Introduction to Numerical Electrostatics Using MATLAB

12The Finite Element Method (FEM)

12.1 Introduction

The finite element method (FEM) is probably the most widely used numerical approximation technique for solving electrostatic problems. This is true, as well as magnetostatic, general electromagnetic, structural (elastic body), and other physical system analyses.

The finite element method is similar to the FD method for electrostatics in that the space where the electric field exists is discretized (meshed, or “zoned up”) and the electrodes, with applied voltages, are the boundary conditions. It is similar to the MoM in that it is readily amenable to complex geometries.

Because of its abilities to handle complex geometries and boundary conditions, the FEM is uniquely suited to structural analysis (airplane wings, bridges, vibrating structures, etc.). Historically, much of the formal development of the FEM occurred in the mechanical engineering community.¹ While electrostatics problems typically have fairly simple boundary conditions, they can have very complex structures, so the applicability and popularity of the FEM is well deserved.

The FEM formalizes and expands several topics that were foreshadowed in previous chapters:

1. The MoM technique employed both square and triangular cells over which the charge was simply approximated. The FD technique employed square cells over which voltage was simply approximated, with the approximation based on voltages at nodes common to several cells. The FEM may use many different shapes, in one, ...