A Galerkin BEM for transient acoustic scattering by an absorbing obstacle

Abstract

SUMMARYThis paper deals with the numerical resolution of the so-called time-domain boundary integral equations. The scattering problem by an absorbing obstacle serves as the model for this discussion. A new system of retarded potential boundary integral equations (RPBIE) is set up to solve the problem. Using an energy identity, we are able to prove the unconditional stability for standard conforming Galerkin approximation of the RPBIE. The actual space-time boundary elements are described in some detail, and their implementation has conÿrmed that stability in a wide range of situations and for large run times never reached before.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…3 The Proposed 'Wave-Matching' BIE Here we propose a new BIE that is, like the time domain BIEs proposed in [47,48], derived from energy principles. The approach here will however differ because we are considering a frequency domain formulation.…”

“…The approach here will however differ because we are considering a frequency domain formulation. In particular, the formulations in [47] and [48] begin by considering the timederivative of the total system energy, but this is necessarily zero in a time-harmonic scheme, including for the cavity eigenmodes the aim is to prevent. Ergin et al [43] showed that their timedomain version of the Burton-Miller method allowed energy to escape from the cavity.…”

“…Chappell et al [46] analysed these schemes using a similar technique to that which was used to analyse frequency domain formulations, but the analysis in [43] focuses instead on the trapping of energy in the fictitious internal cavity. Such ideas were later taken further by deriving the Galerkin BEM formulation directly from an energy identity [47,48]; a similar approach will be adopted in this paper, but for frequency domain problems with oscillatory basis functions.…”

An Energy Interpretation of the Kirchhoff-Helmholtz Boundary Integral Equation and its Application to Sound Field Synthesis

“…can be shown to be equivalent to a boundary condition which permits energy flow out of the hypothetical enclosed cavity but not in [9]. Another time-domain BEM formulation, derived by applying the divergence theorem to instantaneous acoustic energy density in a volume, has been proven to be unconditionally stable [10], again emphasizing that examining acoustic energy flow may have advantages over examining only pressure. The authors have presented some early results showing a BEM method based on matching acoustic energy flow between waves [11]; at the heart of this is the notion that the kernel of the KHBIE can also be understood as a common-energy-flow testing metric.…”

Time-Dependent Lorentz-Mie-Debye Formulation for Electromagnetic Scattering From Dielectric Spheres (Invited Paper)

“…This has been numerically demonstrated in [45]. However, ongoing work is focused on studying late time stability for this system and extending it to analyzing stabilities of TDIEs as applied to electromagnetics -a long standing problem where significant literature exists [56][57][58][59][60][61][62].…”