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postgraduate thesis: Fast and wellconditioned integral equation solvers for lowfrequency electromagnetic problems
Title  Fast and wellconditioned integral equation solvers for lowfrequency electromagnetic problems 

Authors  
Issue Date  2015 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Liu, Q. [刘{274b4d}]. (2015). Fast and wellconditioned integral equation solvers for lowfrequency electromagnetic problems. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5610959 
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 lowfrequency 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 codetermined by both the condition of the matrix and the righthandside excitation. In a current solution, the weighted contributions from different singular vectors are not only decided by the corresponding singular values but also the righthand 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 deltagap source. The dominant charge currents are enough to capture the capacitive physics. Detailed spectral analysis with righthand side effect validates the proposed theory.
Secondly, in order to overcome the lowfrequency inaccuracy problem for open capacitive structures in CMPEFIE, a perturbed CMPEFIE is proposed to extract accurate highorder current at low frequencies. Further study of the capacitive problems in CMPEFIE utilizes a simplified twoterm system by removing the contribution from the hypersingular preconditioned term, which captures the correct physics without doing the perturbation steps. The aforebuilt righthand side analysis theory is applied here to explain the stability and accuracy of the simplified CMPEFIE 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 potentialbased 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 wellsolved 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, wideband and accurate integral algorithms. 
Degree  Doctor of Philosophy 
Subject  Integral equations Electromagnetic fields  Mathematical models 
Dept/Program  Electrical and Electronic Engineering 
Persistent Identifier  http://hdl.handle.net/10722/221183 
DC Field  Value  Language 

dc.contributor.author  Liu, Qin   
dc.contributor.author  刘{274b4d}   
dc.date.accessioned  20151104T23:11:55Z   
dc.date.available  20151104T23:11:55Z   
dc.date.issued  2015   
dc.identifier.citation  Liu, Q. [刘{274b4d}]. (2015). Fast and wellconditioned integral equation solvers for lowfrequency electromagnetic problems. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5610959   
dc.identifier.uri  http://hdl.handle.net/10722/221183   
dc.description.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 lowfrequency 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 codetermined by both the condition of the matrix and the righthandside excitation. In a current solution, the weighted contributions from different singular vectors are not only decided by the corresponding singular values but also the righthand 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 deltagap source. The dominant charge currents are enough to capture the capacitive physics. Detailed spectral analysis with righthand side effect validates the proposed theory. Secondly, in order to overcome the lowfrequency inaccuracy problem for open capacitive structures in CMPEFIE, a perturbed CMPEFIE is proposed to extract accurate highorder current at low frequencies. Further study of the capacitive problems in CMPEFIE utilizes a simplified twoterm system by removing the contribution from the hypersingular preconditioned term, which captures the correct physics without doing the perturbation steps. The aforebuilt righthand side analysis theory is applied here to explain the stability and accuracy of the simplified CMPEFIE 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 potentialbased 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 wellsolved 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, wideband and accurate integral algorithms.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.subject.lcsh  Integral equations   
dc.subject.lcsh  Electromagnetic fields  Mathematical models   
dc.title  Fast and wellconditioned integral equation solvers for lowfrequency electromagnetic problems   
dc.type  PG_Thesis   
dc.identifier.hkul  b5610959   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Electrical and Electronic Engineering   
dc.description.nature  published_or_final_version   