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A Finite Volume scheme for the solution of discontinuous magnetic field distributions on non-orthogonal meshes

Augusto Riedinger, Martín Saravia, José Ramírez

TL;DR

The paper addresses solving discontinuous magnetostatic distributions in media with permeability contrasts and magnetization on non-orthogonal meshes. It develops a Finite Volume formulation of the vector potential equation with a region-wise, multi-region solver, incorporating non-orthogonal gradient corrections and explicit discretization of induced and bound currents. A Block Gauss-Seidel partitioned solver with relaxation stabilizes the coupling across media, and results show good agreement with FEM across diverse geometries, validating accuracy and robustness. This approach enables robust magnetostatic simulations on complex geometries, facilitating practical applications in engineering and geophysics where curved interfaces and material discontinuities are common.

Abstract

We present a Finite Volume formulation for determining discontinuous distributions of magnetic fields within non-orthogonal and non-uniform meshes. The numerical approach is based on the discretization of the vector potential variant of the equations governing static magnetic field distribution in magnetized, permeable and current carrying media. After outlining the derivation of the magnetostatic balance equations and its associated boundary conditions, we propose a cell-centered Finite Volume framework for spatial discretization and a Block Gauss-Seidel multi-region scheme for solution. We discuss the structure of the solver, emphasizing its effectiveness and addressing stabilization and correction techniques to enhance computational robustness. We validate the accuracy and efficacy of the approach through numerical experiments and comparisons with the Finite Element method.

A Finite Volume scheme for the solution of discontinuous magnetic field distributions on non-orthogonal meshes

TL;DR

The paper addresses solving discontinuous magnetostatic distributions in media with permeability contrasts and magnetization on non-orthogonal meshes. It develops a Finite Volume formulation of the vector potential equation with a region-wise, multi-region solver, incorporating non-orthogonal gradient corrections and explicit discretization of induced and bound currents. A Block Gauss-Seidel partitioned solver with relaxation stabilizes the coupling across media, and results show good agreement with FEM across diverse geometries, validating accuracy and robustness. This approach enables robust magnetostatic simulations on complex geometries, facilitating practical applications in engineering and geophysics where curved interfaces and material discontinuities are common.

Abstract

We present a Finite Volume formulation for determining discontinuous distributions of magnetic fields within non-orthogonal and non-uniform meshes. The numerical approach is based on the discretization of the vector potential variant of the equations governing static magnetic field distribution in magnetized, permeable and current carrying media. After outlining the derivation of the magnetostatic balance equations and its associated boundary conditions, we propose a cell-centered Finite Volume framework for spatial discretization and a Block Gauss-Seidel multi-region scheme for solution. We discuss the structure of the solver, emphasizing its effectiveness and addressing stabilization and correction techniques to enhance computational robustness. We validate the accuracy and efficacy of the approach through numerical experiments and comparisons with the Finite Element method.
Paper Structure (21 sections, 64 equations, 15 figures)

This paper contains 21 sections, 64 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Magnetostatic formulation
  3. Conservation laws
  4. Ampere's law
  5. Permeable media and current carrying media
  6. Permanently magnetized media
  7. Combined form of the Ampere's law
  8. Vector potential formulation
  9. Interface boundary condition
  10. Numerical scheme
  11. Finite Volume formulation
  12. Laplacian discretization
  13. Induced and bound current discretization
  14. Solution scheme
  15. Numerical experiments
  16. ...and 6 more sections

Figures (15)

  • Figure 1: Depiction of neighboring elements in a non-orthogonal mesh.
  • Figure 2: Schematic of benchmark Case 1.
  • Figure 3: Benchmark Case 1 mesh.
  • Figure 4: Magnetic field SLIC for Case 1 – FEM solution.
  • Figure 5: Magnetic field SLIC for Case 1 – Present FVM solution.
  • ...and 10 more figures