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Stability and convergence of the penalty formulation for nonlinear magnetostatics

Herbert Egger, Felix Engertsberger, Klaus Roppert

Abstract

The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome. A remedy is to split the magnetic field into a known source field and the gradient of a scalar potential which is governed by a convex minimization problem. The penalty approach avoids the use of artificial potentials and Lagrange multipliers and leads to an unconstrained convex minimization problem involving a large parameter. We provide a rigorous justification of the penalty approach by deriving error estimates for the approximation due to penalization. We further highlight the close connections to the Lagrange-multiplier and scalar potential approach. The theoretical results are illustrated by numerical tests for a typical benchmark problem

Stability and convergence of the penalty formulation for nonlinear magnetostatics

Abstract

The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome. A remedy is to split the magnetic field into a known source field and the gradient of a scalar potential which is governed by a convex minimization problem. The penalty approach avoids the use of artificial potentials and Lagrange multipliers and leads to an unconstrained convex minimization problem involving a large parameter. We provide a rigorous justification of the penalty approach by deriving error estimates for the approximation due to penalization. We further highlight the close connections to the Lagrange-multiplier and scalar potential approach. The theoretical results are illustrated by numerical tests for a typical benchmark problem
Paper Structure (5 sections, 1 theorem, 23 equations, 1 figure, 1 table)

This paper contains 5 sections, 1 theorem, 23 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. Proof of the main theorem
  4. Observations and further results
  5. Numerical tests

Key Result

theorem 1

Let Assumption egger:ass:1 hold. Then the variational problems egger:eq:3 and egger:eq:7 each have a unique solution, and amounts to the unique (weak) solution of egger:eq:1--egger:eq:2. For any $\varepsilon>0$, also problem egger:eq:9 has a unique solution $\mathbf{h}^\varepsilon$ and $\|\mathbf{h}^\varepsilon\|_{H(\operatorname{curl})} \le C \|\mathbf{j}\|_{L^2}$. With $\mathbf{b}^\varepsilon =

Figures (1)

  • Figure 1: Left: Geometry of test problem. Middle: Magnetic field density $|\mathbf{h}|$ (values $0 \text{--} 30.000 \,\unit{\ampere}/\unit{\meter}$) computed by the finite element approxmation of \ref{['egger:eq:14']} on a mesh with $\text{nt}=58.240$ triangles and with polynomial order $\text{p}=2$. Right: Corresponding flux density $|\mathbf{b}| = \widetilde{w}_*'(|\mathbf{h})|$ (values $0$--$2 \, \unit{\tesla}$).

Theorems & Definitions (1)

  • theorem 1