On Domain Decomposition for Magnetostatic Problems in 3D

TL;DR

The paper tackles scalable 3D magnetostatic simulations by extending the dual-primal isogeometric tearing and interconnecting (IETI) framework to subdomains composed of multiple IGA patches and enabling load-balancing via graph partitioning with METIS. It leverages a vector-potential formulation $\operatorname{curl}(\nu\operatorname{curl} \boldsymbol{A}) = \boldsymbol{J}$ and a wire-basket coupling, together with tree-cotree gauging to select primal DOFs and a Dirichlet preconditioner for the interface solve. The main contributions are the formal extension to multipatch subdomains, a detailed gauge and primal DOF selection strategy ensuring local/global invertibility, and numerical evidence of optimal convergence, favorable conditioning, and scalable PCG performance. These results support scalable, accurate magnetostatic simulations for large-scale industrial contexts, including synchronous electric machines.

Abstract

The simulation of three dimensional magnetostatic problems plays an important role, for example when simulating synchronous electric machines. Building on prior work that developed a domain decomposition algorithm using isogeometric analysis, this paper extends the method to support subdomains composed of multiple patches. This extension enables load-balancing across available CPUs, facilitated by graph partitioning tools such as METIS. The proposed approach enhances scalability and flexibility, making it suitable for large-scale simulations in diverse industrial contexts.

Figures (6)

• Figure 1: Exemplary decomposition (exploded view) of a cube $\Omega=\left(0,3\right)^3$ consisting of 27 patches, which includes a visualization of the wire basket. The edges in the wire basket are marked in blue and green. Blue refers to those originating from interfaces while green denotes the parts of the wire basket originating from splitting the remaining subdomain boundaries into multiple facets.
• Figure 2: Weights to construct consistent spanning tree for application in magnetostatic TI problems.
• Figure 3: Exploded view of subdomain graphs corresponding to the decomposition shown in Fig. \ref{['fig:decomp']}.
• Figure 4: Exploded view of subdomain graphs corresponding to the decomposition shown in Fig. \ref{['fig:decomp']}.
• Figure 5: Scalability and convergence results with respect to mesh size for the domain configuration shown in Fig. \ref{['fig:decomp']} with PCG tolerance $\eta_{\mathrm{tol}}=10^{-6}$.