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3D modelling of macroscopic force-free effects in superconducting thin films and rectangular prisms

When the magnetic field has a parallel component to the current density ${\bf J}$ there appear force-free effects due to flux cutting and crossing. This results in an anisotropic ${\bf E}({\bf J})$ relation, being ${\bf E}$ the electric field. Understanding force-free effects is interesting not only for the design of superconducting power and magnet applications but also for material characterization. This work develops and applies a fast and accurate computer modeling method based on a variational approach that can handle force-free anisotropic ${\bf E}({\bf J})$ relations and perform fully three dimensional (3D) calculations. We present a systematic study of force-free effects in rectangular thin films and prisms with several finite thicknesses under applied magnetic fields with arbitrary angle $θ$ with the surface. The results are compared with the same situation with isotropic ${\bf E}({\bf J})$ relation. The thin film situation shows gradual critical current density penetration and a general increase of the magnitude of the magnetization with the angle $θ$ but a minimum at the remnant state of the magnetization loop. The prism model presents current paths with 3D bending for all angles $θ$. The average current density over the thickness agrees very well with the thin film model except for the highest angles. The prism hysteresis loops reveal a peak after the remnant state, which is due to the parallel component of the self-magnetic-field and is implicitly neglected for thin films. The presented numerical method shows the capability to take force-free situations into account for general 3D situations with a high number of degrees of freedom. The results reveal new features of force-free effects in thin films and prisms.