The Virtual Boundary Collocation Method For 3D Laplace Equation

The boundary element method is an effective numerical method for solving potential problem, but it needs to calculate singular integrals. Especially, we will meet the hyper-singular integral when we calculate the normal derivative of a double layer potential. When virtual boundary method is used, we set a closed virtual surface, so called virtual boundary outside the domain under consideration. So the unknown virtual density function distributed on the virtual boundary can be determined by the boundary conditions given on the physical boundary via nonsingular integral equation, since the source points are on the physical boundary, and the integral points on virtual boundary respectively, the integral is no longer singular or hyper-singular.The virtual boundary element method has been used for solving so many problems successfully, but less numerical implementation for three dimensional problems. This paper based on the single and double layer potentials, gives the virtual boundary integral equations for different boundary value problems of 3D Laplace equation with the unknown function of density or moment density on the virtual boundary. The virtual boundary collocation method (VBCM) is applied with constant elements on virtual boundary and equal collocation nodes on real boundary, the integrations are carried out by 4-points Gauss quadrature formulae. Furthermore, we adopt a replace way to transform the double integral of the virtual boundary integral equations based on the single layer potential into the linear integrals on the boundary of each sub-domain, so that this kind of integrals can be calculated analytically.The numerical examples present the efficiency and the feasibility. A good accuracy can be obtained with less boundary nodes.