Fast Multipole Accelerated Boundary Element Method for Problems in an Elastic Half-Space

Abstract eng:
Context. This contribution is concerned with the development of the Fast Multipole accelerated Boundary Element Method (FM-BEM) to simulate the propagation of elastic waves in semi-infinite media. In [1], we previously developed a FM-BEM based on the elastic full-space fundamental solution. This method has been shown to be very efficient to simulate elastic wave in semi-infinite media, but nevertheless suffers from the drawback of requiring a discretization of the free surface. In practice, the free-surface was truncated at a chosen radius "large enough" for achieving a good accuracy in the target domain of study. A large number of BEM degrees of freedom (DOFs) was thus required on the free-surface for the sole purpose of enforcing the traction-free boundary condition. To avoid the truncation issue, one can use a fundamental solution that intrinsically satisfies a tractionfree boundary condition on the free-surface. This formulation thus reduces the overall size of the BE model since the free-surface no longer requires discretization. The derivation and implementation of the corresponding fundamental solution [3] are involved. In particular, unlike its full-space counterpart, the half-space fundamental solution cannot be expressed in terms of simpler kernels (e.g. Laplace or Helmholtz fundamental solutions) having already-known multipole expansions. Multipole expansions of the elastic half-space fundamental solution thus cannot be obtained in a simple way. Methodology. In [2], we proposed an expansion of the elastic half-space fundamental solution in a form which achieves the separation of variables required by the Fast Multipole Method. This expansion is based on a decomposition of the fundamental solution and of the single-layer potential into three terms: the full-space fundamental solution, the image full-space fundamental solution and a complementary term to satisfy the traction-free condition on the free-surface. In this communication, we present a new FM-BEM based on this treatment of the half-space fundamental solution. Numerical efficiency of the new FM-BEM. The accuracy of this new FM-BEM (which does not require meshing the free surface) is compared to that of the FM-BEM based on the elastic full-space fundamental solutions (which does require meshing the free surface). Moreover, the numerical efficiency of both approaches is compared on seismology-oriented examples such as the scattering of plane waves by a cavity embedded in an elastic half-space. Despite the additional computational effort required by the evaluation of the proposed FM-capable form of the half-space fundamental solution, the new approach is shown to reduce several-fold the overall analysis time, thus establishing the overall benefit brought by removing the free-surface mesh.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket