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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-&#x03C6; formulation in COMSOL multiphysics in order to compare the numerical performance of the H and H-&#x03C6; formulations in the context of computing the magnetization of bulk superconductors. We show that the H-&#x03C6; 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-&#x03C6; 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-&#x03C6; formulations to be quartic in 2-D and cubic in 3-D, corresponding to the highest element orders implementable in COMSOL.

ieee transactions on applied superconductivity

Implementation of the H-<inline-formula><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> Formulation in COMSOL Multiphysics for Simulating the Magnetization of Bulk Superconductors and Comparison With the H-Formulation

Implementation of the H-$\phi$ Formulation in COMSOL Multiphysics for Simulating the Magnetization of Bulk Superconductors and Comparison With the H-Formulation