The Comparison Of Virtual Bem And Traditional Bem For 2-d Laplace Equation

Boundary element method (BEM) is a simple and effective numerical method to solve potential problems, but usually requires calculation of singular integral, even hyper-singular integrals. The traditional BEM based on potential theory takes the density function on the real boundary as unknown quantity, while the virtual BEM takes the virtual density function on the virtual boundary for unknown quantity. The virtual boundary elements method(VBEM) based on the idea that selecting a virtual boundary outside the domain under consideration, we can establish a virtual potential and a boundary integral equation, according to the known boundary conditions of the original problem on real boundary, we can get the virtual density on the virtual boundary, then, using the potential integral on virtual boundary we can calculate the solution of the original problem by the discrete form of virtual boundary integral expression. Since the field points and the source points are on different boundary, so that singular or hyper-singular integrals are avoid in the numerical computing process.In this paper, we compared the numerical results of virtual density function, virtual moment density function and the potential function under consideration obtained by traditional BEM and VBEM not only based on single-layer potential and also based on mixed-layer potential which is different from the VBEM based on single-layer potential[58] and on double-layer potential[38]. Through the analysis to the shape of the virtual boundary and the relation with real boundary, the different schemes adopted in our paper based on different virtual potentials validate the consistency of traditional BEM and virtual BEM.In order to compare the numerical results of the VBEM based on mixed-layer potential with the traditional BEM based on the same mixed-layer potential, we devised a program for computing the mixed-boundary value problems of 2-D Laplace equation. In our paper, we adopted a scheme of re-subdivision the element with singularity by reducing its length in power series to carry out the numerical integration of hyper-singular integrals.All computing program are coded by Fortran 90 programming language. The conclusions are verified by several numerical examples.