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Enriched Absorbing Boundary Conditions for Acoustic Waves

Hélène Barucq 1, 2 Julien Diaz 1, 2 Véronique Duprat 1, 2 
2 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : The numerical simulation of scattering problems generally involves the truncation of the propagation domain and boundary conditions on the exterior boundary are then required. In some cases such as scattering problems by elongated obstacles, the size of the computational box can be reduced significantly if the external boundary is adapted to the shape of the obstacle. Hence, it is interesting to use boundary conditions which both approximate the behavior of infinity accurately and can be set on arbitrarily shaped surfaces. These conditions are called Absorbing Boundary Conditions (ABCs) when they satisfy the following properties: ABCs correspond to the approximation of a transparent condition, involve local or pseudo-local operators and minimize the reflections generated by the exterior boundary. It is worth noting that there are few works dealing with high-order ABCs for arbitrarily-shaped surfaces and high-order conditions are generally written for piecewise-flat surfaces and corner conditions must be introduced. Now ABCs are generally obtained from the approximation of the Dirichlet-to-Neumann operator in the propagation cone. It is thus possible that the performance of ABCs might be improved by broadening its spectrum to evanescent and/or grazing waves. Recently, a new ABC has been proposed by Hagstrom et al. It is an Improved Higdon ABC (IHABC) in which a differential operator is included to represent evanescent waves. The IHABC is efficient when coupled with a standard finite element method, but it seems to hamper the Courant-Friedrichs-Lewy (CFL) condition in particular when using a Discontinuous Galerkin Method (DGM) for the space discretization. Moreover, it is not easy to apply on arbitrarily-shaped boundaries because it requires supplementing conditions at each corner of the polyhedral boundary. In [1], we have designed an ABC which takes into account both propagating and evanescent waves and can be applied on arbitrarily-shaped convex boundaries. Numerical experiments have shown that the CFL condition is preserved. In this work, we present a new ABC that is obtained from the approximation of a complete transparent boundary condition modeling propagating, evanescent and grazing waves. It is pseudo-local since it involves a fractional derivative arising from the grazing part of the solution. However, it is easily included into an Interior Penalty Discontinuous Galerkin (IPDG) formulation of the acoustic wave equation. Moreover, it does not generate large computational costs because the band with of the discrete fractional operator involves a very low number of points. Numerical experiments illustrate the efficiency of the new condition both in the time and harmonic domains. They show that the absorption rate is improved when compared to classical ABCs, in particular for harmonic waves. We also perform long- time simulations to illustrate the robustness of the numerical scheme.
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Submitted on : Friday, December 21, 2012 - 7:15:24 PM
Last modification on : Tuesday, November 8, 2022 - 3:55:56 AM


  • HAL Id : hal-00768550, version 1


Hélène Barucq, Julien Diaz, Véronique Duprat. Enriched Absorbing Boundary Conditions for Acoustic Waves. Conference on Frontiers in Applied and Computational Mathematics, Jun 2011, Newark, United States. ⟨hal-00768550⟩



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