Finite Element Analysis of Acoustic Scattering

Total Page:16

File Type:pdf, Size:1020Kb

Finite Element Analysis of Acoustic Scattering Finite Element Analysis of Acoustic Scattering Frank Ihlenburg Springer To Krystyna and Katja Love’s not Time’s fool — William Shakespeare, Sonnett 116 Preface Als ¨uberragende Gestalt . tritt uns Helmholtz entgegen . Seine außerordentliche Stellung in der Geschichte der Naturwissenschaf- ten beruht auf einer ungew¨ohnlich vielseitigen, eindringenden Bega- bung, innerhalb deren die mathematische Seite eine wichtige, f¨uruns nat¨urlich in erster Linie in Betracht kommende Rolle spielt. (Felix Klein, [84, p. 223])1 Waves are interesting physical phenomena with important practical appli- cations. Physicists and engineers are interested in the reliable simulation of processes in which waves are scattered from obstacles (scattering prob- lems). This book deals with some of the mathematical issues arising in the computational simulation of wave propagation and fluid–structure interac- tion. The linear mathematical models for wave propagation and scattering are well-known. Assuming time-harmonic behavior, one deals with the Helm- holtz equation ∆u + k2u = 0, where the wave number k is a physical parameter. Our interest will be mainly in the numerical solution of exte- rior boundary value problems for the Helmholtz equation which we call Helmholtz problems for short. The Helmholtz equation belongs to the classical equations of mathema- tical physics. The fundamental questions about existence and uniqueness 1In Helmholtz we meet an overwhelming personality. His extraordinary position in the history of science is based on his unusually diverse and penetrating talents, among which the mathematical side, which for our present purpose is of primary importance, plays an important role. viii Preface of solutions to Helmholtz problems were solved by the end of the 1950s; cf., e.g., the monographs of Leis [87], Colton–Kress [39], and Sanchez Hubert– Sanchez Palencia [107]. Those results of mathematical analysis form the fundamental layer on which the numerical analysis in this book is built. The two main topics that are discussed here arise from the practical app- lication of finite element methods (FEM) to Helmholtz problems. First, FEM have been conceptually developed for the numerical discreti- zation of problems on bounded domains. Their application to unbounded domains involves a domain decomposition by introducing an artificial boun- dary around the obstacle. At the artificial boundary, the finite element dis- cretization can be coupled in various ways to some discrete representation of the analytical solution. We review some of the coupling approaches in Chapter 3, focusing on those methods that are based on the series represen- tation of the exterior solution. In particular, we review localized Dirichlet- to-Neumann and other absorbing boundary conditions, as well as the recent perfectly matched layer method and infinite elements. Second, when using discrete methods for the solution of Helmholtz prob- lems, one soon is confronted with the significance of the parameter k. The wave number characterizes the oscillatory behavior of the exact solution. The larger the value of k, the stronger the oscillations. This feature has to be resolved by the numerical model. The “rule of thumb” is to resolve a wavelength by a certain fixed number of mesh points. It has been known from computational experience that this rule is not sufficient to obtain re- liable results for large k. Looking at this problem from the viewpoint of numerical analysis, the reason for the defect can be found in the loss of operator stability at large wave numbers. We address this topic in Chap- ter 4, where we present new estimates that precisely characterize the error behavior in the range of engineering computations. We call these estimates preasymptotic in order to distinguish them from the well-known asympto- tic error estimates for indefinite problems satisfying a G˚ardinginequality. In particular, we accentuate the advantages of the hp-version of the FEM, as opposed to piecewise linear approximation. We also touch upon gene- ralized (stabilized) FEM and investigate a posteriori error estimation for Helmholtz problems. Our theoretical results are obtained mainly for one- dimensional model problems that display most of the essential features that matter in the true simulations. In the introductory Chapters 1 and 2, we set the stage for the finite ele- ment analysis. We start with an outline of the governing equations. While our physical application is acoustic fluid–structure interaction, much of the mathematics in this book may be relevant also for numerical electro- dynamics. We therefore include a short section on Maxwell’s equations. In Chapter 2, we first (Section 2.1) review mathematical techniques for the analytical solution of exterior Helmholtz problems. Our focus is on the separation of variables and series representations of the solution (comple- mentary to the integral methods and representations), as needed for the Preface ix outline of the coupling methods in Chapter 3. The second part (Sections 2.2–2.5) of Chapter 2 is a preparation of the finite element analysis in Chapter 4. We first briefly review some necessary definitions and theorems from functional analysis inasmuch as they are needed for the subsequent investigation. Then we consider the variational formulation of Helmholtz problems and discuss variational methods. Computational results for three-dimensional scattering problems are re- ported in Chapter 5. This text is addressed to mathematicians as well as to physicists and computational engineers working on scattering problems. Having a mixed audience in mind, we attempted to make the text self-contained and easily readable. This especially concerns Chapters 3 and 4. We hope that the illus- tration with many numerical examples makes for a better understanding of the theory. The material of the introductory chapters is presented in a more compact manner for the convenience of later reference. It is assumed that the reader is familiar with the basic physical and mathematical concepts of fluid–structure interaction and/or finite element analysis. References to various expositions of these topics are supplied. Acknowledgments Es ist eben viel wichtiger, in welche geistige Umgebung ein Mensch hineinkommt, die ihn viel st¨arker beeinflußt als Tatsachen und kon- kretes Wissen, das ihm geboten wird. (Felix Klein, [84, p. 249])2 Much of this book is a report of my own cognitive journey towards the reliable simulation of scattering problems with finite element methods. My interest in numerical acoustics began while I was working as an associate of Ivo Babuˇska at the University of Maryland at College Park. Many results in this book stem from our joint work, and I have tried my best to put the spir- it of our discussions down on paper. My gratitude goes to J. Tinsley Oden and Leszek Demkowicz, of the Texas Institute for Computational and Ap- plied Mathematics (TICAM). Most of this monograph was written during my appointment as a TICAM Research Fellow, and TICAM’s extraordi- nary working conditions and stimulating intellectual atmosphere were an essential ingredient for its shape and content. I gratefully acknowledge the financial support from the Deutscher Akademischer Austauschdienst and the Deutsche Forschungsgemeinschaft. Thanks to Professors Reißmann and R¨ohrfrom my home University of Rostock, Germany, for their support of my grant applications. With deep gratitude I acknowledge the close co- operation with Joseph Shirron of the Naval Research Laboratory (NRL) in Washington, D.C., who made his program SONAX available to me and was always ready to share his broad and solid experience in numerical acoustics. 2The intellectual environment a person enters is more significant and will be of much greater influence than the facts and concrete knowledge that are offered him. x Preface He was also the first discerning reader of the manuscript. Many thanks also to Oliver Ernst (Freiberg/ College Park), Lothar Gaul (Stuttgart), Jens Markus Melenk (Z¨urich), and Guy Waryee (Brussels), who carefully read later versions of the text. Their remarks led to a considerable improvement in content and style. Thanks to Louise Couchman (NRL) for permission to use SONAX and to Brian Houston (NRL) for providing me with the results of his experiments and for the numerous explanations of the details of his studies. Many thanks go to Achi Dosanjh, David Kramer, and Vic- toria Evarretta from the Springer-Verlag New York, for their professional support in making the book. I am much obliged to Malcolm Leighton who thoroughly checked the language of the manuscript. Finally, I gladly use this opportunity to thank my parents Ingrid and Karl Heinz Ihlenburg for their encouragement of my interests. Hamburg, Germany Frank Ihlenburg February 1998 Contents Preface vii 1 The Governing Equations of Time-Harmonic Wave Propagation 1 1.1 Acoustic Waves ......................... 1 1.1.1 Linearized Equations for Compressible Fluids .... 2 1.1.2 Wave Equation and Helmholtz Equation ....... 3 1.1.3 The Sommerfeld Condition .............. 6 1.2 Elastic Waves .......................... 8 1.2.1 Dynamic Equations of Elasticity ........... 8 1.2.2 Vector Helmholtz Equations .............. 9 1.3 Acoustic/Elastic Fluid–Solid Interaction ........... 11 1.3.1 Physical Assumptions ................. 12 1.3.2 Governing Equations and Special Cases ....... 13 1.4 Electromagnetic Waves ..................... 16 1.4.1 Electric Fields ..................... 16 1.4.2 Magnetic Fields ..................... 17 1.4.3 Maxwell’s Equations .................
Recommended publications
  • A Review of Finite Element Methods for Time-Harmonic Acoustics
    L.L.Thompson: Finite element methods for acoustics, Preprint: J.Acoust.Soc.Am. A review of ¯nite element methods for time-harmonic acoustics Lonny L. Thompson Department of Mechanical Engineering, Clemson University Clemson, South Carolina, 29634-0921, USA Email: [email protected] (Dated: Submitted July 8, 2004; Resubmitted May 28, 2005; Revised Dec 10, 2005; Accepted Dec 12, 2005) PACS numbers: 43.20.Bi,43.20.Fn,43.20.Rz Keywords: acoustics; Helmholtz equation; ¯nite element method; stabilized methods; unbounded domains; absorbing boundary condition; nonreflecting boundary condition; absorbing layer; in¯nite element; complex- symmetric; domain decomposition method; adaptive method; a posteriori error estimate; inverse scattering; optimization 1 Abstract State-of-the-art ¯nite element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the ¯eld are speci¯cally addressed: the e®ective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, in¯nite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, pol- luting resolved scales, and requiring a large computational e®ort; e±cient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems in- cluding sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in ¯nite element methods are described. Stabilized, multiscale and other wave- based discretization methods developed to reduce this error are reviewed.
    [Show full text]
  • Performance Study of Plane Wave Finite Element Methods with a Padé
    Performance study of plane wave finite element methods with a Padé-type artificial boundary condition in acoustic scattering Ryiad Kerchroud, Azzedine Soulaimani, Xavier Antoine To cite this version: Ryiad Kerchroud, Azzedine Soulaimani, Xavier Antoine. Performance study of plane wave finite element methods with a Padé-type artificial boundary condition in acoustic scattering. Advances in Engineering Software, Elsevier, 2009, 40 (8), pp.738-750. 10.1016/j.advengsoft.2008.12.016. hal- 00591438 HAL Id: hal-00591438 https://hal.archives-ouvertes.fr/hal-00591438 Submitted on 9 May 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Performance study of plane wave finite element methods with a Pad´e-type artificial boundary condition in acoustic scattering R. Kechroud∗, A. Soulaimani∗, X. Antoine†‡ Abstract The aim of this paper is to propose and numerically study the performance of coupling a high-order Pad´e-type non-reflecting boundary condition with plane wave finite element formulations for solving high-frequency scattering problems involving elongated scatterers. It is shown on some numerical examples that the approximate solution can be obtained using a small number of degrees of freedom for a suitable accuracy.
    [Show full text]
  • Wave Propagation and Scattering, Inverse Problems, and Applications in Energy and the Environment November 21-25, 2011
    Workshop on Wave Propagation and Scattering, Inverse Problems, and Applications in Energy and the Environment November 21-25, 2011 as part of the Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment The efficient computation of wave propagation and scattering is a core problem in numerical mathematics, which is currently of great research interest and is central to many applications in energy and the envi- ronment. Two generic applications which resonate strongly with the central aims of this special semester are forward wave propagation in heterogeneous media and seismic inversion for subsurface imaging. As an example of the first application, modelling of absorption and scattering of radiation by clouds, aerosol and precipitation is used as a tool for interpretation of (e.g.) solar, infrared and radar measurements, and as a component in larger weather/climate prediction models in numerical weather forecasting. One key numerical component in this modelling is the prediction of the total optical properties and the full scattering matrix from an ensemble of irregular particles. The underlying mathematical problem is that of accurately computing high frequency wave propagation in a highly heterogeneous medium. As an example of the second application, inverse problems in wave propagation in heterogeneous media arise in the problem of imaging the subsurface below land or marine deposits. Solutions to this problem have a number of environmental uses, for example in the location of hydrocarbon-bearing rocks, in the mon- itoring of pollution in groundwater or in earthquake modelling. A seismic source is directed into the ground and the material properties of the subsurface are inferred by analysing the observed scattered field, recorded by sensors.
    [Show full text]
  • 2 Acoustic and Elastic Wave Equations
    ABSTRACT Mönkölä, Sanna Numerical Simulation of Fluid-Structure Interaction Between Acoustic and Elastic Waves Jyväskylä: University of Jyväskylä, 2011, 136 p. (Jyväskylä Studies in Computing ISSN 1456-5390; 133) ISBN 978-951-39-4438-4 (nid.) ISBN 978-951-39-4439-1 (PDF) Finnish summary Diss. This study considers developing numerical solution techniques for the computer sim- ulations of the fluid-structure interaction. The focus is especially on the efficiency of the iterative methods based on exact controllability and spectral element methods. In particular, the thesis concentrates on the coupling between two linear wave equations: the scalar-valued equation concerning the propagation of acoustic waves and the vector- valued equation modeling the propagation of waves in an elastic medium. These funda- mental equations occur in a number of physical applications, such as acoustics, medical ultrasonics, and geophysics. We consider both transient and time-harmonic problems. Traditionally, the com- plex-valued time-harmonic equations and low-order finite elements are used for solving the time-harmonic problems. This leads to large-scale indefinite systems, for which it is challenging to develop efficient iterative solution methods. Taking account of these diffi- culties, we turn to time-dependent equations. It is known that time-dependent equations can be simulated with respect to time until a time-harmonic solution is reached, but the approach suffers from poor convergence. Thus, we accelerate the convergence rate by employing the exact controllability method. The problem is formulated as a least-squares optimization problem, which is solved with the conjugate gradient (CG) algorithm. Com- putation of the gradient of the functional is done directly for the discretized problem.
    [Show full text]
  • Boundary Integral Equations Methods in Acoustic Scattering
    Boundary Integral Equations Methods in Acoustic Scattering A. Bendali Mathematical Institute, INSA, Toulouse and CERFACS, Toulouse M. Fares CERFACS, Toulouse February 28, 2007 1 Abstract The main subject of this contribution is to present some recent methods, specially designed to be implemented on parallel platforms, to deal with acoustic scattering problems involving a bounded zone filled by a heterogeneous medium. The main ap- proach is to couple a Finite Element Method, for handling this zone, with a Boundary Integral Equation, specially adapted to treat the unbounded part of the computational domain. After giving a short review of some alternative methods, we focus on the methods based on this approach and give a framework which makes it possible to construct almost all the standard Boundary Integral Equations. As well-known, each instance of this kind of scattering problems can be solved by a manifold of such equa- tions. This framework allows one to have a good insight into the advantages and the drawbacks of each of them. It is seen next that the above coupling gives rise to non standard linear systems, with a matrix being partly sparse and partly dense. Serious difficulties then arise when the solution of such systems has to be tackled on a parallel platform. It is shown how techniques from domain decomposition methods can be used to efficiently overcome these difficulties. Keywords: Acoustic Scattering, Helmholtz equation, Boundary Integral Equations, Finite Element Method, Coupling, Domain Decomposition Method, Cross-points. 1 Typical scattering problems in acoustics and brief review of some approaches to their numerical solu- tion The first part of this section is devoted to state some examples of scattering problems in acoustics.
    [Show full text]
  • Analysis of a Special Immersed Finite Volume Method for Elliptic Interface Problems
    INTERNATIONAL JOURNAL OF c 2019 Institute for Scientific NUMERICAL ANALYSIS AND MODELING Computing and Information Volume 16, Number 6, Pages 964{984 ANALYSIS OF A SPECIAL IMMERSED FINITE VOLUME METHOD FOR ELLIPTIC INTERFACE PROBLEMS KAI LIU AND QINGSONG ZOU Abstract. In this paper, we analyze a special immersed finite volume method that is different from the classic immersed finite volume method by choosing special control volumes near the interface. Using the elementwise stiffness matrix analysis technique and the H1-norm-equivalence between the immersed finite element space and the standard finite element space, we prove that the special finite volume method is uniformly stable independent of the location of the interface. Based on the stability, we show that our scheme converges with the optimal order O(h) in the H1 space and the order O(h3=2) in the L2 space. Numerically, we observe that our method converges with the optimal convergence rate O(h) under the H1 norm and with the the optimal convergence rate O(h2) under the L2 norm all the way even with very small mesh size h, while the classic immersed finite element method is not able to maintain the optimal convergence rates (with diminished rate up to O(h0:82) for the H1 norm error and diminished rate up to O(h1:1) for L2-norm error), when h is getting small, as illustrated in Tables 4 and 5 of [35]. Key words. Immersed finite volume method, stability, optimal convergence rates, immersed finite element method. 1. Introduction Interface problem is a kind of problem whose domain is typically separated by some curves or surfaces.
    [Show full text]
  • Hybrid Discontinuous Galerkin Methods for the Wave Equation
    DISSERTATION Hybrid Discontinuous Galerkin Methods for the Wave Equation ausgeführt zum Zwecke der Erlangung des akademischen Grades eines Doktors der technischen Wissenschaften unter der Leitung von Univ.-Prof. Dipl.-Ing. Dr.techn. Joachim Schöberl E101 Institut für Analysis und Scientific Computing eingereicht an der Technischen Universität Wien bei der Fakultät für Mathematik und Geoinformation von Dipl.-Ing. Martin Huber Matrikelnummer: 0056265 Wilhelmstraße 5/3 1120 Wien Wien, am 27. Februar 2013 i Kurzfassung Beim Lösen der Wellengleichung mit klassischen Finiten Elementen wächst für hohe Frequenzen ! die Anzahl der für eine vorgegebene Genauigkeit benötigten Unbekannten aufgrund des sogenannten „pollution effects“ stärker als O(!d) mit d als Raumdimension an. Als eine Möglichkeit dieses Problem in den Griff zu bekommen, werden in der vor- liegenden Dissertation hybride Discontinuous Galerkin Methoden für die skalare und die vektorwertige Wellengleichung vorgestellt. Üblicherweise beruhen klassische Finite Element Methoden (FEM) auf stetigen Ansatz- funktionen, während Discontinuous Galerkin Verfahren Basisfunktionen verwenden, die über Elementgrenzen hinweg unstetig sind. Erst zusätzliche Strafterme führen hier zu einer stetigen Lösung. Hybride FEMen basieren auf den selben Ansatzräumen, nur wird jetzt die Stetigkeit über zusätzliche Lagrangevariablen, die nur auf den Elementgrenzen leben, erzwungen. Im Falle der Wellengleichung ermöglicht die Einführung eines zweiten Satzes von Lagrangevariablen eine vollständige Elimination der ursprünglichen Freiheits- grade. Dadurch kann das Problem auf ein viel kleineres, nur für die Lagrangevariablen reduziert werden. Werden Finite Elemente zum Lösen eines auf einem unendlichen Gebiet gestellten Problems verwendet, wird in der Regel das Rechengebiet eingeschränkt, was transpar- ente Randbedingungen am neu entstandenen künstlichen Rand nötig macht. Als Real- isierungsmöglichkeiten für solche Randbedingungen werden sowohl die Hardy-Raum Meth- ode als auch eine Zerlegung des Fernfeldes in ebene Wellen diskutiert.
    [Show full text]
  • Andrés Prieto Aneiros
    Andr´es Prieto Aneiros PhD Dissertation SOME CONTRIBUTIONS IN TIME-HARMONIC DISSIPATIVE ACOUSTIC PROBLEMS Departamento de Matem´atica Aplicada Facultade de Matem´aticas ¿C´omo me vas a explicar, di, la dicha de esta tarde, si no sabemos porqu´e fue, ni c´omo, ni de qu´e ha sido, si es pura dicha de nada? En nuestros ojos visiones, visiones y no miradas, no percib´ıan tama˜nos, datos, colores, distancias. Pedro Salinas Contents Preface v I Porous materials 1 1 Porous models 3 1.1 Introduction .................................... 4 1.2 Rigid porous models ............................... 6 1.2.1 Darcy’s like model ............................ 7 1.2.2 Allard-Champoux model ......................... 13 1.3 Poroelastic models ................................ 15 1.3.1 Classical Biot’s model .......................... 15 1.3.2 Non-dissipative poroelastic model (closed pores) ............ 18 1.3.3 Non-dissipative poroelastic model (open pores) ............. 21 1.3.4 Dissipative poroelastic model (open pore) . .............. 22 2 Finite element solution of acoustic propagation in rigid porous media 25 2.1 Introduction .................................... 26 2.2 Models for fluid-porous vibrations ........................ 27 2.3 Associated nonlinear eigenvalue problems .................... 30 2.4 Statement of the weak formulation ....................... 34 2.5 Finite element discretization ........................... 35 2.6 Matrix description ................................ 37 2.7 Numerical results ................................. 38 2.8 Conclusions .................................... 41 3 Finite element solution of new displacement/pressure poroelastic models in acoustics 43 3.1 Introduction .................................... 44 3.2 Statement of the problem ............................ 44 3.3 Weak formulation ................................. 47 3.4 Finite element discretization ........................... 48 3.5 Matricial description ............................... 51 i ii Contents 3.6 Numerical solution of cell problems ......................
    [Show full text]
  • Beam-Tracing Domain Decomposition Method for Urban Acoustic Pollution
    Beam-tracing domain decomposition method for urban acoustic pollution Guillaume Gbikpi-Benissan∗ Frédéric Magoulès∗ Abstract This paper covers the fast solution of large acoustic problems on low-resources parallel platforms. A domain decomposition method is coupled with a dynamic load balancing scheme to efficiently accelerate a geometrical acoustic method. The geometrical method studied implements a beam-tracing method where intersections are handled as in a ray-tracing method. Beyond the distribution of the global processing upon multiple sub-domains, a second parallelization level is operated by means of multi-threading and shared memory mechanisms. Numerical experiments show that this method allows to handle large scale open domains for parallel computing purposes on few machines. Urban acoustic pollution arrising from car traffic was simulated on a large model of the Shinjuku district of Tokyo, Japan. The good speed-up results illustrate the performance of this new domain decomposition method. Keywords: Domain decomposition methods; Parallel and distributed computing; Acous- tics; Ray-tracing methods; Beam-tracing methods 1 Introduction Important material resources is often a requirement, either for the processing of large volumes of data, or for the fast resolution of large problems. Especially in parallel computing, the effectiveness of the designed algorithms is related to their scaling behavior on massively parallel platforms. However, for a great part of the scientific community, there remains a need for big simulations with very low resources. Then, it could be of major interest to design algorithms with good acceleration properties on very small clusters of machines. In this study, we tackle the simulation of car traffic noise level at a whole district scale.
    [Show full text]
  • Spectral-Infinite-Element Simulations of Gravity Anomalies
    Spectral-infinite-element simulations of gravity anomalies Item Type Article Authors Gharti, Hom Nath; Tromp, Jeroen; Zampini, Stefano Citation Gharti HN, Tromp J, Zampini S (2018) Spectral-infinite- element simulations of gravity anomalies. Geophysical Journal International 215: 1098–1117. Available: http://dx.doi.org/10.1093/ GJI/GGY324. Eprint version Publisher's Version/PDF DOI 10.1093/GJI/GGY324 Publisher Oxford University Press (OUP) Journal Geophysical Journal International Rights This is a pre-copyedited, author-produced PDF of an article accepted for publication in Geophysical Journal International following peer review. The version of record is available online at: https://academic.oup.com/gji/article/215/2/1098/5067876. Download date 05/10/2021 10:39:07 Link to Item http://hdl.handle.net/10754/629951 Geophys. J. Int. (2018) 215, 1098–1117 doi: 10.1093/gji/ggy324 Advance Access publication 2018 August 07 GJI Gravity, geodesy and tides Spectral-infinite-element simulations of gravity anomalies Downloaded from https://academic.oup.com/gji/article-abstract/215/2/1098/5067876 by King Abdullah University of Science and Technology user on 19 November 2018 Hom Nath Gharti,1 Jeroen Tromp1,2 and Stefano Zampini3 1Department of Geosciences, Princeton University, Princeton, New Jersey, USA. E-mail: [email protected] 2Program in Applied & Computational Mathematics, Princeton University, Princeton, New Jersey, USA 3Extreme Computing Research Center Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Accepted 2018 August 3. Received 2018 July 29; in original form 2018 February 02 SUMMARY Gravity anomalies induced by density heterogeneities are governed by Poisson’s equation.
    [Show full text]
  • Toward Hp-Adaptive Solution of 3D Electromagnetic Scattering from Cavities
    An IntemationalJournal Available online at www.sciencedirectcom computers& .~,..~.@°,..~. mathematics with applications ELSEVIER Computers and Mathematics with Applications 49 (2005) 23-38 www.elsevier.com/locate/camwa Toward hp-Adaptive Solution of 3D Electromagnetic Scattering from Cavities A. ZDUNEK* Swedish Defence Research Agency FOI, Aeronautics Division (FFA) Computational Physics Department, SE-172 90 Stockholm, Sweden zka©foi, se W. RACHOWICZ t Institute for Computer Modelling, Section of Applied Mathematics Cracow University of Technology, P1 31-155 Cracow, Poland waldek©mlody, zms. pk. edu. pl N. SEHLSTEDT* Swedish Defence Research Agency FOI, Aeronautics Division (FFA) Computational Physics Department, SE-172 90 Stockholm, Sweden stn~foi.se (Received and accepted September 2003) Abstract--An effective hp-adaptive finite-element (FE) approach is presented for a reliable and accurate solution of 3D electromagnetic scattering problems. The far field is approximated with the infinite-element method. This allows one to reduce the external domain (discretised with finite elements) to a minimum preserving the possibility of arbitrary reduction of the error as the method does not introduce modelling error. The work is focused on scattering from cavity backed apertures recessed in a ground plane. Near optimal discretisations that can effectively resolve local rapid variations in the scattered field can be obtained adaptively by local mesh refinements (so called h- type refinements) blended with graded polynomial enrichments (p-enrichments). The discretisation error can be controlled by a self-adaptive process, which is driven by a posteriori error estimates in terms of the energy norm or in a quantity of interest. The radar cross section (RCS) is an example of the latter, h- and p-adaptively constructed solutions are compared to pure uniform p approximations.
    [Show full text]
  • Development of an Infinite Element Boundary to Model Gravity For
    University of Birmingham Development of an infinite element boundary to model gravity for subsurface civil engineering applications Haji, Twana; Faramarzi, Asaad; Metje, Nicole; Chapman, David; Rahimzadeh, Farough DOI: 10.1002/nag.3027 License: Creative Commons: Attribution (CC BY) Document Version Publisher's PDF, also known as Version of record Citation for published version (Harvard): Haji, T, Faramarzi, A, Metje, N, Chapman, D & Rahimzadeh, F 2019, 'Development of an infinite element boundary to model gravity for subsurface civil engineering applications', International Journal for Numerical and Analytical Methods in Geomechanics. https://doi.org/10.1002/nag.3027 Link to publication on Research at Birmingham portal General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version.
    [Show full text]