An Effective Technique for Reducing Internal Points in RIBEM for Nonhomogeneous Media

Boundary element method (BEM) has been extensively applied to solve various engineering problems. However, conventional BEM is only attractive in solving linear and isotropic material problems. For varying coefficient and nonlinear problems, due to the difficulty of finding fundamental solutions to the governing equations of the problems, researchers have to use the fundamental solutions corresponding to linear isotropic problems. As a result, there are domain integrals appearing in the resulting integral equations. To evaluate the domain integrals, the domain of the problem has to be discretized into internal cells and this eliminates the inherent feature of BEM in that only boundary of the problem needs to be discretized into elements. The author presented a boundary-only element method in 2002 [1, 2], called Radial Integration Boundary Element Method (RIBEM) for solving nonlinear and nonhomogeneous problems. Since this method does not require internal cells and therefore the inherent feature of BEM can be retained, it has been extensively used [3] since it was proposed. Although RIBEM does not require internal cells to solve a nonhomogeneous or a nonlinear problem, it may need some internal points to improve the computational accuracy over the area where large displacement (or potential) gradients may occur. It has been shown that the number and distribution of internal points have certain influence on the computational results. This paper describes a technique to effectively reduce the number of internal points used in RIBEM by further developing the one-dimensional property of the radial integral of RIBEM for a multi-dimensional field quantity. In this technique, the effect of internal points is transferred to the boundary quantities by integration by parts in the radial integral and therefore the nonlinearity of a field quantity can be reduced significantly. Consequently, much few internal points are needed to achieve a satisfactory result. For some problems, even no internal points are needed to obtain an acceptable result. Finally, numerical examples are given to demonstrate the correctness and efficiency of the presented technique.