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A modified singular boundary method for three-dimensional high frequency acoustic wave problems. (English) Zbl 07166586
Summary: The main purpose of this article is to propose a modified singular boundary method using the modified fundamental solution of Helmholtz equation for simulation of three-dimensional high frequency acoustic wave problems. Compared with the standard second-order discretization methods which usually need to place more than 10–12 grid points in one wavelength per direction, the newly proposed modified singular boundary method only needs 2–3 source points in one wavelength of each direction to produce the accurate solutions with relative error around \(1E-3\) level which satisfies most application requirements. It is observed that the present algorithm has similar condition number to the boundary element method and can hereby be solved efficiently by the iterative solver. By adopting the range restricted generalized minimal residual algorithm iterative solver, numerical experiments with \(194,400\) source points have successfully been achieved on a single laptop for three-dimensional high frequency acoustic wave problems with up to wavenumber 440.

MSC:
65-XX Numerical analysis
76-XX Fluid mechanics
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[1] Fu, Z. J.; Chen, W.; Gu, Y., Burton-Miller-type singular boundary method for acoustic radiation and scattering, J. Sound Vib., 333, 3776-3793 (2014)
[2] Li, J. P.; Fu, Z. J.; Chen, W., Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method, Comput. Math. Appl, 71, 381-390 (2016)
[3] Chen, J. T.; Lee, Y. T.; Lin, Y. J., Analysis of mutiplE − shepers radiation and scattering problems by using a null-field integral equation approach, Appl. Acoust, 71, 690-700 (2010)
[4] Aziz, I.; Siraj-ul-Islam; Šarler, B., Wavelets collocation methods for the numerical solution of elliptic BV problems, Appl. Math. Model, 37, 676-694 (2013) · Zbl 1352.65661
[5] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method (1991), McGraw-Hill: McGraw-Hill New York
[6] Ernst, O. G.; Gander, M. J., Why it is difficult to solve Helmholtz problems with classical iterative methods, Lect. Notes Comput. Sci. Eng., 83, 325-363 (2012) · Zbl 1248.65128
[7] Zhang, J. M.; Lin, W. C.; Dong, Y. Q.; Ju, C. M., A double-layer interpolation method for implementation of BEM analysis of problems in potential theory, Appl. Math. Model, 51, 250-269 (2017)
[8] Brebbia, C. A., The birth of the boundary element method from conception to application, Eng. Anal. Bound. Elem., 77, iii-iix (2017) · Zbl 1403.65002
[9] Babuska, I. M.; Sauter, S. A., Is the pollution effect of the fem avoidable for the Helmholtz equation considering high wave numbers?, SIAM Rev., 42, 451-484 (2000) · Zbl 0956.65095
[10] Erlangga, Y. A., Advances in iterative methods and preconditioners for the Helmholtz equation, Arch. Comput. Methods Eng., 15, 37-66 (2008) · Zbl 1158.65078
[11] Ihlenburg, F.; Babuska, I., Finite element solution of the Helmholtz equation with high wavenumber part I: the h-version of the fem, Comput. Math. Appl., 30, 9-37 (1995) · Zbl 0838.65108
[12] Zhao, M. H.; Li, Y.; Yan, Y.; Fan, C. Y., Singularity analysis of planar cracks in three-dimensional piezoelectric semiconductors via extended displacement discontinuity boundary integral equation method, Eng. Anal. Bound. Elem., 67, 115-125 (2016) · Zbl 1403.74266
[13] Li, X. L.; Li, S. L., Analysis of the complex moving least squares approximation and the associated element-free Galerkin method, Appl. Math. Model, 47, 45-62 (2017) · Zbl 1446.65176
[14] Li, X. L., Meshless Galerkin algorithms for boundary integral equations with moving least square approximations, Appl. Numer. Math., 61, 1237-1256 (2011) · Zbl 1232.65160
[15] Cerrato, A.; Rodríguez-Tembleque, L.; González, J. A.; Aliabadi, M. H.F., A coupled finite and boundary spectral element method for linear water-wave propagation problems, Appl. Math. Model., 48, 1-20 (2017) · Zbl 07163389
[16] Giladi, E., Asymptotically derived boundary elements for the Helmholtz equation in high frequencies, J. Comput. Appl. Math, 198, 52-74 (2007) · Zbl 1102.65120
[17] Kim, S.; Shin, C. S.; Keller, J. B., High-frequency asymptotics for the numerical solution of the Helmholtz equation, Appl. Math. Lett, 18, 797-804 (2005) · Zbl 1073.65131
[18] Fan, C. M.; Huang, Y. K.; Li, P. W.; Lee, Y. T., Numerical solutions of two-dimensional stokes flows by the boundary knot method, CMES-Comput. Model. Eng. Sci., 105, 491-515 (2015)
[19] Sun, L. L.; Chen, W.; Cheng, A. H.D., One- step boundary knot method for discontinuous coefficient elliptic equations with interface jump conditions, Numer. Method Part. D. E., 32, 1509-1534 (2016) · Zbl 1354.65260
[21] Chen, W.; Li, J. P.; Fu, Z. J., Singular boundary method using time-dependent fundamental solution for scalar wave equations, Comput. Mech, 58, 717-730 (2016) · Zbl 1398.74457
[22] Sun, L. L.; Chen, W.; Cheng, A. H.D., Singular boundary method for 2D dynamic poroelastic problems, Wave Motion, 61, 40-62 (2016)
[23] Liu, Q. G.; Šarler, B., A non-singular method of fundamental solutions for two-dimensional steady-state isotropic thermoelasticity problems, Eng. Anal. Bound. Elem., 75, 89-102 (2017) · Zbl 1403.74312
[24] Šarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Eng. Anal. Bound. Elem., 33, 1374-1382 (2009) · Zbl 1244.76084
[25] Liu, L., Single layer regularized meshless method for three dimensional exterior acoustic problem, Eng. Anal. Bound. Elem., 77, 138-144 (2017) · Zbl 1403.74310
[26] Gu, Y.; Chen, W., Infinite domain potential problems by a new formulation of singular boundary method, Appl. Math. Model, 37, 1638-1651 (2013) · Zbl 1349.65686
[27] Qu, W. Z.; Chen, W.; Gu, Y., Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime, Comput. Math. Appl, 70, 679-690 (2015)
[28] Wang, F. J.; Chen, W.; Zhang, C. Z.; Lin, J., Analytical evaluation of the origin intensity factor of time-dependent diffusion fundamental solution for a matrix-free singular boundary method formulation, Appl. Math. Model, 49, 647-662 (2017) · Zbl 07163505
[29] Li, J. P.; Chen, W.; Gu, Y., Error bounds of singular boundary method for potential problems, Numer. Method Partial Differ Equation (2017)
[30] Li, J. P.; Chen, W.; Fu, Z. J.; Sun, L. L., Explicit empirical formula evaluating original intensity factors of singular boundary method for potential and Helmholtz problems, Eng. Anal. Bound. Elem., 73, 161-169 (2016) · Zbl 1403.65204
[31] Hansen, P. C., Regularization Tools version 4.0 for Matlab 7.3, Numer. Algorithms,, 46, 189-194 (2007) · Zbl 1128.65029
[32] Calvetti, D.; Lewis, B.; Reichel, L., GMRES-type methods for inconsistent systems, Linear Algebra Appl., 316, 157-169 (2001) · Zbl 0963.65042
[33] Bellalij, M.; Reichel, L.; Sadok, H., Some properties of range restricted GMRES methods, J. Comput. Appl. Math., 290, 310-318 (2015) · Zbl 1321.65046
[34] Dykes, L.; Reichel, L., A family of range restricted iterative methods for linear discrete ill-posed problems, Linear Algebra Appl., 436, 3974-3990 (2012) · Zbl 1241.65045
[35] Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018
[36] Greengard, L. F.; Huang, J. F., A new version of the fast multipole method for screened coulomb interactions in three dimensions, J. Comput. Phys., 180, 642-658 (2002) · Zbl 1143.78372
[37] Huang, J. F.; Jia, J.; Zhang, B., FMM-Yukawa: an adaptive fast multipole method for screened Coulomb interactions, Comput. Phys. Commun., 180, 2331-2338 (2009) · Zbl 1197.81019
[38] Li, W. W.; Chen, W.; Fu, Z. J., Precorrected-FFT accelerated singular boundary method for large-scale three-dimensional potential problems, Commun. Comput. Phys., 22, 460-472 (2017)
[39] Lai, J.; Ambikasaran, S.; Greengard, L. F., A fast direct solver for high frequency scattering from a large cavity in two dimensions, SIAM J. Sci. Comput., 36, B887-B903 (2014) · Zbl 1319.78008
[40] Greengard, L.; Rokhlin, V., The rapid evaluation of potential fields in three dimensions, (Greengard, C.; Anderson, C., Vortex Methods (1988), Springer-Verlag: Springer-Verlag Berlin), 121-141 · Zbl 0661.70006
[41] Greengard, L. F., The Rapid Evaluation of Potential Fields in Particle Systems (1988), MIT Press: MIT Press Cambridge · Zbl 1001.31500
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