A method of harmonic extension for computing the generalized stress intensity factors for Laplace's equation with singularities

The solution of the Dirichlet problem for Laplace's equation on a special polygon is harmonically extended to a sector with the center at the singular vertex. This is followed by an integral representation of the extended function in this sector, which is approximated by the mid-point rule. By using the extension properties for the approximate values at the quadrature nodes, a well-conditioned and exponentially convergent, with respect to the number of nodes algebraic system of equations are obtained. These values determine the coefficients of the series representation of the solution around the singular vertex of the polygonal domain, which are called the generalized stress intensity factors (GSIFs). The comparison of the results with those existing in the literature, in the case of Motz's problem, show that the obtained GSIFs are more accurate. Moreover, the extremely accurate series segment solution is obtained by taking an appropriate number of calculated GSIFs.