CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries

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.

A virtual element method for the solution of 2D time-harmonic elastic wave equations via scalar potentials

“…As a future development, we aim at extending the analysis of this approach to curved VEM as well as to exterior elastic problems, combining the interior VEM with a boundary one, both in the time-harmonic and in the space-time case. In particular, for the space-time case, we aim at making use of the numerical scheme for the classical two dimensional wave equation proposed in [38].…”

Quadrature method and extrapolation algorithm for solving the boundary integral equations of the acoustic wave scattering problems

“…the boundary integral equations (BIEs) method (see [2], [7], [9], [14], [19], [29]- [32], [38]), the method of fundamental solution [25], the method that reduces the infinite domain to the finite domain by introducing the artificial boundary (see [3], [5], [10], [16], [35]- [37]). According to the method of artificial boundary, we shall get a finite computational domain and our problem can be solved in this domain by the finite difference method [16] or the finite element method (see [18], [37]) or the virtual element method [13], while the main disadvantage of this method is that we need much time to arrive at the results. Meanwhile, as for the scattering problem of a smooth open arc Γ in the unbounded domain, the authors considered introducing the energy weak formula and used the Galerkin boundary element method (BEM) to get some propeties of the solution [1].…”

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

“…Both space‐time Galerkin and convolution quadrature methods have been developed, see References 20–22 for an overview. Recent developments in the directions of the current work include stable formulations, 23–25 efficient discretizations, 26–31 compression of the dense matrices 32,33 as well as complex coupled and interface problems 34–41 …”

“…Recent developments in the directions of the current work include stable formulations, [23][24][25] efficient discretizations, [26][27][28][29][30][31] compression of the dense matrices 32,33 as well as complex coupled and interface problems. [34][35][36][37][38][39][40][41] To be specific, the method presented here reduces stochastic boundary value problems for the exterior acoustic wave equation to integral equations on the boundary of the computational domain. Using a polynomial chaos expansion of the random variables, a high-dimensional integral equation is obtained which is then discretized by a Galerkin method in space and time.…”