The Finite Element Method in Electromagnetics, 3rd Edition

CHAPTER 7EIGENVALUE PROBLEMS: WAVEGUIDES AND CAVITIES

In Chapters 3–5 we developed the finite element method in one, two, and three dimensions, respectively, and demonstrated its application to several deterministic problems in which either the governing partial differential equations or the boundary conditions or both were inhomogeneous. As we pointed out in Section 2.3.4, in addition to the deterministic problems, there is another class of boundary-value problems known as eigenvalue problems. In contrast to deterministic problems, both governing partial differential equations and boundary conditions are homogeneous in eigenvalue problems. From the physical point of view, this is to say that there is no source or excitation of any form in an eigenvalue problem. This class of problems can also be dealt with using the finite element method, and the resultant system of equations has the form of the generalized eigenvalue equation

(7.1)

where [A] and [B] are known matrices and λ and {ϕ} are unknowns. Rather than solving for {ϕ} for a nonzero right-hand side as done for a deterministic problem, here we solve for the eigenvalue λ, which makes the system singular, or in other words, which makes the determinant of [A − λB] vanish. As a result, there will be a corresponding nontrivial solution for {ϕ} which is called the eigenvector. For an eigenvalue problem of order N, there are ...