[Paper Review] A globally conservative finite element MHD code and its application to the study of compact torus formation, levitation and magnetic compression

Carl Dunlea, Ivan Khalzov|arXiv (Cornell University)|Jul 31, 2019

Magnetic confinement fusion research16 references4 citations

TL;DR

This paper presents a globally conservative finite element code, DELiTE, for axisymmetric magnetohydrodynamics (MHD) using linear triangular elements and matrix-based discrete differential operators to ensure global conservation of mass, energy, toroidal flux, and angular momentum. The code successfully models compact torus formation, magnetic levitation, and compression, showing strong agreement with experimental diagnostics, particularly in flux conservation and toroidal field evolution during compression.

ABSTRACT

The DELiTE (Differential Equations on Linear Triangular Elements) framework was developed for spatial discretisation of partial differential equations on an unstructured triangular grid in axisymmetric geometry. The framework is based on discrete differential operators in matrix form, which are derived using linear finite elements and mimic some of the properties of their continuous counterparts. A single-fluid two-temperature MHD code is implemented in this framework. The inherent properties of the operators are used in the code to ensure global conservation of energy, particle count, toroidal flux, and angular momentum. The code was applied to study a novel experiment in which a compact torus (CT), produced with a magnetized Marshall gun, is magnetically levitated off an insulating wall and then magnetically compressed through the action of currents in the levitation/compression coils located outside the wall. We present numerical models for CT formation, levitation, and magnetic compression, and comparisons between simulated and experimental diagnostics.

Motivation & Objective

• To develop a finite element MHD code with intrinsic global conservation of mass, energy, toroidal flux, and angular momentum for axisymmetric systems.
• To address numerical instability and unphysical solutions in complex MHD simulations by embedding conservation laws at the discrete operator level.
• To model and simulate the formation, levitation, and magnetic compression of compact tori (CTs) in a novel experiment at General Fusion.
• To enable accurate simulation of plasma-wall interactions and coil-induced field effects by incorporating insulating walls and vacuum field coupling.
• To validate the numerical model against experimental diagnostics, particularly poloidal and toroidal magnetic field measurements.

Proposed method

• The DELiTE framework uses linear finite elements on unstructured triangular meshes to derive discrete differential operators in matrix form.
• Discrete operators are constructed to satisfy mimetic properties, including the discrete product rule and divergence theorem, ensuring conservation laws at the discrete level.
• The single-fluid two-temperature MHD equations are discretized by replacing continuous derivatives with matrix operators, preserving conservation globally.
• The poloidal vacuum field in the insulating region is solved separately and coupled to the plasma domain to maintain toroidal flux conservation.
• Explicit time stepping is used, with artificial diffusion added for numerical stability, limiting time step size and diffusion coefficient ranges.
• The code couples the plasma MHD solution with external coil currents and vacuum field solutions to model magnetic levitation and compression.

Experimental results

Research questions

• RQ1 Can a finite element MHD code be designed to globally conserve mass, energy, toroidal flux, and angular momentum in axisymmetric geometry?
• RQ2 How well can the DELiTE code reproduce experimental diagnostics during compact torus formation, levitation, and magnetic compression?
• RQ3 What is the impact of including an insulating wall and vacuum field coupling on plasma-wall interaction and flux conservation?
• RQ4 Why does the simulation show better poloidal flux conservation than the experiment during compression, and what causes the discrepancy?
• RQ5 How does the artificial diffusion in the scheme affect the accuracy of density and field diagnostics?

Key findings

• The DELiTE code successfully conserves total energy, particle count, toroidal flux, and angular momentum globally through mimetic discrete operators.
• Simulations reproduce the experimental observation that improved coil configuration reduces plasma-wall interaction, as evidenced by reduced wall contact during compression.
• The simulated poloidal field at inner probes continues to rise during compression, consistent with flux conservation, unlike the experimental collapse observed at ~145 μs.
• The simulated toroidal field ($B_\phi$) shows qualitative agreement with experimentally averaged $B_\phi$ measurements, especially during current diversion to lower inductance paths.
• The model captures the general trend of $B_\phi$ rise with increasing crowbarred shaft current, though axisymmetric modeling cannot reproduce toroidally asymmetric instabilities.
• The code's first-order accuracy and explicit time stepping limit simulation speed and stability, particularly for high diffusion coefficients, suggesting a need for implicit time integration in future work.