本文關(guān)鍵詞：FDTD改進(jìn)算法及其理想導(dǎo)體邊界實現(xiàn)　出處：《西南交通大學(xué)》2014年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 時域有限差分方法 減縮時域有限差分方法 交替方向隱式時域有限差分方法 局部一維時域有限差分方法 對稱性 理想導(dǎo)體邊界

【摘要】：計算電磁學(xué)是電磁場理論、數(shù)學(xué)和計算機技術(shù)相結(jié)合的產(chǎn)物。計算電磁學(xué)正向著高精度、高效能、高速度的目標(biāo)快速發(fā)展,很多以前無法解決的疑難電磁場問題找到了很好的解決方法,得到了精確的計算結(jié)果。越來越多的實際工程電磁場疑難問題擺在人們面前,這更促進(jìn)了計算電磁學(xué)向更高水平發(fā)展。基于麥克斯韋方程組的解析方法和數(shù)值方法求解是計算電磁學(xué)的主要任務(wù)。通常,只有經(jīng)典的電磁場問題才有解析解,數(shù)值計算逐漸成為解決復(fù)雜電磁場問題的主要手段,有時甚至是唯一手段。FDTD方法是典型的時域全波分析方法,應(yīng)用范圍非常廣泛,是近些年最受關(guān)注、發(fā)展最迅速的數(shù)值方法之一。麥克斯韋方程是描述電磁現(xiàn)象的基本方程,FDTD方法從其旋度方程,即麥克斯韋微分形式出發(fā),對時間和空間的一階偏導(dǎo)數(shù)采取中心差分近似,直接轉(zhuǎn)換為顯式差分運算,這樣可以在時間和空間上對連續(xù)電磁場數(shù)據(jù)實現(xiàn)抽樣離散。FDTD方法能夠描述時域電磁場的傳播特性,只要給出問題的初始條件和邊界條件,即可應(yīng)用FDTD方法迭代遞推得到各個時間步和空間步的電磁場分布。隨著計算機硬件條件的發(fā)展,與FDTD方法相關(guān)的創(chuàng)新研究不斷涌現(xiàn),FDTD方法將會贏得越來越多的計算電磁學(xué)領(lǐng)域?qū)＜业年P(guān)注和青睞。然而,FDTD方法也有其自身不可忽視的不足：一方面,采用差分法對麥克斯韋方程近似求解時,會在計算網(wǎng)格中引起數(shù)值色散,這種關(guān)系隨數(shù)值波模的傳播方向以及離散化程度不同而改變。因此,FDTD方法受限于數(shù)值色散條件,一般空間步長不大于波長的十分之一,當(dāng)數(shù)值模擬對象的電尺寸較大時,將導(dǎo)致所需內(nèi)存劇增；另一方面,FDTD方法的時間步長和空間步長不獨立,Courant-Friedrich-Levy (CFL)穩(wěn)定條件限制了時間離散,使得時間步長必須隨空間步長變化而變化,如果空間步長變小,則需要增加時間迭代步數(shù)以實現(xiàn)收斂。因此,對于目前普通的PC機而言,FDTD方法在計算效率方面存在著不足。圍繞著電磁場數(shù)值方法的計算效率問題,近年來出現(xiàn)了FDTD的多種改進(jìn)算法：針對內(nèi)存問題的R-FDTD方法,針對計算時間步長問題的ADI-FDTD和LOD-FDTD方法都是較熱門的數(shù)值計算方法。當(dāng)然,在實際應(yīng)用中,會發(fā)現(xiàn)這些算法都或多或少存在一些問題,對這些FDTD方法的改進(jìn)算法作進(jìn)一步的研究,以節(jié)約內(nèi)存使用量和計算時間,提高計算準(zhǔn)確程度為目的,實現(xiàn)更加高效、更加精確的數(shù)值計算,具有理論和應(yīng)用意義。論文首先對課題的研究背景及意義進(jìn)行了闡述,包括計算電磁學(xué)進(jìn)展、發(fā)展時域數(shù)值計算的必要性以及時域有限差分方法的介紹。本文涉及到的FDTD改進(jìn)算法包括R-FDTD方法、ADI-FDTD方法和LOD-FDTD方法。它們能夠很好的解決內(nèi)存使用量大,計算時間長的問題。介紹了幾種FDTD改進(jìn)算法的研究現(xiàn)狀。最后說明了本文所做的主要工作。針對具有對稱結(jié)構(gòu)的計算模型,從理論上分析了采用PEC邊界和PMC邊界截斷的對稱邊界條件,提出了對稱截斷FDTD方法,利用該方法能夠確定截斷邊界以外場分量的值,以實現(xiàn)截斷邊界處的FDTD方法計算。數(shù)值計算驗證了對稱截斷FDTD方法的正確性和可行性。論述了R-FDTD方法中對暫存場分量邊值進(jìn)行補充計算的必要性。論證了三維R-FDTD求解感應(yīng)電荷密度處理導(dǎo)體問題的方法與傳統(tǒng)FDTD方法等價。提出了在內(nèi)存使用量和計算時間上都具有明顯優(yōu)勢的周期對稱結(jié)構(gòu)R-FDTD方法,該方法結(jié)合了R-FDTD方法和第二章對稱截斷方法的優(yōu)點,將計算區(qū)域的內(nèi)存使用量最多降為FDTD算法的1/6。由于每個時間步迭代計算的復(fù)雜度降低,需要計算的網(wǎng)格數(shù)明顯減少,總的迭代計算時間也大幅縮減。數(shù)值計算驗證了該方法的正確性和有效性。為了準(zhǔn)確求解ADI-FDTD方法實現(xiàn)PEC邊界和PMC邊界的待求場分量系數(shù),通過在獲得該系數(shù)前應(yīng)用理想導(dǎo)體邊界條件,推導(dǎo)出了相應(yīng)的修正系數(shù)。計算了單個金屬立方體和對稱的兩個金屬立方體的雙站RCS。結(jié)果表明,理想導(dǎo)體邊界作為理想導(dǎo)體表面,采用修正系數(shù)的計算結(jié)果與FDTD方法計算結(jié)果更為吻合；理想導(dǎo)體邊界作為截斷計算空間對稱面,采用修正系數(shù)的計算結(jié)果與ADI-FDTD方法計算結(jié)果相同,與理論推導(dǎo)結(jié)論一致。證明了LOD-FDTD方法實現(xiàn)PEC邊界和PMC邊界時的待求場分量系數(shù)與傳統(tǒng)的LOD-FDTD方法系數(shù)不同。通過在獲得該系數(shù)前應(yīng)用理想導(dǎo)體邊界條件,得到對應(yīng)的修正系數(shù)。針對將理想導(dǎo)體邊界條件作為理想導(dǎo)體表面和截斷計算空間對稱面的不同情況,討論了修正系數(shù)與傳統(tǒng)LOD-FDTD系數(shù)的區(qū)別。具有統(tǒng)一的表達(dá)式的修正系數(shù)LOD-FDTD方法計算理想導(dǎo)體表面較傳統(tǒng)LOD-FDTD方法誤差更小,并對其進(jìn)行了數(shù)值驗證。
[Abstract]:Computational electromagnetics is the product of electromagnetic theory, mathematics and computer technology. The combination of computational electromagnetics towards high precision, high efficiency, rapid development of high speed target, many can not solve the difficult problem of electromagnetic field before to find a good solution, obtain the calculation precision. The actual engineering electromagnetic problems more and more in front of people, it promotes the computational electromagnetics to a higher level of development. To solve the analytical and numerical methods based on Maxwell's equations is the main task of computational electromagnetics. Usually, only by classical electromagnetic field problems has analytic solution, numerical calculation has become the main means to solve complex electromagnetic problems, sometimes even is the only.FDTD method is a typical time-domain full wave analysis method, application range is very wide, is the most popular in recent years, the rapid development of numerical One of the methods. The Maxwell equation is the basic equation describing the electromagnetic phenomena, the FDTD method from the view that the Maxwell curl equations, the differential form of time and space to take the first derivative central difference approximation is directly converted into explicit difference operation, the propagation characteristics of this can in time and space of continuous electromagnetic data sampling discrete.FDTD method can describe the time-domain electromagnetic field, as long as the problem is given the initial conditions and boundary conditions, the distribution of electromagnetic field can be used FDTD iterative recursive method to get each time step and spatial step. With the development of computer hardware, and FDTD related innovative research methods are constantly emerging, the FDTD method will win the attention and favor more and more experts in the field of computational electromagnetics. However, the FDTD method has its own shortcomings can not be ignored: on the one hand, the difference of Mike The approximate solution of Maxwell equations, will cause the numerical dispersion in computational grid, the relationship with the numerical wave propagation direction and discrete degree of different changes. Therefore, the FDTD method is limited to numerical dispersion conditions, general space step size is not greater than the wavelength of the 1/10, when the electrical large size when the simulation object, will cause to increase memory; on the other hand, the time step and spatial step FDTD method is not independent, Courant-Friedrich-Levy (CFL) stable conditions of discrete time, the time step must change with the change of space, if space is smaller, you will need to increase the time step iteration convergence hundreds. Therefore, for the PC machine at present in general, the FDTD method in computing efficiency shortcomings. Around the computation efficiency of the numerical method, in recent years there has been a variety of improved FDTD algorithm Method: R-FDTD method for memory problems, according to the ADI-FDTD and LOD-FDTD method to calculate the time step problem is numerical calculation of popular methods. Of course, in practice, will find that these algorithms are more or less there are some problems and improvement of these FDTD algorithm for further study, in order to save the use amount and calculation time memory, improve the calculation accuracy for the purpose of achieving more efficient and more accurate numerical calculation, which has theoretical and practical significance. Firstly, the research background and significance are expounded, including the development of computational electromagnetics, computational development and the necessity of time-domain numerical finite-difference time-domain method is introduced. This article relates to the the improved FDTD algorithm including R-FDTD method, ADI-FDTD method and LOD-FDTD method. It can solve the memory usage and long computing time. The research status of several improved FDTD algorithm. Finally the main work done in this paper. According to the calculation model with symmetrical structure, from the theoretical analysis of the symmetric boundary conditions of the PEC and PMC boundary truncation, the symmetric truncated FDTD method, using this method can determine the truncation boundary to field values. In order to implement the FDTD method the truncation boundary calculation. The numerical results verify the correctness and feasibility of truncated FDTD method. The R-FDTD method of temporary field boundary value necessary to supplement calculation. Demonstrates the equivalence method of 3D R-FDTD for induction treatment and charge density of the conductor. The traditional FDTD method put forward cycle symmetrical R-FDTD method has obvious advantages in memory usage and computation time, this method combines the R-FDTD method and the second chapter symmetric truncation method. That reduces the complexity of the computational region of the memory usage of the most reduced FDTD algorithm 1/6. calculation because each time step iteration, the number of grid computing needs significantly reduced, the total computing time is significantly reduced. The numerical results verify the correctness and validity of the method. In order to accurately solve the implementation method of PEC ADI-FDTD and PMC boundary unknown field coefficients, through the application of ideal conductor boundary conditions in the coefficients are derived, corresponding correction factor. The calculation of the two metal cube single metal cube and symmetrical bistatic RCS. results show that the ideal conductor boundary as ideal conductor surface, calculation results using the results correction the coefficient and FDTD method is more consistent; as the perfect conductor boundary to truncate the computational space plane of symmetry, the calculation results of correction coefficient method and ADI-FDTD calculation results At the same time, consistent with the theory. The results show that the LOD-FDTD method to achieve PEC and PMC boundary when the undetermined coefficient method of LOD-FDTD coefficients and the traditional field. Through the application of ideal conductor boundary conditions in the coefficient, the correction coefficient has been obtained. The corresponding ideal conductor boundary conditions as the ideal conductor surface and truncation calculation the spatial symmetry of different situations, different with the traditional correction coefficient LOD-FDTD coefficient is discussed. The modified LOD-FDTD method has uniform coefficient expression of the calculation error of ideal conductor surface than the traditional LOD-FDTD method, and carries on the numerical simulation.

【學(xué)位授予單位】：西南交通大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：TM15

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