Our problem is to find a solution *u* of Laplace's equation

subject to the boundary condition

regularity condition

and an asymptotic condition

where and stand for a solution of Laplace's equation in the whole space and the crack opening displacement, respectively. Physically, the function can be viewed as the ``no crack solution'' of the problem.

The solution *u* to this problem has an integral representation given by

where is the fundamental solution of Laplace's equation expressed as

From the boundary condition (3.12) and (3.14), one obtains the following hypersingular integral equation given by

where stands for the finite part of a divergent integral. Also, using Stokes' theorem, one can regularise (3.15) as (See Appendix L.1)

where stands for the Cauchy principal value of a singular integral. We use (3.16) for a direct computation of (3.15) in the conventional BIEM as well as in FM-BIEM. In the following sections we describe two types of FM-BIEM; one is FM-BIEM with the hypersingular integral equation and the other is FM-BIEM with the regularised integral equation.