Séance Séminaire

Séminaire ACSIOM

(Co)homology in electromagnetic modeling.

In the recent years, reformulating the mathematical description of physical laws in an algebraic form using tools from algebraic topology gained popularity in computational physics. Phyiscal variables are modeled as cochains with real or complex coefficients while physical laws are expressed in a metric-free fashion with the coboundary operator. The metric and the material information are encoded in the discrete counterpart of the constitutive laws of materials, that may be interpreted as discrete Hodge star operators. The discrete Hodge star operators, called also constitutive or material matrices, can be constructed thanks to dualities arising when two ``dual" interlocked cell complexes are introduced. In this framework, the constitutive matrices are geometrically defined just by using the geometric elements of the mesh. That is why this original technique may be considered as a ``Discrete Geometric Approach'' (DGA) to computational physics. The pedagogical and in some cases technical advantages with respect to Finite Elements and the potential for efficient implementations, as the GAME/CDICE/CAFE codes developed from 2003 at University of Udine, have made this approach nowadays attractive. After an introduction to the DGA, this seminar will focus on the solution of eddy-current problems, that are obtained by neglecting the displacement current in the Ampère-Maxwell equation. By swapping the associations of the physical variables to elements of the original complex and its dual, two complementary formulations naturally arise. The most efficient one, based on a magnetic scalar potential in the insulating domain, requires a careful definition of potentials involving the introduction of the so-called cuts. While an intimate connection of this issue with homology theory has been quickly recognized, heuristic and fuzzy definitions of cuts are surprisingly still dominant even in nowadays mathematical literature. We will first survey several definitions of cuts together with their shortcomings. Then, it will be shown that if cuts are defined as generators of the first cohomology group over integers, then all relevant physical laws are satisfied. This provably general definition possesses also the virtue of providing general and efficient algorithms for the automatic computation of cuts. The use of computational cohomology effectively solved what has been considered an open problem for more than 25 years in the low frequency electrodynamics community.