The Finite Element Method in Electromagnetics, 3rd Edition

APPENDIX DSINGULAR INTEGRAL EVALUATION

In the implementation of the finite element–boundary integral method, discussed in Chapter 10, one has to evaluate singular integrals. When triangular elements are used, the singular integrals have the following form:

(D.1)

where Δ denotes a triangular patch. These integrals are difficult to evaluate except for some simple f(r, r’) [1]. A common practice is to evaluate one set of integrals, say the one with respect to dS, using numerical integration such as Gaussian quadrature. The remaining integral becomes

(D.2)

where r_i denotes a point inside Δ. For some simple f(r_i, r’), this integral can be evaluated analytically [1–5]. However, when f(r_i, r’) is complicated, which is often the case with the use of higher-order basis functions and curvilinear elements, the integral must be evaluated numerically.

In this appendix we describe the method proposed by Duffy [6], which is suited for this purpose. The method can give an accurate solution and is relatively easy to implement. To illustrate this method, consider the triangle shown in Figure D.1a. Suppose that we wish to integrate a function that has a first-order singularity at the origin over this triangle:

Figure D.1 (a) Triangle with singularity at the origin. (b) Mapping of a triangle ...