Finite Element Analysis of Acoustic Scattering

Transcription

1 Frank Ihlenburg Finite Element Analysis of Acoustic Scattering With 88 Illustrations Springer

2 Contents Preface vii 1 The Governing Equations of Time-Harmonic Wave Propagation, Acoustic Waves Linearized Equations for Compressible Fluids Wave Equation and Helmholtz Equation The Sommerfeld Condition Elastic Waves Dynamic Equations of Elasticity Vector Helmholtz Equations Acoustic/Elastic Fluid-Solid Interaction Physical Assumptions Governing Equations and Special Cases Electromagnetic Waves Electric Fields Magnetic Fields Maxwell's Equations Summary Bibliographical Remarks 20 2 Analytical and Variational Solutions of Helmholtz Problems Separation of Variables 22

3 xii Contents Cartesian Coordinates Spherical Coordinates Cylindrical Coordinates Atkinson-Wilcox Expansion Far-Field Pattern Computational Aspects References from Functional Analysis Norm and Scalar Product Hilbert Spaces Sesquilinear Forms and Linear Operators Trace of a Function Variational Formulation of Helmholtz Problems Helmholtz Problems on Bounded Domains Helmholtz Problems on Unbounded Domains Weak Formulation for Solid-Fluid Interaction Well-Posedness of Variational Problems Positive Definite Forms The inf-sup Condition Coercive Forms Regularity and Stability Variational Methods Galerkin Method and Ritz Method Convergence Results Conclusions for Helmholtz Problems Summary Bibliographical Remarks 58 3 Discretization Methods for Exterior Helmholtz Problems Decomposition of Exterior Domains Introduction of an Artificial Boundary Dirichlet-to-Neumann Operators Well-Posedness The Dirichlet-to-Neumann Operator and Numerical Applications The Exact DtN Operator Spectral Characterization of the DtN-Operator Truncation of the DtN Operator Localizations of the Truncated DtN Operator Absorbing Boundary Conditions Recursion in the Atkinson-Wilcox Expansion Localization of a Pseudodifferential Operator Comparison of ABC The PML Method The Finite Element Method in the Near Field Finite Element Technology 81

4 Contents xiii Identification of the FEM as a Galerkin Method The h-version and the hp-version of the FEM Infinite Elements and Coupled Finite-Infinite Element Discretization Infinite Elements from Radial Expansion Variational Formulations Remarks on the Analysis of the Finite-Infinite Element Method Summary Bibliographical Remarks 98 Finite Element Error Analysis and Control for Helmholtz Problems Convergence of Galerkin FEM Error Function and Residual Positive Definite Problems Indefinite Problems Model Problems for the Helmholtz Equation Model Problem I: Uniaxial Propagation of a Plane Wave Model Problem II: Propagation of Plane Waves with Variable Direction Model Problem III: Uniaxial Fluid-Solid Interaction Stability Estimates for Helmholtz Problems The inf-sup Condition Stability Estimates for Data of Higher Regularity Quasioptimal Convergence of FE Solutions to the Helmholtz Equation Approximation Rule and Interpolation Error An Asymptotic Error Estimate Conclusions Preasymptotic Error Estimates for the h-version of the FEM Dispersion Analysis of the FE Solution The Discrete inf-sup Condition A Sharp Preasymptotic Error Estimate Results of Computational Experiments Pollution of FE Solutions with Large Wave Number Numerical Pollution The Typical Convergence Pattern of FE Solutions to the Helmholtz Equation Influence of the Boundary Conditions Error estimation in the L 2 -norm Results from 2-D Computations Analysis of the hp FEM ftp-approximation 140

5 ' Contents Dual Stability FEM Solution Procedure. Static Condensation Dispersion Analysis and Phase Lag Discrete Stability Error Estimates Numerical Results Generalized FEM for Helmholtz Problems Generalized FEM in One Dimension Generalized FEM in Two Dimensions The Influence of Damped Resonance in Fluid-Solid Interaction Analysis and Parameter Discussion Numerical Evaluation A Posteriori Error Analysis Notation Bounds for the Effectivity Index Numerical Results Summary and Conclusions for Computational Application Bibliographical Remarks 187 Computational Simulation of Elastic Scattering Elastic Scattering from a Sphere Implementation of a Coupled Finite-Infinite Element Method for Axisymmetric Problems Model Problem Computational Results Conclusions Elastic Scattering from a Cylinder with Spherical Endcaps Model Parameters Convergence Tests Comparison with Experiments Summary 210 References 211 Index 221