基于高阶叠层矢量基函数的快速算法研究
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摘要
现代目标识别、目标隐身技术、微波成像及微波遥感等工程领域均需要对目标的电磁散射特性进行分析，而如何采用数值分析方法精确高效地分析目标的电磁散射特性一直是计算电磁学领域研究工作的重点。本文的研究是以高阶叠层矢量基函数在电磁场积分方程方法中的应用为主要内容，重点研究了如何使用该基函数精确高效地计算目标的电磁响应，主要分为四个部分进行讨论。
本文第一部分从频域角度出发，先将基于修正勒让德多项式的高阶叠层矢量基函数应用于频域电磁场积分方程方法中，并研究了基函数的正交性对其计算性能的影响。通过分析可发现基函数的正交性越好，其计算性能也越好。因此，本文重点研究了对高阶叠层矢量基函数进行正交化而得到的最大正交高阶矢量基函数，并对尺度因子进行修正以提高其计算性能。其次，本文还分析了定义在大尺寸单元上的高阶基函数的辐射特性，提出一种针对采用电磁场积分方程方法求解三维导体目标时得到的阻抗矩阵进行稀疏化的方法。同时，本文还将定义在四边形单元上的高阶叠层矢量基函数进行推广，提出一种定义在三角形单元上的基于修正勒让德多项式的高阶叠层矢量基函数，以解决四边形单元难以精确拟合复杂结构目标的问题。
本文第二部分从时域角度出发，重点研究了准正交高阶叠层矢量基函数在时域电磁场积分方程方法(TDIE)中的算法实现，并通过数值算例详细讨论了该基函数在时域积分方程方法中的性质。其次，本文在简要分析了时域积分方程时间步进算法(MOT)的后时不稳定性问题后，指出引起该算法后时不稳定的主要原因之一是离散TDIE时采用了不精确的数值计算方法。因此，本文提出一种基于曲面单元的卷积积分方法，以精确求解任意曲面单元上的时域阻抗矩阵元素。同时，本文还采用TDIE对运动金属目标的时域电磁响应做了初步研究，并采用时域区域分解算法对空间相对运动的群目标的电磁响应做了简单分析。
本文第三部分研究了时频互推算法。该部分首先说明了当前时域方法的主要困难，指出时频互推算法是解决时域算法后时不稳定的有效手段之一。本文对时频互推算法的互推原理进行了详细的分析，并提出一种自适应算法以确定时域和频域信息的采样宽度。其次，本文还将准正交高阶叠层矢量基函数应用于该算法中计算时频采样信息，从而解决了传统算法中频域算法低频采样时计算量过大的问题。最后本文充分研究了该算法的物理基础，提出一种采用频域滤波对互推结果进行校正的方法，从而有效地提高了互推结果的可靠性，并降低了互推算法的计算复杂度。
本文第四部分针对大尺寸单元难以精确拟合复杂结构目标几何结构的问题，提出一种非均匀剖分的解决方案。该方法通过将目标表面结构分解，采用不同尺寸的单元对不同区域进行剖分，并对交界处公共边上的基函数进行特殊处理，从而使得在保持电流连续性的同时，极大地方便了复杂结构目标的几何建模。同时，本文还将该方法推广，提出一种自适应剖分方法。
通过以上四部分，本文对基于高阶叠层矢量基函数的电磁场积分方程方法进行了系统而深入的研究，并以其在三维电磁散射问题中的优异表现证明了高阶叠层矢量基函数的优势。本文的工作表明基于高阶叠层矢量基函数的电磁场积分方程方法具有解决工程电磁场问题的优势和潜力，必将在未来的研究工作中得到广泛使用。
The electromagnetic scattering is very important in many fields such as modern radar target recognizing, microwave imaging and microwave remote sensing. In order to meet the practical engineering requirement, the field of computational electromagnetics has seen a considerable surge in research on efficient accurate numerical methods. The emphases of this paper are concentrating on the application of the higher order hierarchical vector basis functions in the electromagnetic integral equation method, specially majored on the methods to improve the computational property when those bases are used to get the electromagnetic response of the target. This paper is composed as the following four parts.
The first part is the application of those bases in the frequency domain integral equation. The near orthogonal basis functions are introduced at first, which based on the modified Legendre polynomials. Then, the influences of the orthogonality on the bases have been investigated and the result is that the bases will have a well computational property when they have a better orthogonality. So the following part will emphasize the orthogonality and analyze the maximally orthogonalized higher order vector basis functions. The scaling factor was reformed to speed up the iteration convergence in the numerical solution. Furthermore, the radiating far field of the higher order bases which defined on the large patches has been analyzed. The result is that a new method to sparsify the impedance matrix and relief the memory pressure is introduced when the higher order vector basis functions defined on large patches have been utilized in the numerical solution of integral equations. At last, the idea of near orthogonal basis functions has been extended into the triangular case as a new higher order hierarchical vector bases defined on the triangular elements is introduced.
The second part is the application of the near orthogonal hierarchical vector bases in the time domain integral equation method (TDIE). In this part, the algorithm of the near orthogonal bases in the TDIE is investigated, and the numerical example in this part will show the well property of this method. Then, after briefly discussing several causes of the late-time instability of the TDIE solvers, a novel viewpoint about the instability is proposed. A new method has been invented to calculate the element of the impedance matrix accurately in TDIE. This method is adapted to the higher order geometric model which will lead it universal in TDIE. At last, some work on the temporal response of the moving PEC target by TDIE is introduced. Furthermore, the temporal response of the multi-target with a relative movement is investigated too, which will use the domain decomposition method in time domain.
The interpolation/extrapolation method of the information in time and frequency domain researched in the third part. The difficulties in the time domain method will be listed in this part at first. Then the interpolation/extrapolation method will be introduced as it is a perfect method to delay the instability of explicit MOT algorithm. This method will use the low frequency information in the frequency domain to complement the information of the latter temporal response of the target which will deal the instability. An adaptive sampling method is introduced when the principle of the interpolation/extrapolation method have been investigated. Then, the near orthogonal hierarchical vector basis functions are used to get the sampling information in the frequency and time domain respectively, which will improve the efficiency in frequency domain dramatically. At last, the physical bases of the interpolation/extrapolation method have been emphasized and a new method to consolidate the results by the interpolation/extrapolation method is invented.
The fourth part has deal with the problem of the large size patch is hard to model the surface of the complex structure. This problem is the foremost difficulty in utilizing the higher order bases to solve the engineering problem. The method introduced in this paper will make it flexible in geometry model by separate the surface of the target into slick and subtle areas. Then the different size patches are used to mesh the different areas. This method will make sure the continuity of the normal current along the common edge between the different areas and get a flexible meshing method. Furthermore, a new adaptive meshing method in introduced as the extend of this meshing method.
The studies of this paper demonstrate the high accuracy and efficiency of higher order hierarchical vector bases in the integral equation method. The analysis and the numerical results in this paper display its potential to solutions of electromagnetic engineering problems.
本文第一部分从频域角度出发，先将基于修正勒让德多项式的高阶叠层矢量基函数应用于频域电磁场积分方程方法中，并研究了基函数的正交性对其计算性能的影响。通过分析可发现基函数的正交性越好，其计算性能也越好。因此，本文重点研究了对高阶叠层矢量基函数进行正交化而得到的最大正交高阶矢量基函数，并对尺度因子进行修正以提高其计算性能。其次，本文还分析了定义在大尺寸单元上的高阶基函数的辐射特性，提出一种针对采用电磁场积分方程方法求解三维导体目标时得到的阻抗矩阵进行稀疏化的方法。同时，本文还将定义在四边形单元上的高阶叠层矢量基函数进行推广，提出一种定义在三角形单元上的基于修正勒让德多项式的高阶叠层矢量基函数，以解决四边形单元难以精确拟合复杂结构目标的问题。
本文第二部分从时域角度出发，重点研究了准正交高阶叠层矢量基函数在时域电磁场积分方程方法(TDIE)中的算法实现，并通过数值算例详细讨论了该基函数在时域积分方程方法中的性质。其次，本文在简要分析了时域积分方程时间步进算法(MOT)的后时不稳定性问题后，指出引起该算法后时不稳定的主要原因之一是离散TDIE时采用了不精确的数值计算方法。因此，本文提出一种基于曲面单元的卷积积分方法，以精确求解任意曲面单元上的时域阻抗矩阵元素。同时，本文还采用TDIE对运动金属目标的时域电磁响应做了初步研究，并采用时域区域分解算法对空间相对运动的群目标的电磁响应做了简单分析。
本文第三部分研究了时频互推算法。该部分首先说明了当前时域方法的主要困难，指出时频互推算法是解决时域算法后时不稳定的有效手段之一。本文对时频互推算法的互推原理进行了详细的分析，并提出一种自适应算法以确定时域和频域信息的采样宽度。其次，本文还将准正交高阶叠层矢量基函数应用于该算法中计算时频采样信息，从而解决了传统算法中频域算法低频采样时计算量过大的问题。最后本文充分研究了该算法的物理基础，提出一种采用频域滤波对互推结果进行校正的方法，从而有效地提高了互推结果的可靠性，并降低了互推算法的计算复杂度。
本文第四部分针对大尺寸单元难以精确拟合复杂结构目标几何结构的问题，提出一种非均匀剖分的解决方案。该方法通过将目标表面结构分解，采用不同尺寸的单元对不同区域进行剖分，并对交界处公共边上的基函数进行特殊处理，从而使得在保持电流连续性的同时，极大地方便了复杂结构目标的几何建模。同时，本文还将该方法推广，提出一种自适应剖分方法。
通过以上四部分，本文对基于高阶叠层矢量基函数的电磁场积分方程方法进行了系统而深入的研究，并以其在三维电磁散射问题中的优异表现证明了高阶叠层矢量基函数的优势。本文的工作表明基于高阶叠层矢量基函数的电磁场积分方程方法具有解决工程电磁场问题的优势和潜力，必将在未来的研究工作中得到广泛使用。
The electromagnetic scattering is very important in many fields such as modern radar target recognizing, microwave imaging and microwave remote sensing. In order to meet the practical engineering requirement, the field of computational electromagnetics has seen a considerable surge in research on efficient accurate numerical methods. The emphases of this paper are concentrating on the application of the higher order hierarchical vector basis functions in the electromagnetic integral equation method, specially majored on the methods to improve the computational property when those bases are used to get the electromagnetic response of the target. This paper is composed as the following four parts.
The first part is the application of those bases in the frequency domain integral equation. The near orthogonal basis functions are introduced at first, which based on the modified Legendre polynomials. Then, the influences of the orthogonality on the bases have been investigated and the result is that the bases will have a well computational property when they have a better orthogonality. So the following part will emphasize the orthogonality and analyze the maximally orthogonalized higher order vector basis functions. The scaling factor was reformed to speed up the iteration convergence in the numerical solution. Furthermore, the radiating far field of the higher order bases which defined on the large patches has been analyzed. The result is that a new method to sparsify the impedance matrix and relief the memory pressure is introduced when the higher order vector basis functions defined on large patches have been utilized in the numerical solution of integral equations. At last, the idea of near orthogonal basis functions has been extended into the triangular case as a new higher order hierarchical vector bases defined on the triangular elements is introduced.
The second part is the application of the near orthogonal hierarchical vector bases in the time domain integral equation method (TDIE). In this part, the algorithm of the near orthogonal bases in the TDIE is investigated, and the numerical example in this part will show the well property of this method. Then, after briefly discussing several causes of the late-time instability of the TDIE solvers, a novel viewpoint about the instability is proposed. A new method has been invented to calculate the element of the impedance matrix accurately in TDIE. This method is adapted to the higher order geometric model which will lead it universal in TDIE. At last, some work on the temporal response of the moving PEC target by TDIE is introduced. Furthermore, the temporal response of the multi-target with a relative movement is investigated too, which will use the domain decomposition method in time domain.
The interpolation/extrapolation method of the information in time and frequency domain researched in the third part. The difficulties in the time domain method will be listed in this part at first. Then the interpolation/extrapolation method will be introduced as it is a perfect method to delay the instability of explicit MOT algorithm. This method will use the low frequency information in the frequency domain to complement the information of the latter temporal response of the target which will deal the instability. An adaptive sampling method is introduced when the principle of the interpolation/extrapolation method have been investigated. Then, the near orthogonal hierarchical vector basis functions are used to get the sampling information in the frequency and time domain respectively, which will improve the efficiency in frequency domain dramatically. At last, the physical bases of the interpolation/extrapolation method have been emphasized and a new method to consolidate the results by the interpolation/extrapolation method is invented.
The fourth part has deal with the problem of the large size patch is hard to model the surface of the complex structure. This problem is the foremost difficulty in utilizing the higher order bases to solve the engineering problem. The method introduced in this paper will make it flexible in geometry model by separate the surface of the target into slick and subtle areas. Then the different size patches are used to mesh the different areas. This method will make sure the continuity of the normal current along the common edge between the different areas and get a flexible meshing method. Furthermore, a new adaptive meshing method in introduced as the extend of this meshing method.
The studies of this paper demonstrate the high accuracy and efficiency of higher order hierarchical vector bases in the integral equation method. The analysis and the numerical results in this paper display its potential to solutions of electromagnetic engineering problems.
引文
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