Analytical Solutions of Some Steady-state Electrical Problems in the Rectangular Domain

In potential theory, the boundary element method is recognized as a powerful numerical tool. Usually it works with the simple fundamental solution based on the wholeor half-space scalar Green function. In the case of a uniform and isotropic domain connected only to isolated regions, a Fredholm integral equation of the second kind with respect to the potential can be derived [l, 21 which is solved numerically by using piecewise polynomial trial functions [3-71 kept over every boundary elements. As is presented in this paper, if we practice numerical quadratures in computing the integral equation, then unacceptable results are obtained within the vicinity of the boundary surface. In the case of analytical integrations, on the other hand, we have highly accurate boundary-element solutions everywhere. However, analytical integrations are not always possible, and hence more complicated fundamental solutions should be considered in the boundary integral approach. This paper is devoted to the analytical solutions of electrical potential within the simple rectangular domain. We utilize the image method in conjunction with a conformal mapping, and the solutions are given in the form of infinite series. Examples are then presented under Dirichlet and Neumann boundary conditions. Through the weighted residual formulations, we further derive a general integral equation. Here attachment of the positioning constant to the governing equation is of great significance [8]. Under the Neumann boundary condition, we have the Fredholm integral equation of the second kind with respect to the potential. The problem is then solved numerically based on the whole-space scalar Green function. Typical finite-element solutions are also compared to exact values.