Parallel Coupling of FEM and BEM for 3D Elasticity Problems

Abstract

In this paper we make use of the advantages of mixed variational FE/BE discretization and propose a parallel algorithm for solving the linear system of algebraic equations arise from discretizations of three dimensional linear elastostatic problems. The applied coupling technique was published in 1985 by Schnack [9]. The discretization leads to a linear system of algebraic equations, that can be reformulated in symmetric and indefinite form. This is very advantageous, because several efficient equation solvers can be applied. We can use Bramble/Pasciak’s conjugate gradient method [1], or we can derive a positive definite and symmetric system (with building the Schur complement) and solve it with standard direct or iterative methods. Here, we solve the Schur complement system parallel, based on recursive substructuring and LU factorization. To decompose the mesh into subdomains, we use the simple and efficient method of Farhat [4]. To obtain the BE Schur complement matrices, one has to solve a Neumann boundary value problem in each BE subdomain locally. The BE discretization results in a linear system, where the solution is increasingly sensitive to perturbations (e g roundoff errors) and one has to resort to special techniques. We use the singular value decomposition (SVD), that enables us to intelligently handle the problem and to compute the solution to the linear system.