A Boundary Integral Method for Computing Eddy Currents in Thin Conductors of Arbitrary Topology

Abstract

We present an effective technique to solve eddy current problems in thin conductors of arbitrary topology by a boundary element method based on a stream function. By considering a mesh of the thin conductor, which we assume to be a surface (i.e., an orientable combinatorial two-manifold embedded in R 3 ), the aim of this paper is to introduce a novel technique to render the stream function single valued when the thin conductor is not topologically trivial. In particular, a novel combinatorial algorithm to compute the appropriate cohomology generators in linear time worst case complexity is introduced, providing an effective and rigorous solution for the required topological preprocessing.Index Terms-Boundary element method (BEM), cohomology, eddy currents, stream function, thin shields with holes.

A boundary element method for eddy-current problems in fusion devices

“…1b), i is the array of independent currents [8] and the columns of H store the representatives of the so-called generators of a suitable cohomology group [6]. A fast graph-theoretic algorithm with a linear worst-case complexity to compute matrix H is introduced in [6] together with more insights on the physical interpretation of the cohomology generators. We remark that the algorithm is very easy to implement being based on computations of spanning trees, therefore the knowledge of cohomology theory is not necessary to implement or use it.…”

“…That means one needs additional constraints for such cycles. Bettini and Specogna [6] show that to obtain a square symmetric and positive definite system one has to write additional Faraday's laws enforced on suitable cycles that, thanks to the duality between K andK, are obtained for free by using H T…”

Computation of Relative 1-Cohomology Generators From a 1-Homology Basis for Eddy Currents Boundary Integral Formulations

“…By modeling the conducting sheet as an orientable combinatorial 2-manifold K with boundary ∂K [2], such BI formulations require relative cohomology generators H 1 (K, ∂K) to make the problem well defined [3], [5]. For a formal introduction of algebraic topology please consult [2].…”

“…Nowadays, the state-of-theart technique in the context of BI formulations is to compute directly relative cohomology generators e.g. the combinatorial algorithm introduced in [5], which is general and exhibits a linear worst-case complexity. Yet, a different approach has also been considered in literature: relative cohomology generators may be obtained from homology generators in post-processing.…”

“…Digital Object Identifier XXX A practical advantage of the homology-based technique with respect to [5] is that one may use an off-the-shelf software to compute homology generators and then build the required relative cohomology generators with a simple and cheap procedure. In fact, there are many codes available for computing homology, whereas computing cohomologyespecially the relative one-is much less explored and as far as we know no efficient implementation with linear time complexity is available as open-source software, yet.…”