Abstract
In this paper, we present a numerical method based on the coupling between a Curved Virtual Element Method (CVEM) and a Boundary Element Method (BEM) for the simulation of wave fields scattered by obstacles immersed in homogeneous infinite media. In particular, we consider the 2D time-domain damped wave equation, endowed with a Dirichlet condition on the boundary (sound-soft scattering). To reduce the infinite domain to a finite computational one, we introduce an artificial boundary on which we impose a Boundary Integral Non-Reflecting Boundary Condition (BI-NRBC). We apply a CVEM combined with the Crank–Nicolson time integrator in the interior domain, and we discretize the BI-NRBC by a convolution quadrature formula in time and a collocation method in space. We present some numerical results to test the performance of the proposed approach and to highlight its effectiveness, especially when obstacles with complex geometries are considered.
Funding source: Ministero dell’Università e della Ricerca
Award Identifier / Grant number: CUP E11G18000350001
Funding statement: This work was performed as part of the GNCS-INDAM 2020 research program Metodologie innovative per problemi di propagazione di onde in domini illimitati: aspetti teorici e computazionali. The third author was partially supported by MIUR grant Dipartimenti di Eccellenza 2018-2022, CUP E11G18000350001.
References
[1] B. Ahmed, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013), 376–391. 10.1016/j.camwa.2013.05.015Search in Google Scholar
[2] A. Aimi, L. Desiderio and G. Di Credico, Partially pivoted ACA based acceleration of the energetic BEM for time-domain acoustic and elastic waves exterior problems, Comput. Math. Appl. 119 (2022), 351–370. 10.1016/j.camwa.2022.05.024Search in Google Scholar
[3] A. Aimi, L. Desiderio, M. Diligenti and C. Guardasoni, A numerical study of energetic BEM-FEM applied to wave propagation in 2D multidomains, Publ. Inst. Math. (Beograd) (N. S.) 96(110) (2014), 5–22. 10.2298/PIM1410005ASearch in Google Scholar
[4] A. Aimi, L. Desiderio, P. Fedeli and A. Frangi, A fast boundary-finite element approach for estimating anchor losses in micro-electro-mechanical system resonators, Appl. Math. Model. 97 (2021), 741–753. 10.1016/j.apm.2021.04.002Search in Google Scholar
[5] P. F. Antonietti, G. Manzini and M. Verani, The conforming virtual element method for polyharmonic problems, Comput. Math. Appl. 79 (2020), no. 7, 2021–2034. 10.1016/j.camwa.2019.09.022Search in Google Scholar
[6] E. Artioli, S. Marfia and E. Sacco, VEM-based tracking algorithm for cohesive/frictional 2D fracture, Comput. Methods Appl. Mech. Engrg. 365 (2020), Paper No. 112956. 10.1016/j.cma.2020.112956Search in Google Scholar
[7] A. Bamberger and T. H. Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique. I, Math. Methods Appl. Sci. 8 (1986), no. 3, 405–435. 10.1002/mma.1670080127Search in Google Scholar
[8] L. Banjai, Multistep and multistage convolution quadrature for the wave equation: Algorithms and experiments, SIAM J. Sci. Comput. 32 (2010), no. 5, 2964–2994. 10.1137/090775981Search in Google Scholar
[9] L. Banjai, Implicit/explicit, BEM/FEM coupled scheme for acoustic waves with the wave equation in the second order formulation, Comput. Methods Appl. Math. 22 (2022), no. 4, 757–773. 10.1515/cmam-2021-0186Search in Google Scholar
[10] L. Banjai, C. Lubich and F.-J. Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math. 129 (2015), no. 4, 611–646. 10.1007/s00211-014-0650-0Search in Google Scholar
[11] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199–214. 10.1142/S0218202512500492Search in Google Scholar
[12] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1541–1573. 10.1142/S021820251440003XSearch in Google Scholar
[13] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 727–747. 10.1051/m2an/2015067Search in Google Scholar
[14] L. Beirão da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method, Math. Models Methods Appl. Sci. 27 (2017), no. 13, 2557–2594. 10.1142/S021820251750052XSearch in Google Scholar
[15] L. Beirão da Veiga, A. Russo and G. Vacca, The virtual element method with curved edges, ESAIM Math. Model. Numer. Anal. 53 (2019), no. 2, 375–404. 10.1051/m2an/2018052Search in Google Scholar
[16] S. Berrone, A. Borio and F. Marcon, Comparison of standard and stabilization free virtual elements on anisotropic elliptic problems, Appl. Math. Lett. 129 (2022), Paper No. 107971. 10.1016/j.aml.2022.107971Search in Google Scholar
[17] S. C. Brenner, Q. Guan and L.-Y. Sung, Some estimates for virtual element methods, Comput. Methods Appl. Math. 17 (2017), no. 4, 553–574. 10.1515/cmam-2017-0008Search in Google Scholar
[18] S. Chaillat, L. Desiderio and P. Ciarlet, Theory and implementation of ℋ-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels, J. Comput. Phys. 351 (2017), 165–186. 10.1016/j.jcp.2017.09.013Search in Google Scholar
[19] B. Chen, F. Ma and Y. Guo, Time domain scattering and inverse scattering problems in a locally perturbed half-plane, Appl. Anal. 96 (2017), no. 8, 1303–1325. 10.1080/00036811.2016.1188288Search in Google Scholar
[20] M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements (invited contribution), Boundary Elements IX, Vol. 1 (Stuttgart 1987), Computational Mechanics, Southampton (1987), 411–420. 10.1007/978-3-662-21908-9_26Search in Google Scholar
[21] L. Desiderio and S. Falletta, Efficient solution of two-dimensional wave propagation problems by CQ-wavelet BEM: Algorithm and applications, SIAM J. Sci. Comput. 42 (2020), no. 4, B894–B920. 10.1137/19M1287614Search in Google Scholar
[22] L. Desiderio, S. Falletta, M. Ferrari and L. Scuderi, CVEM-BEM coupling with decoupled orders for 2D exterior Poisson problems, J. Sci. Comput. 92 (2022), no. 3, Paper No. 96. 10.1007/s10915-022-01951-3Search in Google Scholar
[23] L. Desiderio, S. Falletta, M. Ferrari and L. Scuderi, On the coupling of the curved virtual element method with the one-equation boundary element method for 2D exterior Helmholtz problems, SIAM J. Numer. Anal. 60 (2022), no. 4, 2099–2124. 10.1137/21M1460776Search in Google Scholar
[24] L. Desiderio, S. Falletta and L. Scuderi, A virtual element method coupled with a boundary integral non reflecting condition for 2D exterior Helmholtz problems, Comput. Math. Appl. 84 (2021), 296–313. 10.1016/j.camwa.2021.01.002Search in Google Scholar
[25] H. Eruslu and F. J. Sayas, Polynomially bounded error estimates for trapezoidal rule convolution quadrature, Comput. Math. Appl. 79 (2020), no. 6, 1634–1643. 10.1016/j.camwa.2019.09.020Search in Google Scholar
[26] S. Falletta and G. Monegato, An exact non reflecting boundary condition for 2D time-dependent wave equation problems, Wave Motion 51 (2014), no. 1, 168–192. 10.1016/j.wavemoti.2013.06.001Search in Google Scholar
[27] S. Falletta and G. Monegato, Exact non-reflecting boundary condition for 3D time-dependent multiple scattering–multiple source problems, Wave Motion 58 (2015), 281–302. 10.1016/j.wavemoti.2015.06.002Search in Google Scholar
[28] S. Falletta, G. Monegato and L. Scuderi, A space-time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case, IMA J. Numer. Anal. 32 (2012), no. 1, 202–226. 10.1093/imanum/drr008Search in Google Scholar
[29] S. Falletta, G. Monegato and L. Scuderi, A space-time BIE method for wave equation problems: The (two-dimensional) Neumann case, IMA J. Numer. Anal. 34 (2014), no. 1, 390–434. 10.1093/imanum/drs040Search in Google Scholar
[30] S. Falletta and S. A. Sauter, The panel-clustering method for the wave equation in two spatial dimensions, J. Comput. Phys. 305 (2016), 217–243. 10.1016/j.jcp.2015.10.033Search in Google Scholar
[31] G. N. Gatica and S. Meddahi, Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems, J. Numer. Math. 28 (2020), no. 4, 223–245. 10.1515/jnma-2019-0068Search in Google Scholar
[32] C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg. 79 (2009), no. 11, 1309–1331. 10.1002/nme.2579Search in Google Scholar
[33]
H. Gimperlein, C. Özdemir and E. P. Stephan,
A time-dependent FEM-BEM coupling method for fluid-structure interaction in
[34] H. Gimperlein, C. Özdemir and E. P. Stephan, Error estimates for FE-BE coupling of scattering of waves in the time domain, Comput. Methods Appl. Math. 22 (2022), no. 4, 839–859. 10.1515/cmam-2021-0162Search in Google Scholar
[35] D. Givoli, Numerical Methods for Problems in Infinite Domains, Stud. Appl. Math. 33, Elsevier Scientific, Amsterdam, 2013. Search in Google Scholar
[36] H. D. Han, A new class of variational formulations for the coupling of finite and boundary element methods, J. Comput. Math. 8 (1990), no. 3, 223–232. Search in Google Scholar
[37] C. Johnson and J.-C. Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), no. 152, 1063–1079. 10.1090/S0025-5718-1980-0583487-9Search in Google Scholar
[38] C. Lubich, Convolution quadrature and discretized operational calculus. II, Numer. Math. 52 (1988), no. 4, 413–425. 10.1007/BF01462237Search in Google Scholar
[39] G. Monegato and L. Scuderi, Numerical integration of functions with boundary singularities, J. Comput. Appl. Math. 112 (1999), 201–214. 10.1016/S0377-0427(99)00230-7Search in Google Scholar
[40] F.-J. Sayas, The validity of Johnson–Nédélec’s BEM-FEM coupling on polygonal interfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3451–3463. 10.1137/08072334XSearch in Google Scholar
[41] M. Schanz, Fast multipole method for poroelastodynamics, Eng. Anal. Bound. Elem. 89 (2018), 50–59. 10.1016/j.enganabound.2018.01.014Search in Google Scholar
[42] A. Sommariva and M. Vianello, Product Gauss cubature over polygons based on Green’s integration formula, BIT 47 (2007), no. 2, 441–453. 10.1007/s10543-007-0131-2Search in Google Scholar
[43] A. Sommariva and M. Vianello, Gauss–Green cubature and moment computation over arbitrary geometries, J. Comput. Appl. Math. 231 (2009), no. 2, 886–896. 10.1016/j.cam.2009.05.014Search in Google Scholar
[44] O. Steinbach, A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal. 49 (2011), no. 4, 1521–1531. 10.1137/090762701Search in Google Scholar
[45] F. Xie, Y. Qu, M. A. Islam and G. Meng, A sharp-interface Cartesian grid method for time-domain acoustic scattering from complex geometries, Comput. & Fluids 202 (2020), Paper No. 104498. 10.1016/j.compfluid.2020.104498Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Recent Advances in Boundary Element Methods
- Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)
- Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case
- CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries
- Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements
- Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems
- BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering
- Force Computation for Dielectrics Using Shape Calculus
- A Time-Adaptive Space-Time FMM for the Heat Equation
- Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
- High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations
- A Posteriori Error Estimates for Darcy–Forchheimer’s Problem
- Convergence of a Finite Difference Scheme for a Flame/Smoldering-Front Evolution Equation and Its Application to Wavenumber Selection
Articles in the same Issue
- Frontmatter
- Recent Advances in Boundary Element Methods
- Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)
- Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case
- CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries
- Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements
- Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems
- BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering
- Force Computation for Dielectrics Using Shape Calculus
- A Time-Adaptive Space-Time FMM for the Heat Equation
- Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
- High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations
- A Posteriori Error Estimates for Darcy–Forchheimer’s Problem
- Convergence of a Finite Difference Scheme for a Flame/Smoldering-Front Evolution Equation and Its Application to Wavenumber Selection