Implementation of the H-$\phi$ formulation in COMSOL Multiphysics for simulating the magnetization of bulk superconductors and comparison with the H-formulation

TL;DR

This paper compares the H and H-$$ formulations in COMSOL for simulating bulk superconductor magnetization, demonstrating that H-$$ reduces computation time significantly while maintaining similar accuracy.

Contribution

The paper implements the H-$$ formulation in COMSOL and compares its performance with the H-formulation for superconductors, highlighting computational efficiency gains.

Findings

H-$$ formulation reduces degrees of freedom and computation time by nearly 50%.

Both formulations produce similar magnetic field accuracy.

H-$$ formulation is especially advantageous in 3-D simulations.

Abstract

The H-formulation, used abundantly for the simulation of high temperature superconductors, has shown to be a very versatile and easily implementable way of modeling electromagnetic phenomena involving superconducting materials. However, the simulation of a full vector field in current-free domains unnecessarily adds degrees of freedom to the model, thereby increasing computation times. In this contribution, we implement the well known H-$\phi$ formulation in COMSOL Multiphysics in order to compare the numerical performance of the H and H-$\phi$ formulations in the context of computing the magnetization of bulk superconductors. We show that the H-$\phi$ formulation can reduce the number of degrees of freedom and computation times by nearly a factor of two for a given relative error. The accuracy of the magnetic fields obtained with both formulations are demonstrated to be similar. The computational benefits of the H-$\phi$ formulation are shown to far outweigh the added complexity of its implementation, especially in 3-D. Finally, we identify the ideal element orders for both H and H-$\phi$ formulations to be quartic in 2-D and cubic in 3-D, corresponding to the highest element orders implementable in COMSOL.