Fast and well-conditioned integral equation solvers for low-frequency electromagnetic problems

Abstract

Inspired by the important low frequency applications, such as the integrate circuits, the nano electromagnetic compatibility and quantum optics, several aspects of the computational electromagnetic low-frequency problems in surface integral equation (SIE) are carefully investigated in this dissertation.

Firstly a capacitive model is studied that the convergence of the matrix system is co-determined by both the condition of the matrix and the righthand-side excitation. In a current solution, the weighted contributions from different singular vectors are not only decided by the corresponding singular values but also the right-hand side. The convergence of the capacitive problems is guaranteed by the fact that the singular vectors corresponding to the small singular values are not excited under the delta-gap source. The dominant charge currents are enough to capture the capacitive physics. Detailed spectral analysis with right-hand side effect validates the proposed theory.

Secondly, in order to overcome the low-frequency inaccuracy problem for open capacitive structures in CMP-EFIE, a perturbed CMP-EFIE is proposed to extract accurate high-order current at low frequencies. Further study of the capacitive problems in CMP-EFIE utilizes a simplified two-term system by removing the contribution from the hypersingular preconditioned term, which captures the correct physics without doing the perturbation steps. The afore-built right-hand side analysis theory is applied here to explain the stability and accuracy of the simplified CMP-EFIE system.

Thirdly, a point testing system is constructed to eliminate the nontrivial nullspaces of the static MFIE systems by enforcing extra zero magnetic flux conditions at the testing points locations. The projection of the current solution onto the magnetostatic nullspaces is truncated accordingly, thus the system convergence can be much improved without losing any accuracy.

Finally, the electromagnetic solution is obtained from a potential-based integral equation solver, capturing electrostatic physics from the scalar potential formulation and magnetostatic physics from the vector potential formulation. The combination of the two formulations reveals the correct solution and physics at low frequencies. And the equations, formulated with the potential quantities, make it possible to couple with quantum effects theories. The resulting system appears to be a symmetric saddle point problem, where the efficiency of the iterative solver can be well-solved by a typical appropriate constraint preconditioner. The stability and capability of the new system in solving different kinds of electromagnetic problems are validated over a wide range of frequency range.

The research topics in this dissertation cover different aspects of low frequency integral equation solvers, aiming at fast, stable, wide-band and accurate integral algorithms.