体面结合积分方程及快速算法在复杂电磁问题中的分析与应用
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摘要
复杂电磁目标的快速、有效分析始终是计算电磁学领域的研究热点与发展目标。本文首先建立了用以求解金属介质混合问题的体面结合积分方程,然后研究了用以快速求解体面结合积分方程的数值算法,包括预修正快速傅立叶变换(P-FFT)算法、特征基函数方法(CBFM)、CBFM/P-FFT以及P-CBFM/P-FFT等算法,并将这些算法用于复杂电磁目标的散射和辐射分析中。
首先,本文从等效原理和边界条件出发,推导了用于求解金属介质混合目标的体面结合积分方程(VSIE),并根据金属介质交界面处不同的边界条件处理方法将VSIE区分为普通体面结合积分方程(NCVSIE)和体面耦合积分方程(CVSIE),然后首次从散射、辐射以及电路分析等几个应用领域分析并对比了两类积分方程的数值结果,由结果发现,在分析辐射问题和电路问题时,CVSIE的性能要优于NCVSIE,可较好的解决天线或电路计算中遇到的谐振频率偏移问题。
为加速矩量法(MoM)迭代求解过程,研究了P-FFT算法的基本流程,并分别结合面积分方程(SIE)、体积分方程(VIE)和体面结合积分方程(VSIE)分析了典型电磁目标的散射和辐射问题,验证了P-FFT算法的准确性和有效性。
为有效减少矩量法分析中所用到的未知量个数,并优化矩阵方程的条件数,研究了CBFM求解积分方程的基本过程。首次将CBFM应用于VSIE积分方程的求解中,并对比了两种特征基函数(CBF)提取方法的计算精度。然后,利用CBFM算法分析了金属、介质、金属-介质混合目标的散射或辐射特性,说明了算法的准确性与普适性。
为加快CBFM迭代求解速度,本文引入了P-FFT算法,并提出了用于求解非周期结构的CBFM/P-FFT混合算法,然后通过算例验证算法的准确性。尽管由于本文所举算例规模较小,难以体现混合算法的高效性,但是从计算资源对比结果可以说明CBFM/P-FFT算法较P-FFT和CBFM方法所需内存更少,迭代求解速度较P-FFT更快。
针对有限周期结构如光子晶体或相控阵列的准确有效计算,引入了近区单元场值修正模型,研究并改进了用于求解周期金属-介质混合结构的CBFM/P-FFT算法,称之为P-CBFM/P-FFT算法,改进后的算法能够准确考虑邻近单元间的互耦。在此基础上,本文提出了一种可高效分析连续介质周期阵列的算法,并通过算例验证了其准确性。
最后,利用本文快速算法、基函数方法和VSIE积分方程分析了一系列复杂电磁问题,包括任意馈电方式激励的天线或电路互耦分析、基于P-FFT算法的多层微带天线辐射分析及共形微带阵列辐射分析、基于P-CBFM/P-FFT算法的宽带天线阵列辐射特性分析、基于CBFM算法的直升机搭载微带天线辐射分析、基于P-CBFM/P-FFT算法的光子晶体的散射和传输特性分析。同时,通过与理论结果、测试结果或者其它算法计算结果相对比,说明本文算法的准确性、高效性和适用性,为复杂电磁问题的快速、准确数值分析提供了一种有效解决途径。
To analyze complicated electromagnetic objects efficiently and accurately is always one of the challenging problems in computational electromagnetics. Firstly, the volume-surface integral equation (VSIE) is established to solve composite conducting and dielectric problems, then, in order to the solve the VSIE efficiently, numerical methods including the precorrected-fast Fourier transform (P-FFT) method, characteristic basis function method (CBFM), combined CBFM/P-FFT algorithm and P-CBFM/P-FFT algorithm are studied, and applied to solve electromagnetic scattering and radiation problems of complicated bodies.
At first, the VSIE formulation is deduced based on equivalence theorem and boundary condition. Besides, by using different boundary condition at the interface of the conductor and dielectric, two approaches of the VSIE including the non-coupled VSIE (NCVSIE) and the coupled VSIE (CVSIE) are discussed and analyzed. Then, from the compared results using the two approaches to analyze scattering, radiation and circuit problems respectively, it can be concluded that, the performance of the CVSIE is much better than that of NCVSIE for radiation and circuit problems, additionally, the offset problem of resonant frequency can be settled appropriately to an acceptable extent using the CVSIE formulation.
Then, in order to accelerate the iterative procedure for solution, the P-FFT algorithm is presented, and combined with surface integral equation (SIE), volume integral equation (VIE) and VSIE respectively, to analyze typical scattering and radiation problems, which indicates the accuracy and efficiency of the algorithm.
Besides, in order to reduce the number of unknowns availably, CBFM method is introduced to solve the VSIE. Two methods used to retrieve the characteristic basis function (CBF) are discussed and compared. Then, the CBFM is utilized to analyze the scattering and radiation characteristics of conducting, dielectric and mixed conducting-dielectric objects, which shows the accuracy and generality of CBFM.
Additionally, the P-FFT process is introduced to combine with the CBFM algorithm, so as to speed up the iterative procedure. Then the combined CBFM/P-FFT is applied to analyze non-periodic objects, and the accuracy of the combined algorithm has been validated by examples. For the scale of problem analyzed is not large enough, it is difficult to demonstrate the efficiency of CBFM/P-FFT, however, from the calculated results, it can be observed that less memory is required for CBFM/P-FFT than P-FFT and CBFM, and less iterative time is needed for CBFM/P-FFT than P-FFT.
To calculate finite periodic structures efficiently, a combined CBFM and P-FFT algorithm which is called P-CBFM/P-FFT, is presented to solve VSIE. A near correction model is introduced to consider the mutual coupling between nearby units. Some large-scale examples about scattering of mixed metallic-dielectric periodic arrays, are illustrated to demonstrate the efficiency and accuracy of the algorithm, based on which, a novel efficient algorithm is provided to analyze periodic structures with dielectric connected between units, the accuracy of the algorithm has been demonstrated by some examples.
After the studies above, the fast solvers and basis function methods developed in the paper are applied to analyze complicated electromagnetic problems including: mutual coupling between antennas and circuits, radiation of multi-layer microstrip antenna and conformal antenna arrays by P-FFT algorithm, large antenna array with wide bandwidth by P-CBFM/P-FFT algorithm, radiation of microstrip antenna mounted on helicopter by CBFM method, scattering and transmission characteristics of photonic crystals by P-CBFM/P-FFT algorithm. By comparing the calculated results to analytical solution, measurement or results from references, the accuracy, efficiency and universality of the algorithms are demonstrated. It can be concluded that, an efficient and accurate scheme has been provided in the thesis for the solution of complicated electromagnetic problems.
首先,本文从等效原理和边界条件出发,推导了用于求解金属介质混合目标的体面结合积分方程(VSIE),并根据金属介质交界面处不同的边界条件处理方法将VSIE区分为普通体面结合积分方程(NCVSIE)和体面耦合积分方程(CVSIE),然后首次从散射、辐射以及电路分析等几个应用领域分析并对比了两类积分方程的数值结果,由结果发现,在分析辐射问题和电路问题时,CVSIE的性能要优于NCVSIE,可较好的解决天线或电路计算中遇到的谐振频率偏移问题。
为加速矩量法(MoM)迭代求解过程,研究了P-FFT算法的基本流程,并分别结合面积分方程(SIE)、体积分方程(VIE)和体面结合积分方程(VSIE)分析了典型电磁目标的散射和辐射问题,验证了P-FFT算法的准确性和有效性。
为有效减少矩量法分析中所用到的未知量个数,并优化矩阵方程的条件数,研究了CBFM求解积分方程的基本过程。首次将CBFM应用于VSIE积分方程的求解中,并对比了两种特征基函数(CBF)提取方法的计算精度。然后,利用CBFM算法分析了金属、介质、金属-介质混合目标的散射或辐射特性,说明了算法的准确性与普适性。
为加快CBFM迭代求解速度,本文引入了P-FFT算法,并提出了用于求解非周期结构的CBFM/P-FFT混合算法,然后通过算例验证算法的准确性。尽管由于本文所举算例规模较小,难以体现混合算法的高效性,但是从计算资源对比结果可以说明CBFM/P-FFT算法较P-FFT和CBFM方法所需内存更少,迭代求解速度较P-FFT更快。
针对有限周期结构如光子晶体或相控阵列的准确有效计算,引入了近区单元场值修正模型,研究并改进了用于求解周期金属-介质混合结构的CBFM/P-FFT算法,称之为P-CBFM/P-FFT算法,改进后的算法能够准确考虑邻近单元间的互耦。在此基础上,本文提出了一种可高效分析连续介质周期阵列的算法,并通过算例验证了其准确性。
最后,利用本文快速算法、基函数方法和VSIE积分方程分析了一系列复杂电磁问题,包括任意馈电方式激励的天线或电路互耦分析、基于P-FFT算法的多层微带天线辐射分析及共形微带阵列辐射分析、基于P-CBFM/P-FFT算法的宽带天线阵列辐射特性分析、基于CBFM算法的直升机搭载微带天线辐射分析、基于P-CBFM/P-FFT算法的光子晶体的散射和传输特性分析。同时,通过与理论结果、测试结果或者其它算法计算结果相对比,说明本文算法的准确性、高效性和适用性,为复杂电磁问题的快速、准确数值分析提供了一种有效解决途径。
To analyze complicated electromagnetic objects efficiently and accurately is always one of the challenging problems in computational electromagnetics. Firstly, the volume-surface integral equation (VSIE) is established to solve composite conducting and dielectric problems, then, in order to the solve the VSIE efficiently, numerical methods including the precorrected-fast Fourier transform (P-FFT) method, characteristic basis function method (CBFM), combined CBFM/P-FFT algorithm and P-CBFM/P-FFT algorithm are studied, and applied to solve electromagnetic scattering and radiation problems of complicated bodies.
At first, the VSIE formulation is deduced based on equivalence theorem and boundary condition. Besides, by using different boundary condition at the interface of the conductor and dielectric, two approaches of the VSIE including the non-coupled VSIE (NCVSIE) and the coupled VSIE (CVSIE) are discussed and analyzed. Then, from the compared results using the two approaches to analyze scattering, radiation and circuit problems respectively, it can be concluded that, the performance of the CVSIE is much better than that of NCVSIE for radiation and circuit problems, additionally, the offset problem of resonant frequency can be settled appropriately to an acceptable extent using the CVSIE formulation.
Then, in order to accelerate the iterative procedure for solution, the P-FFT algorithm is presented, and combined with surface integral equation (SIE), volume integral equation (VIE) and VSIE respectively, to analyze typical scattering and radiation problems, which indicates the accuracy and efficiency of the algorithm.
Besides, in order to reduce the number of unknowns availably, CBFM method is introduced to solve the VSIE. Two methods used to retrieve the characteristic basis function (CBF) are discussed and compared. Then, the CBFM is utilized to analyze the scattering and radiation characteristics of conducting, dielectric and mixed conducting-dielectric objects, which shows the accuracy and generality of CBFM.
Additionally, the P-FFT process is introduced to combine with the CBFM algorithm, so as to speed up the iterative procedure. Then the combined CBFM/P-FFT is applied to analyze non-periodic objects, and the accuracy of the combined algorithm has been validated by examples. For the scale of problem analyzed is not large enough, it is difficult to demonstrate the efficiency of CBFM/P-FFT, however, from the calculated results, it can be observed that less memory is required for CBFM/P-FFT than P-FFT and CBFM, and less iterative time is needed for CBFM/P-FFT than P-FFT.
To calculate finite periodic structures efficiently, a combined CBFM and P-FFT algorithm which is called P-CBFM/P-FFT, is presented to solve VSIE. A near correction model is introduced to consider the mutual coupling between nearby units. Some large-scale examples about scattering of mixed metallic-dielectric periodic arrays, are illustrated to demonstrate the efficiency and accuracy of the algorithm, based on which, a novel efficient algorithm is provided to analyze periodic structures with dielectric connected between units, the accuracy of the algorithm has been demonstrated by some examples.
After the studies above, the fast solvers and basis function methods developed in the paper are applied to analyze complicated electromagnetic problems including: mutual coupling between antennas and circuits, radiation of multi-layer microstrip antenna and conformal antenna arrays by P-FFT algorithm, large antenna array with wide bandwidth by P-CBFM/P-FFT algorithm, radiation of microstrip antenna mounted on helicopter by CBFM method, scattering and transmission characteristics of photonic crystals by P-CBFM/P-FFT algorithm. By comparing the calculated results to analytical solution, measurement or results from references, the accuracy, efficiency and universality of the algorithms are demonstrated. It can be concluded that, an efficient and accurate scheme has been provided in the thesis for the solution of complicated electromagnetic problems.
引文
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