The Singular Function Boundary Integral Method for Solving Three-Dimensional Laplacian Problems with Conical Vertex Singularities

Abstract

The Singular Function Boundary Integral Method (SFBIM) is extended to solve three-dimensional Laplacian problems with conical vertex singularities. The solution is approximated by the leading terms of the local asymptotic series in spherical coordinates. In order to calculate the unknown singular coefficients, i.e., the vertex stress intensity factors, the Laplacian problem is discretized by applying Galerkin’s principle. The governing equation is weighted by the local functions over the domain and the volume integrals are then reduced to surface ones by means of Green’s second identity. Given that the local solution satisfies identically the boundary conditions over the conical surface causing the vertex singularity, the dimension of the problem is reduced by one and the boundary integrals need to be calculated only far from the vertex singularity, which yields a considerable reduction of the computational cost. Neumann boundary conditions are weakly imposed and Dirichlet conditions are applied by means of Lagrange multiplier functions. The latter are approximated by means of finite elements over the corresponding boundary parts and the corresponding nodal values are thus additional unknowns. Preliminary results for a model are presented. The advantages and the limitations of the method are also discussed.