Advanced finite element methodology for low-frequency and static electromagnetic modeling

Abstract

The design of state-of-the-art microelectronic devices poses unprecedented challenges to computational electromagnetics (CEM), which is cursed by the null space of curl operator. Both the low-frequency catastrophe for dynamic electromagnetic problems and non-uniqueness for magnetostatic problems originate from the null space. Although a few remedies are proposed during the last decade, a theoretically rigorous and numerically efficient solution is still on its way.

Toward this end, this thesis constructs a finite element framework, which consists of generalized gauge condition, compatible finite element discretization, sparse approximate inverse (SAI) technique and static incomplete LU (ILU) preconditioned iterative solution.

The generalized gauge condition introduces a gauge operator, which is comparable in magnitude and complementary in space with the double curl operator, into the original governing equations. The null space is removed and the combined operator becomes positive definite. However, the combined operator is so complicated that its discretization and matrix representation are unclear. Thanks to the theory of differential forms, the mapping of the quantity of interest from one form to another becomes distinct. Hence, the compatible discretization can be carried out based on the versatile Whitney elements. The resultant matrix system is much better conditioned than that of the ungauged one, whereas more treatment is still necessary to make it less sparse and faster convergent.

The SAI and ILU preconditioning techniques provide an excellent solution to this difficulty. The former approximates the inverse of a mass matrix by a nearly-diagonal matrix, which greatly reduces the sparsity of the matrix system. The later shifts all the eigenvalues to the neighborhood of 1 and thus achieves an extremely fast convergence. Moreover, the static incomplete LU (ILU) preconditioning scheme is well suited to wideband analysis, because the preconditioner is calculated just once for a wide range of frequency.

This framework is verified, by low-frequency circuit problems as well as magnetostatic ones, to be accurate and efficient.

In addition, more effort is devoted to explore other possibilities to solve the aforementioned problem. The application of loop basis functions is also a promising solution, provided that the redundant loops in the mesh can be removed.

Finally, the displacement current effect is studied in depth by a full-wave semianalytical solution of wireless power transfer into dispersive layered media. The comparison between the results with and without the displacement current advocates the full-wave electromagnetic modeling for multi-scale problems and wideband analysis.