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Structure-Preserving Transfer of Grad-Shafranov Equilibria to Magnetohydrodynamic Solvers

Rushan Zhang, Golo Wimmer, Qi Tang

TL;DR

This work tackles the problem of transferring axisymmetric Grad–Shafranov equilibria to 3D MHD solvers while preserving force balance and the divergence-free condition. It develops a structure-preserving transfer using compatible finite element spaces that form a discrete de Rham complex, enabling consistent loading of Ψ and f from the GS solver into B_p, B_t, J_p, and J_t for the MHD solver. The study identifies three key error sources—discretization incompatibility, mesh misalignment, and under-resolved separatrix gradients—and demonstrates through MFEM-based experiments that alignment and refinement, together with compatible element choices, minimize equilibrium perturbations. The findings provide practical guidance for initializing MHD simulations in fusion contexts and suggest future work on fully curl-conforming representations to further improve divergence preservation and force balance.

Abstract

Magnetohydrodynamic (MHD) solvers used to study dynamic plasmas for magnetic confinement fusion typically rely on initial conditions that describe force balance, which are provided by an equilibrium solver based on the Grad-Shafranov (GS) equation. Transferring such equilibria from the GS discretization to the MHD discretization often introduces errors that lead to unwanted perturbations to the equilibria on the level of the MHD discretization. In this work, we identify and analyze sources of such errors in the context of finite element methods, with a focus on the force balance and divergence-free properties of the loaded equilibria. In particular, we reveal three main sources of errors: (1) the improper choice of finite element spaces in the MHD scheme relative to the poloidal flux and toroidal field function spaces in the GS scheme, (2) the misalignment of the meshes from two solvers, and (3) possibly under-resolved strong gradients near the separatrix. With this in mind, we study the impact of different choices of finite element spaces, including those based on compatible finite elements. In addition, we also investigate the impact of mesh misalignment and propose to conduct mesh refinement to resolve the strong gradients near the separatrix. Numerical experiments are conducted to demonstrate equilibria errors arising in the transferred initial conditions. Results show that force balance is best preserved when structure-preserving finite element spaces are used and when the MHD and GS meshes are both aligned and refined. Given that the poloidal flux is often computed in continuous Galerkin spaces, we further demonstrate that projecting the magnetic field into divergence-conforming spaces is optimal for preserving force balance, while projection into curl-conforming spaces, although less optimal for force balance, weakly preserves the divergence-free property.

Structure-Preserving Transfer of Grad-Shafranov Equilibria to Magnetohydrodynamic Solvers

TL;DR

This work tackles the problem of transferring axisymmetric Grad–Shafranov equilibria to 3D MHD solvers while preserving force balance and the divergence-free condition. It develops a structure-preserving transfer using compatible finite element spaces that form a discrete de Rham complex, enabling consistent loading of Ψ and f from the GS solver into B_p, B_t, J_p, and J_t for the MHD solver. The study identifies three key error sources—discretization incompatibility, mesh misalignment, and under-resolved separatrix gradients—and demonstrates through MFEM-based experiments that alignment and refinement, together with compatible element choices, minimize equilibrium perturbations. The findings provide practical guidance for initializing MHD simulations in fusion contexts and suggest future work on fully curl-conforming representations to further improve divergence preservation and force balance.

Abstract

Magnetohydrodynamic (MHD) solvers used to study dynamic plasmas for magnetic confinement fusion typically rely on initial conditions that describe force balance, which are provided by an equilibrium solver based on the Grad-Shafranov (GS) equation. Transferring such equilibria from the GS discretization to the MHD discretization often introduces errors that lead to unwanted perturbations to the equilibria on the level of the MHD discretization. In this work, we identify and analyze sources of such errors in the context of finite element methods, with a focus on the force balance and divergence-free properties of the loaded equilibria. In particular, we reveal three main sources of errors: (1) the improper choice of finite element spaces in the MHD scheme relative to the poloidal flux and toroidal field function spaces in the GS scheme, (2) the misalignment of the meshes from two solvers, and (3) possibly under-resolved strong gradients near the separatrix. With this in mind, we study the impact of different choices of finite element spaces, including those based on compatible finite elements. In addition, we also investigate the impact of mesh misalignment and propose to conduct mesh refinement to resolve the strong gradients near the separatrix. Numerical experiments are conducted to demonstrate equilibria errors arising in the transferred initial conditions. Results show that force balance is best preserved when structure-preserving finite element spaces are used and when the MHD and GS meshes are both aligned and refined. Given that the poloidal flux is often computed in continuous Galerkin spaces, we further demonstrate that projecting the magnetic field into divergence-conforming spaces is optimal for preserving force balance, while projection into curl-conforming spaces, although less optimal for force balance, weakly preserves the divergence-free property.

Paper Structure

This paper contains 27 sections, 1 theorem, 69 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Grad--Shafranov equation
  4. 2D component-wise conversion of the magnetic potential to MHD quantities
  5. Finite element discretization
  6. Compatible finite element spaces
  7. Vector-CG spaces
  8. Numerical discretization
  9. Discretization of the magnetic field
  10. Discretization of the divergence of the magnetic field
  11. Discretization of the current density field
  12. Discretization of the Lorentz force
  13. Cylindrical coordinate projections
  14. Implementation
  15. Mesh
  16. ...and 12 more sections

Key Result

Proposition 4.1

Let $\mathbf{B}_p \in H(\textup{curl}, \mathcal{T}_{2D})_m$ be defined by discr_B_p_Hcurl. Further, assume the boundary of our 2D planar domain to be piecewise linear. Then $\mathbf{B}_p$ admits as discrete divergence $D_b \in CG(\mathcal{T}_{2D})_{m}$ such that $D_b \equiv 0$.

Figures (11)

  • Figure 1: First-order and second-order continuous Galerkin spaces on a 2D quadrilateral element, along with their corresponding curl- and divergence-conforming spaces, and discontinuous Galerkin spaces.
  • Figure 2: Illustration of forming 3D finite element spaces from tensor products of 1D finite element spaces and 2D finite element spaces, using the example of a 3D divergence-conforming space based on a 2D divergence-conforming space and a 2D DG space.
  • Figure 3: First-order continuous Galerkin space on a 3D hexahedral element, along with its corresponding curl- and divergence-conforming spaces, and discontinuous Galerkin space.
  • Figure 4: 2D meshes and input poloidal flux function used in the numerical experiments: (a) original GS solver mesh, (b) GS solver mesh with refinement along the separatrix, and (c) $\Psi$ on the original GS solver mesh. The visualizations of the meshes are colored differently by the regions, of which the white region, which is inside the plasma-facing wall, is our MHD equilibrium's plasma region of interest.
  • Figure 5: Comparison of the current density from different projection paths, for $\Psi, f \in CG(\mathcal{T}_{2D})_m$ and $m = 1$. Top row: Magnitude of poloidal component $\mathbf{J}_p$. (a) reference $\mathbf{J}_p \in H(\text{div}, \mathcal{T}_{2D})_m$ from \ref{['J_p_direct']}, (b) $\mathbf{J}_p \in H(\text{curl}, \mathcal{T}_{2D})_m$ from \ref{['eq:proj_path_A']}, (c) $\mathbf{J}_p \in H(\text{div}, \mathcal{T}_{2D})_m$ from \ref{['eq:proj_path_B']}, (d) $\mathbf{J}_p \in CG(\mathcal{T}_{2D})_m^2$ from \ref{['eq:proj_path_C']}. Bottom row: Toroidal component $J_t$. (e) reference $J_t \in CG(\mathcal{T}_{2D})_{m}$ from \ref{['J_t_direct']}, (f) $J_t \in CG(\mathcal{T}_{2D})_{m}$ from \ref{['eq:proj_path_A']}, (g) $J_t \in DG(\mathcal{T}_{2D})_{m-1}$ from \ref{['eq:proj_path_B']}, (h) $J_t \in CG(\mathcal{T}_{2D})_{m}$ from \ref{['eq:proj_path_C']}.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 4.1
  • proof