Non-Abelian Symmetry in the 3-D Eddy-Current Analysis

Abstract

Geometrical symmetry is found in many electrical engineering problems. It is often the case in computational electromagnetics with finite element type methods. Symmetry can be taken into account in a rational way by using the Group Representation Theory that is scarcely used in this context. Briefly, it consists in reducing an original problem to a set of sub-problems, or “modes”, defined on a symmetry cell of the domain under study; the global solution is obtained from superposition of the partial ones. Recent experiences1,2 exist and show that there is much to be gained by way of economy in computation. Symmetry groups are of two types according as the composition of the isometries happens to be commutative or not so that they can be either abelian or non-abelian. This property is of great importance because the modes described above involve tedious coupled problems in the latter case. Further to a previous paper3 that only considered the abelian case, we propose the implementation of the 3-D eddy-current analysis in the general non-abelian case4. A mixed FEM-BEM method is used, the degrees of freedom are the H-circulations along the inner edges of the conducting regions (edge elements) and the nodal values of a scalar magnetic potential on their boundaries (boundary elements). An example points out that it is worth managing the non-abelian case in spite of a more difficult implementation since some substantial improvement is brought in the performances.