The Virtual Boundary Element Method Based On Double Layer Potential

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,the singularity of integral can be avoided. We set a closed virtual curve, 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. In calculation, since the source points are on the physical boundary, and the integral points on virtual boundary respectively, so that the integral is no longer singular or hyper-singular. The formulations used in virtual boundary are the indirect boundary integrals.The virtual boundary method used before is based on single layer potential, in which the indirect unknown function is the density function distributed on the virtual boundary. And then the potential and its derivative on the physical boundary can be determined by virtual density function, which can be obtained according to the boundary conditions of the original problem.In this paper, we use another kind of virtual boundary elements method, which is based on double layer potential by using of the moment density distributed on the virtual boundary. Since there is a distance between the virtual boundary and physical one, the integral is no longer singular. It can be applied into various elliptic boundary condition problems.Firstly, we write the virtual boundary integral equation according to three boundary conditions for Laplace problem and solve it by constant elements. Secondly, for generalized Poisson equation, we combine virtual boundary method with RBFs to solve it. We use RBF approximation is employed to construct particular solution and VBCM for homogenous solution instead of domain discretization. Thirdly, for harmonic equation,we divide it into two Poisson equations by introducing intermediate function. And then, we can solve it by use of virtual boundary method and RBF, this approach can be used to solve nonlinear harmonic equation also. Finally, we write a program in Matlab and did some numerical tests, the numerical results of examples demonstrate that the scheme presented is effective and accurate.