Effective Computational Methods for Wave Propagation (Numerical Insights)

707 Pages·2008·14.86 MB·English

by  Ed: Nikolaos A. Kampanis, Vassilios Dougalis, John A. Ekaterinaris

EFFECTIVE
COMPUTATIONAL
METHODS FOR WAVE
PROPAGATION
Numerical Insights
Series Editor
A. Sydow, GMD-FIRST, Berlin, Germany
Editorial Board
P. Borne, École de Lille, France; G. Carmichael, University of Iowa, USA;
L. Dekker, Delft University of Technology, The Netherlands; A. Iserles, University
of Cambridge, UK; A. Jakeman, Australian National University, Australia;
G. Korn, Industrial Consultants (Tucson), USA; G.P. Rao, Indian Institute of
Technology, India; J.R. Rice, Purdue University, USA; A.A. Samarskii, Russian
Academy of Science, Russia; Y. Takahara, Tokyo Institute of Technology, Japan
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Volume 5
Effective Computational Methods for Wave Propagation
Nikolaos A. Kampanis, Vassilios A. Dougalis, and John A. Ekaterinaris
EFFECTIVE
COMPUTATIONAL
METHODS FOR WAVE
PROPAGATION
Edited by
Nikolaos A. Kampanis
Vassilios A. Dougalis
John A. Ekaterinaris
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Kampanis, Nikolaos A.
Effective computational methods for wave propagation / Nikolaos A.
Kampanis, John A. Ekaterinaris, Vassilios Dougalis.
p. cm. -- (Numerical insights)
Includes bibliographical references and index.
ISBN 978-1-58488-568-9 (alk. paper)
1. Wave-motion, Theory of--Data processing. 2. Electromagnetic waves--Data
processing. 3. Numerical analysis. I. Ekaterinaris, John A. II. Dougalis, Vassilios.
III. Title. IV. Series.
QA927.K255 2008
530.12’4--dc22 2007030361
Visit the Taylor & Francis Web site at
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Contents
Preface 1
I Nonlinear Dispersive Waves 5
1 Numerical Simulations of Singular Solutions of the Nonlinear
Schro¨dinger Equations, Xiao-Ping Wang 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Dynamic Rescaling Method . . . . . . . . . . . . . . . . . . 8
1.2.1 Radially Symmetric Case . . . . . . . . . . . . . . . . 8
1.2.2 Anisotropic Dynamic Rescaling . . . . . . . . . . . . 9
1.3 Adaptive Method Based on the Iterative Grid Redistribution
(IGR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Grid Distribution Based on the Variational Principle 13
1.3.2 An Iterative Grid Redistribution Procedure . . . . . 16
1.3.3 Adaptive Procedure for Solving Nonlinear Schro¨dinger
Equations . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Singular Solutions with Multiple Blowup Points . . . 18
1.4.2 Numerical Simulations of Self–Focusing of Ultrafast
Laser Pulses . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Ring Blowup Solutions of the NLS . . . . . . . . . . . 23
1.4.4 Keller-Seigel Equation: Complex Singularity . . . . . 28
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References 32
2 Numerical Solution of the Nonlinear Helmholtz Equation, G.
Fibich and S. Tsynkov 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.2 The Nonlinear Helmholtz Equation . . . . . . . . . . 38
2.1.3 Transverse Boundary Conditions . . . . . . . . . . . . 40
2.1.4 Paraxial Approximation and the Nonlinear Schro¨dinger
Equation . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.5 Solitons and Collapse . . . . . . . . . . . . . . . . . . 41
2.2 Algorithm — Continuous Formulation . . . . . . . . . . . . 42
v
vi
2.2.1 Iteration Scheme . . . . . . . . . . . . . . . . . . . . . 43
2.2.2 Separation of Variables and Boundary Conditions . . 43
2.3 Algorithm — Finite-Difference Formulation . . . . . . . . . 47
2.3.1 Fourth Order Scheme . . . . . . . . . . . . . . . . . . 47
2.3.2 Transverse Boundary Conditions . . . . . . . . . . . . 48
2.3.3 Discrete Eigenvalue Problem . . . . . . . . . . . . . . 50
2.3.4 Separation of Variables . . . . . . . . . . . . . . . . . 52
2.3.5 Properties of Eigenvalues . . . . . . . . . . . . . . . . 53
2.3.6 Nonlocal ABCs . . . . . . . . . . . . . . . . . . . . . 54
2.4 Results of Computations . . . . . . . . . . . . . . . . . . . . 55
2.4.1 Critical Case . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.2 Subcritical Case . . . . . . . . . . . . . . . . . . . . . 57
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
References 61
3 Theory and Numerical Analysis of Boussinesq Systems: A
Review, V. A. Dougalis and D. E. Mitsotakis 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Derivation and Examples of Boussinesq Systems . . . . . . . 64
3.3 Well-Posedness Theory . . . . . . . . . . . . . . . . . . . . . 72
3.4 Solitary Waves . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 84
3.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . 89
3.7 Boussinesq Systems in Two Space Dimensions . . . . . . . . 99
References 105
II The Helmholtz Equation and Its Paraxial Approx-
imations in Underwater Acoustics 111
4 Finite Element Discretization of the Helmholtz Equation in
an Underwater Acoustic Waveguide, D. A. Mitsoudis, N. A.
Kampanis and V. A. Dougalis 113
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . 116
4.2.1 Reformulation of the Problem in a Bounded Domain 118
4.2.2 Construction of the Nonlocal Conditions at the Artifi-
cial Boundaries . . . . . . . . . . . . . . . . . . . . . 119
4.3 The Finite Element Method . . . . . . . . . . . . . . . . . . 122
4.3.1 The Finite Element Discretization . . . . . . . . . . . 123
4.3.2 Implementation Issues . . . . . . . . . . . . . . . . . . 124
4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . 126
References 131
vii
5 Parabolic Equation Techniques in Underwater Acoustics, D.
J. Thomson and G. H. Brooke 135
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Parabolic Approximations . . . . . . . . . . . . . . . . . . . 140
5.2.1 Standard PE . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.2 Exact PE . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.3 Propagator Approximations . . . . . . . . . . . . . . 142
5.2.4 Finite-Difference Scheme . . . . . . . . . . . . . . . . 144
5.2.5 Profile Interpolation . . . . . . . . . . . . . . . . . . . 145
5.2.6 Energy Conservation . . . . . . . . . . . . . . . . . . 146
5.2.7 Equivalent Fluid . . . . . . . . . . . . . . . . . . . . . 147
5.2.8 Initial Field . . . . . . . . . . . . . . . . . . . . . . . 148
5.2.9 Perfectly Matched Absorber . . . . . . . . . . . . . . 148
5.2.10 Propagation Example . . . . . . . . . . . . . . . . . . 149
5.3 PE-Based Matched Field Processing . . . . . . . . . . . . . 150
5.3.1 Standard Processor . . . . . . . . . . . . . . . . . . . 150
5.3.2 Backpropagated Processor . . . . . . . . . . . . . . . 152
5.3.3 Localization Example . . . . . . . . . . . . . . . . . . 153
5.4 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . 155
5.4.1 Relation between PE Modes and Normal Modes . . . 157
5.4.2 Modal Excitations . . . . . . . . . . . . . . . . . . . . 158
5.4.3 Modal Phases . . . . . . . . . . . . . . . . . . . . . . 160
5.4.4 Modal Decomposition Example . . . . . . . . . . . . 160
5.4.5 Modal Beamforming . . . . . . . . . . . . . . . . . . . 162
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
References 165
6 Numerical Solution of the Parabolic Equation in Range–
Dependent Waveguides, V. A. Dougalis, N. A. Kampanis,
F. Sturm and G. E. Zouraris 175
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2 Initial–Boundary Value Problems in Axially Symmetric
Range-Dependent Environments . . . . . . . . . . . . . . . . 178
6.3 Finite Element Solution of the 2D PE in a General Stratified
Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.3.1 Horizontal Interface . . . . . . . . . . . . . . . . . . . 187
6.3.2 Sloping Interface . . . . . . . . . . . . . . . . . . . . . 193
6.4 Finite Element Solution of the 3D Standard PE in a General
Stratified Waveguide . . . . . . . . . . . . . . . . . . . . . . 198
6.4.1 The Initial–Boundary Value Problem for the 3D Stan-
dard PE . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.4.2 The Transformed Initial–Boundary Value Problem . . 200
6.4.3 The Numerical Scheme . . . . . . . . . . . . . . . . . 202
6.4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . 203
viii
References 205
7 Exact Boundary Conditions for Acoustic PE Modeling Over
2
an N -Linear Half-Space, T. W. Dawson, G. H. Brooke and
D. J. Thomson 209
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.2.1 PE Theory . . . . . . . . . . . . . . . . . . . . . . . . 212
7.2.2 Solution in the Lower Half-Space . . . . . . . . . . . . 214
7.2.3 Solution Details . . . . . . . . . . . . . . . . . . . . . 216
7.2.4 Complex Half-Space Profile–Attenuation . . . . . . . 217
7.3 Non-Local Boundary Conditions . . . . . . . . . . . . . . . . 217
7.3.1 General Considerations . . . . . . . . . . . . . . . . . 217
7.3.2 Narrow Angle (Tappert) PE . . . . . . . . . . . . . . 219
7.3.3 Simple Wide-Angle (Claerbout) PE . . . . . . . . . . 220
7.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . 223
7.5 First-Order Claerbout Examples . . . . . . . . . . . . . . . 225
7.5.1 Modified AESD [13] Case . . . . . . . . . . . . . . . . 225
7.5.2 Modified Norda Test Cases . . . . . . . . . . . . . . . 227
7.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . 229
References 237
III Numerical Methods for Elastic Wave Propaga-
tion 239
8 Introduction and Orientation, P. Joly 241
9 The Mathematical Model for Elastic Wave Propagation, P.
Joly 247
9.1 Preliminary Notation . . . . . . . . . . . . . . . . . . . . . . 247
9.2 The Equations of Linear Elastodynamics . . . . . . . . . . . 249
9.2.1 The Unknowns of the Problem . . . . . . . . . . . . . 250
9.2.2 Useful Differential Operators and Green’s Formulas . 250
9.2.3 The Equations of the Problem . . . . . . . . . . . . . 251
9.3 Variational Formulation and Weak Solutions . . . . . . . . . 253
9.4 Plane Wave Propagation in Homogeneous Media . . . . . . 257
9.5 Finite Propagation Velocity . . . . . . . . . . . . . . . . . . 263
10 Finite Element Methods with Continuous Displacement, P.
Joly 267
10.1 Galerkin Approximation of Abstract Second Order Variational
Evolution Problems . . . . . . . . . . . . . . . . . . . . . . . 267
10.2 Space Approximation of Elastodynamics Equations with La-
grange Finite Elements . . . . . . . . . . . . . . . . . . . . . 274
ix
10.3 On the Use of Quadrature Formulas . . . . . . . . . . . . . 287
10.4 The Mass Lumping Technique . . . . . . . . . . . . . . . . . 305
10.5 Time Discretization by Finite Differences . . . . . . . . . . . 313
10.6 Computational Issues . . . . . . . . . . . . . . . . . . . . . . 321
10.6.1 General Considerations . . . . . . . . . . . . . . . . . 321
10.6.2 An Efficient Algorithm for General Lagrange Elements 322
10.6.3 A Link with Mixed Finite Element Methods . . . . . 327
11 Finite Element Methods with Discontinuous Displacement,
P. Joly and C. Tsogka 331
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
11.2 Mixed Variational Formulation . . . . . . . . . . . . . . . . 332
11.3 Space Discretization . . . . . . . . . . . . . . . . . . . . . . 334
11.3.1 Choice of the Approximation Space for the Stress Ten-
sor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11.3.2 Choice of the Approximation Space for the Velocity . 337
11.3.3 Extension to Higher Orders and Mass Lumping . . . 339
11.4 Theoretical Issues . . . . . . . . . . . . . . . . . . . . . . . . 340
div
11.4.1 The Q
k+1 − Pk+1 Element . . . . . . . . . . . . . . . 340
div
11.4.2 The Q
k+1 − Qk Element . . . . . . . . . . . . . . . . 340
11.5 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . 355
12 Fictitious Domains Methods for Wave Diffraction, P. Joly
and C. Tsogka 359
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
12.2 The Continuous Formulation . . . . . . . . . . . . . . . . . . 361
12.3 Finite Element Approximation and Time Discretization . . 363
12.4 Existence of the Discrete Solution and Stability . . . . . . . 367
12.5 About the Convergence Analysis . . . . . . . . . . . . . . . 370
div
12.5.1 FDM with the Q
k+1 − Qk Element . . . . . . . . . . 370
div
12.5.2 FDM with the Q
k+1 − Pk+1 Element . . . . . . . . . 372
12.5.3 An Abstract Result . . . . . . . . . . . . . . . . . . . 377
12.6 Illustration of the Efficiency of FDM . . . . . . . . . . . . . 382
13 Space Time Mesh Refinement Methods, G. Derveaux, P.
Joly and J. Rodr´ıguez 385
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
13.2 The Domain Decomposition Approach . . . . . . . . . . . . 386
13.2.1 Velocity Stress Formulation . . . . . . . . . . . . . . . 386
13.2.2 Transmission Problem . . . . . . . . . . . . . . . . . . 387
13.2.3 Variational Formulation . . . . . . . . . . . . . . . . . 388
13.3 Space Discretization . . . . . . . . . . . . . . . . . . . . . . 390
13.3.1 Semi–Discretized Variational Formulation . . . . . . . 390
13.3.2 Matrix Formulation . . . . . . . . . . . . . . . . . . . 391
13.3.3 Choice for the Approximation Spaces . . . . . . . . . 393