Finite Element Method for Electromagnetics on Riemannian Manifolds: Topology and Differential Geometry Toolkit

Abstract

This thesis applies new branches of mathematics in computational electromagnetics software. Namely, we consider the application of algebraic topology and differential geometry in finite element modeling. We conclude that from this approach, one can draw benefits to practical electromagnetic modeling. For example, more efficient numerical formulations, field-circuit coupling, and metric and coordinate free modeling techniques.

We present efficient methods for homology and cohomology computation of finite element meshes together with their software implementation. The presented homology and cohomology solver is a part of finite element mesh generator Gmsh. Therefore, its use can be easily incorporated into finite element modeling workflow.

We demonstrate the use of homology and cohomology computation results in static and quasistatic electromagnetic field problems. We describe finite element formulations which can be used in lumped parameter extraction from field problems and which can be naturally coupled to electronic circuit problems. Importantly, cohomology computation enables the use of magnetic scalar potential in eddy current problems without any topological restrictions, leading to more efficient and robust field computations.

Lastly, we present a finite element programming environment, where the language of differential geometry has the main role. We interpret the finite element model as a Riemannian manifold, and the fields of interest as differential forms. Using the environment, one can give the computational instructions in metric and coordinate free manner, as the used metric and coordinate system are provided separately. Then, the environment translates the instructions to the actual floating-point operations, which ultimately depend on the used metric and coordinate system. The programming environment implementation builds on top of the Gmsh API. That is, we implement tools from differential geometry which utilize an existing finite element framework.

The main contribution of this thesis is the development of these tools to the point where they can be readily exploited in computationally demanding engineering problems. Also, this thesis offers a unified exposition of the needed mathematical concepts and their relation to the electromagnetic field problem formulations.