Tree-Cotree Decomposition of Isogeometric Mortared Spaces in H(curl) on Multi-Patch Domains

TL;DR

This paper introduces a novel tree-cotree decomposition method for mortared isogeometric spaces in H(curl) on multi-patch domains, improving Maxwell's equations solutions in complex geometries.

Contribution

It presents a new approach to remove discrete kernel subspaces in mortared spline spaces using graph-theoretical tree-cotree decomposition, applicable to non-contractible domains.

Findings

Effective removal of kernel subspaces in mortared spaces

Applicable to complex, multi-patch geometries

Validated on a realistic electric machine model

Abstract

When applying isogeometric analysis to engineering problems, one often deals with multi-patch spline spaces that have incompatible discretisations, e.g. in the case of moving objects. In such cases mortaring has been shown to be advantageous. This contribution discusses the appropriate B-spline spaces needed for the solution of Maxwell's equations in the functions space H(curl) and the corresponding mortar spaces. The main contribution of this paper is to show that in formulations requiring gauging, as in the vector potential formulation of magnetostatic equations, one can remove the discrete kernel subspace from the mortared spaces by the graph-theoretical concept of a tree-cotree decomposition. The tree-cotree decomposition is done based on the control mesh, it works for non-contractible domains, and it can be straightforwardly applied independently of the degree of the B-spline bases. Finally, the simulation workflow is demonstrated using a realistic model of a rotating permanent magnet synchronous machine.