Restriction matrices for SGBEM applications

Abstract

In the context of linear elasticity, we consider a symmetric boundary integral formulation associated with a mixed boundary value problem defined on a domain 2 IR m , m = 2, 3, with piecewise smooth boundary . We assume that is mapped onto itself by a finite groupG of congruences having at least two distinct elements. It is easy to take advantage of this when the source fields and the boundary conditions share the symmetry, as it is the case for an even or odd parity with respect to a reflection plane, but the intuitive reasoning fails for complex geometry. However, it is still possible to take advantage of the symmetry owing to the group representation theory and a decomposition theorem. From the latter, using the symmetry one can reduce the original problem to a family of smaller ones defined on a reduced geometry. The aim of this paper is to present a systematic technique for exploiting symmetry in the numerical treatment of boundary integral equations with the symmetric Galerkin BEM (SGBEM). This technique will be based upon suitable restriction matrices strictly related to the mesh defined on the boundary. Hence, we can decompose the related symmetric Galerkin BEM problem into independent subproblems of reduced dimension with respect to the complete one. Shape functions for each subproblem can be obtained from classical BEM basis, ordered as a vector, applying suitable restriction matrices constructed starting from group representation theory.