A direct method for obtaining the critical state in two and three dimensions

A method for obtaining the critical state in three dimensions is described. This uses an extension of a previous 1D model based on flux line motion in which the equations are not based on an E-J curve, which leads to time dependence. In order to make clear the connection between the scalar potential, the particular vector potential derived and the electrostatic surface charges, it has proved necessary to start from eddy currents in a normal conductor. The eddy current solutions for a normal conductor are the same as those for a London superconductor in a DC field, except that although a scalar potential is needed there are no electrostatic charges. The problem of a superconducting puck in a field parallel to the faces is solved. It is assumed that the electric field is parallel to the current density which is quite probable for high Tc superconductors, but other criteria could be used. Like other 3D solutions the computation takes a long time; even this relatively simple case takes 7 h with a 1.3 GHz PC. However this was using a standard PC with default parameters in the finite element package so there is room for optimization. All the results were obtained with FlexPDE.