Inversion Transformation for the Finite Element Solution of Three Dimensional Exterior-Field Problems

Abstract

A simple and efficient method for the finite-element solution of 3-D unbounded region field problem is presented in this paper. The proposed technique consists in a global mApplng of the original unbounded region onto a bounded domain by applying a standard inversion transformation to the spatial coordinates. Same numerical values of the potential function are assigned to the transformed points and the functional associated to the field problem, which incorporates the boundary condition, and has the same structure in the transformed domain as that in the original one. This allows the implementation of the standard Finite Element Method (FEM) in the bounded transformed domain.The finite-element solution is obtained on the basis of a complete discretization of the bounded, transformed domain by standard finite elements, with no approximate assumption made for the field behavior at infinity, other than that introduced by the finite-element idealization. This leads to an improved accuracy of the numerical results, as compared to those obtained in the original region, for the same number of nodes. Application to three test problems illustrates the efficacy of the proposed method In terms of both accuracy and computational error. The technique presented is particularly recommended for exterior-field problems in the presence of material inhomogeneties and anisotropies.