Lean Cohomology Computation for Electromagnetic Modeling

Abstract

Solving eddy current problems formulated by using a magnetic scalar potential in the insulator requires a topological pre-processing to find the so called first cohomology basis of the insulating region, which may result being very time consuming for challenging industrially driven problems. The physics-inspired Dłotko-Specogna (DS) algorithm was shown to be superior to alternatives in performing such a topological pre-processing. Yet, the DS algorithm is particularly fast when it produces as output not a regular cohomology basis but a so called lazy one, which contains the regular one but it keeps also some additional redundant elements. Having a regular basis may be advantageous over the lazy basis if a technique to produce it would take about the same time as the computation of a lazy basis. In literature such a technique is missing. This paper covers this gap by introducing modifications to the DS algorithm to compute a regular basis of the first cohomology group in practically the same time as the generation of a lazy cohomology basis. The speedup of this modified DS algorithm with respect to the best alternative reaches more than two orders of magnitudes on challenging benchmark problems. This demonstrates the potential impact of the proposed contribution in the low-frequency computational electromagnetics community and beyond.

Cuts for 3-D magnetic scalar potentials: Visualizing unintuitive surfaces arising from trivial knots

“…[5,22]). Efforts have been made for implementing general, automatic and efficient algorithms for such needs [6]. Indeed, utilization of cuts combined with their efficient automatic computation can, for instance, be a very significant difference-maker in efficiency when solving non-linear MQS problems; for a recent example, see e.g.…”

T-Ω Formulation with Higher-Order Hierarchical Basis Functions for Nonsimply Connected Conductors

“…In principle, one can use rigorous methods from standard algebraic topology, like the ones presented in [7], to compute representatives of a cohomology basis. These approaches are based on reduction of the complex K followed by a Smith Normal Form (SNF) computation on the boundary matrix of the reduced complex.…”

“…Yet, the set of representatives provided by the DS algorithm is a lazy cohomology basis [4,13]: the provided set of representatives span the needed cohomology group but contains additional, dependent elements. The size of the lazy basis is no more than twice the size of a standard cohomology basis and with moderate effort one may produce a standard cohomology basis (see [4] or the lean cohomology computation described in [14]) given a lazy one. However, it has been verified that these techniques which produce a standard cohomology basis do not provide any speedup in the solution of the electromagnetic problem while giving exactly the same solution in terms of induced currents up to linear solver tolerance.…”

Geometric construction of bases of $H_2(\overlineΩ, \partialΩ, \mathbb{Z})$

“…Thus, before the Smith normal form procedure is employed, the problem size is reduced using fast algorithms (usually algorithms that run in linear time) that remove homologically irrelevant parts of the triangulation (see e.g. [9], [20]). An implementation of these techniques have been integrated in the finite element mesh generator Gmesh by Pellikka et al (see [22]).…”