March 2014
Intermediate
876 pages
28h 17m
English
Content preview from The Finite Element Method in Electromagnetics, 3rd Edition
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CHAPTER 5THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS
So far, we have formulated the finite element method in one and two dimensions and demonstrated its application to a number of electromagnetic problems. While some physical problems can be represented or approximated by a one- or two-dimensional mathematical model, all physical problems are three dimensional in nature. When a one- or two-dimensional representation or approximation is not possible, a three-dimensional treatment becomes necessary. In this chapter we extend the finite element method to three dimensions in a very straightforward manner. First, we briefly formulate the finite element method for a general three-dimensional scalar problem. We then formulate useful higher-order and isoparametric elements and demonstrate application of the finite element method to specific problems in electromagnetics. We show that although the present finite element method is accurate and robust for solving the scalar potential in electrostatic problems, it is severely challenged for solving vector fields such as the magnetic vector potential and the electric and magnetic fields. These challenges and the resulting limitations of the current formulation are discussed extensively together with some potential solutions.
5.1 BOUNDARY-VALUE PROBLEM
The boundary-value problem under consideration is defined by the second-order differential equation
in conjunction with the boundary conditions
and
where S (= S1 + ...
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