| Revealed by the Caldero′n relation and the Caldero′n identities in electromagnetictheory, the properties and relations of different integral operators in the computation-al electromagnetics (CEM) are utilized to construct the Caldero′n preconditioning tech-niques, which are applied in the integral-equation-based methods in this dissertation. Athorough and systematic research has been accomplished to cover the Caldero′n precon-ditioning techniques for the perfect electric conductor (PEC) and the dielectric cases. Forthe PEC case, the Caldero′n preconditioners for the electric-field integral equation (EFIE)at mid, low, and high frequencies are constructed and studied. For the dielectric case, theCaldero′n preconditioners for the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT)integral equation are investigated, and the Caldero′n technique for the N-Müller integralequation is developed. Moreover, the accuracy improving technique for the second-kindFredholm integral equations for both PEC and dielectric cases is also studied in this dis-sertation.First, the integral equations in CEM, including the surface integral equations and thevolume integral equations, are constructed. The general principle and major steps of themethod of moments (MoM) are described. Then the Caldero′n relation and the Caldero′nidentities are introduced, and the Caldero′n preconditioner for the EFIE at mid frequenciesis reviewed. All these serve as the theoretical foundations of the entire dissertation.In order to overcome the low-frequency breakdown problems of the EFIE and theCaldero′n preconditioner at low frequencies, the loop-star basis functions based on thecurvilinear Rao-Wilton-Glisson (CRWG) and the Buffa-Christiansen (BC) functions areconstructed and applied to the numerical discretization of the EFIE and the Caldero′npreconditioner. This leads to the Caldero′n preconditioner at low frequencies, which iscapable of alleviating the low-frequency breakdown problem effectively and convergingat an arbitrarily low frequency rapidly and independently with respect to the mesh con-figurations.In the high-frequency region, both the EFIE and the Caldero′n preconditioner sufferfrom the spurious interior resonance problem. To alleviate this problem, an augmented EFIE with the Caldero′n preconditioner is proposed in this dissertation. It is demonstrat-ed by several numerical examples that this high-frequency Caldero′n preconditioner caneliminate the spurious interior resonance problem effectively and result in a fast conver-gent and accurate formulation which can be used in the electromagnetic calculations oflarge complex problems.In the research of Caldero′n preconditioning techniques for the dielectric case, wehave first developed and investigated three different Caldero′n preconditioners for thePMCHWT equation. Their numerical performances at different frequencies are stud-ied and compared thoroughly, through both the theoretical analysis and the numericalexperiments.Then the operator property of the N-Müller equation is studied, and the N-Müller e-quation is derived by preconditioning the EFIE and the MFIE (magnetic-field integralequation) for the dielectric case with the Caldero′n preconditioners and by using theCaldero′n relation. The derivation introduced in this dissertation provides a brand newexplanation for the excellent spectrum property of the N-Müller integral operator.At last, the numerical accuracies of the second-kind Fredholm integral equations arethoroughly studied and effectively improved. After the discussions of the discretizationschemes of different surface integral operators, the n×BC functions are used as thetesting function to suppress the major error source of the second-kind integral equations,the numerical error related to the identity operators. The proposed scheme is able togenerate much more accurate numerical solutions to the second-kind integral equations.The reasons of the accuracy improvement are also analyzed theoretically, which providethe proposed scheme with theoretical foundations.The research presented in this dissertation covers the major topics of the Caldero′npreconditioning techniques for the integral-equation-based methods, including the itera-tive convergence acceleration for the first-kind Fredholm integral equations and the nu-merical accuracy improvement for the second-kind Fredholm integral equations. It pro-vides us with solid theory foundations and technique approaches for the accurate andfast iterative solution of the integral equations in computational electromagnetics, andtherefore, is a powerful tool for the fast and accurate solution to the various engineeringproblems.
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