无条件稳定的快速时域算法及应用研究

发布时间：2018-06-10 09:29

本文选题：加权Laguerre多项式 + 时域有限差分　；参考：《电子科技大学》2014年博士论文

【摘要】：本文研究了计算电磁学领域中一种新的无条件稳定的快速时域数值计算方法——基于加权Laguerre多项式(weighted Laguerre polynomials,WLPs)的时域有限差分(finite-difference time-domain,FDTD)方法——的基本原理及应用。WLP-FDTD算法在空间域采用Yee氏网格划分和中心差分技术离散,在时间域采用加权Laguerre多项式作为基函数、Galerkin过程作为权函数处理时间变量。这样,WLP-FDTD算法的电磁场分量在空间域和时间域分别计算,按照Laguerre多项式的阶数步进求解,不受Courant-Friedrich-Levy(CFL)时间稳定性条件的限制。WLP-FDTD算法特别适合分析计算宽频带、复杂结构和多尺度结构的电磁特性问题,相比传统的时域计算方法,在计算效率方面有较大的优势。本文在已有WLP-FDTD算法框架下,进一步完善了其基本理论、提出了改进技术和扩展了其应用范围:一、通过Laguerre域麦克斯韦方程中引入电磁场的傅立叶形式展开式,对二维WLP-FDTD算法的数值色散进行了分析,并从理论上分析了与数值色散有关的关键参数选取方法,导出了时间尺度因子与工作频率的关系。通过分析多项式最大零根的特性,可以计算出为保证算法计算的准确性所需要的步进阶数。然后把二维WLP-FDTD方法的数值色散分析推广到三维,丰富了WLP-FDTD方法的基本理论。把包含有增长因子的电磁场的傅立叶形式展开式引入到Laguerre域麦克斯韦方程,从理论上证明了WLP-FDTD算法按阶数步进是无条件稳定的。二、把具有四阶精度的中心差分公式引入到WLP-FDTD算法中,推导得到高阶WLP-FDTD算法,对高阶WLP-FDTD的数值色散关系和稳定性进行了分析,并对算法的关键参数的确定进行了定量描述。通过与低阶WLP-FDTD算法的比较,高阶WLP-FDTD算法具有数值色散误差小、计算精度高和计算效率高的特点。三、把表征色散媒质特性的辅助差分方程(auxiliary differential equation,ADE)运用到WLP-FDTD算法中,得到了适合分析广义色散媒质模型的ADE-WLP-FDTD方法。同时,把近似完全匹配层(nearly perfectly matched layer,NPML)引入到ADE-WLP-FDTD算法中,得到比传统PML更好的吸收效果。提出的ADE-WLP-FDTD方法及其NPML,有效地扩展了传统WLP-FDTD算法的使用范围,能模拟复杂色散媒质的电磁特性。四、将两种快速求解WLP-FDTD算法的技术——因式分解技术和区域分解技术——引入到ADE-WLP-FDTD算法中,有效地提高了ADE-WLP-FDTD算法的计算效率。五、把广义曲线坐标系引入WLP-FDTD算法中,得到了适合模拟任意复杂曲面的非正交WLP-FDTD算法的计算格式。采用这种计算格式模拟复杂结构的电磁特性问题时,可以在不增加计算量的情况下提高计算精度。
[Abstract]:In this paper, we study the basic principle and application of a new unconditionally stable fast time-domain numerical method in computational electromagnetics, a finite-difference time-domain FDTDmethod based on weighted Laguerre polynomials and weighted Laguerre polynomialsWLPs-and its application. WLP-FDTD algorithm In spatial domain, Yee's mesh division and central difference technique are used to discretize. In time domain, the weighted Laguerre polynomial is used as the basis function and the Galerkin process is used as the weight function to deal with the time variable. In this way, the electromagnetic field components of WLP-FDTD algorithm are calculated in the space domain and the time domain, respectively. According to the order step solution of the Laguerre polynomial, the time-stability condition of Courant-Friedrich-Levyn CFL is not restricted. WLP-FDTD algorithm is especially suitable for the analysis and calculation of wide frequency band. Compared with the traditional time-domain calculation method, the electromagnetic characteristics of complex structures and multi-scale structures have more advantages in computational efficiency. In this paper, the basic theory of WLP-FDTD algorithm is further improved, and the improved technique and its application are proposed. Firstly, the Fourier expansion of electromagnetic field is introduced into the Maxwell equation in Laguerre domain. The numerical dispersion of the two-dimensional WLP-FDTD algorithm is analyzed. The method of selecting the key parameters related to the numerical dispersion is theoretically analyzed and the relationship between the time scale factor and the working frequency is derived. By analyzing the property of the maximum zero root of the polynomial, the step number needed to ensure the accuracy of the algorithm can be calculated. Then, the numerical dispersion analysis of two-dimensional WLP-FDTD method is extended to 3D, which enriches the basic theory of WLP-FDTD method. The Fourier form expansion of electromagnetic field with growth factor is introduced into Maxwell equation in Laguerre domain. It is proved theoretically that the WLP-FDTD algorithm is unconditionally stable in order. Secondly, the central difference formula with fourth-order precision is introduced into the WLP-FDTD algorithm, and the high-order WLP-FDTD algorithm is derived. The numerical dispersion relation and stability of the high-order WLP-FDTD are analyzed, and the key parameters of the algorithm are quantitatively described. Compared with the low-order WLP-FDTD algorithm, the higher-order WLP-FDTD algorithm has the advantages of small numerical dispersion error, high accuracy and high efficiency. Thirdly, the auxiliary difference equation which characterizes the properties of dispersive media is applied to WLP-FDTD algorithm, and the ADE-WLP-FDTD method suitable for the analysis of generalized dispersive media model is obtained. At the same time, the approximate perfectly matched layer (NPML) is introduced into ADE-WLP-FDTD algorithm, and the absorption effect is better than that of traditional PMLs. The proposed ADE-WLP-FDTD method and its NPMLs effectively extend the application range of the traditional WLP-FDTD algorithm and can simulate the electromagnetic properties of complex dispersive media. Fourthly, two techniques of fast solving WLP-FDTD, factorization technique and domain decomposition technique, are introduced into ADE-WLP-FDTD algorithm, which can effectively improve the computational efficiency of ADE-WLP-FDTD algorithm. Fifthly, the generalized curvilinear coordinate system is introduced into the WLP-FDTD algorithm, and the non-orthogonal WLP-FDTD algorithm suitable for simulating arbitrary complex surfaces is obtained. When using this scheme to simulate the electromagnetic characteristics of complex structures, the calculation accuracy can be improved without increasing the computational complexity.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TM15

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