be extended as a shock, no extension is possible after the blow-up of a solution.

2 Waves in an incompressible isotropic hyperelastic medium

In a three-dimensional medium, if the Cartesian coordinates of a particle in the initial state are Xi, and its coordinates at time t are xi , i = 1,2, 3, the displacement field can be written xi = xi&, t) and the strain becomes

with the strain invariants ZI= tr C, Z2 = $ [(tr q2- tr C 2] , Z, = det C. For an isentropic incompressible isotropic hyperelastic material, the energy function .Z (the mean internal energy per unit volume at constant entropy) may be considered to be a function of the strain measure C . Then, as the medium is incompressible, its displacement field must satisfy the incompressibility condition det C = 1 and we can write (see, for example, [5])C= ~ZI, 12). In the case of such a material, the stress tensor can be expressed in terms of the strain t, = -p&j+ @ cij + y(Ckkcij - Cikckj), where @ = 2dYd I], Y = 2dYd 12 andp is the pressure field. When no body force is present and the material density is po the equations of motion become 195

, j=1,2,3. PO 'xi

Considering the half space XI 2 0 and a plane deformation field for t > 0 we have x1 = X,,x2 = X2 + u2(Xl, t), x3 = X3 + u3(X1,t), where u, with s = 2, 3 is the displacement of a particle in the X, direction caused by a wave propagating in the XI- direction. Proceeding in this manner it can be shown (see for example [6])that for the hyperelastic medium under consideration the equations of motion become

2u3 --a2u2 - A22-+aZu2 A23-a2u3 a2u a2u2 a -- 23 -A227+A337 at2 ax,' ax12' at 3x1 ax, where

These equations form a coupled quasilinear system of second order equations, and using the equality of mixed derivatives we now re-cast them as the equivalent first order system U, + A(U)U = 0, XI

0 0 -A2' -A23 0 0 -A23 -A33 U= and NU) = -10 0 0 0-1 0 0

Matrix A(U) has two invariant sub-spaces of order two, so for an energy function with the property that A(U) has real eigenvalues, the system is hyperbolic, and so describes wave propagation. The eigenvalues Ai and the eigenvectors I satisfy I (A - Air) = 0. So partitioning A(U)gives 196

A(U) =[ 0 -A ] where A=[::: :::]and I=[o10 l]. -I 0

As a result the corresponding left eigenvectors may also be partitioned. The eigenvalues of A(U) are g)= .( -712 d2C and Ap) = *[ --).2 dC Clearly Po d9 Po4 dq these eigenvalues will be real when d ’Zldq2 > 0 and (l/q)&/dq > 0, with the plus and minus signs indicating waves propagating to the right and left, respectively. The left eigenvectors 1 are

2 d2Z 1+(2)=- [1 ’ -p3pz’ T (-- );” +3(-- “”)I. Po dq2 P2 Po d4 and the right eigenvectors r are defined in similar fashion. Following the arguments in’*637,and using generalized Riemann invariants, it is easily shown that waves characterized by the eigenvalues 1:) and Ay are generalized simple waves in the

(,) - q d3C d2C hyperelastic medium. It follows that(V,Af))r, -- [3z , and in P2 dq d4 general this expression is non-zero, showing the A$) characteristic fields are not exceptional, so shock wave formation can occur when these waves propagate. The situation is different however, when the A?) are considered, because then (V,q))r J2) = 0, showing that the A:) characteristic fields are exceptional, and consequently shock wave formation cannot occur on the wavef’ront when these waves propagate.

3 Acceleration wave propagation

An acceleration wave in a hyperelastic medium can be initiated by a loading program where at time t = 0 the medium, previously subject to a constant traction TO,has it changed to T = T(t), where T(0) = To. This problem is now a particular case of the general problem considered and the arguments developed there can be used to show the only waves that can lead to shock wave formation are the 1:) waves. 197

As the acceleration wave advances into a constant state, the characteristic along which it propagates is a straight line starting from the origin, so if a shock wave forms at the time tc, it will occur at a distance xc = from the plane face of the medium. This argument extends to a layered medium, as indicated in the next section. A totally different, though completely equivalent way of studying the development of an acceleration wave is to be found in the work of Chen' and the references contained in its bibliography. It amounts to showing that the intensity of an acceleration wave as it propagates along a ray is governed by a Bernoulli equation, with one of its coefficient depending on both the state of the medium and the geometry, while another depends only on the properties of the medium itself. The equivalence of the approach by Chen, and the one developed in [l] and subsequently used involving a system of linear equations and a nonlinear coordinate transformation, was established by Boillat and Rugged'. A generalization of the work by Chen, detailing precise conditions for the evolution and non-evolution of shocks was found by Jeffrey, Menon and Sharma".

4 Wave propagation through a layered half-space

A discussion of the transmission and reflection of acceleration waves in a layered hyperelastic medium, based on the work of Jeffrey and Suhubi6 , is to be found in7 . It is shown there how such a discontinuity advancing through a layered medium will give rise to transmitted and reflected waves. A criterion for the development of a shock can be obtained, together with expressions for the strengths of the reflected and transmitted parts of the wave at each interface, Knowing the strength of the transmitted wave at an interface, the criterion for the development of a shock can then be applied to each layer in turn. In this way each layer can be treated in the same way and it is a simple matter to determine the layer in which the wave develops into a shock.

5 General remarks

The presence of dissipative and dispersive effects modifies wave propagation by allowing global solutions to exist, and in certain cases soliton propagation becomes possible. A detailed discussion of general soliton propagation is to be found in Jeffrey and Kawahara". More general aspects of nonlinear dissipative wave propagation have been examined recently by Jeffrey and Zhao where global existence and temporal decay estimates are given in both one and many space dimensions. 198

References

1. Jeffrey, A., Quasilinear Hyperbolic Systems and Waves, Research Notes in Mathematics 5, Pitman, London (1976). 2. Smoller J., Shock Waves and Reaction-DiffUsion Equations, (Springer, New YorkJ983). 3. Lax P.D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, (Regional Conference Series in Applied Mathematics, 11, Society for Industrial and Applied Mathematics, Philadelphia, 1973). 4. John F., Solutions of quasilinear wave equations with small initial data. The third phase, (Lecture Notes in Mathematics 1402, Springer Verlag, ed. by C.Carasso, 1989) pp. 155-173. 5. Eringen A.C. and Suhubi E.S., Elastodynamics 1. Finite Motions, (Academic Press, New York, 1974). 6. Suhubi E and Jeffrey A., Propagation of weak discontinuities in a layered hyperelastic half-space, Proc.Roy.Soc.Edinburgh 75A (1 976) pp. 209-221. 7. Donato A and Fusco D., Nonlinear wave propagation in a layered half- space, Znt. J Nonlin. Mech. 15 (1980) pp. 497-503. 8. Chen P.J., Selected Topics in Wave Propagation, (Noordhoff International Publications ,1976). 9. Boillat G., and Ruggeri T., On the evolution law for weak discontinuities for quasilinear hyperbolic systems, Wave Motion 1(1979) pp. 149-152. 10. Jeffrey A., Menon V.V. and Sharma V.D., On the general behaviour of acceleration waves, Applicable Analysis 16 (1983) pp. 101-120. 11. Jeffrey A. and Kawahara T., Asymptotic Method in Nonlinear Wave Theory, (Pitman, London, 1982). 12. Jeffrey A. and Zhao H., Global existence and optimal temporal decay estimates for systems of parabolic conservation laws I: The one-dimensional case. Applic. Anal. 70 (1998) pp. 175-193. 13. Jeffrey A. and Zhao H., Global existence and optimal temporal decay estimates for systems of parabolic conservation laws 11: The multi-dimensional case, J. Math. Anal. Appl. 217 (1998) pp. 597-623. 14 Jeffrey A. and Zhao H., A remark on optimal temporal decay estimates for systems of multi-dimensional parabolic conservation laws. Rendiconti di Matematica, Series VZZ 20 (2000) pp. 1-34. SUMMABILITY OF SOLUTIONS OF DIRICHLET PROBLEM

A. KOVALEVSKY Institute of Applied Mathematics and Mechanics, Rosa Luxemburg St. 74, 83114 Donetsk, Ukraine E-mail: kovalevskyOirnath.kieu.ua

F. NICOLOSI Dipartimento di Maternatica e Informatica, Cittd Universitaria, Kale A. Dom'a 6, 95125 Catania, Italia E-mail: fnicolosiOdmi.zlnict.it

We consider a class of degenerate nonlinear high-order equations. Supposing that right-hand sides of the equations belong to a logarithmic class, we prove existence of solutions of Dirichlet problem under consideration with improved summability properties.

1. Introduction In this paper we consider a class of degenerate nonlinear elliptic high-order equations with coefficients which satisfy a strengthened ellipticity condition. Existence of solutions of Dirichlet problem for equations of the given class in the case where these equations have L1-right-hand sides has been established in 7. In the case where right-hand sides of the same equations belong to Lebesgue spaces L' with T > 1 dependence of summability of solutions of corresponding Dirichlet problem on T and other parameters involved has been studied in 9. In the present work supposing that right-hand sides of the equations belong to a logarithmic class we prove existence of solutions of Dirichlet problem under consideration with improved summability properties as compared with those described by the main result of 7. We note that analogous results on improvement of summability of solutions have already been obtained in 235,6 for non-degenerate nonlinear elliptic second- order equations and in for a class of non-degenerate nonlinear elliptic fourth-order equations. Some ideas of 516 are used in this paper. For close questions on existence and properties of solutions of nonlinear equations with L1-data we refer to 1,3941778.

199 200

2. Statement of the problem and the main result Let m E N, m 2 3, n E N, n > 2m, and let R be a bounded open set of R". We shall use the following notation: A is the set of all n-dimensional multiindices a such that (a1 = 1 or la1 = m; R"lm is the space of all functions < : A + Iw; if a function u E Lto,(R) has the weak derivatives D"u, a E A, then Vmu : R + Rngm is the mapping such that for every x E R and a E A, (Vmu(x))~ = Dau (5). Let q be a number such that 2m < q < n and let v : 52 + R be a positive function. Let c1, c2, c3 be positive constants, g : R + R be a non-negative function, g E L'(R), and let for every ct E A, A, : R x + R be a Carathbodory function. We shall suppose that for almost every x E R and every <,<' E R">",

Let f E L'(R2). We shall consider the following problem:

D'% = 0, la1 < m - 1 on 80. (4)

Definition 2.1. A W-solution of problem (4) is afunction u €4WmJ(R) such that: 1) for every a E A, A,(x,V,u) E L1(R); 201

Existence of a W-solution of problem (4) and its summability properties have been established in under certain conditions on the weighted function v. The main result of the present paper (see Theorem 2.2 below) keeping the same conditions on the weight u as in shows that the increasing of summability of the function f implies existence of a W-solution of problem (4) with improved summability properties as compared with those obtained in ?.

Theorem 2.2. Let r, t be positive numbers such that -+-<-,11 4 rtn

Let

1 v E LT(R), - E Lt(S1). U Let the function v have weak derivatives of the first order and let

Let

n-1 1 Let u be a number such that -+ - < u < 1 and let n qt

f[ln (1 + If1 1 I" E L'(fl2) . (9) Then there exists a W-solution u of problem (2.4) such that: (i) u E LQ(fl); (ii) for every n-dimensional multiindex a, la1 = 1, vl/qDOru E Lv(R); (iii) for every n-dimensional multitndex a, (a(= m, u'/~D%E Lp(R).

3. Proof of Theorem 2.2 202

We denote by W1*q(v,52) the set of all functions u E L'J(52)such that for every n-dimensional multiindex a, la1 = 1, there exists the weak derivative D"u and vID"uIq E L1(R). W1,q(v,52) is a Banach space with the norm

We denote by 4 W1y'J(v,sZ) the closure in W1$g(v,52) of the set Cr(52). Define ij = nq/(n-q+n/t). Since by (5) and (7) t > n/q and 1/v E Lt(R), we have: 4 W1>q(v,52) c Lg(52) and there exists a positive constant c such that for every u & W19q(v, 52),

We denote by WkP,(v,52) the set of all functions u E W1gq(v,52) such that for every n-dimensional multiindex a, 1011 = m, there exists the weak derivative D"u and VID"U~~E L'(52). Wk5(v,52)is a Banach space with the norm

We denote by 4 WkP,(v,SZ)the closure in W$,\(v,52) of the set C,"(R). By ci, i = 4,5, ..., we shall denote positive constants which depend only on m, n, q, r, t, (T, c, c1, c2, c3, meas52, and the norms in L1(52) of the 2(m--l)47/(47--2) functions f,g, v, v ($10~1) and f[ln (1 + If1 1 I". Let {fl} C CT(52) be a sequence such that

lim llfl - fllLqn) = 0 7 (11) l+CC V1 E N, llfiIlu(n) 6 IlfII~l(n).Due to (l),(2), (10) and well known results of the theory of monotone operators we have: if Z E N, there exists u1 €4W$;(v,52) such that for every v €4W;;(v,52), lk &(x, Vrnu~)D"v}dx = fivdx In it has been proved that there exist a W-solution u of problem (4) and an increasing sequence {Zi} C N such that 2uli + u a.e. in 52 , (13)

Va E A, D%li + D"u a.e. in 52 , (14) 203

Va E A, Dffuli+ D"u strongly in L'(i-2) . (15) The proof of this fact is closely connected with the use of integral identity (12),inequalities (1)-(3),(5), (6) and inclusions (7),(8). Let {Xk} C C" (R) be a sequence of functions such that for every k E N, 2Xk(s) = s if Is1 < k , (16)

1 xy'~< c4 XS' in i = 2, ..., m . (19) Due to (16)-(19)and Lemmas 3.1-3.3 of for every k,Z E N we have Xk(u1) €A wk,\(v,n) and llxk(ul)ll < cgk. Hence taking into account (13) we deduce that for every k E N, xk(U) €4Wk%(v,n). We set

Lemma 3.3 of and (13), (14) for every k € k we have Fk E L'(0). The following assertions hold: (*I) for every k E N,

(*2) for every n-dimensional multiindex a, Icy) = 1, and every k,kl E N,

meas { v'/~~D%I k } < cakTd ( 4,dx)'" + k-q 9,dx ; (21)

(*3) for every n-dimensional multiindex a, la1 = m, and every k,kl E N,

In fact, let k E N. By using Lemma 3.1 of and (13),(15), (18)we establish that

C vlD"xk(u)lq< ~k a.e. in 0. (22) l+l Applying (3.1) to the function xk(u) and using (22) we obtain that 204

Hence taking into account (16) and (18) we derive inequality (20). There- fore, assertion (*I) holds. Now let cy be an arbitrary n-dimensional multiindex of the length 1 and let k,kl E N. Denote G = { IuI < kl, vl/qlDauI 2 k }. It is clear that meas { Y~/~~D%I2 k } 6 meas { IuI 3 kl } + meas G . (24)

Due to (16) we have kq 6 4, in G and therefore, measG 6 k-qJn Fkl dx. This inequality, assertion (*I) and (24) imply (21). Therefore, assertion (*2) holds. Analogously we establish that assertion (*3) holds. We observe that by analogy with Lemma 3.3 of for every k,1 E N we have

This result and (ll), (13), (14), (17) imply that for every k E N,

Let us prove that the following assertion holds: (q) for every k E N, k > e,

Fk dx 6 csk (In k)-" .

In fact, let k E N, k > e. We set y = %. jhom (25) it follows that

(27) By using Holder and Young inequalities and (23) we estimate the first addend in the right-hand side of (27) as follows:

If1 Ixk(u)I dx 6 Fk dx+ c6/ {IflQk-ll 9n/ -9-1 [ cc6(l + measQ)]q/(q-') k1/2 . (28) 9 Taking into account (17) and (9) for the second integral in the right-hand side of (27) we obtain the next estimate: 205 jFrom (3.18)-(3.20) we deduce that soFk ds < cg [ k1j2+k (In k)-" 1. This inequality implies (26). Therefore, assertion (Q) holds. We note that proving assertion (*4) we used a method of 5. jFrom assertions (*1) and (Q) it follows that for every k E N, k > e, meas { 1u1 2 k } 6 c1&$ (In k)-"Q/q. Hence taking into account that ad/q > 1 and using Lemma 2 of we conclude that u E LB(s2). Thus, assertion (i) of the conclusion of the theorem holds. Now let Q be an arbitrary n-dimensional multiindex such that IQI = 1. We fix k E N, k > e($+l)/q. It is easy to see that there exists r] > e such that

r]d+l (In r])"(Q/q-I) = k4 . (30) Let k1 E N be a number such that

~ e, by (31) kl > e, and then by assertion (*4) s, F,,, ds 6 cgk1( In kl)-". This inequality and assertion (*2) imply that meas { v'/~~D%J2 k } 6 cllk,d ( In kl)-"Q/q+ c~k-~k1( In kl)-" . (32) Let us estimate the right-hand side of inequality (32). Due to (32)we have

qp4 ( In r] )-uQ/q = k-q r] ( In 7 )-" , (33)

kq < r]@ . (34) jFrom (30) and (34) it follows that - k-q r] ( In r] )-" 6 c12kPq ( In k )-"T/(q-l) . (35)

Using (31), (33), (35) we establish that kr" (In kl)-"C/q 6 c12k-7 (In k)-"T/(q-') and k-q kl( In kl)-" 6 2~12k-V (In k)-"T/(q-l). These inequalities and (32) imply that meas{d/qI(D"u( 2 k} 6 - c13k-Q (In k)-"T/(q-'). Hence taking into account that @/(q - 1) > 1 and using Lemma 2 of we deduce that d/qD'% E LT(s2). Therefore, assertion (ii) of the conclusion of the theorem holds. Finally, let Q be an arbitrary n-dimensional multiindex such that la/ = m. We fix k E N, k > e(d+l)/'. It is clear that there exists 8 > e such that ed+1 ( )u(Q/q--l)= k2 . (36) Let k2 E N be a number such that e < k2 < 28. (37) 206

By virtue of assertion (*4) we have J, Fk2 dx 6 c8k2 (In kZ)-". This inequality and assertion (*3) imply that

meas { ul/'IDau( 2 k } 6 c14kae (In k2)-"4/9 +c8K2k2( In k2)-" . (38) We note that due to (36) - 8-e ( In 8 = k-28 ( In 8 )-" 6 ~15k-*( In k )-@/(q-') . (39) jFrom (37)-(39) it follows that meas{ U'/~~D'%I 2 k} 6 cl&-P(ln k)-"T/(q-'). Hence taking into account that cq/(q - 1) > 1 and using Lemma 2 of we deduce that U'/~D~UE LB(R). Therefore, assertion (iii) of the conclusion of the theorem holds.

The proof of the theorem is complete.

References 1. Ph. BBnilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J. L. Vazquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995), 241-273. 2. L. Boccardo, T. Gallouet, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655. 3. A. Kovalevsky, Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L1-data, Nonlinear Boundary Value Problems 9 (1999), 46-54. 4. A. A. Kovalevsky, Entropy solutions of the Dirichlet problem for a class of nonlinear elliptic fourth- order equations with right-hand sides in L', Izvestiya: Mathematics 65 (2001), 231-283. 5. A. A. Kovalevsky, Integrability of solutions of nonlinear elliptic equations with right-hand sides from classes close to L1, Math. Notes 70 (2001), 337-346. 6. A. A. Kovalevsky, Summability of solutions of the Dirichlet problem for some classes of nonlinear elliptic equations, Preprint Inst. Appl. Math. Mech., NAS of Ukraine, no. 2002.02, Donetsk, 2002. 7. A. Kovalevsky, F. Nicolosi, Solvability of Dirichlet problem for a class of degenerate nonlinear high- order equations with L1 -data, Nonlinear Analysis, Theory Methods Appl. 47 (2001), 435-446. 8. A. Kovalevsky, F. Nicolosi, Entropy solutions of Dirichlet problem for a class of degenerate anisotropic fourth-order equations with L1-right-hand sides, Non- linear Analysis, Theory Methods Appl. 50 (2002), 581-619. 9. A. Kovalevsky, F. Nicolosi, Summability of solutions of Dirichlet problem for a class of degenerate nonlinear high-order equations, Applicable Analysis, to appear. ON ISOPHONIC SURFACES *

ROLAND0 MAGNANINI Dipartimento di Matematica “U.Dini” Universitd di Firenze Viale Morgagna 67/A 50134 Firenze, Italy E-mail: magnaninOmath.unifi.it

DEDICATED TO ROBERT P. GILBERT ON THE OCCASION OF HIS 70TH BIRTHDAY

We present some remarks about the conjecture Drums in the night.

1. Introduction In this note we will present some remarks on a conjecture that was posed by L. Zalcman under the title Drums in the night. Drums an the night A thin elastic membrane M of uniform ared density u is stretched to a uniform tension T and held fixed at its boundary r, a simple closed curve. The small transverse vibrations of M can be modeled as solutions u(x,t)of the wave equation in D, the region bounded by I’, which vanish on : 1 AU = p Utt x E D, t > 0, (1) +t) = o x E r, t > 0. (2) Here c = is the wave velocity and A is the Laplacian with respect to x = (xI,x~). Suppose some solution u of (1) and (2) has the property that Vu vanishes identically on a simple closed curve y c D U I?. Must r be a circle? In case I? is a circle (of radius R, say, about the origin), the function u(x,t) = Jo(k 1x1) eicktwill satisfy (1) and (2) if kR is a zero of the Bessel function Jo. Since JA = -J1, Vu = -kJl(k 1x1) eicktf$.

*This work is supported by a 1999-2000 grant of the italian MURST.

207 208

Thus if J~(kr)= 0, Vu will vanish on the circle of radius T concentric with I’. Choosing k sufficiently large yields solutions of (1) and (2) which vanish on a family of such circles.

2. Drums in the night, Schiffer’s conjecture and Pompeiu’s problem A solution of (1)-(2) can be written as a series expansion,

n= 1 where u,, n = 1,2,. . . are the Dirichlet eigenfunctions of the Laplacian, and A,, n = 1,2,. . . the corresponding eigenvalues, that is, u, and A, satisfy the problem: Au+Xu=O on D, u=O on I‘. (4) If we require that the gradient of the function u defined in (3) vanishes identically on y,

Vu(s,t)= 0 2 E y, t > 0, (5) then we obtain that

Vu,(z) = 0 2 E y, n EN, (6) where N = {n E N : (a,, b,) # 0). (7) The set N can be finite or infinite. If R denotes the interior of y, then each u,, n E N, satisfies Schifler’s overdetermined boundary value problem: Au+Au=O in R, u = constant on y, (8) &=OOU on 7, where v is the exterior normal unit vector to y. It is well-known that if a non-trivial solution of problem (8) exists, then the set R does not enjoys the Pompeiu property 7, that is there exists a function f : R2 + R, f not identically zero, such that

1 f(s)dx = 0 for all rigid motions 0. (9) 4n) Viceversa, if f # 0 exists such that (9) holds, then a non-trivial solution of (8) exists. 209

An old conjecture states that the only domain not enjoying the Pom- peiu property is the disk. Although this conjecture has not been proved or disproved up to now, a great variety of results are known on domains not satisfying the Pompeiu property. Having established a connection between the overdetermined problems (1)-(2)-(5) and (B), we can claim, for instance, that, if u is a solution of (l), (2) satisfying (5), then y is a real analytic curve, by invoking Williams’s result 6. Moreover, a symmetry result can be drawn.

Proposition 2.1. Let a solution u of (I) and (2) satisfy condition (5) and suppose that y is a simple closed curve of class C2’e,E > 0. If the set N defined in (7) is infinite, then D is a disk.

Proof. Proposition 2 * states that if the eigenvalue problem (8) has infinitely many solutions, then R must be a disk. Hence, each u, is a Neumann eigenfunction for the disk R. By continuing analytically u, to D, we infer that I? is a circle. CI If Schiffer’s conjecture for the domain R were true, we could also settle down the case where the set N is finite. It should be noticed though that, even in the least favourable case where set N is made of a single element no, the overdetermined problem (1)-(2)-(5) gives more information than Schiffer’s eigenvalue problem (8). In fact, in problem (1)-(2)-(5), we assume the existence of a Dirichlet eigenfunction u,, in a domain D that contains fl. In the following result, we try to exploit this observation.

Proposition 2.2. A solution u of (1) and (2) satisfies condition (5) if and only if, for every positive number r with r < dist (y,I?), we have that I u,(y) (y - x) dS, = 0, for every x E y and n EN. (10) Iv-xl=r

Proof. Consider the function

h(x,t)= c, u,(x) eCxnt, nEN where the numbers c,, n E N, are arbitrarily chosen; h(x,t) is a solution of the heat equation 210

By Theorem 2 or Corollary 2.2 3, we have that Vh(x,t) = 0 for every t > 0 if and only if

J h(y, t)(y- x) dS, = o for every o < r < dist (x,r) and t > 0. (13)

I,-Xl=T Therefore, the assertion of Proposition 2.2 follows from (13) and the definition (11) of h by the arbitrary choice of the en's. U

3. Drums in the night and isophonic curves We observe that, if the gradient of a solution u of (1) and (2) vanishes on y,then, in particular, y is a stationary isophonic curve for u, i. e.

u(x,t)= U(t),x E y, t > 0, (14) where U is some real-valued function. Of course, the requirement that y be a stationary isophonic curve for u is less strict than asking that the gradient of u vanishes on y;hence, it is less likely that the existence of a stationary isophonic curve for u imply that D is a disk. In order to get symmetry, we need some additional information on u, as the following result shows.

Proposition 3.1. Let u be a solution u of (1) and (2) such that

u(x,O) = 0 and ut(x,O) = 1, x E D. (15) Assume that y is a simple closed curve such that R Satisfies the interior cone condition. If u satisfies condition (Id), then D must be a disk.

We recall that 0 satisfies the interior cone condition if for every x E y there exists a finite right spherical cone K, with vertex at x such that Kx c and Eny = {z}. Proof. If we extend u by -u(q -t) for t < 0, then u satisfies (1) and (2) in D x (--00, +m). The function defined for (x,t) E D x (0,+m) by

then satisfies the Cauchy-Dirichlet boundary value problem:

ht = Ah in R x (O,+m), h=O on 80x (O,+m), h=l onRx{O}. 21 1

Moreover, if y is a stationary isophonic curve for w, then I' is a stationary isothemic curve for h, since

1 +O0 e-s2/4tU's h(x,t) = H(t) := =Lm ( ) ds XEy.

The conclusion then follows from Theorem 1.1 '. 17

References 1. C. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Analyse Math. 37, 128-144 (1980). 2. R.Magnanini and S. Sakaguchi, The spatial critical points not moving along the heat flow, J. Analyse Math. 71, 237-261 (1997). 3. R. Magnanini and S. Sakaguchi, Matzoh ball soup: heat conductors with a stationary isothermic surface, Ann. Math., to appear. 4. D. Pompeiu, Sur certains systhmes d'hquations linhaires et sur une proprihtk int4grale des fonctions de plusieurs variables, C. R. Acad. Sci. Par%s, 188, 1138-1 139 (1929). 5. S.A. Williams, A partial solution to the Pompeju problem, Math. Ann. 223, 183-190 (1976). 6. S.A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J. 30, 357-369 (1981). 7. N.B.. Willms and G.M.L. Gladwell, Saddle points and overdetermined problems for the Helmholtz equation, 2. Angew. Math. Phys. 45, 1-26 (1994). 8. L. Zalcman, Some inverse problems in potential theory, Contemporary Math- ematics 63, 335-350 (1987). A SURVEY OF POINTWISE INTERPOLATION INEQUALITIES FOR INTEGER AND FRACTIONAL DERIVATIVES

VLADIMIR MAZ'YA AND TATYANA SHAPOSHNIKOVA Department of Mathematics, University of Linkoping, SE-581 83 Linkoping, Sweden E-mail: vlmazOmai.Eiu.se, tashaQmai.liu.se

DEDICATED TO ROBERT P. GILBERT ON THE OCCASION OF HIS 70TH BIRTHDAY

We survey of our recent results concerning pointwise interpolation inequalities for derivatives of integer or frxtional order.

1. Introduction The idea to obtain information concerning intermediate derivatives using properties of a higher derivative and the function itself goes back to Hadamard4 (1897), Kneserg (1897), and Hardy and Littlewood5 (1912). It was developed in various directions by Kolmogorov'O (1938), Sdkefalvy- Nagy24 (1941), Gagliard~-Nirenberg~~~~(1959) et aE. In its simplest form, this idea is expressed by the Landau inequality on the real line'' (1913)

br(~)125 2 IIuIItm(~)IIu"II~,(~). (1) Under the additional asumption u 2 0 on R, one can readily verify the estimate

lur(Z)12 5 u(X)llu"llL,(R) (2) which proved to be useful in various topics of the theory of differential and pseudodifferential operators (22912,131817). Variants and extensions of (2) were treated by Maz'ya and Kufner in 14. They proved, in particular, the following generalization of (2) with the sharp constant

where u 2 0 and 0 < a < 1.

212 213

Clearly, (2) can fail for some smooth functions u changing sign. How- ever, one may ask whether it is possible to replace the L,-norms in (1) by values at the point x of certain operators acting on u. In what follows, we describe different ways to give an affirmative answer to this question.

2. Some sharp pointwise inequalities Let M" be the maximal operator defined by

where u is a locally summable function on Rn, n 2 1, Br(x) is the ball {y E Rn : Ix - yI < r}, and the bar stands for the mean value of the integral. Clearly, MOu(x)does not exceed the sharp maximal function of Fefferman and Stein2

M'u(X)= :i fB,.(z) IU(Y> - fB,,($)Wd+Y and it is dominated by the Hardy-Littlewood maximal function

Mu(x)=sup T >o f,,( .) Iu (Y) Id We introduce the mean value of the vector-valued function v : Rn + Rn over the sphere aBT(x)as follows: - (4) v(x;r) = f aB,.(x;(Y)dSy' We also set

In particular, for a function v of one variable we have l2v(x)- w(5 + r) - W(X - .)I D, (v;x) = sup , r>O 24r) Henceforth it is assumed that w is a continuous nondecreasing function on [O,co) such that w(0) = 0 and w(co) = co. We formulate a new pointwise inequality involving the gradient Vu(x), the maximal function M"u(z),and the function Du(Vu;x)and show that this inequality is sharp. The extremal function, whose form is rather complicated, was found by guess. There is no standard approach to such a construction for the time being. 214

Theorem 1. 2o (i) Let the function

be strictly increasing on [0,CQ) and let R-' be the inverse function for O. Further let

Then for any u E C'(R")

IVu(2)l 5 n(n 4- l)D,(Vu; z)Q-'( Mou(2) ) , nD, (Vu;2) where Q-' is the inverse function for Q. (ii) Let w E C'(0,oo).Suppose the function tw'(t) is nondecreasing on (0, GO) and that, for n > 1, the function tO'(t) is nondecreasing on (0,~). Let R be a unique root of the equation

n(n+ l)O(t)= 1. (9) Inequality (8) becomes an equality for the function

If one changes the definition of D,(v;z) replacing the mean value (4) by

f BT(z)V(Y)dsy one can arrive at another sharp inequality of type (8) (see 20).

For the particular case W(T) = P,a > 0, Theorem 1 is as follows.

Corollary 1. Let u E C'(Rn), and let cy > 0. Then 215 with the best constant ff+l an c = (n+ 1)-( a (n+ .)(n + a + 1) Inequality (11) becomes an equality for the function

where (72 + .)(n + a + 1) t R=( (a + l)n(n+ 1) >- Corollary 2. (Local version of Corollary 1.) Let MY denote the modified maximal operator given by

and let IVu(2) - vu(x;r)l D1,,(Vu;x) = sup O 0, the inequality

(V~(z)lI (C(D1,a(Vu;r))m + (n + l)(M~u(z))*)(M;)u(x))* (13) holds with the best constants C defined by (12). For the one-dimensional case we use the notation (6) and note that M" is defined by 1 x+r M"u(z)= sup - sign(y - x)u(y)dyl r>o 2r II_, The next two corollaries immediately follow from Theorem 1.

Corollary 3. Let u E C'(R). Then the inequality

holds, where W1is the inverse function for t q(t)= 1 Q-'(T)dT 21 6 with 0-' standing for the inverse function for 0(t)= Jo' aw(at)da. Suppose tw'(t)is nondecreasing on (0,m). Then inequality (15) becomes an equality for the odd function u given on the semi-axis x 2 0 by the formula 1 x(1- 1 w(0x)da) for 0 5 x 5 R u(x)= (2R - s) r(20- l)w(a(2R- z))da for R < x < 2R 0 0 for x 2 2R, where R is a unique root of the equation 2R(t) = 1.

Set

and note that D,(u'; x) 5 D,(u'; x). Moreover, if u is odd, then D, (u';0) = D, (21'; 0). Therefore, Corollary 3 implies the following assertion.

Corollary 4. Let u E G1(R). Then

Inequality (16) becomes an equality for the same function as in Corollary 3. As in Corollary 3, we here assume that rw'(r) is nondecreasing on (0, m).

In the special case w(t)= t", a > 0, Corollaries 3 and 4 can be stated as follows.

Corollary 5. Let u E C'(R), and let Q > 0. Inequality

)2u'(s)- ~'(6:+ r) - u'(x - r)I 2m 1u'(x)1 5 C(M'u(x))* (sup r" ) (17) r>O holds with the best constant

The rougher inequality 217 follows from (17). Inequality (17) and even (19) becomes an equality for the odd function u whose values for x 2 0 are given by (a+ l)x - X"+l for o 5 2 5 (q)'/" a +2 '/" -x)"+' for (~>1/"< x < 2(~>'/"(20) (2(-) for x 2 2(9)'/" For a = 1, (19) implies the sharp estimate 8 1u1(x)125 3 M*u(x) IIU"IIL,(R)* (21)

3. The Gagliardo-Nirenberg and other inequalities Diverse pointwise interpolation inequalities for derivatives of integer or fractional order were obtained in 14,1,6,15,16,17,18,19 without best constants. We give several examples. Let m be a positive noninteger, p 2 1, and let

--n-P{m)dy , (DP,rnU)(2) = Iv[mlu(x)- v[mIu(Y)lP1% - YI (Ln ) llp where [m]and {m}are the integer and fractional parts of m and

Vk= {8z1-'.. .a::}, a1 + . . . + an = k. We remind that the operator DP,,,is used in the definition of the fractional Sobolev space Wp7n(Rn) introduced as the completion of G,"(R") in the norm ~IDp,mUllLP(Rn)+ 1b11LP(Rn)- Theorem 2. [MSl] (i) Let k, 1 be integers and let m be noninteger, 0 5 15 k < m. Then

IVkU(X)l 5 C[(MVlU)(x)]=+ [(27p,mu) (x)]% for almost all x E R". (ii) Let k, m be integers, and let 1 be noninteger, 0 < 1 < k 5 m. Then IVku(x)I 5 c(D~,~u(x))~(MV,U(X))~ for almost all x E R". (iii) Let k be integer and let l,m be noninteger, 0 < 1 < k < m. Then lVk~(x)I5 ~(D~,lu(x))~(Dp,mu(x))S for almost all x E Rn. (iv) Let 0 < s < 1 < 1 and let p 2 1. Then (Dp,8u)(x)5 c(MIu - u(x)Ip) (D,,lu(x))' 218

Remark 1. Theorem 2 leads to the Gagliardo-Nirenberg inequality

where 1 < q 5 00, 1 < p 5 00 and -+--=(l---);.1 kl k1 s mP Indeed, by (i) with I = 0 the left-hand side does not exceed

c( / ((Mu(x))s(l-k'm) (DP,"U(Z)) ek'mdx)lIs R" which by Holder's inequality is majorized by and it remains to refer to the boundedness of the operator M in L,(R"), q> 1.

In l6 we applied pointwise inequalities from Theorem 2 to describe the maximal Banach algebra AFT' imbedded in the space of multipliers M(Wr(R") + WL(Rn)) which map the Sobolev space Wr(R") to WL(Rn) with noninteger m and I, m 2 I, p E (1,~).We showed that AT" is isomorphic to M(W,n"(Rn) + Wi(R")) n L,(R") and gave a precise description of all imbeddings AF'l c The following lemma became a crucial tool of our elementary proof of the Bezis and Mironescu theorem on the boundedness and continuity of the composition operator W,(Rn) n W,',(Rn) 3 u + f(u)E W,S(Rn), where s is noninteger, 1 < s < 00 and p 2 1.

Lemma 1. l9 Suppose a E (0,l), p 2 1, and u E Wpl,loc(R"). Then for almost all x E R"

(DP>QW)5 c((Mlu- u(41P)(x))(l--cu)lP ((MIVuIP)(x)) where M is the Hardy-Littlewood maximal operator.

The paper by A. Kalamajska6 is dedicated to some integral representation formulas for differentiable functions and pointwise interpolation inequalities on bounded domains with the operator M both in the right- and left-hand sides. Kdamajska proved, in particular, that if

lim RPkf lu(z)ldx = 0, where a E Rn and T > 0, R-t, B,R(aR) 219 then, for any polynomial P of degree less than j, MVbu(x) I(M(u(x)- P(z)))F(MVp(x))J.

In l7 we obtained inequalities of a similar structure for Riesz and Bessel potentials. Let z be a complex number with 0 < Rz < n and let f be a complex valued function in L1(Rn). By I,f we denote the Riesz potential of order z:

with the constant factor chosen in such a way that

Izf(x) = F;f,lSl-”Fx+&), where F is the Fourier transform and F-’ is its inverse.

Theorem 3. Let 0 < 8% < R< < n and let f E Ll,lOc(Rn).Then Ml,f(z) 5 c (MIc~(x))”/’~(Mf(x))l-R”/’c for almost all x E R”. The proof is based on the estimate

M(k * u) I2n+111kIILI(R-)MU, where k(lx1) is a nonnegative and nonincreasing function. This estimate refines that given in 23, Th. 2, Ch.3, Sect. 2.2:

k * u IcIlkll~~(~-)Mu. An assertion analogous to Theorem 3 holds for Bessel potentials. Let z be a complex number. The Bessel potential J, is defined by J, = (-A + I)-”/~,i.e.

2 -2/2F Jzf(x) = F;-f,(1+ IEl 1 x+€f. Another formula for J, is Jzf(x) = c s, G& - !/)f(Y)dY, where

G,(x) = c Izl(”-n)/2K(n-,)/2(1ZI), K, is the modified Bessel function of the third kind. 220

Theorem 4. Let 0 < RRZ< 8fZc' and let f E L1,ioC(Rn).Then

M Jzf (x) 5 c (MJc f (x))"/'~ (Mf (x)) l-rJzz/lRc for almost all x E R". Immediate corollaries of Theorems 3 and 4 are Gagliardo-Nirenberg inequalities for Ftiesz and Bessel potentials which are derived in the same way as in Remark 1.

Remark 2. Let 1 < q 5 00, 1 < p 5 00, 0 < 8.z < BC < n, and let

Then szz/sJz~ I--Rz/RC I112 f I I L, (R" 5 c I IIS f I IL, (Rn IIf l I L, (Rn1 . A similar estimate is valid with Bessel potentials in place of Riesz potentials.

References 1. B. Bojarski, P. Hajlasz, Pointwise inequalities for Sobolev functions, Studia Math. 106 (1993), 77-92. 2. C.Fefferman, E. Stein, Hp-spaces of several variables, Acta Mathernatica, 129 (1972), 137-193. 3. E. Gagliardo, Ulteriori proprieti di alcune classi di funzioni in pi^ variabili, Ric. Mat. 8 (1959), 24-51. 4. J. Hadamard, Sur certaines propriCtCs des trajectoires en dynamique, J. Math. Se'r. 5, 3 (1897), 331-387. 5. G.H. Hardy, J.E. Littlewood, Contributions to the arithmetic theory of series, Proc. London Math. SOC.Ser. 2, 11 (1912-1913), 411-478. 6. A. Kalamajska, Pointwise interpolative inequalities and Nirenberg type estimates in weighted Sobolev spaces, Studia Math. 108 (1994), 275-290. 7. Y.Kannai, Hypoellipticity of certain degenerate elliptic boundary value problems, Trans. Amer. Math. SOC.217 (1976), 311-328. 8. Y.Kato, Mixed type boundary conditions for second order elliptic differential equations, J. Math. SOC.Japan 26 (1974), 405-432. 9. A. Kneser, Studien uber die Bewegungsvorgiinge in der Umgebung instabiler Gleichgewichtslagen, J. fur die reane und angew. Math. 118 (1897), 186-223. 10. A.N. Kolmogorov, Une g6nCralisation de l'inCgalit6 de M. J. Hadamard entre les bornes superieures des d6rivCes successives d'une fonction, C. R. Acad. Sci. Paris 207 (1938), 764-765. 11. E. Landau, Einige Ungleichungen fur zweimal differenzierbare Funktionen, Proc. London Math. SOC.13 (1913), 43-49. 12. P.D. Lax, L. Nirenberg, On solvability of difference schemes, a sharp form of Girding's inequality, Comm. Pure Appl. Math. 19 (1966), 473-492. 22 1

13. V. Maz’ya, The degenerate problem with an oblique derivative, Mat. Sb. 87 (1972), 417-454. 14. V. Maz’ya, A. Kufner, Variations on the theme of the inequality (f’)2 5 2fsup[f”(, Manuscripta Math. 56 (1986), 89-104. 15. V. Maz’ya, T. Shaposhnikova, On pointwise interpolation inequalities for derivatives, Mathematica Bohemica 124 (1999), 131-148. 16. V. Maz’ya, T. Shaposhnikova, Maximal algebra of multipliers between fractional Sobolev spaces, Proceedings of Analysis and Geometry, S.K. Vodop’yanov (Ed.), Sobolev Institute Press, Novosibirsk, 2000, pp. 387-400. 17. V. Maz’ya, T. Shaposhnikova, Pointwise interpolation inequalities for Riesz and Bessel potentials, Analytical and Computational Methods in Scattering and Applied Mathematics, Chapman and Hall, London, 2000, pp.217-229. 18. V. Maz’ya, T. Shaposhnikova, Maximal Banach algebra of multipliers between Bessel potential spaces, Problems and Methods in Mathematical Physics, The Siegfried Prossdorf Memorial Volume, J. Elschner, I. Gohberg, B. Sil- bermann (Eds.), Operator Theory: Advances and Application, Vol. 121, Birkhauser, Basel, 2001, pp. 352-365. 19. V. Maz’ya, T. Shaposhnikova, An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces, J. Evol. Equat. 2 (2002), 113-125. 20. V. Maz’ya, T. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives, Functional Analysis and its Applications, 36 (2002) , 30-48. 21. L. Nirenberg, On elliptic partial diffrential equations: Lecture 2, Ann. Sc. Norm. Sup. Pisa, Ser. 3 13 (1959), 115-162. 22. L. Nirenberg, F. TrBves, Solvability of the first order linear partial differential equation, Comm. Pure Appl. Math. 16 (1963), 331-351. 23. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. 24. B. Szokefalvy-Nagy,Uber Integralungleichungen zwischen einer Funktion und ihrer Ableitung, Acta Sci. Math. Szeged 10 (1941), 64-74. NON-UNIQUENESS IN CONNECTION WITH METHODS FOR THE RECONSTRUCTION OF THE SHAPE OF CYLINDRICAL BODIES FROM ACOUSTIC SCATTERING DATA

ERICK OGAM, THIERRY SCOTTI AND ARMAND WIRGIN Laboratoire de Mbcanique et dgcoustique, UPR 7051 du CNRS, 31 chemin Joseph Aiguier. 13402 Marseille cedex 20, France E-mail: wirgin@lma. cnrs-mrs.J?

It is shown that the combined use of: 1) data relating to the response, on a circle circumscribing the object in its cross-section plane, to two monochromatic probe plane waves having different frequencies, 2) the intersecting canonical body approximation (ICBA) of the wave-object interaction, and 3) an asymptotic analysis of cost fhctionals of the discrepancy between the data and the ICBA estimation thereof, enables a substantial reduction of the ambiguity of the identification of the boundary of the scattering body.

1 Introduction

To solve the inverse problem, we employ an estimator which appeals to the Intersecting Canonical Body Approximation (ICBA)”*. The ICBA assumes that the amplitudes in the partial wave representation of the scattered field are nearly those of a canonical body (circular cylinder in the 2D problem treated herein), this being true locally for each observation angle, and the canonical body having the same local radius at this angle as that of the real body. The reconstruction of the shape of the body, represented at a given angle by its local radius, then proceeds by minimizing the discrepancy between the measured or simulated data and the estimation thereof, this being done at each observation angle. In one of its forms, the procedure enables the reconstruction of the local radius of the body, for a given polar angle, by solving a single non-linear equation2. Another variant consists in finding this local radius by minimizing the L2 cost functional of the aforementioned discrepancy. It is shown: i) that the reconstruction of the boundary by these methods is not unique for both (synthetic) simulated and (real) experimental data, and ii) how to single out the correct solution by employing data at two frequencies.

2 Forward and inverse scattering problems

Let ui(x;~) be an incident plane-wave monochromatic (pressure) wavefield (the exp(-iwt) time (t) factor, with w the angular frequency, is hereafter implicit) at point x = (r,€J) of the xOy plane (i.e., the field in the absence of the object), U(X;W)

222 223

the total field in response to u'(x;o), R the subdomain of xOy occcupied by the sound-hard cylindrical object in its cross-section (xOy) plane, and r I asZ the trace in xOy of the boundary of the object, assumed to be representable by the parametric equation r = p(8) ( p = p(8) a continuous, single-valued hction of 8 ). u'(x;o) and u(x;o)satisfy:

wherein c is the sound speed in the medium outside of the object, 8' the angle of

incidence, p I dp(8)ldO and s = (lj2+ p2)" . The forwardscattering (measurement)probIem is : given o , c, u'(x;o)and r ; determine u(x;w) at all points on the circumscribing circle r, of radius b>p=Maxp. eE[o.zn[ The problem of particular interest here is to solve the inverse scattering problem : given o,c, u'(x;o),b, and the simulated or measured field on r, ; determine the location, size and shape of the object, embodied by the so-called shape function p(19) , knowing a priori that : i) the origin 0 is somewhere within the object, and ii) poO, b>O. (5)

wherein pa is assumed to be a known positive real constant (as is b). Note that in the inverse problem u(x;w) is unknown everywhere except on r, .

3 A method for the reconstruction of the shape of the body using the ICBA as the estimator

The estimator appeals to the so-called 'Intersecting Canonical Body Approximation ' (ICBA)Is2,whose mathematical expression is:

~a,(p(8).8)W~"(kr)e'"'; r 2 p(8), 8~ [0,2n], (6) "=-I. 224 wherein

HA') is the n-th order Hankel function of the first kind, J, the n-th order Bessel function, and Z,(s) = dZn(s)/ds. The ICBA furnishes the exact solution for scattering from a sound-hard circular cylinder of radius a and center 0 (i.e., the case p(8) = a ) provided L + - . Let ti', Zi" Zi designate synthetic (simulated) or real (experimental) data pertaining to the incident, scattered and total field respectively for the (real) body whose (real) shape (defined by the hction p(0) ) is unknown and to be determined. Let 2, 2, ii designate the estimated incident, scattered and total field for a trial body with trial shape z(8). To reconstruct the entire shape of the body requires finding p(8) for all values of 8 . In principle, we can identify the reconstructed value of p(8) with that z(0) for which a discrepancy hctional between the measured and estimated fields vanishes. Practically speaking, this is done at M measurement angles { 8, ; m = 1,2,..., M }, so that the discretized version of r, embodied in the set {z(8_); m=1,2, ..., M}, is recovered from the set of M equations (i.e., discretized form of the discrepancy functional)

(8) wherein it is observed, with the help of (6)-(7),that the m-th equation depends only on the m-th trial boundary shape parameter z(8,). Although these equations are uncoupled in terms of {z(Om); m = 1,2,..., M}, each one is nonlinear because each member of the set { a,, (z(8,) } is a nonlinear function of z(em).

4 Use of the K discrepancy functional and a perturbation technique

We assume that 225 where a, d and E are positive real constants, f(8) has the same functional properties as z(8) , g(8) has the same hctional properties as p(8) , If(8]I 1, lg(e)l 1, and ks<

To zeroth order in ks, Z;"(z(8,),8,) =Z;'(b,8,) is just the field scattered by a circular cylinder of radius a and center at 0, which, if it were simulated or measured in exact manner, would be of the form (see 53):

whereas the ICBA representation of the estimated scattered field takes the same form with d replacing a, so that the continuous form of (8) yields:

wherein If:"(<) = J,(<) + i<(<) I An obvious solution of (12) is d=a, which is the correct solution to order (kE)' for the shape function, since z(8) = d and p(8) = a to zeroth order in k& , but this doesn't mean that it is the only solution. First consider the subset of the equations (12) for which and 1.1 << ka . Making use of the asymptotic forms of the Bessel and Hankel functions :

gives rise (via (12)) to sin(kd - ku) - 0 ; Vlnl<< kd,ka , whose solutions are

d=a+- I€Z . k'

This shows that there exist an infinite number of solutions of (12) in the asymptotic regime In1 << kd,ka , but this, by itself, does not mean that this set of equations does not possess a unique solution. To pursue this question, let us turn to the case In1 = kd, 1.1 = ka . As written previously, an obvious solution of (12) is then d=a ,but this probably is not the only 226 solution in this regime. What this means is, that for some or all of the values of n in this regime, we may find a set { df' } of roots of (12) such that d;'' = a . Finally turn to the case 1.1 >> kd, 1.1 >> ka . It suffices to look at what happens for n 2 0. We make use of of the following asymptotic forms

to find (m)-'[(d/a)"-(dla)-"I- 0; 1.1 + - , the only solution of which, for positive real d and a, is d=a, so that if L = - , we would have to conclude that the only solution satisfylng (12) for all n E 2 is d = a , which happens to be the right zeroth-order perturbation solution for the inverse problem. The value of L used in the ICBA is more like L<-kd and L<-h (imitating what is done for scattering by the canonical circular cylinder; it may be that the ICBA partial wave series even diverges for L = - at certain points on or outside a body of arbitrary shape), so that it is inappropriate to use the result of uniqueness of the solution d=a in the regime In( >> kd 1.1 >> ka simply because this regime is impossible to attain with the indicated (finite) values of L. The conclusion of this is that if L+kd and L<-ka, then the solution for d is not unique since it is given by (1 3) plus other eventual solutions of (12) corresponding to the regime 1.1 = kd , 1.1 = ka .

5 Non-uniqueness in connection with least squares method for solving the inverse problem using the ICBA as the estimator

Instead of actually solving (i.e.7looking for the roots of) each member of the set in (8), we can rather search for the set of shape parameters {z(Om); m = 42, ..., M} by minimizing the L2 cost functionals

and again apply the perturbation method to zeroth order in k& so as to find 227

(wherein { } is finite for e = 0) whence J(a,B;w)=O; VOE [0,2n[,which indicates that the zeroth-order perturbation L2 cost functional possesses a global minimum at the right solution d=a for all angles. This is a reassuring result, but doesn't mean that there do not exist local minima for other values of d. In fact, J(a + 4n I k,O;o) probably does not vanish for values of 4 other than 0.

6 Method for eliminating or at least reducing the non-uniqueness problem

The principal result of the preceding sections is the zeroth-order perturbation prediction: d=a+lnlk; IEL , (14) wherein L is the finite subset of Z for which d is within the bounds of p written in (5). We hypothesize (admittedly encouraged by the results of numerical experiments3) that if the perturbation analysis were carried out to higher order it would give rise to a relation of the sort

This relation signifies that the inversion process should lead to a series of nearly- homothetic reconstructed bodies, one of which (i.e., the one for which 4 = 0) is nearly the real bcdy. Thus, we are faced with the problem of eliminating or at least reducing the non-uniqueness expressed by (1 5). Let us term the curves for which l! # 0 (( artifacts D. Eq. (1 5) tells us that the curves relative to the artifacts, contrary to the one relative to the boundary of the real body, depend on the wavenumber of the probe radiation. This suggests that the boundav we are looking for is the only one that doesn 't change when thefiequency of the probe radiation is changed. It is then a relatively simple matter to devise a scheme that filters out the artifacts from two sets of data obtained at two frequencies4. The data we employed was both synthetic (computed by a boundary element method) and real (obtained at audio frequencies 14 kHz and 16 kHz for a monostatic arrangement 8, = 8' = 0; Vm E [1,M] with a rotating (about its axis) cylindrical body in an anechoic chamber). A representative result is depicted in Fig.1. 228

Figure 1. Real data case. Top: polar plot of the minima of the cost functionals J at the two fkequencies 14 lcHz (circles) and 16 !dIz (points). Bottom: filtered reconstruction (squares) of the shape function of a hexagonal cylinder (continuous curve). The reconstruction (not shown) is less ambiguous with synthetic data.

References

1. Wirgin A. and Scotti T., Wide-band approximation of the sound field scattered by an impenetrable body of arbitrary shape . JSound Vibr. 194 (1996) pp. 537-572. 2. Scotti T. and Wirgin A., Shape reconstruction using diffracted waves and canonical solutions. Inverse Probs. 11 (1995) 1097-111 1. 3. Ogam E., Scotti T. and Wirgin A., Non-ambiguous boundary identification of a cylindrical object by acoustic waves. C.R.Acad.Sci.IIb 329 (2001) pp. 61-66. DISPERSION IDENTIFICATION USING THE FOURIER ANALYSIS OF RESONANCES IN ELASTIC AND VISCOELASTIC RODS

R. OTHMAN AND G. GARY Laboratoire de Mkcanique des Solides, Ecole Polytechnique, 91 128, Palaiseau. France. E-mail :[email protected].~

R. H. BLANC Trans Waves, 150 le Corbusier, 13008, Marseille. France.

M.N. BUSSAC AND P. COLLET Centre de Physique Thkorique, Ecole Polytechnique, 91 128, Palaiseau, France.

A new method for identifylng the dispersion relation in elastic and viscoelastic rods is presented. It relies on the multiple resonance of a strain measurement. The wave velocity is related to the resonance position whereas the damping is related to the resonance bandwidth. Applied to an aluminum bar, the method provides wave dispersion for kequencies up to 60 kHZ.

1 Introduction

Wave dispersion in rods is due to geometric and (or) viscoelastic reasons. When the cross-section is circular the dispersion relation is governed by the Pochhammer- Chree eq~ation~.~"~,which was extended to viscoelastic rods by Zhao & Gary 15. In the Split Hopkinson bar apparatus (SHB), experimental results are improved when wave dispersion is taken into ac~ount~'~-'.Wave dispersion in rods is also used to investigate the viscoelastic properties of materials*"'. Dispersion may be experimentally determined by comparing the wave Fourier components measured at two points on the rod129 Recently, Hillstrom & a1.I' developed a multi-point method using least squares. In the present paper, a one-point method is developed using a spectral analysis of the resonant frequencies of the rod.

2 Theory

We consider a finite rod of length L (see Figure 1). E denotes the longitudinal strain. A perturbation is generated at the left end of the rod. The right end of the bar is kept free from stress. Henceforth, we assume that only the first longitudinal mode of propagation is excited5-'. In this case, the Fourier transform of the strain can be expressed by : E(a,o)= A(o)e-"(")"+ B(w)e'*'"'", (1)

229 230 where <(a)=k(@)+ ia(w)=m/c(w)+ia(w) , (2) <(o)is the complex wave number of the first mode of propagation and it takes account of geometric effects and (or) viscoelastic effects, c(w) is the phase velocity and a(@) the attenuation. The Fourier components of the strain at the left side (striker side, abscissa -L) of the bar are denoted by O(w).Considering the boundary conditions, i.e. E(0,o)= 0 and E(- L,o)= 8(o),it follows that:

Introducing (3) into (1) enables the strain to be expressed as a function of the complex wave number and of the boundaw conditions at the left side of the rod

In a first step, wave attenuation is neglected (ia(w] << 1). With low damping materials, this hypothesis is valid for most experimental applications. A correction must be made for high damping materials. Denoting by S (a,w ) the strain spectrum, (4) and (5) yield :

(51

The spectrum denominator becomes zero for the frequencies onsuch that I3,l4: k(on)=na / L, (6) where n is a relative integer. For each n , unless the angular frequency onnullifies the numerator as well as the denominator, a resonance occurs. The corresponding resonant frequencies are associated with the local maxima of the spectrum, so they are easily assessed. Therefore, when 8(wn)#0and VpEZN,a/L#p/n, (7) the wave velocity is given by : C(O, ) = wnL/nz , (8) Henceforth, we reconsider (4). The numerator and the denominator are developed as follows : sin (e (w )x) = sin (k(w)x)cos (ia(w )x) + cos (k (w )x )sin (ia(w ).) . (9) We develop the sine and cosine to the second order of a (w )x : 23 1

sin(ia(w)x)= ia(o)x+o(a2(w>x2) . (1 Ob) Eqns. (5), (9) and (1 0) yield : (w)oz + sin (k(w)a) sz(a,@)= p(wl' a (1 1) a (w)L' + sin (k(w)~)' We assume that o is close to a resonance value w, . The attenuation is considered to be constant in this vicinity: a(o)=a,,=a(on). The sine is developed in the vicinity of a)": sin(k(w)x)= sin(k(wn).)+ xp, COS(~(W~)x)(w - wm)

-+x2an sin(k(w")x)(o-wnr +o((w-o,r) ? (12)

Let 60, be half the bandwidth at the half-height. A development to the second order, with respect to &on ,yields a simple expression for the wave damping coefficient: a; =a'(o,)=p:6w: . (13) Since the waves attenuate as they advance, damping is negativel3.l4: an= -p",6wn . (14)

3 Experimental validation

3.1 Experimental set-up A 3.019-m-long and 40-mm-diameter aluminium bar was considered. One strain gauge was cemented at x = -1.567 m , Six tests were carried out. Three tests were camed out using a 15-cm-lOng aluminium striker having the same section as the bar and the other three tests were carried out using a 120-cm-long aluminium striker having also the same section as the bar. The striker velocity was about 16m/s. The measured strain in test 1 and its spectrum are shown in Figure 2. Signals were sampled with a frequency f, =5OOkHz. Fast Fourier Transforms (FFT) were performed using 106-point long signals.

X .O ...... "...... "...... :*

Figure 1. Simplified sketch of the experimental set-up. 232

3.2 Experimental results For each test, the spectrum of the strain was computed using theFFT. Firstly, we determined the local maxima, which correspond to the resonant frequencies. Considering (6), the wave celerity is therefore assessed. Results of the five tests are superimposed in Fig.2. The observed sensitivity to the noise of the measurements is very small. The wave number was interpolated over the interval 0-60kHz as a polynomial function. Secondly, we determined the bandwidth of the resonance at the half-height of the peak. Knowing the derivative of the wave number with respect to the angular frequency, which was easily determined from the wave celerity, attenuation was calculated using (14). The values were also interpolated over the same interval as a polynomial function. Results for damping obtained from the five tests show a stronger sensitivity to noise than those obtained for wave celerity : they are more scattered.

7 -5000 2? 5100r- 4900 YE 0) (Dm c - sa4800 e v) - .- 4700 P'

4x10 I I f 20 30 40 I 6C "4~frequence (kHr)

Figure 2. Longitudinal wave celerity 233

.n MS

6.04 1 , I I I I I 0 10 20 30 40 50 60 frqU8RCe(kHs) Figure 3. Longitudinal wave damping

3.3 Methodprecision In this section, the quality of the dispersion relation determined in the above section is evaluated. The first incident wave of the measured strain (the first 0.06 ms, see Fig. 4) was extracted from the complete signal. The strain, after N round-trip in the bar, was then reconstructed, by transportation of the first incident wave, using the following equation : First incident wave First incident wave N-I N-I -tit (+L $) (x, -tic (a)((k+ 1 )L+X1 3 (N)(x, = ‘$1 (x , - j k=O k=O Incident wave afer N round-tips Rejlected waw ajer N round-trips This equation is easily derived by developing the fraction in (4). The measured and the rebuilt strains are then synchronised. The ascendant and descendant fronts start at the same time. The tail oscillations also appear as synchronised (Fig. 5). The phase difference between the two signals is very small. The wave celerity is therefore of a high accuracy. Damping is less accurate. The amplitude of the re&nstructed strain is slightly greater than the amplitude of the measured one. However, the results are still satisfactory. The maximum relative error is less than 3.3% after the five wave round- trip in the bar (i.e. after 30 m shifting). This approach is then sufficient for classical SHPB applications where waves are shifted less than a length of the bar. 234

1 I 0 1 2 8 4 5 6 7 temps (s) XlbJ Figure 4. Method accuracy

4 Conclusion

A new method was presented to measure experimentally the longitudinal wave dispersion relation in finite elastic and viscoelastic rods. It is based on the spectral analysis of the measured strain at one point of the bar. We checked the validity of the method on an aluminium bar. The damping coefficient is slightly sensitive to noise, much more than the wave number. However, the method shows a high accuracy when waves are shifted by less than 3Om.

References

Bacon, C. An experimental Method for Considering Dispersion and Attenuation in a Viscoelastic Hopkinson Bar. Exper. Mech 38 (1998) pp. 242-249. Blanc, R. H. DCtermination de l'kquation de comportement des corps viscoClastqiues linkaires par une mkthode &impulsion. Symposium franco- polonais, Problzmes de Rhdologie, Varsovie, IPPT Pan, W. Nowacki Ed., (1971) pp. 65-85. Bussac, M. N., Collet, P., Gary, G. & Othman, R. An optimisation method for separating and rebuilding one-dimensional dispersive waves from multi-point measurements. Application to elastic or viscoelastic bars. J. Mech. Phys. Solids 50 (2002) pp. 321-349. 235

4. Chree, C. The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications. Cambridge Phil. SOC. Trans. 14 (1889) pp. 250-369. 5. Davies, R.M. A critical study of the Hopkinson pressure bar. Philos. Trans. A 240 (1948) pp. 375-457. 6. Follansbee, P. S. & Frantz, C. Wave Propagation in the Split Hopkinson Pressure Bar. J. Engng. Muter. Tech. 105 (1983) pp. 61-66. 7. Gong, J.C. Malvem, L.E. & Jenkins, D. A. Dispersion investigation in the split Hopkinson pressure bar. J. Engng. Muter. Tech. 112 (1990) pp. 309-314. 8. Gorham, D. A. A numerical method for the correction of dispersion in pressure bar signals. J. Phys. E :Sci. Instrum. 16 (1983) pp. 477-179. 9. Gorham, D. A. & Wu, X. J. An empirical method for correcting dispersion in pressure bar measurements of impact stress. Meas. Sci. Technol., 7 (1996) pp. 1227-1233. 10. Hillstrom, L. Mossberg, M. & Lundberg, B. Identification of complex modulus from measured strains on an axially impacted bar using least squares. J. Sound Vibration 230 (2000) pp. 689-707. 11. Lundberg, B. & Blanc, R. H. Determination of mechanical material properties from the two-point response of an impacted linearly viscoelastic rod specimen. .ISound Vibration 126 (1988) pp. 97-108. 12. Pochhammer, L. Uber die Fortpflanzungsgeschwindigkeiten kleiner Schwingung in einem ubegrenzten isotropen Kreiscylinder. J. fir die Reine und Angewande Mathematik 81 (1 876) pp. 324-336. 13. Othman, R., Blanc, R. H., Bussac, M. N., Collet, P. & Gary, G. A spectral method for wave dispersion analysis. Application to an aluminium rod. Proceedings of the 4Ih International Symposium of Impact Engineering, Vol. I, A. Chiba & S. Tanimura (2001) pp. 71-76. 14. Othman, R. R., Blanc, R. H., Bussac, M. N., Collet, P. & Gary, G. Identification de la relation de la dispersion dans les barres. C. R. Acad. Sci. Strie IIb (2002) accepted for publication. 15. Zhao, H. & Gary, G. A three dimensional analytical solution on of the longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar, application to experimental techniques. J. Mech. Phys. Soli& 43 (1995) pp. 1335-1348. APPLICATION OF THE LIKELIHOOD METHOD TO THE ANALYSIS OF WAVES IN ELASTIC AND VISCOELASTIC RODS

R. OTHMAN AND G. GARY Laboratoire de Mbcanique des Solides, Ecole Polytechnique.91128, Palaiseau, France. E-mail :[email protected]?

M.N. BUSSAC AND P. COLLET Centre de Physique Thborique, Ecole Polytechnique, 91 128, Palaiseau. France.

In this paper, we are interested in separating waves in elastic and viscoelastic rods propagating in opposite direction. N strain and P velocity measurements are taken into account. This application of the likelihood method gives a solution in the fiequency domain. Using the inverse Fourier transform, one can recover the strain, stress, displacement and velocity at any section of the rod. In experimental conditions, the results are stable against noise when N+D2 and NP # 0.

1 Introduction

In the classical configuration, the loading time in the SHB (Split Hopkinson Bar) system is limited by the length of the bars together with the maximum measured strain in the specimen, because of the need to separate opposite waves propagating in the bar. Hence, for many materials, it is of no interest to carry out tests with the SHB apparatus at medium strain-rates. As mechanical testing machines are limited at much lower strain rates because of sensor oscillations,-alternative solutions have been already investigated, in particular the wave separation technique. They are based on a two strain measurement and they take account of wave dispersion', 7-8 or not3". Bussac and a1.2 showed that the noise is amplified on the reconstructed signals when using only two measurements. In this paper, a new separation method using N strain and P velocity measurements is presented. It is based on the Maximum of Likelihood prin~iple~~~~~.

2 Theory

Let us consider an L-long elastic or viscoelastic bar. In the case of single mode propagating longitudinal waves, the Fourier transform of stress, strain, displacement and velocity are expressed as follows : ~(x,m)=A(m)e-i"m'x + B(o)ei'(a)x, a(x,m)= E * (m) (A(m)e-t'(m)x+ B(m)eit(m)z),

236 237

w(- A(0)e-"(")" + B(o)e"'"'") qx,0) = , 5 (4 i (A (0 ) -'l<(")x -B(m)e"(")')

C(X,W)' 9 5 (4 where A(@) and B(w) are the Fourier components of the ascendant and descendant waves at origin, respectively. E*(w) is the complex Young's modulus and t(o)= k(w)+ ia(o) is the complex wave number. The two parameters E * (0)and < (0)are only related to the bar properties (geometry and material). In the following, it is assumed that they are known. From strain and/or speed measurements, we want to recover A(o) and B(w) so that strain, stress, displacement and velocity can be calculated at any point of the bar, in particular at both ends. We perform N strain and P velocity measurements on the bar. The corresponding record is modelled as the superposition of the exact measurement and a Gaussian white noise: z,(t)= ~(x,,t)+ W, (t), J = l,.., N , $K(t)=v(xJ,t)+WJ(t), J=N+l,..,N+P . The N+P white noises are supposed to be two-by-two independent. The amplitudes of the noise concerning strain and velocity are denoted 1 I a: and 1 I a; , respectively. In order to estimate the two hnctions A(@) and B(w), the Maximum Likelihood Method is used2. We denote X,, J =1,.., N+ P and K = 1,.., M, the random variable corresponding to the noise stored on the measurement made at the station J at the time t = K/f,,* , where M is the maximum measured points and f,,, is the sampling frequency. The likelihood function is given by :

V(E^J(tX i,(tX A(+(@ >)=P ,;,?N+pcxx=w , (K / Ye* >) iK=LM 1 Since all the noises are white and two-by-two independent, the likelihood fimction is then expressed as follows :

K=I, ... M Noises are also Gussiens, hence : 238

where v, = 11 a, is the standard deviation and pJis the mean of the noise W, . This method consists in writing that what is measured corresponds to the most probable event (a particular application is the least-square method). This leads to maximize the function V(2,(t);, (t)~(w),B(w)) which is equivalent to minimizing the following function:

According to Parseval's theorem:

The function F is minimized when:

where:

A(@)and B(u)are then calculated. Assuming that mechanical values are constant in a bar cross-section, formulas 1 provide forces and displacements at any point, and in particular at bar ends. 239

3 Numerical simulation

In this section we propose to validate the method with a numerical test. The bar simulated is elastic. It is 3m long and 40 mm in diameter. The Young’s modulus is E, = 70GPa, the Poisson’s ratio is v = 0.34, and volumic mass is p = 2800kg/m3 . A I .2m long striker, having the same chracteristic as the bar, is launched at one end of the bar ( x = 0) at a speed V, = 12 m/s. Five strain measurements at sections

x, = 0.5~1, x, = 1.02~1, X, = 1.4~1, x, = 1.78~~1,X, = 2.2m and two velocity measurements at sections X, = 0.8m, X, = 1.4m are simulated. Gaussian noise with amplitude 2 % of the maximum strain or the maximum velocity is added to each measurement. We suppose also that the mean of each noise on strain measurements is not zero and equals 5 %o of the maximum strain. We compare results provided by the method developed in section 2. to exact simulated signals. We denote med the maximum relative error on strain and meu the maximum relative error of reconstructed displacements. Figs. 1 and 2 compare the error on strain and displacement for different values of Net P. The results show that it is sufficient to use three strain and one velocity measurement to have good accuracy on reconstruct strain and displacement. For experimental application we choose this solution (Tab. 1).

-2 0 0.02 0.04 0.06 0.08 rmllpe (61

x TO*

PO %o

-1 -1 -1

-2 -2 0 0.02 OW 006 0.06 -‘ +-wan (.)

Figure 1. Error on strain (a) N=2 and P=O - (b) N=3 and P= 1 - (c) N=3 and P=2 -(d) N=3 and P=O - (e) N=4 and P=O - fi N=5 and P=O 240

a0 B

-0 1

-0 2

-0.3

a0 'd

-0 1

-0 2

-0.3

Pigure2. Error on displacement (a) N=2 and P=O - (b) N=3 and P=l - (c) N=3 and P=2 -(d) N=3 and P=O - (e) N=4 and P=O - If) N=5 and P=O

Table 1. Maximum error on strain and displacement

4 Application to a Nylon bar

The validity of the method is checked using a nylon bar. A nylon striker is launched at the left end of the bar at a speed of 3.03 mh. The right end is free. Three strain and one-velocity measurements are recorded on the bar. We use the method developed in section 2. to reconstruct the stress at two ends of the bar and the displacement at the free end. 24 1

+shsrss at striker side dirtctly neasureddisplacement

0 10 30 0 10 30 the tm) stress displacement

Figure 3. Reconstructed stresses at the ends of the bar and displacement at the free end (The stress at the free end has been shifted down by 0.2 MPa to make the figure more readable).

The stress at the left end was almost zero as expected. At the right end, the stress became almost zero after the first incident wave. Compared to the amplitude of the impact stress, the error on the reconstructed stress was less than 3.5% (Fig 3). The reconstructed displacement was similar to the directly measured one. The relative error was less than 2.5% (Fig 3).

5 Conclusion

A multi-point method (multi-strain and/or multi-velocity measurements) is presented for reconstructing one-dimensional waves in bars. This method is exact when used with the single-mode dispersive propagation model commonly applied to Hopkinson bars. It yields consistent results (the inaccuracy due to imprecise measurements does not increase with time). It is illustrated here by applying it successfully to the analysis of a real test on a Nylon bar. It provides a significant increase in the observation time available when using measuring techniques based on the use of bars such as SHPB set-ups. The method would make it possible to obtain precise measurements at 242 medium strain rates in a test range in between that of mechanical testing machines and that of Hopkinson bars.

References

1. Bacon, C. Separating waves propagating in an elastic Hopkinson pressure bar with three-dimensional effects. Int. J. Impact Engng. 28 (1999) pp. 55-69. 2. Bussac, M. N., Collet, P., Gary, G. and Othman, R. An optimisation method for separating and rebuilding one-dimensional dispersive waves from multi-point measurements. Application to elastic or viscoelastic bars. J. Mech. Phys. Solids 50 (2002) pp. 321-349. 3. Lundberg, B. and Henchoz, A. (1977) Analysis of elastic waves from two-point strain measurement. Exper. Mech. 17 (1977) pp. 213-218. 4. Park, S. W. and Zhou, M. Separation of elastic waves in split Hopkinson bars using one-point strain measurements. Exper. Mech. 39 (1999) pp. 287-294. 5. Othman, R., Bussac, M. N., Collet, P. and Gary, G. SBparation et reconstruction des ondes dans les barres Blastiques et viscoklastiques A partir de mesures redondantes. C. R. Acad. Sci. Skrie ZZb. 329 (2001) pp. 369-376. 6. Othman, R., Bussac, M. N., Collet, P. and Gary, G. Application du maximum de vraisemblance A la skparation des ondes dans les barres. HmeCongr2s Franqais de Mkcanique (2001). 7. Zhao, H. and Gary, G. Une nouvelle mBthode de scparation des ondes pour I'analyse des essais dynamiques. C. R. Acad. Sci. Skrie ZZ 319 (1994) pp. 987- 992. 8. Zhao, H. and Gary, G. A new method for the separation of waves. Application to the SHPB technique for an unlimited measuring duration. J. Mech. Phys. Solids 45 (1997) pp. 1185-1202. ON THE CONTROLLED EVOLUTION OF LEVEL SETS AND LIKE METHODS: THE SHAPE AND CONTRAST RECONSTRUCTION

C. RAMANANJAONA, M.LAMBERT AND D. LESSELIER Dkpartement de Recherche en EIectrornagn6tisme, Laboratoire des Signaux et SystBmes, 3, rue Joliot-Curie, Plateau de Moulon, 91192 Gif-sur-Yvette cedex, fiance E-mail: christophe.ramananjaonaQlss.supelec.fr, marc.larnbertQlss.supelec.fr, dominique.lesselierOlss.supelec. fr

J.-P. ZOLESIO INRIA 2004, route des Lucioles, B.P. 93, 06902 Sophia-Antipolis cedex, fiance E-mad: jean-pauE.zolesioQsophia.inria.fr

The controlled evolution of level sets provides a successful framework to solve time-harmonic inverse scattering problems for objects whose contrast with the environment is known but whose shape is unknown ’. New solution tools extend this framework to the retrieval of objects whose both shape and contrast are unknown.

1. A short introduction to the level set technique The level set representation of domains enables to describe the evolution of fronts l, and is now used in a variety of applications 2, including inverse scattering problems 33415,6. The idea of using level sets to retrieve location and shape of an unknown homogeneous and penetrable object embedded in a portion (some search domain 0)of a known space by the observation of the scattered field on a set of receivers M which is resulting from its interaction with an impinging time-harmonic pressure wave lies in the combination of two techniques: (i) the representation of a bounded domain 52 (singly- or multiply- connected as well) within D by a function @ whose zero-levels correspond to the boundary contours 852; (ii) the speed method, which consists in the construction of a velocity field ensuring a deformation of 52, a properly chosen objective or cost

243 244

functional J being minimized 7.

Like with many inverse problems, the solution is obtained by iterative minimization of J, the latter measuring the discrepancy between the field us scattered by R and data < collected by the sensors at the frequency(ies) w of operation for probing field(s) UI. The scattering problem itself is approached via a (scalar) contrast-source domain integral formulation and the application of a method of moments at the discretization stage 8. The formulation is similar to the one developed earlier in the TM-polarization case in 2-D electromagnetics 315. State (coupling) and observation (data) equations respectively read as

(2)

In the above Green functions GOTand Goowith source point < and observation point 2 are used, index o standing for a point in the search domain D and index r for a point in the measurement domain M;the wave number of the host medium D is ko(w), and the scatterer is of wave number k(w); contrast q reads as q = a - 1 = $ - 1, where co and c are the velocities of the pressure wave insicfe the host medium and the scatterer, respectively, both media being fluid and with unit density. When the contrast q is known, the problem comes down to an optimal shape design problem in some pseudo-time t (in the discrete model, the number of iterations), with the associated objective functional 1 JW = ,Ilus(fl) - <11i2(M). (3) The gradient of J with respect to the variation of shape is handled via a Lagrangian formulation 5, and reads as

dJdt = %e (ki 1,qu(i')p(J)c(i').ii(i')dZ) , (4) where u is the total field in D; p the adjoint field, computed as usual by assuming that the measurement set M radiates an incident field of complex- valued amplitude us (0)- <, the overbar denoting complex-conjugation; ii the normal to the boundary contour dR (assumed regular enough); and $ the speed of evolution of this boundary. The choice of

9 = -Re (~cgqup)ii (5) 245 along d52 provides in theory a strictly negative derivative of J. Upon extending the above speed to the whole of D,the motion of every level of @ -the level 0 is 852- is described by the Hamilton-Jacobi equation I, d@ + F(Z).?T(Z)pP(Z)ll = 0,VZ E D. -(Z)dt (6) Examples of such reconstructions are available in different electromagnetic and acoustic configurations 5,9.

2. Shape and contrast reconstruction The simultaneous retrieval of contrast q and shape 52 is a demanding task. Variations of the cost functional (3) now read as where x is the support of 52. The partial derivation of J with respect to x is dealt with as is usual within the level set method (4). As for the partial derivation of J with respect to 17, it can be performed either in full, which means the variations of the total field vs. q are accounted for in the calculation, or in approximate fashion, which means the total field u is kept constant in the estimation. (Details of the calculations are given next.) Once suitable derivatives in the direction of q and x are available, an alternating procedure can be built: shape contours are updated by the level set method until convergence, then contrasts are updated by an optimization algorithm adapted to non-linear problems (such as the Levenberg- Marquardt method lo) until convergence, and the procedure is repeated from the new updates until a satisfactory solution is obtained. The full calculation of the derivative with respect to the contrast proceeds as follows. The cost functional (3) is cast into a constrained La- grangian: J = min max L, wEH'(n) 4EH'(St)

the saddle-point (u, ii, $, 4)of which is verified to be unique by considering the change of variable, VZ E 52,

$(Z) = -k;qp(Z). (9) 246

Then, application of Cuer-Zol6sio's theorem yields the derivative of J as

Now, in order to use the Levenberg-Marquardt algorithm, the complex- valued contrast is written as v(a,b) = a + ib, and the derivations are performed with respect to the real-valued coefficients a and b. By derivating (8) with respect to a, one obtains

Noticing that VZ~,Z~EC, Re(zlZ2) = Re(21~2),Eq. (11) is rewritten as

Performing the change of variable defined in (9), applying F'ubini's theorem, and using Eq. (2), one gets

and finally, by definition of p,

da 247

The derivation with respect to b can be carried out rather similarly: starting with

using the property Vz1,zz E C, 3m(z1.&) = -3m(%z2), and the same properties as those used for g,one finally gets dJ -(a, b) = -3m ki u(Z)p(Z)d? db [S, 1 The use of such derivatives requires the availability of the total field u and of its adjoint p at each iteration step of the q-optimization procedure, and this might become computationally costly. A less computationally demanding calculation of an approximated derivative is as follows. Derivating the cost functional (3), u being taken as a constant, yields

- 3m(&(Z) S, GoT(Z,gu(J)dg)}dZ. (19) Let us notice that the approximated derivatives are similar to the exact ones in Eq. (11) and Eq. (16), the last terms taken out. For each of those expressions, one could have obtained it directly by calculating $$ 12, which enables to write, for the full derivative, dJ 1 dJ dJ -dq = -2 (-da - ia) = $ s, u(Z)p(Z)dZ, and for the approximated derivative, 248

3. Numerical examples Using proper derivatives (4), (15), (17), (18) and (19), various algorithms can be devised by choosing the order and the accuracy of each optimization procedure g. Requiring a minimal value of the cost functional separately in the search of shape 17 and in the one of contrast x, and alternating one search of each kind is the option taken from now. For simplicity, only the case of lossless media will be considered, which means that only the derivative with respect to a is considered. Let us consider for example the configuration sketched in Table 1.

Table 1. Incident waves 36 line sources, in circle, radius 3 mm Frequencies 500,1000, 1500, 2000 kHz Receivers 64, on a circle of radius 2.5 mm Domain D 2 x 2 mm2, co = 1470 m.s-' Defect 2 rectangles, c = 1800 m.s-l

The scatterer domain is initialized as a centered disc the radius of which is equal to one-fourth of the side of the search domain, the first guess of its acoustic velocity being 1600 m.s-l. Figure 1 displays the evolution of the cost functional J (here, the sum of the costs at all observation frequencies) and of the retrieved velocity c of the scatterer as a function of the number of iterations; the correspondingly retrieved domains are displayed in Figure 2.

One observes that the approximated calculation of the derivative yields a correct solution (the two rectangles) after not too high a number of iterations (with a cost functional reaching its least magnitude at iteration 41), but this result is not stable, since the procedure diverges beyond (at iteration 100 the velocity is unrealistically huge and the two rectangles are reduced almost to point scatterers . . . ). In contrast, with the full calculation of the derivative, the retrieved domains stay almost the same beyond the iteration at which the minimum of the cost functional is reached (at iteration 39), being said that the velocity fluctuates somewhat around the true value. Notice that both methods however provide the same velocity (about c = 1770 m.s-') at the minimum of the cost functionals. 249

0.1

0.01

I 0.001 ' 0 20 40 60 80 100

2800

0 20 40 60 80 100

Figure 1. Top: evolution of the cost functional as a function of the number of iterations; bottom: corresponding evolution of the scatterer velocity c using the approximated calculation of the derivative with respect to the contrast (-) and the full one (- -).

4. Conclusion Results shown so far are still provisional. Complementary experiments are needed, with the help of both approximated and full evaluation of the derivatives of the cost functional with respect to the contrast, for more complicated configurations (such as an object confined within a layer of a stratified space or lossy media), but proper retrieval of both shape and contrast with a level set method is certainly promising.

References 1. S. Osher and J. A. Sethian, J. Comput. Phys., 79, 12-49 (1988). 2. J. A. Sethian, Levelset Methods and Fast Marching Methods, Cambridge Uni- versity Press, 2nd edition (1999). 3. A. Litman, D. Lesselier, and F. Santosa, Inverse Problems, 14,658-706 (1998). 4. 0. Dorn, E. L. Miller, and C. M. Rappaport, Inverse Problems, 16, 1119-1156 (2000). 250

I--- /--- ‘4,; , ‘4. / ’f\

I--- /--- ‘., / ‘4,, / , \ I I I I / - - .. /

Figure 2. Refer to Fig. 1. Black-and-white representations of the retrieved domains using the approximated calculation of the derivative with respect to the contrast (top row, left at iteration 41, right at iteration 100) and the exact one (bottom row, left at iteration 39, right at iteration 100). The initial contour is displayed as a dashed line.

5. C. Ramananjaona, M. Lambert, D. Lesselier, and J.-P. ZolBsio, Inverse Prob- lems, 17, 1087-1111 (2001). 6. H. Feng, D. A. Castafion, and W. C. Karl, IEEE Bans. Image Process., to appear. 7. J. Sokolowski and J.-P. ZolBsio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Verlag, (1992). 8. R. F. Harrington, Field Computation by Moment Method, Macmillan (1968). 9. C. Ramananjaona, Me‘thodes d’ensembles de niveaux pour la re‘solution de problimes inverses des ondes, These de Doctorat, Universith de Versailles- Saint-Quentin-en-Yvelines (2002). 10. A. Franchois and C. Pichot, IEEE Bans. Antennas Propagat., 45, 203-215 (1997). 11. M. Cuer and J.-P. ZolBsio, Systems and Control Letters, 11 151-158 (1988). 12. D. H. Brandwood, Proceedings ofthe IEE-H, 130 11-16 (1983). RECENT PROGRESS IN THE THEORETICAL AND NUMERICAL MODELING OF THIN-LAYER FLOW

L. W. SCHWARTZ Departments of Mechanical Engineering and Math Sciences, University of Delaware, Newark, DE 19716, USA E-mail: schwartzOudel. edu

The motion of liquids is of pervasive importance throughout the natural and industrial worlds. Of particular interest is the motion of liquids in thin layers, with obvious application to paints and other liquid coatings, but also to manufactur- ing operations. The thinness of the liquid layers can be exploited to simplify the mathematical description of these motions and allow inclusion of other important effects, icluding Marangoni forces associated with surfactants and variable composition, non-Newtonian rheology and gravitational and centrifugal body forces. Foci of our work include the construction of detailed process models which may be used, for example, to better understand the causes of observed defects in industrial coating processes and their remediation, and also fundamental studies which may lead to new applications of the model.

1. Introduction Over the past two decades we have had collaborations with a number of industrial groups. Our most important affiliation has been with ICI, especially their Paints Division. With the flow behavior of paints as our central interest, we have also treated related slow-flow problems including the behavior of soap films, foams, and lava. Much of our published work can be downloaded from www. coats'ng.udel.edu where numerical algorithms, not discussed here, can also be found. Virtually all manufactured products require coating, for both decorative and protective reasons. Because the industry is fragmented, it is difficult to assign to it a precise yearly value, but it is surely several hundred billion dollars worldwide. Most commonly, coatings are applied as liquids; these liquids continue to flow until the coating is fully dry. Thus fluid mechanics modeling is potentially of great benefit in helping to improve the flow performance of these thin liquid films. Our mathematical models employ the low-speed, small-surfaceslope approximation, referred to commonly as the lubrication approximation. Here we discuss several of our recent projects.

251 252

2. Thin-layer Flows in the Presence of Surfactant We present the full mathematical problem for slow flow of a viscous liquid with a free surface. The problem is then specialized to consider the presence of surfactant. Slow flow satisfies the Stokes momentum equation

vp = pv2v (la) and the incompressible continuity equation V*V=Q within the liquid region. In addition to the neslip condition V = 0 on solid surfaces (“substrates”) we have stress and kinematic conditions on the free surface. With II, s, and n signifying the stress tensor and unit tangent and normal vectors, respectively, these are

rI-n.n=utc, (1c) and n-(V-k)=O where k is the surface velocity. Here p , p , u,and tc represent pressure, viscosity, surface tension and surface curvature, respectively. The surfactant, assumed to be insoluble in the bulk of the fluid, is conserved; thus

aC - V,-(CU) = DViC at + Here u is particle velocity on the surface, D is a surface diffusion coefficient and 0, = (I - nn)V is the surface gradient operator. c is assumed to be dimensionless without loss of generality. The surface tension changes due to changes in surface concentration c. The simplest assumption is the linear law

0 = go + r(i - c) (2) where uo and I’ are constants. The lubrication approximation is obtained by a systematic expansion in the assumed small parameter hok where ho is the average liquid depth and k is a typical wave number for free-surface undulations. For a twe dimensional problem, for example, the equation for surface concentration becomes

Ct + (u(’)c), = A c,, + O(h~k)~, (3) 253 where u(~)is the surface velocity that is given by 3 u(’) = Zh2hz+, - Rhc,. (4) The evolution equation for the surface profile is

Here subscripts signify partial differentiation. The two parameters that appear in the model are 3r R=- and A = k2DT* . uohik2 ’ The characteristic time is T*= 3p/(aohik4).

Figure 2.1. Calculated shape of a “crater” defect in a paint layer. Rendered graphics of the numerical solution is used here.

The model system (3)-(6)has been applied to the problem of the leveling of surface ripples in the presence of surfactant ’. Since a painted surface requires a very smooth coating, one depends on surface tension to contract the surface to produce the necessary leveling. Since atmospheric dust acts as a surfactant, and surfactants lower surface tension, it is standard wisdom that painting must be done in a very clean environment. While this is true in general, we found the surprising result that there is a significant parameter regime in which leveling can be accelerated by adding, rather than removing, surfactant. This surprising conclusion is validated in three different ways: (i) by linearly perturbing an almost flat surface, (ii) by solving the coupled system (3)-(6)numerically, and (iii) by solving the unapproximated Stokes flow problem numerically. Details may be found in the original paper l. An important industrial application is the identification and remediation of contaminant-induced paint defects. One common defect is the so-called 254 crater. We have demonstrated, by solving the above coupled system in an axisymmetric geometry, that the precise shape of these defects can be reproduced by the model ’. A calculated result is shown in Fig. 2.1.

3. Flow on Partially-wetting Substrates A liquid droplet at rest on a plane solid surface meets the surface at a well-defined angle. This angle, measured within the liquid, is called the equilibrium contact angle 8,. Good wetting refers to situations where 8, < < 1. The term partial wetting is used for the more common case where

8, N O(1). When this is true, often the total system energy can be reduced by spontaneously dewetting portions of the substrate. A common example is water “beading up’’ on a freshly waxed car. The simulation results given here describe these spontaneous flows. Information about the contact angle is included in a contribution to the total pressure called disjoining pressure. The thickness profile of the liquid coating layer on a planar substrate is represented by the function h(z,y, t) where z and y are orthogonal coordinates measured along the substrate. Mass conservation yields ht = -V . Q - E (7) where E(z,y, t)represents evaporation. Applying the no-slip boundary condition on the substrate and assuming zero stress on the liquid-air interface, the flux is h3 Q= --vp 3P The pressure p includes capilllary and disjoining contributions, i.e. p = -aV2h - II . (9) The first term on the right of (9) is the capillary pressure where the curvature of the liquid-vapor interface has been simplified using the small-slope approximation. I3 is the disjoining pressure as introduced in the 1930’s by F’rumkin and Deryaguin 3. We use the two-term model II=B [(a)”-(+)“I . B and the exponents n and m are positive constants with n > m > 1. h, is a very thin “precursor” film thickness that permits contact-line motion. In dynamic simulations h, plays the role of a “slip coefficient” allowing motion of the apparent contact lines where a thick coating layer meets a “dry” portion of the substrate. It may be shown that (n - l)(m - 1) (n- l)(m - 1) B= - case,) M d,Z. (11) h,(n - m) 2h,(n - rn) 255

The final form of the evolution equation for the coating thickness, including capillarity, substrate energetics, and evaporation, is found by combining equations (7) to (11).

Figure 3.1. Dewetting and liquid beading up on surface; experiment (left) and simulation (right) are compared. the bottom row pictures are taken about 3 sec after the top row pictures. The letters refer to certain features seen in both the experiments and the simulations, such as (A) nucleated holes and (B) liquid filaments.

Figure 3.1 shows a comparison of experimental and simulation results showing the patterns that form when liquid beads up on a surface that exhibits a moderately large contact angle. In this work, evaporation and drying are also included; thus a coupled system of equations is actually solved. Details may be found in the published paper '.

4. Spin Coating Spin coating is a technique for producing a thin uniform layer of a liquid on a substrate. It finds wide application in the electronics industry where uniform layers of solidified coating are needed in the fabrication of various devices. We develop an approximate mathematical model for the three- dimensional time-dependent flow of a viscous liquid as it is spun upon a flat substrate. The initial liquid configuration is an almost symmetric sessile drop of liquid that is positioned near the axis of rotation. The drop is then driven outward by centrifugal force. 256

Figure 4.1. A portion of an expanding spin-coating profile showing liquid “fingering.” The simulation result (left) is compared with an experimentally measured profile. Note the uniform central regions and the characteristic “wall and tower” frontal shapes in both pictures.

Within the model, the evolving shapes of spinning drops depend primarily on three independent dimensionless parameters

as well as the level of “noise” in the system. The spin speed parameter S is seen to depend on the density p and surface tension u of the liquid, the rotation rate w, and the initial radius & and volume V of the drop. The Bond number Bo measures the relative importance of gravity and surface tension on a horizontal substrate while 8, is the equilibrium or static contact angle. A dimensionless evolution equation for the surface profile h(z,y, t) is ah - = -V [h3(VV2h - BO Vh+A VlI(h;h,))] at - (h3x)+ -a (h3y)]. aY Here (2, y) = (0,O) is the center of rotation. The parameter A is proportional to 82. Figure 4.1 shows a numerical result and an experimentally- derived profile 5.

5. A Model for Foam Expansion Foams can be found in both the industrial and the natural worlds. Solid polyurethane foams, for example, are used for upholstery, building products, car parts, etc. A so-called “blowing agent” causes the gas fraction to increase dramatically during the foam production process. We have modeled the process in order to ascertain the importance of surfactant and 257 other mixture constituents (see Fig. 5.1). The coupled governing equations include Marangoni and disjoining effects 6t7.

Figure 5.1. Four adjacent bubbles in a periodic two-dimensional foam at early and late times. Thin liquid films separate the almost-hexagonal gas bubbles on the right. These pictures are constructed from reflections and repetitions of a “unit-cell.”

Acknowledgments Portions of this work were supported by The NASA Microgravity Program, the ICI Strategic Research Fund, and the State of Delaware.

References 1. L. W. Schwaxtz, R. A. Cairncross and D. E. Weidner, Physics of Fluids 8, 1693 (1996). 2. P. L. Evans, L. W. Schwartz and R. V. Roy, J. Colloid Interf. Sci. 227, 191 (2000). 3. N. V. Churaev, V. D. and Sobolev, Adu. Colloid Interf. Sci. 61, 1 (1995). 4. L. W. Schwaxtz, R. V. Roy, R. R. Eley and S. Petrash, J. Colloid Interf. Sci. 234, 363 (2001). 5. R. Haze and J. Lammers, Philips Report UR 819/99, (1999). 6. L. W. Schwartz and R. V. Roy, J. Colloid Interf. Sci. 218, 309 (1999). 7. L. W. Schwartz and R. V. Roy, A mathematical model for an expanding foam, J. Colloid Interf. Sci. (submitted), (2002). SEISMIC RESPONSE IN A CITY

C. TSOGKA AND A. WIRGIN Laboratoire de Mkcanique et d %coustique, UPR 7051 du CNRS, 31 chemin Joseph Aiguier, 13402 Marseille cedex 20, France E-mail: tsogka@lma. cnrs-mrsf

Extremely large effects in terms of intensity and duration characterized the recent tremors in cities of Mexico, Japan and Turkey. Studying the responses of complex substratum models, while considering the ground to be flat, is the traditional approach in the analysis of such earthquakes. However, these models ignore buildingkoilbuildmg interactions. Evidence exists that explicitly integrating the buildings into the theoretical models will lead to a better understanding and prediction of the tremor's causes and effects. Such a model is considered in this paper wherein the response of a simplified city with ten buildings, non-equally sized and spaced, located on a substratum with a low velocity layer, is studied. Our results display very strong responses, both inside the buildings and on the ground, which qualitatively match the responses observed in the above-mentioned cities.

1 Introduction

The tremendous amount of energy released in earthquakes is much larger than that in most other mechanical phenomena. This, in itself, should be a good reason to analyze the causes and effects of seismic disturbances. Surprisingly, only a small number of mathematical/numerical studies has been devoted to one of the aspects of earthquakes having the greatest social and economic impact: their effect in cities'. Usually, when such an analysis is made, the motion of each building is considered to be independent of that of all the others and is treated separately from that of the substratum, although, as an elementary examination of the governing equations of the problem shows, the motions of the different components of the city (buildings and substratum) are in fact coupled the motion of each building is coupled to that of the substratum and to that of other buildings via the substratum. Herein, a different approach is adopted: all couplings are taken into account a priori by the fact that the governing equations are solved exactly (to within numerical errors that can be reduced at will). The reason for doing this is that many aspects of seismic response in cities, especially in those cities built on a substratum with soft superficial layers (as is unfortunately the case for many large, earthquake- prone, cities in the world) seem to be anomalous' (unusually long codas with characteristic beating, motion of the ground that is larger than for a site without buildings, unusually-large motion and attendant damage in the buildings, and pronounced spatial variability of response), in that they are not well-predicted by classical models.

258 259

Most of these models assume the ground to be flat. It is implicitly suggested that if buildings were present, their motion would be that of simple mechanical systems dynamically loaded via the motion of the ground, with the corollary that the ground motion is unaffected, or hardly affected, by the presence of the buildings. It has been inferred from this that, if extreme response features are to be obtained, they must result from irregularities of the sub~tratum~-~.When these irregularities are large-scale, they give rise to responses, whose duration and peak amplitude of motion, are smaller than necessary to account for the large destructions in recent earthquakes in cities such as Kobe, Mexico City',234and the Los Angeles area6. Small-scale irregularities in the substratum give rise to better fits to observed response, but are in general unknown at a typical urban When the predictions of the seismic response of a city employs a model that aplicitly includes the city's buildings, usually one or another feature of the city model or of its interaction with the incoming wave is not realistic. For instance, in7 a study is made of the response of a city composed of several (i.e., one to five) rigid- base buildings. Other investigations deal with cities composed of either a periodic' or random/periodic9 sets of buildings. The periodic city gives rise to ground and building response that is closest to the observed response in real-life cities. However, periodic distributions of buildings appear only over parts of any contemporary city. Thus, the periodic model8 is of limited practical use, all the more so, that it requires the seismic disturbance to be delivered to the structure in the form of a plane wave (i.e., the source is very far from the city). A more realistic model of a city submitted to a seismic disturbance should be able to accommodate any type of source and any number of buildings of different sizes and spacings, with or without foundations. Such a model is employed herein, together with a rigorous theory of interaction of the incident wave with the structure. The time-domain numerical method we employ avoids spurious reflections from fictitious boundaries such as those occurring in". Moreover, in contrast to the method used in9, it does not necessitate periodization to limit the computational domain to finite size.

2 Basic ingredients of our approach

Our 2D configuration is depicted in Fig. 1. The buildings, which extend into the substratum, are connected to the latter by an interface on which we impose the continuity of displacement and normal stress. These conditions are implicit when the heterogeneous medium, constituted by the half-space underneath the ground, is considered to be a continuum. 260

Figure 1. Sagittal plane view of the 2D city The thick black curve is the stress-free boundary. The grey regions are (from top to bottom): the buildings with foundations, a surficial soft layer and the bedrock. The dashed black lines are the foundation boundaries.

The governing equations, with or without the buildings, are those of linear elastodynamics for a heterogeneous, isotropic medium with upper stress-free boundary. When buildings are present, the formerly-flat ground becomes the portions of the ground in between the buildings plus the boundaries of the buildings in contact with the air. The city is considered to be invariant in the y-direction with x, y, z being the Cartesian coordinates, and z increasing with depth. The seismic source is a line in the y-direction, radiating a Ricker pulse cylindncal shear-horizontal (SH) displacement field. Thus, only the y-component of this field is non-vanishing and invariant with respect to y. The total field underneath and on the free surface is also SH-polarized and invariant with respect toy. In Fig. 1 we denote by h, w and d, the height, width and space interval between buildings. Contrary to what is assumed in the computations of 9910 we consider that these parameters are different from one building to another. The half-space underneath the irregular stress-free surface is occupied by a linear, isotropic, heterogeneous medium, characterized by mass density p(x) and shear modulus ,u(x), x=(x,z). Both p(x) and p(x) are considered to be positive real, piecewise constant, time-invariant functions. In addition, no intrinsic medium losses are taken into account and the buildings are considered to be homogeneous. The impact of these simplifications on the overall response will be evaluated in the near future. Based on previous studies', it can be anticipated that incorporating realistic values of material losses in the model, will result in a reduction of the intensity and duration of our predicted motion by a factor of about 1.5. In", 2D models are considered to underestimate the intensity of shaking. If this rule is adopted herein, our model should provide a lower estimate of the seismic response of real 3D cities, devoid of attenuation (note that our computational method is applicable to 3D cities). Nevertheless, in order to establish the applicability of our results to real cities, it will be necessary to carry out a statistical analysis, which will evaluate the influence on the response of various distributions of building sizes, aspect ratios and separations, as well as of source 26 1 types and locations. This has been done, to some extent, in a recent paper12, although with a non-rigorous interaction model, and leads to conclusions similar to ours.

3 Methods

The propagation of 2D SH-waves in heterogeneous solids is governed by the same equations as those of 2D pressure acoustic waves in compressible fluids, provided that the excess pressure p of the fluid is associated with the y-component of the displacement u of the solid and the adiabatic bulk modulus K of the fluid is associated with the shear modulus p of the solid. The governing equations in the fluid, in the so-called mixed velocity-pressure form, are:

We search for p, v via (1) in a bounded sub-domain SZ of R2with some initial condition at t = 0 . Considering the mixed velocity-pressure formulation instead of the 2"d order wave equation presents two main advantages. Firstly, it can be coupled with the fictitious domain methodI3 for taking into account the free surface boundary condition. Secondly, it permits us to model wave propagation in infinite domains, the case of interest here, by using the Perfectly Matched absorbing Layer (PML)14. The fictitious domain method consists in extending the wave propagation problem in a domain with simple geometry (typically a rectangle in 2D), which enables the use of regular meshes. The free surface boundary condition is then enforced with the introduction of a Lagrange multiplier. This new unknown lives only on the free surface and can be discretized with a non-uniform mesh, different in general from the mesh in the rest of the computation domain . For the space discretization we use a finite element method, whereas for the time discretization we employ a centered second order finite difference scheme. The finite elements are compatible with mass-lumping, which leads to explicit time dlscretization schemes. For the velocity, we use a new finite element method15 and for the pressure we use P' discontinuous functions (this is a different choice from the one inI5).The Lagrange multiplier is discretized with P' continuous function. The incident wave is created by a line source located at x8 = (Om,3000m) emitting a pulse with characteristic frequency 0.25 Hz. The densities in the bedrock, soft layer, and buildings+foundations were: 2000 Kg/m3, 1300 Kg/m3 and 325 Kg/m3 respectively, whereas the bulk shear wave velocities in these three media were 600 m/s, 60 m/s and 100 m/s, respectively. The foundation depth of the buildings was 262

10m and the soft layer thickness 50m. The building widths, heights and separations ranged over 30-60m, 50-70m and 60-1 OOm respectively. These parameters are based ons, and are fairly representative of typical buildings and the substratum at sites such as Mexico City. The computational domain was a 3500mx3500m square discretized by a gnd of 351 nodes in each dimension. This domain was surrounded by a PML layer 30 nodes thick, and 465 nodes where placed on the free surface. To give a measure of the probability of destruction of the n-th building we introduce a so-called vulnerability index R, . Let T be the time interval of significant shaking (in the computations this was 240 sec). We then define Rn as the ratio of the integral over T of the modulus squared particle velocity at the center of the summit of the n-th building and the integral over T of the same quantity measured on the ground in absence of all buildings. The subsurface configuration and excitation are the same with or without the buildings in this computation.

4 Results

It can be observed in the left and right columns of Fig. 2 that the duration of the shaking inside the buildings is much longer than the ground motion in the no- building configuration (termed 1D configuration here after). This is also true for the duration of the shaking on the ground between successive buildings (see middle columns of Fig. 2). This behavior on the ground and in the buildings is in agreement with what was found ins312,and of the same nature as what was observed in various sites in Mexico City. From Fig.2 we can also infer the following: i) strong buildinghoivbuilding interaction results in very large duration of shaking (-3min) even for a short input pulse, ii) a beating phenomenon occurs, similar to the one observed in time records of Mexico City, iii) the peak amplitude of building and ground response is larger in the configuration with buildings than the peak response on the ground in the 1D configuration, iv) the response at the top of the buildings varies significantly from one building to another, corresponding to vulnerability indices ranging from -5 to -50 for the ten-building set which suggests that the probability of destruction of some of the buildings of this set is large, v) the response on the ground between the buildings is generally less than the response inside the buildings and vi) the various Rq are quite different regarding intensity and duration, which indicates considerable spatial variability of ground response. All of these features are in qualitative agreement with what was observed during earthquakes in cities such as Kobe and Mexico City’”2. 263

IS! I I

Figure 2. Time records of response of a 'city' with ten buildings. Each row of the figure depicts the particle velocity (in dsec): at the center of the top of the j-th building (left), the center of the ground segment between thej-th and (j+l)-th building (middle) and the (j+l)-th building (right). Here j ranges &om 6 to 9 (top to bottom). The solid curves in all the subfigures represent the particle velocity at ground level in the absence of buildings. The vulnerability indices Rj at the top of the j-th building and Rq on the ground between the i-th andj-th buildings, are indicated at the top of each subfigure. The abscissas designate time, and range fiom 0 to 250 sec. The scales of the ordinates vary fiom one subfigure to another. 264

References

1. Singh S.K., Mori A., Mena E., Kriiger F. and Kind R., Evidence for anomalous body-wave radiation between 0.3 and 0.7 Hz from the 1985 September 19 Michoacan, Mexico earthquake, Geophys.J.Znt. 101 (1990) pp. 37-48. 2. Chavez-Garcia F.J. and Bard P.-Y., Site effects in Mexico City eight years after the September 1985 Michoacan earthquakes, Soil Dyn.Earthqu. Engrg. 13 (1994) pp. 229-247. 3. Campillo M., Sanchez-Sesma F.J. and Aki K., Influence of small lateral variations of a soft surficial layer on seismic ground motion, Soil Dyn. Earthqu. Engrg. 9 (1 990) pp. 284-287. 4. Bard P.-Y., Eeri M., Campillo M., Chavez-Garcia F.J. and Sanchez-Sesma F.J., The Mexico earthquake of September 19, 1985-a theoretical investigation of large-and small-scale amplification effects in the Mexico City valley, EarthquaSpectra 4 (1988) pp. 609-633. 5. Bard P.-Y, and Bouchon M., The two-dimensional resonance of sediment-filled valleys, Bull.Seism.Soc.Am. 75 (1985) pp. 519-541. 6. Olsen K.B., Site Amplification in the Los Angeles basin from three-dimensional modeling of ground motion, Bull.Seism.Soc.Am. 90 (2000) pp. 77 - 94. 7. Wong H.L. and Trifunac M.D., Two-dimensional, antiplane, building-soil- building interaction for two or more buildings and for incident plane SH waves, Bull.Seism.Soc.Am. 65 (1975) pp. 1863-1885. 8. Wirgin A. and Bard P.-Y., Effects of buildings on the duration and amplitude of ground motion in Mexico City, Bull.Seism.Soc.Am. 86 (1996) pp. 914-920. 9. Clouteau D. and Aubry D., Modifications of the ground motion in dense urban areas. J.Comput.Acoust. 9 (2001) pp. 1659-1675. 10. Hill N.R. and Levander A.R., Resonances of low-velocity layers with lateral variations, Bull.Seism.Soc.Am.,74 (1984) pp. 521-537. 11. Wolf J.P., Vibration Analysis Using Simple Physical Models (Prentice-Hall, Englewood Cliffs, 1994). 12. Gueguen P., Bard P.-Y. and Chavez-Garcia F. J., Site-city seismic interaction in Mexico City like environments : an analytic study, Bull.Seism.Soc.Am., 92 (2002) pp. 794-804 13. Btcache E., Joly P. and Tsogka C., Application of the fictitious domain method to 2D linear elastodynamic problems, J.Comput.Acoust. 9 (2001) pp.1175-1202 14. Collino F. and Tsogka C., Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophys. 66 (2001) pp. 294-307. 15. BCcache E., Joly P. and Tsogka C., An analysis of new mixed finite elements for the approximation of wave propagation problems, SIAM J Numer.Ana1. 37 (20001 00.1053-1084. Acoustics, Mechanics, and the Related Topics of Mathematical Analysis Editor: A. Wigin

Errata to Seismic Response In A City C. Tsogka and A. Wigin (pp. 258-264)

There are some misprints.

On p. 261, (a) third line of Section 3:

Replace “...the excess pressure p of the fluid is associated with the y-component of the displacement u of the solid and the adiabatic bulk modulus IC of the fluid is associated with the shear modulus ,u of the solid.” by

“...the excess pressure p of the fluid is associated with the y-component of the displacement u of the solid, IC-I with the density of the solid, and p-’ with the shear modulus of the solid.”.

(b) fourth line from bottom:

Replace “. ..pulse with characteristic frequency 0.25 Hz .. .” by “. ..pulse with characteristic frequency 0.5 Hz .. .”.

On p. 262, seventh line from bottom:

Replace “...vulnerability indices ranging from -5 to -50.. .” by

“. .. vulnerability indices ranging from -2 to -7.. .”.

On p. 263, Figure 2:

Replace the vulnerability indices by:

R5 = 5.0727, Ry, = 5.1444, Rh = 6.7008, Rfi7 = 4.0369, R7 = 4.4470, R7X = 2.2293, Rx = 3.0211, Rxu= 2.7362 and Ru = 2.0196. TRANSMISSION OF ULTRASONIC WAVES IN CANCELLOUS BONE AND EVALUATION OF OSTEOPOROSIS*

YONGZHI XU Department of Mathematics University of Tennessee at Chattanooga Chatanooga, TN 37403, USA E-mail: [email protected] c. edu

This is a summary of the talk given in AMRTMA. Cancellous bone is known to be poroelastic in structure. Ultrasonic wave propagation in cancellous bone can be described by Biot’s equations. In this paper we present some results of our ongoing research on the transmission of ultrasonic waves in cancellous bones and its application to evaluation of osteoporosis.

1. Introduction Ultrasonic techniques for the non-invasive detection of osteoporosis have received considerable attention. The clinical assessment of bones by ultrasound is based on measurement of the attenuation and speed of sound. So far, this technique is basically empirical. Its accuracy is questionable due to its inability to measure directly bone properties. To alleviate these problems, a propagation model for ultrasound in bone is necessary, together with the development of mathematical tools and the implementation of numerical simulations. In this paper we summarize some results of our ongoing research on the reflection and transmission of ultrasonic waves in cancellous bone. More details may be found in and ’. Bone tissues can be classified into two types. Bones with a low volume fraction of solid (less than 70%) are called cancellous bones. Bones with above 70% solid are called cortical bones. Cancellous bone is a two- component material consisting of a calcified bone matrix with fatty marrow in the pores. Hence, the mathematical models of poroelastic media are

*Research supported in part by NSF grant BES9820813 and grants from UC Foundation and CECA of University of Tennessee at Chattanooga. aThis paper contains joint work with J. Buchanan, R. Gilbert, W. Lin and A. Wirgin.

265 266 applicable43 I ’3 9. Biot developed a general theory for the propagation of acoustic waves in fluid-saturated porous media1>2s3.Recently, Hosokawa and Otani7, Mc Kelvie and Palmer’ and Williamsg discussed the application of Biot’s model for a poroelastic medium to cancellous bones. Following them in this paper we consider a slab of cancellous bone submerged in fluid. An ultrasonic wave is radiated from a transducer on one side and received by a hydrophone on the other side. We apply Biot’s system of compressional wave equations for a dispersive dissipative fluid-saturated porous medium in the time domain to this model.

2. One dimensional Biot model of wave transmission in cancellous bone The Biot model treats a poroelastic medium as an elastic frame with inter- stitial pore fluid. We consider first a one-dimensional dynamic model. The motion of the frame and fluid within the bone axe tracked by displacements u and U respectively. In the bone slab (0 < x < L), the ultrasonic wave satisfies the one- dimensional dynamic equations:

dU e=- c=- dX’ ax . We assume that dl parameters in (1) are constants. Equation (1) is equivalent to (for 0 < x < L)

There are seven parameters in the one dimensional Biot model. The complex frame shear modulus p is not included. The other parameters are calculated from the measured or estimated values. p11 and p22 are density parameters for the solid and fluid, p12 is a density coupling parameter, and b is a dissipation parameter which depends on the wave frequency. In the water, (x < 0 or x > L) let po be the acoustic pressure and Uo the displacement. If the transducer is located at x = x, < 0 with waveform 267 f(t), then

(3)

-dP0 = Po- a2uo dX dt2 . (4) At the interface (20 = 0 or xo = L), the specific flux in bone &(,!?U(x,t) + (1 - p)u(x,t)) equals the corresponding quantity in the fluid where ,!? = 1, &Uo(x,t),and the normal stress and the pore stress in the bone are equal to the acoustic pressure in the fluid. Therefore, the displacements, pressure and stresses satisfy: d d a --uo(xo)dt = P,,U(Z,+) + (1 - 8)$4x,+), (5)

PO(X,) = a&,+> + 4x,'>, (6)

PO(X,) = a(.$)/P. (7) Here x; = 0- or L+, and x: = Of or L-. From the transient property of the wave it follows that limlzl+ooUo = 0.

2.1. Reflection and transmission of waves

Let PO, 60,&, and Z be the Fourier transform of po, UO,e and E respectively. Then from (3) and (4), in the water (x < 0 or x > L)

2 A 2d2P0 ,. -w po - Co- = fb(X - xs), 622

--- -pow260. dX In the bone, (0 < x < L),from (2)

d2 -[XG622 + QU] = -w2(p11G + ~126)+ i~b(G- 6) (10) d2 -[QG + RU] = -w2(p12G + ~220)- id(G- . dX2 0) On the interface, (xo = 0 or xo = L), from (5)-(7), and recalling that oxx= Xe + QE,e = g,E = g,we have as0 - Uo(X,) = --(xo ) = p6(x,+)+ (1 - p)qxo+), dX (11) dG 86 lio(x0) = + Q)G(x$) + (Q + R)K(X$), (12) 268

Here xi = 0- or L+, and x$ = O+ or L-, respectively. Solving the system of equations, we obtain the reflected and transmitted waves as follows: reflected wave:

transmitted wave: t>= c4(x,w)e-iwx/c0eiwtb loo[, ,x>L. (15) where c1 and c4 are determined explicitly by the coefficients of the Biot model. For details, see ’.

2.2. Determination of porosity from transmitted waves We consider the recovery of the porosity p by measuring the transmitted field arising from a point source placed in a tank of water containing a specimen of bone. As suggested in 4, the following parameters a,re least certain: porosity p, permeability k, pore size a, structure factor a, the bulk and shear frame moduli Kb and p. In the one- dimensional case, p is not included. In this paper we consider using only the transmitted wave to determine the porosity p. For a point source incident wave f(t) radiated from a source at x = x, < 0, the transmitted wave at x > L is

J -, If the transmitted wave received at x = x, > L is p* (x,, t,p), then

Since the data is measured at x, > L, we use the interface condition at x=Lto determine p. If we assume that A, R,Q, p11 ,p12, p22, and b are approximated by polynomials or rational functions of p, then 6,U, g, are composites of exponential functions and rational functions of /3 (ref. ”. Define 269

66 a6 Fz(P) = (A + Q)z(L)+ (Q + R)z(L)- ~4e~~~’~~,(19) and

If we allow P to be a complex variable in a region containing the set

then Fl,F2, and F3 are analytic functions of ,8 with at most a finite number of isolated singularities. If Po > 0 is a zero of Fi(P) (i = 1,2,3), then it must be an isolated zero. Hence, we have Proposition 1: (1) For a specimen of bone with porosity PO > 0, if po(z,t;Po)is the corresponding transmitted wave defined by (16), and 6,~, Ox,U, are the corresponding quantities, then

(2) (Local uniqueness) There exists a constant ri > 0, if IP - Pol < ri and Fi(P)= 0 for some i = 1,2 or 3, then P= PO. (3) (Local stability) There exists constants 6i > 0, Ni > 0 and integer mi 2 1, (i = 1,2 or 3,) such that

Based on Proposition 1, if we have a good initial guess of PO,the local stability will ensure that we can recover PO by numerically solving one of the equations

Fi(P)= 0, for i = 1,2 or 3.

Another possibility is to determine PO by minimizing a cost function, such as

or, alternatively, the cost function 270

3. Two dimensional Biot model for cylindrical symmetric ultrasonic field in cancellous bone In general, the dynamic equations are given by1121314 a2 pV2u + V[(A + p)e + QE] = ~(~11~+mu)+ b&(u - U) (22) V[Qe+ RE]= g(p12u+ p22U) - b&(u - U). For a focused ultrasonic wave incident from one side of the bone, the propagating field is symmetric about the axis along the incident wave direction. In this case, it is appropriate to work in cylindrical coordinates and suppress the dependence upon the angular variable whence the displacement vectors are now denoted as u(r,z) = (u,(r, z), uz(r,z)), U(r,z) = (Ur(r,z), U,(r, z)). Assume the z direction is the incident wave direction. Then in the bone, (T > 0,O < z < L), the relevant constitutive equations and strain-displacement relations are

and

respectively. The dynamicd equations (22) are reduced to a two- dimensional system, with the angularly-independent Laplacian and the dilatations: 1 v2:= -8, 822, + r + 1 e = V u = (ar -)u, &u,, - + r + 1 E = V * u = (ar -)Ur azu,. + r + In the fluid, (T > 0,z < 0 or T > 0,z > L) let the density po be constant. The differential equations for the acoustic pressure Po(r,z) and displacement in the z direction, Uo,(r, z), are then given by

woa2 P - ~v’P, = f(r, t)~(z - zo) (26)

azpo =Po%. (27) On the interface between the bone and the fluid (r > 0,zo = 0 or r > 0, zo = L), the aggregate normal stress IS,^ + IS and the pore fluid 27 1 pressure ts/P both equal to acoustic pressure in the fluid Po

PO(T, z;) = Ozz(T, z,') + O(T, .if) (28) PO(T, 2;) = O(T, z,')/P. (29) The specific flux in bone &(~U,(T,z,t) + (1 - ,8)uz(~,z,t)) equals the corresponding quantity in the fluid where ,8 = 1, &Uoz (T, z, t). That is,

Finally, the tangential stress uTzvanishes at the bone-fluid interface

(T, 2,') = 0. (31) In the above equations, z0 = 0- or L+, and z$ = O+ or L-, respectively. At r = 0, it follows from continuity that = 0, = 0, and = 0. Using this model we construct the transmitted waves and conduct numerical simulations of evaluation of osteoporosis. The details are presented in 6.

References 1. Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range, J.Acoust.Soc.Arn. 28, 168-178 (1956). 2. Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Higher-frequency range, J.Acoust.Soc.Am. 28, 179-191 (1956). 3. Biot, M. A., General theory of acoustic propagation in porous dissipative media, J.Acoust.Soc.Am. 34, 1254-1264 (1962). 4. Buchanan, J.L., R. P. Gilbert and K. Khashanah, Determination of the parameters of cancellous bone using low frequency acoustic measurements, preprint , (2001). 5. Buchanan, J.L., R. P. Gilbert, A. Wirgin and Y. Xu, Transient reflection and transmission of ultrasonic wave in cancellous bone, to appear in Mathematical and Computer Modelling, (2002). 6. Gilbert, R. P., W. Lin and Y. Xu, Focused ultrasonic waves in submerged cancellous bone, preprint, (2002). 7. Hosokawa, A. and T. Otani, Ultrasonic wave propagation in bovine cancellous bone, J.Awust.Soc.Am. 101, 558-562 (1997). 8. McKelvie, T.J. and S.B.Palmer, The interaction of ultrasound with cancellous bone, Phys.Med.Bio1. 36, 1331-1340 (1991). 9. Williams, J.L. Ultrasonic wave propagation in cancellous and cortical bone: prediction of some experimental results by Biot's theory, J.Acoust.Soc.Am. 91, 1106-1112 (1992). HADAMARD SINGULAR INTEGRAL EQUATIONS FOR THE STOKES PROBLEM AND HERMITE WAVELETS

LIANGSHUN ZHU AND WE1 LIN Department of Mathematics, Zhongshan University Guangzhou, 510275, P. R. China E-mail: stslw@zsu. edu.cn

In this paper, we apply the wavelet-Galerkin method to solve the natural boundary integral equations of the Stokes equation in the interior circular domain. The simple computational formulae of entries in the stiffness matrix are obtained and only 2J f3J + 7 elements need to be computed for a 2J+3 x 2J+3 stiffness matrix. Error estimates for the approximate solutions are established and two numerical examples are provided.

1. Introduction Many boundary value problems (BVP) connected with partial differential equations (PDE) can be reduced to boundary integral equations (BIE). The main advantage of this method is that the dimensionality of the problem is reduced by one. The Cauchy singular integral equation is a kind of BIE. In 1956, J. Hadamard introduced the concept of the finite part for the divergent integral with high order singularity and discussed the corresponding strongly-singular integral equations which are the generalization of the Cauchy singular integral equation and has many special advantages in the research on BVP. Only few researchers worked on this subject because of the difficulty of the strong singularity. The method of natural boundary elements (NBE) applied in this paper was first introduced by Kang Feng. The main idea of NBE is that the BVP of PDE can be converted, via Green’s functions and Green’s formula, into equivalent strongly- singular BIE and the corresponding equivalent variational problem can then be solved by using some discrete techniques. The natural boundary integral equation is determined uniquely from the original BVP regardless of the choice of conversion technique: Green’s function, Fourier series method or complex analysis etc. The method of NBE can keep the energy functional unchanged and the many useful properties of the PDE, such as the symmetry and coerciveness of the bilinear form etc., are preserved, so that

272 273 the unique existence and stability of the solution of the natural integral equation (NIE) are obtained in straightforward manner. From the point of view of numerical computation, the NBE method has many advantages, such as the symmetry and positive definiteness of the stiffness matrix, the stability of the approximate solutions and high accuracy of the solution in unbounded or crack domains. In a disk, the corresponding stiffness matrices of the NIE are circulant so that we only need to calculate about one half of the row entries of the stiffness matrices. The NBE method also has obvious limitations since it is difficult to find the Green’s function for the general domain, similar to what occurs with other methods of obtaining NIE. In recent years, wavelet methods have been applied extensively to the numerical anaIysis of PDE and integral In this paper, we use the Hermite wavelets introduced by Quak to study the NIE of the second boundary problem for Stokes’ equations in interior circular domains (the problem in the exterior domain of unit disk was investigated in 3). In general, the singularity of NIE is of high order and the integ-rals are of the Hadamard finite part type. We reduce this kind of integral to a lower order singular integral by expanding the strongly-singular kernel into an infinite series in the sense of generalized functions.

2. Hermite interpolant wavelet We first briefly introduce Quak’s work on interpolatory Hermite-type scaling functions4. For all n E N,the Dirichlet kernel Dn(8) and the conjugate Dirichlet kernel Bn(8)are defined as

It is obvious that Dn(8),dn(8)E T,, with T, denoting the linear space of trigonometric polynomials of degree not exceeding n. We define the equally- spaced nodes on the interval [0,2n] as ej,n = 5,jE No,n = 0,1,2,.--,2j+1- 1. Definition 2.1 (scaling functions) For j E No, the scaling junctions cpg,o(8), ~pj,~(8)are defined as: 274

For n = 1,2,..-,2j+1- 1 , define ~$,,(9)= cpi,o(8- ej,,),i = 0,l. fir- themore, let & = cpf,, mod 2j+, (9),i = 0,1, and n E N.

Theorem 2.1. For j E No, and k,n = 0,1,. . ,2j+l - 1, it holds that

'pjo,n(dj,k)= bkn, (Pj"ln(ej,k) = 0, (3) cp;,,(ej,k) = 0, (P;:n(ej,k) = gkn. (4) For j E NO,we define the space of scaling functions as

1 vj = span{p!jn((8), cpj,,(e)[n = 0~1,.. . ,2j+l - 11 then we have dim vj = 2j+' and V, = span{l,~os8,~~~,cos(2~+~- i)e,sine,...,~i~2j+~e) which implies that V, c V,+l(j 20)and if let V-l = (0) ,then L2[0,2.rr]= cx) 00 n V, = (0). Hence {V,}g-,forms a multiresolution analysis j=-1u V,, j=-1 (MRA) of Hermite type in L2[0,2~1. Definition 2.2 (wavelet functions) For j E NO,the wavelet functions +;,o(6), +j,o(6) are defined as: 275

3. Wavelet -Galerkin met hods Consider the second boundary value problem in a plane domain R with smooth boundary r: -qAii+ gradq = 0, in R

divii = 0, in 0 (6) {+t = g, on where the unknown function vector ii = (u1,u2) and unknown function q are flow velocity and pressure respectively, 3 is a given function on the boundary, q > 0 is the dynamic viscosity coefficient of the flow and 1 aui Eij(G) = - + , i, j = 1, 2 2 (-axj 2) (7)

I ti = &(G, q)nj, i = 1, 2. (9) j=1 with n' = (nl,n2) the outer normal unit vector on I?. Yu5 expressed the NIE of the second BVP of Stokes' equations in the unit disk as :

which is equivalent to the following variational problem5:

find Go E H4(l?)2 s.t. (11) { fi(~~,~~)= iy~~),v8~ E d(q2 where a(iio,v;) = s,'" 80 . tciiOdt9, $'(GO) = s,""ij- God0 and the natural operator td0is defined as (10). Let V(r)= {80 E VJ x VJI s,'" 80 . n'ds = 0). We consider the approximate variational problem:

find iii E V(r),i.e.,G$ E VJ x VJ and s,""di .n'ds = 0, s.t. * (12) b(c&go) = P(c0),vgo E v(r) 276

Let %(I?) = {(GI -C3 sine, C2 +C3 ~osB)}lc~,~~,c,~~;then from we know that the variational problem (11)has a unique solution in V(I')/!J?(I') under the compatibility conditions:

s,'" gi(e)de = 0, i = i,2, ( s,"" [g2(8) cos 8 - g1 (8) sin Old8 = 0. From the base BJ of VJ we get a base of VJ x VJ

i=l ,j=J-l u {(~~,,(8),0),(O,$j,,(8))10I k I 2j+l - 1) i=o,j=o i= 1 u u{('p;,k(w), (o?cp;,kw)P = 0,1). i=O From the properties of Hermite interpolatory wavelets presented above, we have 11 1 1 cp&(B) = - + -(-l), cos8, PA,,(@) = -(-l), sin8 + - sin28. 22 2 4 (13) It is clear that (~f,,(e),O),(O,~f,,(8))E V(I'),i = 0,1,0 5 j I J - 1,0 I k I 2j+l-L and (0, cp:,,,, (PA,,,0) E v(r),k = 0,1 but (cp:,,, 01, (0, cpQ 4 V(r)so we cannot choose all basis in VJ x VJ for 50. From (13) we find that (cp:,,, -+(-l),sine), (-+(-l),cos8,cp;,,) E V(I'). If we substitute these vectors for (cp&,O) and (O,cpA,,) that belong to the base BJ but don't belong to V(I'), we find that

i=l,j=J-1 vm= wan{ u {(@j,k(@), O), (0, $,k(e>>lo I k I 2j+l - 1) i=o,j=o

U{(O,cp:,k)r (PA,,, 0)lk = 0, 1) 1 1 u{(cp:;,,-p)k sine), (-pYCOS~,cp~,k)lk = 071)). For J E NO,let the projection of fl; = (uf,ui) in VJ(l?)2 be

J-1 Ij &(e) = C C[C&@,(~)+~j,$;,(e)],i = 1,2 (14) j=-1 ,=o where Ij = - j = -1 and $fl,,(8) = cp;,,(O),p,k = 0,l. Sub- where 278 then we have

(ii)OO (ii)ll (ii)OO (ii)ll (ii)OO, A(/)ll (ii)OO Aii = diW(AJ-1 7 AJ-1 7 AJ-2 7 AJ-2 7.. . ,7 A0 A_, ,A(i";)"}.

Theorem 3.2. A12 = (12)pq and A21 = (21)PP are sparse matrices with only the following nonzero elements

Furthermore A12 and A21 are symmetric with respect to the whole stiffness matrix.

The Stokes second boundary interior problem satisfies the compatibility condition: 2T 1 $(O) . iidO = 0 so that using the projection of $(O) on VJ x VJ,we get

From the above results for the stiffness matrix, we find that the application of the properties of the Hermite trigonometric wavelets properly results in the extremely simple computation of the elements of the stiffness matrix. In addition, in solving the numerical solution, we can improve the accuracy and convergence by using fully the symmetry, circulant property and sparsity of the matrix elements.

4. Error estimates and examples For our fast algorithm we have the following error estimates.

Theorem 4.1. Ift&(O) E and di(O) are the solutions of the variational problem (11) and the approximate variational problem (12) respectively, then

/I'&J - '$llh 5 c * 2-(k+ti)J.&J (cf+"), where 11. llfi is the energy norm in the quotient space H(l?)/?R(I')which is derived from the bilinear fi(c7, d) ,z.e. llc7llfi = [b(a,a)];, &J (uo+(k+l) ) = 2 [CE~J (u1"+")]+ and c does not depend on J. i= 1 279

Theorem 4.2 (estimate of the L2 norm) If Go E C"'(I')' and JtT(Go -Gi) .8d8 = O,WE R(r), then where constant C does not depend on J. We now present numerical results for the Stokes problem in a disk with q = 1 . We take u'(1,O) = (O,l),u'(l,$) = (1,O) to make the solution unique. Example 1 Consider (6) in the unit disk for 9' = (2sin8,2cos8), the exact solution of which is G(r,8) = (T sin 8, r cos O), q(r,8) = 0 . If we select J = -1 , then by our method, we find -1 1 u1 (i,e) = $Jllo(e)-&l(o) = sin8,u; (i,e) = $tlo(e)-$f!ll(e) = COSO, which is the exact solution. Example 2 We address (6)in the unit disk for 3 = (4 cos 28,4 sin 28),the exact solution of which is u'(r,8) = (r2cos 28, r2sin 28), q(r,8) = -8r cos 8 If J = 0, we find uy(1,8) = &o(~)+& (8) = cos 28, u:(1, 8) = 2qtlo(8)+2$!,, (8) = sin 28, which is the exact solution of the problem.

Acknowledgments This work was partially supported by NSF of China and NSF of Guangdong.

References 1. R.P.Gi1bert and W. Lin, J.Cornput.Acoust. l(1) (1993). 2. J. C. Xu and W. C. Shann, Numer.Math.Ana1. 24 (1993). 3. Wensheng Chen and Wei Lin, in Proc. of The International Conference on Wavelet Analysas and its Applications, AMS /IP Studies in Advanced Math- ematics, 25, International Press (2002). 4. E.Quak, Math.Comput. 65 (1996). 5. Dehao Yu, Mathematical Theory of Natural Boundary Element Methods, Sci- ence Press (in Chinese), Beijing (1993). 6. Wensheng Chen and Wei Lin, Applied Math. and Cornput. 121 (2001). 281

List of Communications given at AMRTMA

No. Author (s) Title e-mail of presenting author 1 Begehr H. Orthogonal decompositions of LZ beaehr® math, fitbfirlin.de -> Ben Belgacem Methode d' iterations de sous domaines nabil.gmati @ ipein.rnu-fn F., Gmati N. pour un probleme de propagation dans un and Zrelli N. guide d'ondes 3 Bonafede S. Quasilinear degenerate parabolic [email protected], if and Nicolosi F. equations in unbounded domains '• 4 Bonnet-Ben Resonances d'une plaque elastique dans [email protected] Dhia A.S. and un conduit en presence d'ecoulement Mercier J.-F. 5 Borcea L., Target identification in noisy [email protected] Tsogka C., environments Papanicolaou G. and Berryman J. 5 Bourgeat A., Asymptotic modelling of an underground [email protected] Gipouloux O. waste disposal and Boursier I. 5 Buchanan J.L., Determination of the parameters of [email protected] Gilbert R.P. cancellous bone using low frequency and Khashanah acoustic measurements K. 5 Caputo J.G. Modelling of bore propagation in Seine caputo® insa-roucn.fr and Stepanyants 5 Cardoulis L., An inverse spectral problem for a [email protected] Cristofol M. Schrodinger operator with an unbounded and Gaitan P. potential 10 Carl S. Trapping regions for discontinuously scarl@fi(,edu coupled dynamic systems FT Carroll R. Quantum calculus rcarroll @ math.uiuc.edu 12 Chandezon J., Reconstruction problem for a periodic [email protected] Poyedinchuk boundary between two media bpclermont.fr A.Ye., Yashina N.P. 13 Chang D.C., Weighted holomorphic functions and [email protected] Gilbert R.P. Cesaro operators and Wang G. IT Cianci P. Basic lemma of Moser's method in cianci@ rjrni.unict.it _. anisotropic case 5 Cocou M. and Approximation of a dynamic contact [email protected] Scarella G. problem for a cracked viscoelastic body 282

16 Cristini P. and Le concept d'Imagerie Signal Compatible [email protected] De Bazelaire et son application en geophysique E. 17 D'Asero S. Regularity up to the boundary for a class [email protected] of solutions of a functional-differential system 18 Delanghe R. Hardy spaces of harmonic and monogenic [email protected] functions 19 Engl H.W. Iterative regularization of nonlinear [email protected] inverse problems 20 Fellah Z.E.A., Time domain wave equations for lossy [email protected] Lauriks W. and media obeying a frequency power law: Depollier C. application to the porous materials 21 Flandi L. and A model for porous ductile viscoplastic leblond® lmm.jussieu.fr Leblond J.-B. solids including void shape effects 22 Gilbert R.P. Acoustic wave propagation in a composite miou@ ima.umn.edu and Ou M.-J. of two different poro-elastic materials Y. with a very rough periodic interface — a homogenisation approach 23 Gilbert R.P. Homogenization of the acoustic properties [email protected] and Panchenko of the seabed A. 24 Hackl K. On the calculation of material hackl @ am.bi.ruhr-uni-bochum.de microstructures using relaxed energies 25 HiJe G. Heat polynomial analogs [email protected] 26 Jeffrey A. Blow-up of hyperbolic solutions, shock [email protected] formation, and acceleration waves in layered hyperelastic media 27 Kovalevsky A. Summability of solutions of Dirichlet [email protected] and Nicolosi F. problem for some degenerate nonlinear high-order equations with right-hand sides in a logarithmic class 28 Lin W. and Wavelet method for Hadamard singular [email protected] ZhuL. integrals 29 Magnanini R. On isophonic surfaces [email protected] and Sakaguchi S. 30 Makrakis G. Wignerization of caustics [email protected] 31 Maz'ya V. Boundary singularities of solutions to [email protected] quasilinear elliptic equations 32 Mickelic A. Homogenizing the acoustic properties of a [email protected] porous matrix containing an incompressible inviscid fluid 33 OgamE., Reduction of the ambiguity of shape [email protected] Scotti T. and reconstruction of cylindrical bodies using Wirgin A. both real and synthetic acoustic scattering data 34 Othman R., Determination de la dispersion a partir de othman® lms.polvtechnique.fr Blanc R., 1' analyse de Fourier des resonances dans Bussac M.-N., les barres elastiques et viscoelastiques Collet P. and Gary G. 3S Othman R., Analyse des ondes dans les barres [email protected] 283

Bussac M.-N., Clastiques et viscoClastiques par la Collet P. et rnkthode du maximum de vraisemblance Gary G. Ramananjaona On the controlled evolution of level sets [email protected] C., Lambert and like methods in scalar inverse M., Lesselier scattering D. and Zolesio J.-P. Ramm A.G. Dynamical systems method for solving [email protected] linear and nonlinear ill-posed problems RammA.G. Property C for ODE and PDE and [email protected] applications to inverse scattering and other inverse problems Schwartz L. Recent progress in the theoretical and [email protected] numerical modelling of thin-layer flow Shaposhnikova On the Brezis and Mironescu coniecture [email protected] T. about a Gagliardo-Nirenberg inequality I for fractional Sobolev norms Taroudakis On the use of the parabolic approximation [email protected]&.g M.I. and for time domain solutions of the acoustic Makrakis G.N. equation in shallow water Tsogka C. and Seismic response of a series of buildings tsoekaQIma.cnrs-mrs.fr Wirgin A. (city) anchored in soft soil xu Y. Transmission of ultrasonic wave in yxuQcecasu n.utc.edu I cancellous bone and evaluation of I I I osteoporosis This page intentionally left blank Author Index

Barrett, T.M. 34 Gilbert, R. 92, 157 Begehr,H. 8 Gilbert, R.P. 41, 164 Berger, S. 143 Gipouloux, 0. 28 Berryman, J. 14 Gmati, N. 171 Blanc, R. 229 Gutman, S. 178 Bonnet-Ben Dhia, A.S. 2 I Hile, G.N. 185 Borcea, L. 14 Hsiao, G.C. 48 Bourgeat, A. 28 Jeffrey, A. 192 Boursier, I. 28 Jerome, J.W. 71 Broadridgem, P. 34 Khashanah, K. 41 Buchanan, J.L. 41 Kovalevsky, A. 199 Bussac, M.N. 229,236 Lambert, M. 243 Cakoni, F. 48 Lauriks, W. 143 Caputo, J.G. 55 Leblond, J.B. 150 Cardoulis, L. 64 Lesselier, D. 243 Carl, S. 71 Lin, W. 272 Carroll, R. 1,78 Magnanini, R. 207 Chandezon, J. 85 Marusic-Paloka, E. 28 Chang, D.C. 92 Maz’ya, V. 2 12 Cianci, P. 100 Mercier, J.-F. 21 Clopeau, T. 108 Mickelic, A. 108 Cocou, M. 116 Nicolosi, F. 199 Collet, P. 229,236 Ogam,E. 222 Cristini, P. 123 Othman, R. 229,236 Cristofol, M. 64 Ou, M.J. 157 D’Asero, S. 130 Panchenko, A. 164 De Bazelaire, E. 123 Papanicolaou, G. 14 Delanghe, R. 137 Poyedinchuk, A.Ye. 85 Depollier, C. 143 Ramananjaona, C. 243 Fellah, Z.E.A. 143 Ramm, A.G. 178 Flandi, L. 150 Scarella, G. 116 Gaitan, P. 64 Schwartz, L. 25 1 Gary, G. 229,236 Scotti,T. 222

285 Shaposhnikova, T. 2 12 Xu, Y. 265 Stanoyevitch, A. 185 Yashina, N.P. 85 Stepanyants, Y.A. 55 Zhu,L. 272 Tsogka, C. 14,258 Zolesio, J.-P. 243 Wang,G. 92 Zrelli, N. 171 Wirgin, A. v, 4,222, 258

286