Energy methods for the evaluation of global quantities and integral parameters in a finite elements analysis of electromagnetic devices

For almost all finite element analysis of electromagnetic devices the engineer or the designer is much more interested in the knowledge of the global quantities (flux, active and reactive powers, forces, torques ...) and the electric or magnetic design parameters (reluctances, inductances, capacitances...) than the display of fields map. In many current software these quantities are generally computed by numerical quadrature formulae over the values of fields or flux densities. For instance the computation of forces and torques is usually made by quadrature of the Maxwell stress tensor along an arbitrary integration path. The results depend in fact strongly of the choice of the path and may in specific cases be very inaccurate. An other reason of this lack of accuracy is the fact that quadrature formulae are applied to quadratic expressions of fields, wich are themselves obtained from scalar or vector potential interpolated values by a numerical derivation scheme. But it is well known that numerical differentiation is source of a lot of inaccuracy although the subsequent integration procedure smoothes the results. The paper will introduce methods based on the property of optimization of an energy functional. These methods suit particularly well the finite element method which is based on the same principle.