本文選題：加權Laguerre多項式 + 時域有限差分　； 參考：《電子科技大學》2014年博士論文

【摘要】：本文研究了計算電磁學領域中一種新的無條件穩(wěn)定的快速時域數值計算方法——基于加權Laguerre多項式(weighted Laguerre polynomials,WLPs)的時域有限差分(finite-difference time-domain,FDTD)方法——的基本原理及應用。WLP-FDTD算法在空間域采用Yee氏網格劃分和中心差分技術離散,在時間域采用加權Laguerre多項式作為基函數、Galerkin過程作為權函數處理時間變量。這樣,WLP-FDTD算法的電磁場分量在空間域和時間域分別計算,按照Laguerre多項式的階數步進求解,不受Courant-Friedrich-Levy(CFL)時間穩(wěn)定性條件的限制。WLP-FDTD算法特別適合分析計算寬頻帶、復雜結構和多尺度結構的電磁特性問題,相比傳統(tǒng)的時域計算方法,在計算效率方面有較大的優(yōu)勢。本文在已有WLP-FDTD算法框架下,進一步完善了其基本理論、提出了改進技術和擴展了其應用范圍:一、通過Laguerre域麥克斯韋方程中引入電磁場的傅立葉形式展開式,對二維WLP-FDTD算法的數值色散進行了分析,并從理論上分析了與數值色散有關的關鍵參數選取方法,導出了時間尺度因子與工作頻率的關系。通過分析多項式最大零根的特性,可以計算出為保證算法計算的準確性所需要的步進階數。然后把二維WLP-FDTD方法的數值色散分析推廣到三維,豐富了WLP-FDTD方法的基本理論。把包含有增長因子的電磁場的傅立葉形式展開式引入到Laguerre域麥克斯韋方程,從理論上證明了WLP-FDTD算法按階數步進是無條件穩(wěn)定的。二、把具有四階精度的中心差分公式引入到WLP-FDTD算法中,推導得到高階WLP-FDTD算法,對高階WLP-FDTD的數值色散關系和穩(wěn)定性進行了分析,并對算法的關鍵參數的確定進行了定量描述。通過與低階WLP-FDTD算法的比較,高階WLP-FDTD算法具有數值色散誤差小、計算精度高和計算效率高的特點。三、把表征色散媒質特性的輔助差分方程(auxiliary differential equation,ADE)運用到WLP-FDTD算法中,得到了適合分析廣義色散媒質模型的ADE-WLP-FDTD方法。同時,把近似完全匹配層(nearly perfectly matched layer,NPML)引入到ADE-WLP-FDTD算法中,得到比傳統(tǒng)PML更好的吸收效果。提出的ADE-WLP-FDTD方法及其NPML,有效地擴展了傳統(tǒng)WLP-FDTD算法的使用范圍,能模擬復雜色散媒質的電磁特性。四、將兩種快速求解WLP-FDTD算法的技術——因式分解技術和區(qū)域分解技術——引入到ADE-WLP-FDTD算法中,有效地提高了ADE-WLP-FDTD算法的計算效率。五、把廣義曲線坐標系引入WLP-FDTD算法中,得到了適合模擬任意復雜曲面的非正交WLP-FDTD算法的計算格式。采用這種計算格式模擬復雜結構的電磁特性問題時,可以在不增加計算量的情況下提高計算精度。
[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.
【學位授予單位】：電子科技大學
【學位級別】：博士
【學位授予年份】：2014
【分類號】：TM15

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