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 .................