基于Calderón技术的计算电磁学积分方程方法研究
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摘要
本文基于电磁理论中的Caldero′n关系与Caldero′n恒等式所揭示的不同积分算子之间的关系,系统地研究了Caldero′n预条件技术及其在计算电磁学积分方程方法中的应用。本文研究内容全面覆盖了求解理想导电体目标和均匀或分层均匀介质目标电磁散射与辐射问题的积分方程方法中的Caldero′n预条件技术。在导体积分方程方面,研究了电场积分方程在中频,低频,以及高频区的Caldero′n预处理方法。在介质积分方程方面,则研究了PMCHWT积分方程的Caldero′n预处理方法,和N-Müller积分方程的Caldero′n技术。本文也对金属问题中的第二类Fredholm积分方程和介质问题中的第二类Fredholm积分方程的精度改善进行了深入详尽的研究。
首先,文章回顾了电磁场积分方程方法中常见积分方程,包括面积分方程和体积分方程的构造方法,并描述了对积分方程进行数值求解的矩量法的基本原理和关键步骤。接着,文章阐述了Caldero′n预条件技术在求解金属目标中频区域内的电磁问题时的理论方法与关键技术。这为全文的研究打下了理论基础。
为了克服电场积分方程及Caldero′n预条件在低频下的低频崩溃问题,本文构造了基于曲面Rao-Wilton-Glisson (CRWG)函数和Buffa-Christiansen (BC)函数的loop-star基函数,并将其分别应用于对电场积分方程和Caldero′n预条件的数值离散上。由此构造出的低频Caldero′n预条件能够有效克服电场积分方程的低频崩溃问题,并且能够在任意低的频率下,任意形式的几何离散下无差别地快速收敛。
在高频区域中,电场积分方程和Caldero′n预条件都有很严重的伪内谐振问题。为了克服这一问题,本文提出了使用Caldero′n预条件的增广电场积分方程方法。并通过数值算例证明这种高频Caldero′n预条件方法能够有效克服电场积分方程的伪内谐振问题,并且具有很高的计算精度和很快的收敛速度,因此可以被用于电大尺寸复杂目标的电磁仿真计算。
在介质目标的Caldero′n预条件方法方面,本文首先研究了PMCHWT方程的预处理方法。分别构造了三种不同的Caldero′n预条件对其进行处理,并从理论分析和数值实验两个方面对这三种预条件在各个频段的性能进行了深入的比较研究。
接下来,本文研究了N-Müller积分方程的算子性态,并且通过使用Caldero′n关系与Caldero′n恒等式,从介质EFIE和MFIE推导出了N-Müller积分方程。这一推导过程从一个全新的角度对N-Müller积分方程具有良好矩阵性态的原因做出了解释。
最后,本文对第二类Fredholm积分方程的数值求解精度问题进行了深入的研究。在分析探讨了面积分方程中各个算子的离散方法之后,文章使用n?×BC作为权函数成功减小了第二类Fredholm积分方程的主要数值误差源,即单位算子的数值计算误差。这使得整个第二类积分方程的数值计算精度都得到了非常显著的提高。文中也对计算精度得到提高的原因进行了理论分析,为本文方法提供了理论依据。
本文的研究工作系统而完整地探讨了Caldero′n预条件技术在积分方程方法各个方面的应用,为电磁场积分方程的快速迭代求解提供了坚实的理论基础与技术手段,因此也成为快速精确求解各种实际工程问题的有力工具。
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.
首先,文章回顾了电磁场积分方程方法中常见积分方程,包括面积分方程和体积分方程的构造方法,并描述了对积分方程进行数值求解的矩量法的基本原理和关键步骤。接着,文章阐述了Caldero′n预条件技术在求解金属目标中频区域内的电磁问题时的理论方法与关键技术。这为全文的研究打下了理论基础。
为了克服电场积分方程及Caldero′n预条件在低频下的低频崩溃问题,本文构造了基于曲面Rao-Wilton-Glisson (CRWG)函数和Buffa-Christiansen (BC)函数的loop-star基函数,并将其分别应用于对电场积分方程和Caldero′n预条件的数值离散上。由此构造出的低频Caldero′n预条件能够有效克服电场积分方程的低频崩溃问题,并且能够在任意低的频率下,任意形式的几何离散下无差别地快速收敛。
在高频区域中,电场积分方程和Caldero′n预条件都有很严重的伪内谐振问题。为了克服这一问题,本文提出了使用Caldero′n预条件的增广电场积分方程方法。并通过数值算例证明这种高频Caldero′n预条件方法能够有效克服电场积分方程的伪内谐振问题,并且具有很高的计算精度和很快的收敛速度,因此可以被用于电大尺寸复杂目标的电磁仿真计算。
在介质目标的Caldero′n预条件方法方面,本文首先研究了PMCHWT方程的预处理方法。分别构造了三种不同的Caldero′n预条件对其进行处理,并从理论分析和数值实验两个方面对这三种预条件在各个频段的性能进行了深入的比较研究。
接下来,本文研究了N-Müller积分方程的算子性态,并且通过使用Caldero′n关系与Caldero′n恒等式,从介质EFIE和MFIE推导出了N-Müller积分方程。这一推导过程从一个全新的角度对N-Müller积分方程具有良好矩阵性态的原因做出了解释。
最后,本文对第二类Fredholm积分方程的数值求解精度问题进行了深入的研究。在分析探讨了面积分方程中各个算子的离散方法之后,文章使用n?×BC作为权函数成功减小了第二类Fredholm积分方程的主要数值误差源,即单位算子的数值计算误差。这使得整个第二类积分方程的数值计算精度都得到了非常显著的提高。文中也对计算精度得到提高的原因进行了理论分析,为本文方法提供了理论依据。
本文的研究工作系统而完整地探讨了Caldero′n预条件技术在积分方程方法各个方面的应用,为电磁场积分方程的快速迭代求解提供了坚实的理论基础与技术手段,因此也成为快速精确求解各种实际工程问题的有力工具。
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.
引文
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