本文選題：時(shí)域有限差分法 + 空間濾波��； 參考：《安徽大學(xué)》2017年碩士論文

【摘要】：近年來,隨著計(jì)算電磁學(xué)的蓬勃發(fā)展,計(jì)算電磁學(xué)方法不斷的更新,在社會(huì)的眾多領(lǐng)域做出了杰出的貢獻(xiàn),例如在現(xiàn)代無線電通訊、航天航空、超材料、色散媒質(zhì)等領(lǐng)域得到了很好的應(yīng)用和發(fā)展。期間,有關(guān)計(jì)算電磁學(xué)的學(xué)術(shù)項(xiàng)目,學(xué)術(shù)實(shí)驗(yàn)室,學(xué)術(shù)會(huì)議還有相應(yīng)的學(xué)術(shù)論文大量的增加。同時(shí),以計(jì)算電磁學(xué)為基礎(chǔ)的商業(yè)軟件和實(shí)際應(yīng)用產(chǎn)品在社會(huì)的眾多領(lǐng)域不斷的涌現(xiàn)出來。計(jì)算電磁學(xué)可以說是取得了很高的成就。計(jì)算電磁學(xué)算法層出不窮,主要的杰出代表包括:離散麥克斯韋方程的時(shí)域有限差分法,離散矢量波動(dòng)方程的有限元法,離散積分方程的矩量法。從1966年到Y(jié)ee網(wǎng)格提出至今,經(jīng)典的時(shí)域有限差分法以操作簡單,便于并行,概念清晰,通用性強(qiáng)等優(yōu)點(diǎn)在各個(gè)領(lǐng)域得到了很廣泛的運(yùn)用。經(jīng)典的時(shí)域有限差分法最主要有一下有兩大缺點(diǎn):一,在網(wǎng)格方向上演進(jìn),所以在模擬一些不連續(xù)材質(zhì)中存在一定的困難。二,由于各向異性、數(shù)值色散性、數(shù)值穩(wěn)定性的影響,隨著時(shí)間的積累時(shí)域有限差分法的數(shù)值誤差會(huì)越來越大,導(dǎo)致了計(jì)算結(jié)果的失真。同時(shí),傳統(tǒng)的時(shí)域有限差分法必須始終滿足時(shí)間和空間上的穩(wěn)定性條件。為了打破傳統(tǒng)穩(wěn)定性條件的約束,實(shí)現(xiàn)顯式的高穩(wěn)定,本文采用空間濾波的方法。時(shí)域有限差分法在空間頻域上對數(shù)值精度,數(shù)值色散有影響的是在低頻部分,而高頻部分的影響很小,可以忽略不計(jì),所以通過在空間頻域中濾除高頻的部分,只保留低頻部分,這樣可是實(shí)現(xiàn)時(shí)間步的任意取值,完成顯式的高穩(wěn)定,極大的加快了運(yùn)行速度和效率,并且不會(huì)影響到計(jì)算結(jié)果的精確度。為了解決上述問題,本文對高穩(wěn)定度的辛?xí)r域有限差分法進(jìn)行了系統(tǒng)研究。辛?xí)r域有限差分法依然必須要滿足穩(wěn)定性條件,時(shí)間和空間的交替差分依然受到嚴(yán)格的限制。本文在時(shí)間方向上,通過高階辛積分和濾除空間高次諧波的方法,打破了傳統(tǒng)的穩(wěn)定性條件,實(shí)現(xiàn)顯式的高穩(wěn)定,同時(shí)能在長期仿真中保持麥克斯韋方程的辛結(jié)構(gòu),極大的提高了運(yùn)行的效率,大大的縮減了運(yùn)行時(shí)間,并且提高了計(jì)算結(jié)果的精度;在空間方向上,采用了高階交錯(cuò)差分,提高了數(shù)值精度,減小數(shù)值色散;在網(wǎng)格剖分上,采用亞網(wǎng)格模型結(jié)合空間濾波的方法解決材質(zhì)的不連續(xù)性問題。通過上述的算法和模型,來建立更加快速,高精度的時(shí)域解決方案。
[Abstract]:In recent years, with the vigorous development of computational electromagnetics, computational electromagnetics methods have been constantly updated, and have made outstanding contributions in many fields of society, such as modern radio communications, aerospace, metamaterials, Dispersive media and other fields have been well applied and developed. During this period, there has been a substantial increase in academic projects, laboratories and conferences on computational electromagnetism. At the same time, commercial software and practical application products based on computational electromagnetics are emerging in many fields of society. Computational electromagnetism can be said to have made great achievements. Computational electromagnetic algorithms emerge in endlessly. The main outstanding examples include the finite difference time-domain method for discrete Maxwell equation, the finite element method for discrete vector wave equation, and the moment method for discrete integral equation. From 1966 to now, the classical finite difference time-domain method has been widely used in various fields because of its advantages of simple operation, easy to parallel, clear concept and strong versatility. The classical finite-difference time-domain method has two main disadvantages: first, it evolves in the grid direction, so it is difficult to simulate some discontinuous materials. Secondly, due to the influence of anisotropy, numerical dispersion and numerical stability, the numerical error of the finite difference time-domain (FDTD) method will increase with the accumulation of time, resulting in the distortion of the calculation results. At the same time, the traditional FDTD method must always satisfy the stability conditions in time and space. In order to break the constraint of traditional stability condition and realize explicit high stability, the spatial filtering method is used in this paper. The finite-difference time-domain (FDTD) method has an effect on the numerical accuracy and dispersion in the spatial frequency domain, but the influence of the high-frequency part is too small to be ignored, so the high-frequency part is filtered out in the spatial frequency domain. Only the low frequency part is retained, which can realize the arbitrary value of the time step, complete the explicit high stability, greatly speed up the running speed and efficiency, and will not affect the accuracy of the calculation results. In order to solve the above problems, the symplectic finite-difference time-domain method with high stability is studied systematically in this paper. The symplectic finite-difference time-domain method still has to satisfy the stability condition, and the alternating difference between time and space is still strictly restricted. In this paper, by means of high order symplectic integration and filtering high order harmonics in time direction, the traditional stability condition is broken, and the explicit high stability is realized, and the symplectic structure of Maxwell equation can be maintained in the long term simulation. The efficiency of the operation is greatly improved, the running time is greatly reduced, and the accuracy of the calculation results is improved. In the spatial direction, the high order staggered difference is used to improve the numerical accuracy and reduce the numerical dispersion. The subgrid model combined with spatial filtering is used to solve the material discontinuity problem. Through the above algorithms and models, a faster and more accurate time domain solution is established.
【學(xué)位授予單位】：安徽大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O441

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