Tree-Cotree-Based Tearing and Interconnecting for 3D Magnetostatics: A Dual-Primal Approach

TL;DR

This work presents a tree-cotree gauged, dual-primal Isogeometric Tearing and Interconnecting framework for 3D magnetostatics. By embedding the curl-curl problem into a domain-decomposition setting with high-order IGA spaces and a graph-based edge representation, the authors derive an explicit tree construction and a principled selection of primal DOFs to guarantee local invertibility and global solvability. The method couples local subdomain problems via Lagrange multipliers and uses Schur-complement techniques to form an efficient, scalable interface problem, with coarse-space size growing linearly with the number of subdomains. Numerical experiments on spherical and toroidal geometries demonstrate accuracy, optimal convergence, and scalable performance, including handling of nontrivial topology and mixed boundary conditions. The results indicate practical potential for large-scale magnetostatic simulations on parallel architectures, with avenues for extending to inhomogeneous materials and eddy-current problems.

Abstract

The simulation of electromagnetic devices with complex geometries and large-scale discrete systems benefits from advanced computational methods like IsoGeometric Analysis and Domain Decomposition. In this paper, we employ both concepts in an Isogeometric Tearing and Interconnecting method to enable the use of parallel computations for magnetostatic problems. We address the underlying non-uniqueness by using a graph-theoretic approach, the tree-cotree decomposition. The classical tree-cotree gauging is adapted to be feasible for parallelization, which requires that all local subsystems are uniquely solvable. Our contribution consists of an explicit algorithm for constructing compatible trees and combining it with a dual-primal approach to enable parallelization. The correctness of the proposed approach is proved and verified by numerical experiments, showing its accuracy, scalability and optimal convergence.

Figures (10)

• Figure 1: Visualization of wire basket (blue) and $\partial\Omega^{(i)}$ (orange) for one hexahedral subdomain in \ref{['fig:subGs_a']}. In \ref{['fig:subGs_b']} and \ref{['fig:subGs_c']}, the corresponding subgraphs for two connected subdomains are shown. There, blue indicates $\mathcal{G}_{\mathrm{\lambda}}^{(i)}$ and $\mathcal{G}_{\mathrm{\lambda}}$. Furthermore, orange together with blue indicates $\mathcal{G}_{\mathrm{\Gamma}}^{(i)}$ and $\mathcal{G}_{\mathrm{\Gamma}}$. The inner black nodes and edges belong to neither wire basket nor boundary graphs.
• Figure 2: Weight distribution for Kruskal's algorithm to generate a consistent tree. Tree generation for brown boxes can be parallelized for every subdomain. The blue boxes need global context and no parallelization is possible.
• Figure 3: Local graphs with prioritization, tree edges (blue) and Dirichlet sides marked in gray.
• Figure 4: Local graphs for a problematic configuration with tree edges (blue), cross-edges (orange) and Dirichlet facets marked in gray.
• Figure 5: Geometry and exemplary control mesh for spherical geometry including tree DOFs (blue), and primal DOFs (orange).

Theorems & Definitions (12)

• Remark 1
• Theorem 1
• proof
• Corollary 2
• proof
• Lemma 3
• proof
• Remark 2
• Theorem 4
• proof