Linear-Time Solution of 3D Magnetostatics With Aggregation-Based Algebraic Multigrid Solvers

Abstract

Two different approaches for the linear-time solution of three-dimensional magnetostatic field problems encompassing large-scale linear systems with millions of degrees of freedom are presented. The $\varphi $ –method is formulated in terms of nodal variables, i.e., magnetic scalar potentials, and involves the solution of a curl-curl linear system for pre-processing and of a div-grad linear system. The A–method is formulated in terms of edge variables, i.e., line integrals of the magnetic vector potential, and involves the solution of a curl-curl linear system only. It is shown that these linear systems can be solved by flexible conjugate gradient in combination with the aggregation-based multigrid preconditioner (AGMG), tailored for div-grad problems, and with the aggregation-based (auxiliary-space) multigrid preconditioner (AGMG_CC), tailored for curl-curl problems. The robustness and efficiency of these solvers is illustrated using magnetostatic problems of practical interest, with linear or nonlinear media and having parts with complex geometry. Numerical results show that both AGMG and AGMG_CC are able to attain linear solution time in all cases considered, while being both faster and more robust than state-of-the-art algebraic multigrid solvers and preconditioned Krylov subspace solvers, typically adopted in commercial finite element software for electromagnetic analysis.

Keywords

• Finite element method
• cell method
• computer-aided engineering
• iterative solver
• algebraic multigrid
• auxiliary space preconditioner