Conformal Mapping and the Finite Element Method

One of the interesting properties of the two-dimensional potential problem is that solutions of the Laplace equation remain solutions of the Laplace equation when subjected to a conformal transformation. While this result was established long ago, the consequences within computational mechanics have not been fully explored. Here, we demonstrate for the first time that in a finite element formulation of the potential problem, the stiffness matrix remains invariant under a conformal mapping. This holds even when the mapped domain extends to infinity. Furthermore, by introducing the local flux in a finite element method, we find that the fundamental boundary eigensolutions also are invariant under a conformal mapping transformation by using a special weight function related to the Jacobian of the transformation. A series of computational examples is presented to emphasize the most important characteristics of conformal mappings within the finite element method and to demonstrate convergence of the computational results. Included are two exterior problems, the latter of which permits determination of the tearing stress intensity factor for a crack in an infinite plate.