Adaptive Cross Approximation in an elastodynamic Boundary Element formulation

Abstract

Elastodynamic phenomena can be effectively analyzed by using the Boundary Element Method (BEM), especially in unbounded media. However, for the simulation of such problems, beside others, two difficulties restrict the BEM to rather small or medium–sized problems. Firstly, one has to deal with dense matrices and secondly the treatment of the kernel functions is very costly. Several approaches have been developed to overcome these drawbacks. Approaches, such as Fast Multipole and Panel Clustering etc. gain their efficiency basically from an analytic kernel approximation. The main difficulty of these methods is that the so called degenerate kernel has to be known explicitly.Hence, the present work focuses on a purely algebraic approach, the adaptive cross approximation (ACA). By means of a geometrical clustering and a reliable admissibility condition, first, a so called hierarchical matrix structure is set up. Then each admissible block can be represented by a low–rank approximation. The advantage of the ACA is based on the fact that only a few of the original matrix entries have to be generated.As will be shown numerically, the presented approach is suitable for an efficient simulation of elastodynamic problems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)