Immersed boundary-conformal isogeometric methods for magnetostatics

TL;DR

This paper introduces and evaluates three non-conformal isogeometric discretization strategies for magnetostatics, reducing geometric complexity and preprocessing effort in complex electromagnetic simulations.

Contribution

The work extends isogeometric analysis to non-conformal discretizations with boundary-conformal quadrature, enabling efficient simulation of multi-material electromagnetic devices.

Findings

Union methods achieve high accuracy in magnetostatic problems.

Fully immersed approach struggles with field discontinuities.

Methods significantly reduce geometric preprocessing effort.

Abstract

Isogeometric analysis was proposed to bridge the gap between computer-aided design and numerical discretization. However, standard multi-patch isogeometric analysis mandates conformal discretizations across patch interfaces, posing challenges for multi-material domain problems. In the context of electric machines, this requirement becomes evident in a large number of patches needed to represent machines consisting of several domains and materials. In this work, we adopt, extend, and evaluate three non-conformal discretization strategies for magnetostatic problems: a fully immersed approach, the union with non-conformal patches, and the union with conformal layers. In all three methods, boundary-conformal high-order quadrature rules are employed for integration over trimmed boundary and interface elements. In the two union approaches, material regions are, as far as possible, represented by independent non-conformal spline patches that are embedded within a background patch and coupled weakly through Nitsche's method. In the latter framework, critical interfaces are additionally surrounded by conformal layers that enable the strong imposition of boundary conditions and improved resolution of interface behavior. The proposed approaches are assessed through several magnetostatic benchmark problems and an industrial application. The numerical results show that the union methods achieve highly accurate solutions, while the fully immersed approach struggles with discontinuities in field gradients across material interfaces. Nevertheless, these methods significantly reduce the geometric preprocessing effort compared to conventional, conformal multi-patch analysis and require substantially fewer patches. Overall, this demonstrates that our immersed boundary-conformal isogeometric framework possesses great potential for efficient simulation of complex electromagnetic devices.