Finite Element Analysis of Acoustic Scattering (Applied Mathematical Sciences)

Contents......Page 12
Preface......Page 8
1.1 Acoustic Waves......Page 16
1.1.1 Linearized Equations for Compressible Fluids......Page 17
1.1.2 Wave Equation and Helmholtz Equation......Page 18
1.1.3 The Sommerfeld Condition......Page 21
1.2.1 Dynamic Equations of Elasticity......Page 23
1.2.2 Vector Helmholtz Equations......Page 24
1.3 Acoustic/Elastic Fluid–Solid Interaction......Page 26
1.3.1 Physical Assumptions......Page 27
1.3.2 Governing Equations and Special Cases......Page 28
1.4.1 Electric Fields......Page 31
1.4.2 Magnetic Fields......Page 32
1.4.3 Maxwell's Equations......Page 33
1.5 Summary......Page 34
1.6 Bibliographical Remarks......Page 35
2 Analytical and Variational Solutions of Helmholtz Problems......Page 36
2.1.1 Cartesian Coordinates......Page 37
2.1.2 Spherical Coordinates......Page 39
2.1.3 Cylindrical Coordinates......Page 44
2.1.4 Atkinson–Wilcox Expansion......Page 46
2.1.6 Computational Aspects......Page 47
2.2.1 Norm and Scalar Product......Page 50
2.2.2 Hilbert Spaces......Page 51
2.2.3 Sesquilinear Forms and Linear Operators......Page 53
2.2.4 Trace of a Function......Page 54
2.3.1 Helmholtz Problems on Bounded Domains......Page 55
2.3.2 Helmholtz Problems on Unbounded Domains......Page 56
2.3.3 Weak Formulation for Solid–Fluid Interaction......Page 58
2.4.1 Positive Definite Forms......Page 61
2.4.2 The inf–sup Condition......Page 63
2.4.3 Coercive Forms......Page 66
2.5.1 Galerkin Method and Ritz Method......Page 68
2.5.2 Convergence Results......Page 70
2.5.3 Conclusions for Helmholtz Problems......Page 72
2.7 Bibliographical Remarks......Page 73
3 Discretization Methods for Exterior Helmholtz Problems......Page 76
3.1.1 Introduction of an Artificial Boundary......Page 77
3.1.2 Dirichlet-to-Neumann Operators......Page 78
3.1.3 Well-Posedness......Page 79
3.2.1 The Exact DtN Operator......Page 80
3.2.2 Spectral Characterization of the DtN-Operator......Page 82
3.2.3 Truncation of the DtN Operator......Page 84
3.2.4 Localizations of the Truncated DtN Operator......Page 85
3.3 Absorbing Boundary Conditions......Page 86
3.3.1 Recursion in the Atkinson–Wilcox Expansion......Page 87
3.3.2 Localization of a Pseudodifferential Operator......Page 89
3.3.3 Comparison of ABC......Page 91
3.3.4 The PML Method......Page 93
3.4 The Finite Element Method in the Near Field......Page 95
3.4.1 Finite Element Technology......Page 96
3.4.2 Identification of the FEM as a Galerkin Method......Page 101
3.5.1 Infinite Elements from Radial Expansion......Page 102
3.5.2 Variational Formulations......Page 104
3.5.3 Remarks on the Analysis of the Finite–Infinite Element Method......Page 108
3.6 Summary......Page 112
3.7 Bibliographical Remarks......Page 113
4 Finite Element Error Analysis and Control for Helmholtz Problems......Page 116
4.1 Convergence of Galerkin FEM......Page 117
4.1.2 Positive Definite Problems......Page 118
4.1.3 Indefinite Problems......Page 120
4.2 Model Problems for the Helmholtz Equation......Page 121
4.2.1 Model Problem I: Uniaxial Propagation of a Plane Wave......Page 122
4.2.2 Model Problem II: Propagation of Plane Waves with Variable Direction......Page 123
4.2.3 Model Problem III: Uniaxial Fluid–Solid Interaction......Page 124
4.3.1 The inf–sup Condition......Page 125
4.3.2 Stability Estimates for Data of Higher Regularity......Page 128
4.4.1 Approximation Rule and Interpolation Error......Page 131
4.4.2 An Asymptotic Error Estimate......Page 134
4.4.3 Conclusions......Page 136
4.5.1 Dispersion Analysis of the FE Solution......Page 137
4.5.2 The Discrete inf–sup Condition......Page 139
4.5.3 A Sharp Preasymptotic Error Estimate......Page 140
4.5.4 Results of Computational Experiments......Page 143
4.6 Pollution of FE Solutions with Large Wave Number......Page 147
4.6.1 Numerical Pollution......Page 148
4.6.2 The Typical Convergence Pattern of FE Solutions to the Helmholtz Equation......Page 149
4.6.3 Influence of the Boundary Conditions......Page 151
4.6.4 Error estimation in the L[sup(2)]-norm......Page 152
4.6.5 Results from 2-D Computations......Page 153
4.7.1 hp-Approximation......Page 155
4.7.2 Dual Stability......Page 160
4.7.3 FEM Solution Procedure. Static Condensation......Page 162
4.7.4 Dispersion Analysis and Phase Lag......Page 164
4.7.5 Discrete Stability......Page 166
4.7.6 Error Estimates......Page 168
4.7.7 Numerical Results......Page 170
4.8.1 Generalized FEM in One Dimension......Page 173
4.8.2 Generalized FEM in Two Dimensions......Page 177
4.9.1 Analysis and Parameter Discussion......Page 185
4.9.2 Numerical Evaluation......Page 186
4.10.1 Notation......Page 189
4.10.2 Bounds for the Effectivity Index......Page 190
4.10.3 Numerical Results......Page 194
4.11 Summary and Conclusions for Computational Application......Page 200
4.12 Bibliographical Remarks......Page 202
5.1.1 Implementation of a Coupled Finite–Infinite Element Method for Axisymmetric Problems......Page 204
5.1.2 Model Problem......Page 206
5.1.3 Computational Results......Page 209
5.1.4 Conclusions......Page 216
5.2.1 Model Parameters......Page 217
5.2.2 Convergence Tests......Page 218
5.2.3 Comparison with Experiments......Page 221
5.3 Summary......Page 225
References......Page 226
D......Page 236
L......Page 237
S......Page 238
Z......Page 239