Chapter 8 - Finite-Element Time Domain Method

Abstract

In this chapter, a differential-equation-based time domain (TD) method, the finite-element time domain (FETD) method, is developed. Unlike the finite-difference time domain (FDTD) and transmission line modeling methods, the FETD can be used with unstructured grids that improves the modeling capabilities of arbitrary geometries. The finite-element method was originally applied for the approximate solution of boundary value problems. Much of the original stimulus in the development of finite-element methods came from energy or variational principles. Many of the finite-element methods implemented for initial value problems have been motivated by variational principles, but very few are strictly variational. Most of them either use variational principles in the space variables only, introducing the time variable at later stages, or use variational principles in space and time as a guide for the derivation of a functional that is not, in fact, stationary in the sense of the classical calculus of variations. These quasi-variational principles are well-known in other contexts such as heat conduction and thermodynamics. The advantage of the method of weighted residuals is that it can be applied to general equations, but it is not always clear how to choose the optimum weight factors. Stationary principles and quasi-variational principles help us to choose suitable weighting functions. These considerations are to some extent independent of the fact that finite-element methods usually involve weighting functions that are nonzero in only parts of the range of interest, which tends to produce stable numerical procedures.