March 2014
Intermediate
876 pages
28h 17m
English
Content preview from The Finite Element Method in Electromagnetics, 3rd Edition
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CHAPTER 15SOLUTION OF FINITE ELEMENT EQUATIONS
For many problems in electromagnetics as well as in other engineering fields, the end result of the finite element formulation is a set of linear algebraic equations that can be written as
(15.1)
or more compactly as
in which A is an n × n square matrix, x represents the unknown vector that is to be determined, and b represents the known vector. (For convenience of description, the notations for matrices and vectors used in this chapter are different from those in all other chapters.) When the dimension of the matrix or the number of unknowns, denoted by n, is under several hundred thousand, there are many algorithms available for solving (15.2). However, for many practical applications, the dimension of the matrix can be as large as several million or even several billion, which is particularly true for three-dimensional applications. Consequently, when we seek a numerical solution with the aid of a digital computer, we often encounter two problems: a huge demand for computer memory and excessively long computing time. The former usually outweighs the latter, and to help understand its seriousness, note that it takes eight terabytes of memory to store a full matrix with a dimension of 1,000,000. Thus a question arises, ...
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