Index
42, VIII
absorbing boundary conditions (ABC),
21–23, 38, 89, 101, 102, 123, 133,
145–166, 168, 197, 198, 231
approximate local, 156
high–order, 23, 145, 159, 161–164
non–local, 227, 359
absorption–coefficient, 499
admittance matrix, 359, 360, 530
aeroacoustics, 115–142
anechoic boundary condition, see absorbing
boundary conditions (ABC)
Arnoldi method, 106, 335–337, 339
Atkinson–Wilcox theorem, 209, 210, 213
balance of momentum, 2–4, 7, 120, 123,
373, 529
BEM discretization
collocation, 1, 11–21, 23, 28, 309–386,
389, 411–435, 448, 458, 477, 506,
507, 520, 522
Galerkin, 1, 11–21, 28, 341, 345, 347,
418, 435–458, 507, 519–546
Hermitian, 16, 21, 435–458, 519–546
symmetric, see Hermitian
Nystr¨om, 13, 345
benchmark, 22–23, 99, 219–222, 435, 437,
453, 489
Bessel function, 128, 347, 415, 467, 471,
502
boundary conditions
admittance, 8, 12–16, 21, 26, 28, 61, 66,
103, 111, 112, 156, 200, 201, 234, 275,
277, 284, 317, 326, 359, 360, 396,
398, 401, 412, 436, 442, 452–455,
457–494, 523, 529, 533, 535, 550,
552, 560, 567
Dirichlet, 8, 50, 51, 70, 71, 182, 187, 189,
263, 412, 497, 498, 552
impedance, see admittance
Neumann, 8, 38, 49, 70, 177, 234, 323,
412, 413, 415, 477, 497, 498, 523,
526, 531, 552
Robin, 8, 111, 323, 466, 497, 498, 552
boundary elements, 310, 311
continuous, 309–332, 370, 371, 418, 419,
423, 425, 427–429, 432, 506
discontinuous, 12, 309–332, 360, 370,
371, 418, 423, 425, 427, 428
Bubnov–Galerkin formulation, 205
bulk–reacting material, 367–375, 382
bulk–reacting sound absorbing material, see
bulk–reacting material
Burton and Miller formulation, 310, 340–
345, 411–434, 478, 519, 522–523,
531, 535
CBEM, 460, 480, 488, 489, 491
Chebychev approximation, 153, 154
CHIEF, 21, 411–434, 453, 457, 477, 478
enhanced, 417–418
SuperCHIEF, 417, 418
weighted residual, 417, 418
with square matrix, 416–417, 427, 431
cluster tree, 354–362, 525, 526
Combined Helmholtz Integral Equation
Formulation, see CHIEF
574
Index
complex distance, 472, 473, 481
complex eigenfrequencies, 227, 278, 453
complex–source–point–BEM, see CBEM
complexity estimate, VIII, 22, 23, 71–74, 76,
82–84, 95, 97, 99, 108–109, 320, 333,
336–340, 345, 346, 352–362, 381,
392–395, 402–404, 515, 519, 523, 526
condition number, 96, 100, 184, 185, 220,
222, 223, 235, 237, 239, 240, 417,
418, 531, 537, 555
conservation of mass, 2–4, 6, 7, 171
constitutive equation, 5–7, 498
convergence behavior, 96, 105, 333, 337,
340–344, 460, 466–467, 531
convergence rate, 66, 191, 254, 274, 276,
322, 421
exponential, 70, 176
convolution integral, 224, 495, 497, 500,
507, 511, 514
convolution quadrature method (CQM),
495–497, 507–509, 511, 512, 514–515
CPU time, see complexity estimate
Deep Thought, VIII
Dirac delta function, 9, 13, 17, 438, 463,
468, 500, 504
Dirichlet problem, see Dirichlet boundary
conditions
Dirichlet–to–Neumann boundary condition,
see DtN map
Dirichlet–to–Neumann map, see DtN map
Dirichlet–to–Neumann operator, see DtN
map
discontinuous
BEM, see discontinuous boundary
elements
enrichment method, 52
FEM, 39, 52
Galerkin, 52, 77, 81, 119
dispersion, VII, 37–56, 66, 67, 116, 162,
399–402
displacement formulation of the FEM, 20,
254
displacement potential, 253–256, 258, 259
DtN boundary, 104
DtN condition, 102
DtN map, 102–105, 111, 156, 168, 198, 310
dynamic stiffness matrix, 391–394, 403,
404, 528, 530, 531, 536
eigenvalue problem, 96, 254, 269–271, 387,
388, 392, 394, 401, 402, 406
elastoacoustic problem, see fluid structure
interaction
elements per wavelength, 23, 71, 72, 84,
105, 239, 309, 310, 318, 329, 330
energy boundary element method (EBEM),
288, 290, 304
energy finite element method (EFEM),
287–306
error
dispersion, 37–56, 66, 67
pollution, 37–56, 58, 198, 309, 321, 323,
330
relative, 64, 65, 100, 101, 105–107,
109–112, 184, 235, 274, 316, 392,
421, 422
error estimate, 45–47, 105–109, 169, 176,
193, 557
error estimation, see error estimate
Euler equation, 4–8, 79, 81, 119, 199, 523
examples
acoustic horn, 198, 200–202, 236
acoustic lining, 359, 360, 375, 380,
396–400, 406
aircraft interior, 559
amphitheatre, 22, 495, 497, 509–513
open elliptic, 185
anechoic chamber, 22, 359, 360
auditorium, 341–343
blowout coil, 71, 72
cat’s eye, 411, 413, 420, 422–428, 431
copying machine, 559
cylinders (flow induced), 130–140
double–slit interference experiment,
185–187
duct, 200, 220, 221, 309, 311, 316–325,
328–330, 380, 387, 390, 392, 405,
406, 508, 509, 548
engine, 22, 241–247, 343–345, 411, 413,
428–432, 559–565, 568
expansion chamber, 390, 396–406, 519,
532–534
foothills model, 84, 85
hearing simulation, 22, 110–112
heavy equipment cap, 288
front window, 302–304
rear window, 300–303
L–shape test model, 531, 532
Index
muffler, 367–387, 532–534
printer, 559
refrigerator compressor, 559
reverberation chamber, 22, 58, 65–67
sedan cabin compartment, 22, 309, 311,
316, 326–330, 559
silencer, 367–386, 389, 395–404
sphere, 17, 218–226, 411, 415, 418–422,
431, 453–455, 464, 476, 478, 488,
490, 491, 502
spherical piston, 225–227
submarine, 22, 145, 146, 519, 540–542
vacuum cleaner, 22, 559–568
vehicle interior, see sedan cabin
compartment
vibrating box with rib, 455–457
exponential integral, 475, 476, 490
fast multipole analysis (FMA), 21, 38,
333–366, 519–546
fast multipole method (FMM), see fast
multipole analysis (FMA)
finite elements
hierarchical, 58, 66–73
Lagrangian, 23, 57, 68, 217
mixed spectral, 58, 73–84
natural cubic splines, 64
flow effects, 121, 395
flow induced noise, 119, 122, 123, 127,
130–140
flow noise, 116, 567
fluid structure interaction, 2, 21, 253–286,
389, 436, 461, 519–546
mass coupling, 258, 259, 262, 267
Mortar coupling, 538–539
potential/displacement formulation, 259
pressure/displacement formulation, 257,
259, 262, 274
pure displacement formulation, 253, 255,
258, 260, 262, 264, 268, 274, 275,
279–284
stiffness coupling, 259
strong coupling, 519, 521, 529–530, 535,
542
weak coupling, 519, 531–532, 534
Fourier transformation, 126, 162, 224, 356,
357, 499–501, 504, 508, 515, 526,
549, 550
Fourier–Laplace transform, 193
575
galaxy, VIII
Galerkin–gradient/least squares method
(GGLS), 51
Galerkin/least squares method (GLS), 51
Gaussian elimination, 334, 338, 394
general Kirchhoff–equation for a moved
body, 505
Generalized finite element method (GFEM),
51
Green’s function, 17, 50, 287, 373, 459, 462,
486–491, 551
complex source point, see CBEM
free–space, 9, 119, 370, 413, 415, 419,
462, 465
half–space, 459–476, 480, 491, 492
hard–wall, 119
modified, 419, 420
normal derivative, 413, 482–486
h–FEM, 71–73
half–space, 151, 179, 406, 459–494, 497
fundamental solution in time–domain, 502
Hankel function, 102, 156, 208, 347, 460,
469, 524
Hankel transform, 467, 468
hierarchical finite element basis functions,
67–70
hitchhiker’s guide, VIII
holography, 547–572
identification, 28, 547–572
ill–conditioned matrix, 91, 96, 235, 237,
338, 341, 362, 415, 424, 531, 555
ill–conditioning, 51, 96, 240, 322, 414, 422
impedance matrix, 26, 367, 369, 371,
376–379, 381, 382, 384
impedance plane, 459–494
masslike, 464–468, 478–480, 486–489
springlike, 467–471, 488, 489
incidence
angle of, 150, 168, 181, 182
close to normal, 153
grazing, 481
oblique angle, 154, 167, 170, 179, 180
perpendicular, 460, 474–476, 488–491
table, 528
incident wave, 1, 2, 16–20, 28, 146, 180,
396, 440, 459, 461
infinite boundary elements, 420
576
Index
infinite elements, 21, 23, 38, 145, 151, 155,
160–161, 168, 197–250, 420
Astley–Leis elements, 27, 205, 207,
214–215, 218, 221, 222, 231–234,
236–241, 243, 246, 247, 406
Bettess–Burnett elements, 205, 217–218,
221, 222, 232
conjugated, 27, 205–207, 212, 214–215,
218–224, 227, 228, 232, 406
mapped wave envelope elements,
215–219, 231, 232, 236
unconjugated, 205–206, 212–215,
217–219, 221–223, 228, 232
insertion loss, 395
interface acoustic damping material, 253,
255
inverse acoustics, 28, 548
inverse boundary element method (IBEM),
22, 547–572
inverse Fourier transformation, 162, 163,
224, 225, 234, 357
inverse Laplace transform, 158, 496, 502
inverse problem, 548, 568
irregular frequencies, 21, 23, 241, 311,
411–436, 452–453, 457
iterative solver, 21, 23, 57, 73, 89, 96, 105,
231, 232, 235, 239, 241, 246, 247,
316, 330, 333–366, 423, 424, 519,
520, 531, 537, 539, 541
Jacobi polynomial, 24, 231–250
Krylov subspace methods, 232, 235,
333–344, 361
Arnoldi method, 106, 335–337, 339
BiCG, 337, 339
BiCG–Stab, 235, 341, 343, 523
BiCGStab, 337
BiCGStab(l), 337
BiCGStab2, 337
CG, 336, 337
CGNE, 337
CGNR, 337
CGS, 337, 340, 344
GMRes, 235, 336, 337, 340, 344, 345,
361, 423, 424, 523, 531, 532, 537–541
GPBiCG, 337, 342, 344
Lanczos method, 21, 91–95, 100, 108,
109, 335–337, 339
QMR, 105, 235, 339
TFQMR, 235, 243–247, 337
L–curve criterion, 557
Lagrangian elements, 20, 23, 57, 68, 217,
310–312, 329
Lanczos basis, 91, 93, 100
Lanczos type algorithm, 94, 100, 107
Landweber iteration method, 557
Laplace transform, 158, 476, 497, 499, 500,
504, 507, 508, 514
Legendre function, see Legendre polynomial
Legendre polynomial, 24, 68–70, 76, 208,
223, 237, 238, 240, 241, 243–245,
309, 311, 314–316, 319, 323, 326,
329, 330, 347, 423, 524
Lighthill’s acoustic analogy, 21, 115–142
line integral, 213, 214, 460, 465–472, 491
Mach number, 116, 121, 127, 128, 139, 220,
221, 503
matched asymptotic expansion (MAE), 128,
131
mean flow, 121, 221, 389, 520, 533
medium frequency range, see mid–frequency
range
memory consumption, see memory
requirement
memory requirement, 22, 27, 57, 58, 74,
76, 80–83, 96, 220, 246, 310, 320,
333, 336–339, 343, 344, 358, 360,
362, 367, 381, 384, 395, 436, 520,
531–541, 568
mid–frequency range, VII, 23–25
mirror source method, 460, 464
modal synthesis, 253, 255, 270–275, 284
model reduction, 21, 27, 520
momentum conservation law, see balance of
momentum
momentum equation, see balance of
momentum
moving sound source, 502, 503, 505
multi–domain BEM, 367–386, 420
multifield analysis, 26, 28
multifrequency analysis, 26, 89–114
multilevel fast multipole algorithm, see
multilevel fast multipole analysis
(MLFMA)
Index
multilevel fast multipole analysis (MLFMA),
21, 333–366, 519–546
nearfield acoustic holography (NAH),
547–572
Neumann problem, see Neumann boundary
conditions
Newmark scheme, 125, 226, 234, 246, 280,
284
non–existence difficulty, 412
non–reflecting boundary conditions, see
absorbing boundary conditions (ABC)
non–uniqueness difficulty, 411, 412, 477
operations, see complexity estimate
optimization
absorbing function (of PML), 169
sound package, 288
structural–acoustic, 2, 25–27
order of convergence, see convergence rate
p–FEM, 57, 58, 66–73
Pad´e approximation, 21, 89–93, 95, 96, 100,
105, 153, 154, 162
Pad´e–via–Lanczos, 89–114
panel clustering, 346
Partition of unity method (PUM), 51, 86,
228
perfectly matched layer (PML), 21, 23, 39,
72, 84, 85, 123, 124, 134, 138, 139,
145, 157–161, 164, 167–196, 198, 231
in Cartesian coordinates, 158, 167–186
in polar coordinates, 169, 186–192
Petrov–Galerkin formulation, 50, 233, 336,
337
phase lag, 47, 48
plane wave solution, 19, 490, 491
pollution effect, 20, 37–56, 58, 198, 309,
321, 323, 330
polynomial approximation, 76, 387
incomplete linear, 264
Lagrange, 58, 62, 65–68, 74, 77, 217, 312
orthogonal, 228
Jacobi, 231–250
Legendre, 68–70, 76, 208, 223
Spline, 58, 65, 66
polynomial chaos approximation, 24, 25
poroelastic material, 28, 168, 497, 498, 512,
513
577
porous media, 28, 61, 275, 498
preconditioner, 235–236, 338–344, 531–532
approximate inverse (AIP), 531, 532
block diagonal, 537
block incomplete LU–factorization, 235,
244
diagonal, 338, 539
incomplete Cholesky factorization (IC),
338
incomplete LU–factorization (ILU), 338,
531, 532, 537, 539
ILU(0), 235, 246, 339
ILU(k), 339
ILUT(τ, p), 339, 342, 344
multigrid methods, 96
sparse approximate inverse (SPAI), 339,
537, 539
SSOR, 96, 105
pseudo inverse, 550, 553
quadrature, 486–487
adaptive multigrid, 460, 486, 487, 491
Gauss–Laguerre, 460, 468, 478–480, 486,
487
Gauss–Legendre, 63, 218, 329, 421
Gauss–Lobatto, 73, 76, 77, 80, 81
Newton–Cotes, 76
radial basis function, 232, 233
radial element order, 197, 220–225, 227,
237, 239, 241, 243–247
radial PML method, see perfectly matched
layer in polar coordinates
radiation modes, 27
Raviart–Thomas elements, 20, 253, 254,
264, 267, 268, 276, 280
reconstruction
inverse holographic, 549
of impedance matrix, 26, 27
of sound field, 549
of source parameters, 549, 559, 565, 568
of velocity field, 553
regular grid method (RGM), 340
regularization, 550, 555–560, 565
resolution, 37–56, 65, 66, 83, 117, 128, 135,
137, 198, 207, 241, 360, 392, 550, 565
Reynolds number, 116, 117, 130
rigid plane, 463, 464, 479, 488, 489, 502
578
Index
scattering, 20, 38, 146, 147, 161, 167–169,
176–193, 197–199, 209, 220, 310,
333, 346, 367, 368, 411, 417, 420, 422,
432, 435–439, 453, 457, 459–464,
470, 491, 548, 549, 564
Shannon’s sampling theorem, 310
singular value decomposition (SVD), 550,
552, 556, 557
soft plane, 463, 479, 488
Sommerfeld
identity, 467
mirror source method, 464–466
radiation condition, 8, 9, 18, 21, 145–149,
168, 199–201, 210, 232, 233, 237,
238, 246, 411, 439, 459, 520
sound power, 26, 27, 72, 395, 411, 430, 431,
461, 560, 561, 563, 564
source
dipole, 21, 116, 121, 477, 534, 551, 552
mirror, 463–465
monopole, 10, 16–19, 21, 121, 470, 473,
477, 488, 491, 532
quadrupole, 21, 116, 127, 128
with complex source points, 469–473,
480, 483
source identification, 548, 550, 559, 568
spectral finite element method (SFEM), 21,
57, 58, 73–84, 387–408
spherical Bessel function, 347, 415
spherical Hankel function, 102, 208, 347
spherical harmonics, 347, 351, 352, 356
statistical energy analysis (SEA), 25, 288,
289, 294–298, 302
storage, see memory requirement
superconvergence, 223, 309, 311, 319–322
thin–body BEM, 367–386
Tikhonov regularization, 557
time domain
BEM, 21, 495–516
classical approach, 495–497, 505–509
convolution quadrature method (CQM),
495–497, 507–509, 511, 512, 514–515
FEM, 21, 75
ABC, 149, 150, 152
aeroacoustics, 117, 125, 130, 133, 134
DtN ABC, 156
FSI, 253, 255, 279–284
infinite elements, 224–228, 231, 232,
234, 246–247
PML, 158, 170–176, 193
SFEM, 58, 73–75
radiation condition, 147–149
transient, see time domain
transmission loss, 379, 389, 395–399
Trefftz method, 52
ultra weak variational formulation (UWVF),
52, 86, 228
uncertainty analysis, 23–25
variational multiscale framework (VMS),
48–50
viscoelastic material, 28, 275, 284, 512
viscous damping, 157
viscous effects, 254
viscous fluid, 495, 497–502, 511, 513
viscous stress tensor, 120
wave
compression, 497
creeping, 226
cylindrical, 39, 147
elastic, 152, 153, 160
evanescent, 150, 159, 554
plane, 19, 39–42, 51, 52, 86, 147, 157,
159, 167, 170, 179–181, 185, 200,
220, 228, 347, 378, 395, 490, 491
moderately damped, 290, 298, 304
pressure, 152, 177
radio, 467
seismic, 77
shallow water, 168, 221
shear, 152
spherical, 39, 147, 200, 207, 213, 419,
470, 549
spurious, 116
standing, 25
surface, 221, 467, 469, 472, 473
surface gravity, 204
volume, 469
waveguide boundary spectral finite elements,
see spectral finite element method
(SFEM)
This is the end (JM), 578

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