Mean curl formulation on quadrilaterals with application to implicit magnetics diffusion equations in Alegra 2D.

This report proposes a mean curl on quadrilateral elements for Alegra 2D magnetics calculations. The resulting quadrilateral element is an extension of the work [1, 4] to include the curl operator. The resulting quadrilateral element implementation is applicable to implicit calculations in Cartesian coordinates which use either the vector potential or B formulation. It is useful to note that in the vector potential formulation, the mean curl produces a mean flux density on the element, whereas in the B formulation the mean curl produces a mean current on the element. In both cases, the mean curl is, by construction, divergence free. The mean curl is further developed and applied to the nonlinear permeability capability that currently exists in Alegra 2D for the vector potential formulation. Using the proposed mean curl, a Jacobian operator is developed and presented which is applicable to nonlinear iterations associated with the implicit problem. Note that the existing nonlinear permeability does not include hysteresis; also, the current nonlinear implementation uses a Jacobian free solution algorithm because a Jacobian operator has neither been developed or implemented. A Jacobian for the full quadrature implementation that currently exists in Alegra 2D can be similarly derived based upon the results presented.