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Although the H-formulation has proven to be one of the most versatileformulations used to accurately model superconductors in the finite elementmethod, the use of vector dependent variables in non-conducting regions leadsto unnecessarily long computation times. Additionally, in some applications ofinterest, the combination of multiple magnetic components interacting withsuperconducting bulks and/or tapes leads to large domains of simulation. Inthis work, we separate the magnetic field into a source and reaction field anduse the H-$\phi$ formulation to efficiently simulate a superconductorsurrounded by magnetic bodies. We model a superconducting cube between a pairof Helmholtz coils and a permanent magnet levitating above a superconductingpellet. In both cases, we find excellent agreement with the H-formulation,while the computation times are reduced by factors of nearly three and four in2-D and 3-D, respectively. Finally, we show that the H-$\phi$ formulation ismore accurate and efficient than the H-A formulation in 2-D.

Efficient modeling of high temperature superconductors surrounded by magnetic components using a reduced H-$\phi$ formulation