×

Convolution quadrature for the wave equation with impedance boundary conditions. (English) Zbl 1375.35395

Summary: We consider the numerical solution of the wave equation with impedance boundary conditions and start from a boundary integral formulation for its discretization. We develop the generalized convolution quadrature (gCQ) to solve the arising acoustic retarded potential integral equation for this impedance problem. For the special case of scattering from a spherical object, we derive representations of analytic solutions which allow to investigate the effect of the impedance coefficient on the acoustic pressure analytically. We have performed systematic numerical experiments to study the convergence rates as well as the sensitivity of the acoustic pressure from the impedance coefficients. Finally, we apply this method to simulate the acoustic pressure in a building with a fairly complicated geometry and to study the influence of the impedance coefficient also in this situation.

MSC:

35Q35 PDEs in connection with fluid mechanics
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35C05 Solutions to PDEs in closed form
76Q05 Hydro- and aero-acoustics

Software:

DLMF; HyENA
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Hagstrom, T.; Hariharan, S. I., A formulation of asymptotic and exact boundary conditions using local operators, Appl. Numer. Math., 27, 4, 403-416 (1998) · Zbl 0924.35167
[2] Alpert, B.; Greengard, L.; Hagstrom, T., Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal., 37, 4, 1138-1164 (2000) · Zbl 0963.65104
[3] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31, 139, 629-651 (1977) · Zbl 0367.65051
[4] Grote, M. J.; Sim, I., Local nonreflecting boundary condition for time-dependent multiple scattering, J. Comput. Phys., 230, 8, 3135-3154 (2011) · Zbl 1316.74024
[5] Berenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 2, 185-200 (1994) · Zbl 0814.65129
[6] Lubich, C., Convolution quadrature and discretized operational calculus I, Numer. Math., 52, 129-145 (1988) · Zbl 0637.65016
[7] Lubich, C., Convolution quadrature and discretized operational calculus II, Numer. Math., 52, 413-425 (1988) · Zbl 0643.65094
[8] Hackbusch, W.; Kress, W.; Sauter, S., Sparse convolution quadrature for time domain boundary integral formulations of the wave equation by cutoff and panel-clustering, (Schanz, M.; Steinbach, O., Boundary Element Analysis: Mathematical Aspects and Applications. Boundary Element Analysis: Mathematical Aspects and Applications, Lect. Notes Appl. Comput. Mech., vol. 18 (2006), Springer), 113-134 · Zbl 1298.65185
[9] Lopez-Fernandez, M.; Sauter, S. A., Generalized convolution quadrature with variable time stepping, IMA J. Numer. Anal., 33, 4, 1156-1175 (2013) · Zbl 1283.65091
[10] Banjai, L.; Lubich, C.; Melenk, J. M., Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math., 119, 1, 1-20 (2011) · Zbl 1227.65027
[11] Falletta, S.; Monegato, G.; Scuderi, L., A space-time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case, IMA J. Numer. Anal., 32, 1, 202-226 (2012) · Zbl 1242.65196
[12] Bamberger, A.; Ha-Duong, T., Formulation variationelle espace-temps pour le calcul par potentiel retardé d’une onde acoustique, Math. Methods Appl. Sci., 8, 405-435 (1986), and 598-608 · Zbl 0618.35069
[13] Ha-Duong, T., On retarded potential boundary integral equations and their discretization, (Ainsworth, M.; Davies, P.; Duncan, D.; Martin, P.; Rynne, B., Computational Methods in Wave Propagation, vol. 31 (2003), Springer: Springer Heidelberg), 301-336 · Zbl 1051.78018
[14] Ha-Duong, T.; Ludwig, B.; Terrasse, I., A Galerkin BEM for transient acoustic scattering by an absorbing obstacle, Int. J. Numer. Methods Eng., 57, 1845-1882 (2003) · Zbl 1062.76534
[15] Sauter, S.; Veit, A., A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions, Numer. Math., 123, 145-176 (2013) · Zbl 1262.65125
[16] Sauter, S.; Veit, A., Adaptive time discretization for retarded potentials, Numer. Math., 132, 3, 569-595 (2016) · Zbl 1336.65159
[17] Stephan, E. P.; Maischak, M.; Ostermann, E., Transient boundary element method and numerical evaluation of retarded potentials, (Bubak, M.; van Albada, G.; Dongarra, J.; Sloot, P., Computational Science - ICCS 2008. Computational Science - ICCS 2008, Lect. Notes Comput. Sci., vol. 5102 (2008), Springer: Springer Heidelberg), 321-330
[18] Lopez-Fernandez, M.; Sauter, S., Generalized convolution quadrature with variable time stepping. Part II: algorithm and numerical results, Appl. Numer. Math., 94, 88-105 (2015) · Zbl 1325.65184
[19] Lopez-Fernandez, M.; Sauter, S., Generalized convolution quadrature based on Runge-Kutta methods, Numer. Math., 133, 4, 743-779 (2016) · Zbl 1348.65181
[20] Banjai, L.; Rieder, A., Convolution quadrature for the wave equation with a nonlinear impedance boundary condition · Zbl 1419.65048
[21] Graber, P. J., Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J. Evol. Equ., 12, 1, 141-164 (2012) · Zbl 1250.35134
[22] Beale, J. T.; Rosencrans, S. I., Acoustic boundary conditions, Bull. Am. Math. Soc., 80, 1276-1278 (1974) · Zbl 0294.35045
[23] Diao, H., Adaptive Convolution Quadrature for the Wave Equation (2017), Inst. f. Mathematik, Universität Zürich, Ph.D. thesis
[24] Franck, A.; Aretz, M., Wall structure modeling for room acoustic and building acoustic FEM simulations, (Proc. of 19th International Congress on Acoustics (2007))
[25] Nagler, L.; Rong, P.; Schanz, M.; Estorff, O.v., Sound transmission through a poroelastic layered panel, Comput. Mech., 53, 4, 549-560 (2014) · Zbl 06327102
[26] Stratton, J., Electromagnetic Theory (1941), McGraw-Hill: McGraw-Hill New York · JFM 67.1119.01
[27] Friedman, M.; Shaw, R., Diffraction of pulses by cylindrical obstacles of arbitrary cross section, J. Appl. Mech., 29, 40-46 (1962) · Zbl 0108.40204
[28] Sayas, F.-J., Retarded Potentials and Time Domain Boundary Integral Equations: A Road Map (2016), Springer Verlag · Zbl 1346.65047
[29] Lubich, C., On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math., 67, 365-389 (1994) · Zbl 0795.65063
[30] Lubich, C.; Ostermann, A., Runge-Kutta methods for parabolic equations and convolution quadrature, Math. Comput., 60, 201, 105-131 (1993) · Zbl 0795.65062
[31] Lubich, C., Convolution quadrature revisited, BIT Numer. Math., 44, 503-514 (2004) · Zbl 1083.65123
[32] Laliena, A. R.; Sayas, F.-J., Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numer. Math., 112, 4, 637-678 (2009) · Zbl 1178.65117
[33] Filipe, M.; Forestier, A.; Ha-Duong, T., A time dependent acoustic scattering problem, (Mathematical and Numerical Aspects of Wave Propagation. Mathematical and Numerical Aspects of Wave Propagation, Mandelieu-La Napoule, 1995 (1995), SIAM: SIAM Philadelphia, PA), 140-150 · Zbl 0877.76067
[34] Lopez-Fernandez, M.; Sauter, S., Fast and stable contour integration for high order divided differences via elliptic functions, Math. Comput., 84, 293, 1291-1315 (2015) · Zbl 1311.65028
[35] Sauter, S.; Veit, A., Retarded boundary integral equations on the sphere: exact and numerical solution, IMA J. Numer. Anal., 34, 2, 675-699 (2013) · Zbl 1292.65103
[36] Veit, A., Numerical Methods for Time-Domain Boundary Integral Equations (2012), Inst. f. Math., Universität Zürich, Ph.D. thesis
[37] Kress, R., Minimizing the condition number of boundary integral operators in acoustics and electromagnetic scattering, Q. J. Mech. Appl. Math., 38, 323-341 (1985) · Zbl 0559.73095
[38] Nédélec, J. C., Acoustic and Electromagnetic Equations (2001), Springer: Springer New York · Zbl 0981.35002
[40] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Tables of Integral Transforms, vol. I (1954), McGraw-Hill Book Company, Inc.: McGraw-Hill Book Company, Inc. New York-Toronto-London, based, in part, on notes left by Harry Bateman · Zbl 0055.36401
[41] Erichsen, S.; Sauter, S. A., Efficient automatic quadrature in 3-d Galerkin BEM, Comput. Methods Appl. Mech. Eng., 157, 3-4, 215-224 (1998) · Zbl 0943.65139
[42] Messner, M.; Messner, M.; Rammerstorfer, F.; Urthaler, P., Hyperbolic and elliptic numerical analysis BEM library (HyENA) (2010), [Online; accessed 22-January-2010]
[43] Banjai, L.; Sauter, S., Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47, 1, 227-249 (2008) · Zbl 1191.35020
[44] (Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press: Cambridge University Press New York, NY), print companion to [39] · Zbl 1198.00002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.