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.

中文翻译：

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在 COMSOL Multiphysics 中实现 H-$\phi$ 公式以模拟体超导体的磁化并与 H-公式进行比较

大量用于高温超导体模拟的 H 公式已被证明是一种非常通用且易于实现的模拟涉及超导材料的电磁现象的方法。然而，在无电流域中模拟全矢量场会不必要地增加模型的自由度，从而增加计算时间。在这个贡献中，我们实现了著名的 H-$\phi$ COMSOL multiphysics 中的公式，以比较 H 和 H-$\phi$计算体超导体磁化强度的公式。我们证明 H-$\phi$对于给定的相对误差，公式化可以将自由度数和计算时间减少近两倍。用两种公式获得的磁场精度被证明是相似的。H-的计算优势$\phi$公式显示远远超过其实施增加的复杂性，尤其是在 3-D 中。最后，我们确定了 H 和 H- 的理想元素阶数$\phi$ 公式在 2-D 中为四次，在 3-D 中为三次，对应于 COMSOL 中可实现的最高单元阶数。