Backward Euler method for single time step finite element simulation of the magnetization of high-temperature bulk superconductors with non linear conductivity

We consider a three dimensional (3D) finite element modeling of high temperature superconductors (HTS) submitted to a uniform magnetic field varying linearly with time. HTS are characterized by a non linear conductivity described by a power law with the critical exponent, n. We implement an A- phi weak formulation into the opensource solver GetDP. Backward Euler method is used for the temporal resolution and is coupled to a non linear Picard iteration scheme for dealing with the non linear resistivity of HTS.We consider single- and multiple-time step methods and show that the single time step method produces an accurate result in a drastically reduced calculation time for HTS with large pinning strength (or similarly, with large values of n). The finite element formulation is validated by comparing the magnetization of a HTS tube calculated either with our finite element formulation or with other well-known techniques. The numerical method is then applied for comparing the magnetization of cylinders that are drilled by different periodic arrangements of columnar holes parallel to the cylinder axis.