Degenerate scale problem in antiplane elasticity or Laplace equation for quadrilaterals with arbitrary configuration

This paper provides numerical solutions of the degenerate scale for shapes of quadrilaterals with arbitrary configuration in an exterior boundary value problem of antiplane elasticity or Laplace equation. The first step is to find the parameters in the Schwarz–Christoffel mapping. The first prevertex on the unit circle can be placed in a particular position, or at −1. From the single-valued condition of the mapping function, only one prevertex is independent. The real preverteces can be found from the condition that the computed ratio of two edges is equal to a ratio of two real edges assumed beforehand. An iteration is suggested to obtain the preverteces numerically. After those parameters are obtained, the degenerate sizes of four edges can be evaluated by a numerical integration. Several numerical examples and the computed results were provided.