子全域基函数法及其应用研究
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摘要
本文介绍并研究了矩量法中一种近来提出的方法——子全域(sub-entiredomain)基函数法的原理及其应用。
传统的矩量法中，内存需求和CPU时间分别与未知量数目的平方和立方成正比。因而它一般只能应用于分析电小尺寸以及中等尺寸的电磁问题。子全域基函数法利用了周期结构的物理特性，能够极大地降低未知量的数目，从而被用于分析大规模周期结构的电磁特性。
第一章介绍了当前分析周期结构问题的较为流行的几类数值方法，如时域有限差分法、有限元法和矩量法，包括他们的优劣势也被介绍了。然后描述了本文的主要研究工作以及意义。
第二章介绍了子全域基函数法并提出了几种改进技术。首先，介绍了矩量法的基本原理以及相关问题。然后利用子全域基函数法分析了一维周期结构，并证实了其精度。接着，提出了两类扩展子全域基函数法：扩展的精确子全域基函数法和扩展的简化子全域基函数法。最后提出了基于降维技术的子全域基函数法，它的精度也被数值算例加以验证。这些方法的优势是进一步降低了未知量的数目。
第三章提出了子全域基函数中阻抗矩阵的快速填充技术。首先，将特征函数技术引入到阻抗矩阵的计算中。它的优点是避免了耗时的二重面积分。接着，提出了一维周期结构的阻抗矩阵快速填充。最后给出了二维问题的简化子全域基函数法中阻抗矩阵的快速填充。
第四章将子全域基函数应用于分析混合周期结构(一维情形)、平板以及周期天线阵列。用该基函数得到的结果与传统矩量法得到的结果吻合很好。
第五章对全文的主要工作进行总结，并展望了子全域基函数法中一些能够继续改进的地方以及进一步扩大其应用的领域。
In this dissertation, principle and applications of the sub-entire domain (SED) basis function method that has been recently proposed are introduced and studied.
In conventional method of moments (MoM), memory and CPU time are proportional to N~2 and N~3, respectively, where N is the number of unknowns. Therefore, it is only suitable to analyzing electrically small and/or moderate electromagnetic problems. The sub-entire domain basis function method can dramatically reduce the number of unknowns because the physical properties of periodic structures are considered. And it has been used to analyze electromagnetic performance of large-scale periodic structures.
In Chapter One, some numerical methods, such as the finite-difference time-domain method, finite element method, the MoM, that are popular in analyzing periodic structures are introduced. Advantages and disadvantages of these methods are also given. Then, main work and meaning of this dissertation are given.
In Chapter Two, the SED basis function method is introduced and some improved techniques are proposed. Firstly, the basic principle and related problems of the MoM are introduced. Secondly, the SED basis function method is applied to analyze one-dimensional periodic structures. The accuracy of the SED basis function is validated. Thirdly, two extended SED basis function methods are proposed. They are extended accurate method and simplified SED basis function methods. Finally, dimension-reduction-based SED basis function method is presented. Its accuracy is also validated by several numerical examples. The number of unknowns can be further reduced by using the improved methods.
In Chapter Three, several filling techniques of impedance matrix are presented. The characteristic function technique is introduced to computation of impedance matrix. Its advantage is that it can avoid the double surface integration, which is very time consuming. Then, fast fill-technique is proposed in the one-dimensional periodic structures. Finally, fast generation technique of moment matrices in the simplified SED basis function method for two-dimensional probelms is proposed.
In Chapter Four, the SED basis function method is extended to analyze the hybrid periodic structures (1-D cases), planar plates, and periodic antenna arrays. Results obtained by using the SED basis function method and the conventional MoM are in good agreement.
In Chapter Five, the dissertation is summarized. In addition, improvability and potential applications of the SED basis function method are forecasted.
传统的矩量法中，内存需求和CPU时间分别与未知量数目的平方和立方成正比。因而它一般只能应用于分析电小尺寸以及中等尺寸的电磁问题。子全域基函数法利用了周期结构的物理特性，能够极大地降低未知量的数目，从而被用于分析大规模周期结构的电磁特性。
第一章介绍了当前分析周期结构问题的较为流行的几类数值方法，如时域有限差分法、有限元法和矩量法，包括他们的优劣势也被介绍了。然后描述了本文的主要研究工作以及意义。
第二章介绍了子全域基函数法并提出了几种改进技术。首先，介绍了矩量法的基本原理以及相关问题。然后利用子全域基函数法分析了一维周期结构，并证实了其精度。接着，提出了两类扩展子全域基函数法：扩展的精确子全域基函数法和扩展的简化子全域基函数法。最后提出了基于降维技术的子全域基函数法，它的精度也被数值算例加以验证。这些方法的优势是进一步降低了未知量的数目。
第三章提出了子全域基函数中阻抗矩阵的快速填充技术。首先，将特征函数技术引入到阻抗矩阵的计算中。它的优点是避免了耗时的二重面积分。接着，提出了一维周期结构的阻抗矩阵快速填充。最后给出了二维问题的简化子全域基函数法中阻抗矩阵的快速填充。
第四章将子全域基函数应用于分析混合周期结构(一维情形)、平板以及周期天线阵列。用该基函数得到的结果与传统矩量法得到的结果吻合很好。
第五章对全文的主要工作进行总结，并展望了子全域基函数法中一些能够继续改进的地方以及进一步扩大其应用的领域。
In this dissertation, principle and applications of the sub-entire domain (SED) basis function method that has been recently proposed are introduced and studied.
In conventional method of moments (MoM), memory and CPU time are proportional to N~2 and N~3, respectively, where N is the number of unknowns. Therefore, it is only suitable to analyzing electrically small and/or moderate electromagnetic problems. The sub-entire domain basis function method can dramatically reduce the number of unknowns because the physical properties of periodic structures are considered. And it has been used to analyze electromagnetic performance of large-scale periodic structures.
In Chapter One, some numerical methods, such as the finite-difference time-domain method, finite element method, the MoM, that are popular in analyzing periodic structures are introduced. Advantages and disadvantages of these methods are also given. Then, main work and meaning of this dissertation are given.
In Chapter Two, the SED basis function method is introduced and some improved techniques are proposed. Firstly, the basic principle and related problems of the MoM are introduced. Secondly, the SED basis function method is applied to analyze one-dimensional periodic structures. The accuracy of the SED basis function is validated. Thirdly, two extended SED basis function methods are proposed. They are extended accurate method and simplified SED basis function methods. Finally, dimension-reduction-based SED basis function method is presented. Its accuracy is also validated by several numerical examples. The number of unknowns can be further reduced by using the improved methods.
In Chapter Three, several filling techniques of impedance matrix are presented. The characteristic function technique is introduced to computation of impedance matrix. Its advantage is that it can avoid the double surface integration, which is very time consuming. Then, fast fill-technique is proposed in the one-dimensional periodic structures. Finally, fast generation technique of moment matrices in the simplified SED basis function method for two-dimensional probelms is proposed.
In Chapter Four, the SED basis function method is extended to analyze the hybrid periodic structures (1-D cases), planar plates, and periodic antenna arrays. Results obtained by using the SED basis function method and the conventional MoM are in good agreement.
In Chapter Five, the dissertation is summarized. In addition, improvability and potential applications of the SED basis function method are forecasted.
引文
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