Finite Element Analysis in Unbounded Two-Dimensional Regions Using Schwarz-Christoffel Transformation

Abstract

The finite element method is one of the most widely used methods available to educators to compute electric potential fields in two dimensional geometries that can be described by partial differential equations such as Laplace's equation. However, it can not easily be applied to unbounded regions. This paper describes a hybrid method that relies on conformal transformations, including the Schwarz-Christoffel (S-C) transformation (without having to compute the transformation functions) to map the original boundaries, including those at infinity, to a bounded region and only then applies the finite element method. Testing this approach by means of an example for which an exact solution is obtainable, the hybrid method is applied to determine the electrical potential at a specific point in the field of an integrated circuit (IC) microstrip line. The results are in agreement with analytically derived results and can be used in graduate research as well as by professional engineers.