An Illustration of the Boundary Element Approach

Two boundary element methods for two-dimensional problems governed by Laplace’s equation are presented in this chapter. The first method is based on the solution for a constant strength simple source P along a straight line segment in an infinite homogeneous region, and the second on the solution for a constant strength dipole source D along the segment. In both cases, it is assumed that the boundary of the region of interest can be approximated by N line segments—boundary elements—joined end to end. If the potential, say, is prescribed for each boundary element, a system of N simultaneous linear algebraic equations can be formed to find the N sources. Detailed calculations are provided for obtaining the boundary influence coefficients and several examples are worked to demonstrate the accuracy of the methods. The two boundary element methods in this chapter are called indirect, because the sources P and D are generally not physically meaningful. It is shown that the methods can also be viewed as numerical representations of integral expressions for single and double layer potentials.