A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration inv...

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Vydané v:	Numerische Mathematik Ročník 156; číslo 2; s. 463 - 532
Hlavní autori:	Caetano, A. M., Chandler-Wilde, S. N., Gibbs, A., Hewett, D. P., Moiola, A.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024 Springer Nature B.V
Predmet:	Acoustic scattering Acoustic waves Acoustics Basis functions Boundary element method Boundary integral method Convergence Estimates Fractals Function space Integral equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Regularity Screens Simulation Theoretical 45A05 28A80 46E35 65R20 65N38
ISSN:	0029-599X, 0945-3245
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Abstract	Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in R n + 1 ( n = 1 , 2 ) when the scatterer, assumed to be a compact subset of R n × { 0 } , is a d -set for some d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d . For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R 2 by Cantor sets, and in R 3 by Cantor dusts.
AbstractList	Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in $$\mathbb {R}^{n+1}$$ R n + 1 ( $$n=1,2$$ n = 1 , 2 ) when the scatterer, assumed to be a compact subset of $$\mathbb {R}^n\times \{0\}$$ R n × { 0 } , is a d -set for some $$d\in (n-1,n]$$ d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d . For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in $$\mathbb {R}^2$$ R 2 by Cantor sets, and in $$\mathbb {R}^3$$ R 3 by Cantor dusts. Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in Rn+1 (n=1,2) when the scatterer, assumed to be a compact subset of Rn×{0}, is a d-set for some d∈(n-1,n], so that, in particular, the scatterer has Hausdorff dimension d. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R2 by Cantor sets, and in R3 by Cantor dusts. Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in R n + 1 ( n = 1 , 2 ) when the scatterer, assumed to be a compact subset of R n × { 0 } , is a d -set for some d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d . For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R 2 by Cantor sets, and in R 3 by Cantor dusts.
Author	Hewett, D. P. Chandler-Wilde, S. N. Caetano, A. M. Gibbs, A. Moiola, A.
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Cites_doi	10.1007/BF02476031 10.1016/S0045-7949(96)00137-X 10.1016/j.jfa.2021.109019 10.1080/03605309108820799 10.1002/mma.1670130403 10.1007/s00211-021-01189-5 10.1142/S0218202522500075 10.1017/CBO9780511618635 10.1007/s00211-021-01182-y 10.1007/BF01396766 10.1007/978-0-387-09432-8 10.1006/jfan.1998.3383 10.1006/jcph.1994.1116 10.1137/17M1131933 10.1007/978-3-642-66451-9 10.1080/17476933.2011.557154 10.1080/00036818408839520 10.1017/S0013091500019076 10.1007/s10543-021-00859-y 10.1007/s00041-011-9202-5 10.1017/CBO9780511623738 10.1093/imanum/drab074 10.1007/s11075-022-01378-9 10.1142/S021953051650024X 10.1007/978-0-387-68805-3 10.1090/S0025-5718-03-01583-7 10.1093/imanum/8.1.105 10.1016/j.jmaa.2004.10.059 10.1017/S0962492912000037 10.1007/978-3-540-68093-2 10.1115/1.3424474 10.1007/s00020-017-2342-5 10.1007/978-3-0348-0034-1 10.1109/MAP.2003.1189650 10.1007/s11075-023-01705-8 10.4171/019 10.1016/j.jmaa.2019.01.036 10.1007/BF01199079 10.1112/mtk.12155 10.1080/02786820117868 10.1090/gsm/019 10.1007/s00020-015-2233-6 10.1112/S0025579314000278
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Snippet	Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be...
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SubjectTerms	Acoustic scattering Acoustic waves Acoustics Basis functions Boundary element method Boundary integral method Convergence Estimates Fractals Function space Integral equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Regularity Screens Simulation Theoretical
Title	A Hausdorff-measure boundary element method for acoustic scattering by fractal screens
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