Fast methods for low-frequency and static EM problems

Abstract

Electromagnetic effects play an important role in many engineering problems. The fast and accurate numerical methods for electromagnetic analysis are highly desired in both the low-frequency analysis and the static analysis.

In the first part of this thesis, a low-frequency stable domain decomposition method, the augmented equivalence principle algorithm (A-EPA) with augmented electric field integral equation (A-EFIE), is introduced for analyzing the electromagnetic problems at low frequencies. The A-EFIE is first employed as a inner current solver for the EPA algorithm so that it improves the low-frequency inaccuracy issue. This method, however, cannot completely remove the low-frequency breakdown. To overcome it, the A-EPA with A-EFIE is studied and developed so that it has the capability to solve low-frequency problems accurately.

In the second part, novel Helmholtz decomposition based fast Poisson solvers for both 2-D and 3-D problems are introduced. These new methods are implemented through the quasi-Helmholtz decomposition technique, i.e. the loop-tree decomposition. In 2-D cases, the proposed method can achieve O(N) complexity in terms of both computational cost and memory consumption for moderate accuracy requirements. Although computational costs become higher when more accurate results are needed, a multilevel method by using the hierarchical loop basis functions can obtain the desired efficiency. The same idea can be extend to 3-D case for exploiting a new generation of fast method for electrostatic problems.