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

2024-07-31

arXiv.org

AI Summary

Isogeometric domain decomposition methods (IGA-DDM) for large-scale 3D magnetostatic problems suffer from gauge non-uniqueness and poor parallel scalability. Method: We propose a graph-theoretic tree–cotree decomposition algorithm that explicitly constructs a gauge-free formulation compatible with parallel computation; integrated with a dual-primal strategy, it yields a theoretically sound, numerically robust, and highly scalable parallel solver. The approach ensures well-posedness of all local subdomain problems, guaranteeing optimal convergence and high accuracy. Results: Numerical experiments demonstrate strong scalability on problems with over ten million degrees of freedom, significantly accelerating magnetostatic simulations of complex electromagnetic devices while preserving solution fidelity.

Technology Category

• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Distributed Problem Solving

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Isogeometric Tearing and Interconnecting
• Non-uniqueness Issue
• Parallel Computing Environment

Innovation

• Isogeometric Tearing and Interconnecting
• Tree-Cotree Decomposition
• Parallel Computing