Combined finite element-modal solution of three-dimensional eddy current problems

The reliability of finite-element methods for modal analysis of two- and three-dimensional eddy-current problems is addressed. Separation of variables is used to convert transient-eddy-current problems into an ordinary differential equation in time and linear combination of normal modes in space. The eigensolution of the vector wave equation by the usual finite-element basis functions usually results in numerical instabilities that render the procedure worthless. It has been found that the root cause of these instabilities is the improper approximation of the null space of the curl operator. Three different methods that eliminate the instabilities completely have been developed. The first method uses C¹ or derivative continuous finite elements; the second uses tangential vector basis functions developed in a companion paper; and the third uses ordinary Lagrangian finite elements but places them in a special mesh pattern so that C¹ continuous polynomials are possible, although C¹ continuity is not imposed