Integration of singular integrals for the Galerkin-type boundary element method in 3D elasticity

Abstract

The mixed boundary value problem in three-dimensional linear elasticity is solved via a system of boundary integral equations. The Galerkin approximation of the singular and hypersingular integral equations leads to (hyper)singular and regular double integrals. The numerical cubature of the singular integrals is discussed in the case of domains which have piecewise smooth surfaces with a Riemann metric structure. We give a method for reducing finite part integrals to at most Cauchy singular integrals. The presented integration method applied to Cauchy singular integrals leads to explicitly given regular integrand functions which can be integrated by standard Gaussian product quadrature rules. The number of necessary Gaussian knots depends on the shape of the boundary elements and on the curvature of the surface. We give estimates of the quadrature error including constants, which mark the properties of the boundary elements. The error estimates can be taken to implement adaptive cubature methods. The numerical example is a mixed boundary value problem with corner and edge singularities.