A reduced scalar potential approach for magnetostatics avoiding the coenergy
Herbert Egger, Felix Engertsberger, Bogdan Radu
TL;DR
The paper addresses efficient numerical solution of nonlinear magnetostatic problems by contrasting vector-potential and scalar-potential formulations and introducing a modified scalar-potential approach. It presents a mixed formulation that uses the scalar potential as the primary unknown while aligning the Newton linearization and Schur-complement structure with the vector-potential method; it eliminates the flux variable after linearization, yielding a saddle-point system with a Poisson-like Schur complement. Numerical results show that the proposed method reduces Newton iterations by about half compared to the pure scalar-potential approach and achieves comparable linear-solve costs to the standard scalar-potential formulation, while using 3–6x fewer degrees of freedom than the vector-potential method. Tests on COMPUMAG TEAM problem 13 confirm the practical efficiency gains and demonstrate the approach's viability for large-scale nonlinear magnetostatic problems.
Abstract
The numerical solution of problems in nonlinear magnetostatics is typically based on a variational formulation in terms of magnetic potentials, the discretization by finite elements, and iterative solvers like the Newton method. The vector potential approach aims at minimizing a certain energy functional and, in three dimensions, requires the use of edge elements and appropriate gauging conditions. The scalar potential approach, on the other hand, seeks to maximize the negative coenergy and can be realized by standard Lagrange finite elements, thus reducing the number of degrees of freedom and simplifying the implementation. The number of Newton iterations required to solve the governing nonlinear system, however, has been observed to be usually higher than for the vector potential formulation. In this paper, we propose a method that combines the advantages of both approaches, i.e., it requires as few Newton iterations as the vector potential formulation while involving the magnetic scalar potential as the primary unknown. We discuss the variational background of the method, its well-posedness, and its efficient implementation. Numerical examples are presented for illustration of the accuracy and the gain in efficiency compared to other approaches.