INVERSION TRANSFORMATION FOR THE FINITE‐ELEMENT SOLUTION OF THREE‐DIMENSIONAL EXTERIOR‐FIELD PROBLEMS

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Abstract
A simple and efficient method for the finite‐element solution of three‐dimensional unbounded region field problems is presented in this paper. The proposed technique consists of a global mapping 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. The functional associated to the field problem, which incorporates the boundary conditions, has the same structure in the transformed domain as that in the original one. This allows the implementation of the standard finite‐element method 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 behaviour of the field at infinity, other than that introduced by the finite‐element idealization. This leads to improved accuracy of the numerical results, compared to those obtained in the original region, for the same number of nodes. Application to three test problems illustrates the high efficiency of the proposed method in terms of both accuracy and computational effort. The technique presented is particularly recommended for exterior‐field problems in the presence of material inhomogeneities and anisotropies.

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