Cell Based Volume Integration for Boundary Integral Analysis
The evaluation of volume integrals that arise in boundary integral formulations for non-homogeneous problems is considered. Using the 'Galerkin vector' to represent the Green's function, the volume integral is decomposed into a boundary integral plus a simpler volume integral wherein the source function is everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near-singular integrals are handled by partial analytic integration. The method employs a Galerkin approximation and is presented in terms of the 3D Poisson problem. An axi-symmetric formulation is also presented, and in this setting, the solution of a nonlinear problem is considered.