New development of time-domain numerical algorithms for electromagnetic field
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摘要: 电磁场时域计算方法由于一次计算可以获得目标的时域响应，结合傅里叶变换得到宽带信息等的优势越来越受到关注.本文介绍了近年来时域有限差分（finite-difference time-domain，FDTD）法和时域有限元（finite element time-domain，FETD）无条件稳定算法方面的研究进展以及FETD算法的更新方案——时域非连续伽辽金（discontinuous Galerkin time-domain，DGTD）方法的新进展.
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关键词:
- 时域非连续伽辽金(DGTD)法 /
- 时域有限元(FETD)算法 /
- 时域有限差分(FDTD)方法 /
- 无条件稳定 /
- 数值通量
Abstract: The time-domain computation of electromagnetic field has attracted more and more attention because it can obtain the time-domain response of the target by one calculation, and combines the advantages of Fourier transform to obtain broadband information. In this paper, the research progress of finite-difference time-domain and unconditional stability algorithm of finite element time-domain in recent years is introduced. In addition, the new development of discontinuous Galerkin method time-domain, which is an updated scheme of finite element method time-domain, is introduced. -
引言
在计算电磁学的时域数值算法中时域有限差分(finite-difference time-domain, FDTD)方法是最为流行的算法之一.传统的FDTD法受稳定性条件的限制, 空间离散网格尺度和时间步长绑定, 当计算域内目标细节要求的空间离散尺度远远小于计算上限频率准确计算所需的空间离散尺度时, 按照最小空间离散尺度和稳定性条件选取时间步长将造成极大的计算资源浪费.时域有限元(finite element time-domain, FETD)法是另外一种电磁场时域主流算法.该方法采用非结构网格离散计算域, 对曲面目标的拟合更好.但是, FETD算法需要处理大型的矩阵问题.随着目标复杂度的增加, 多尺度和精细结构的存在, 该算法的弊端也越来越凸显.时域非连续伽辽金(discontinuous Galerkin time-domain, DGTD)法采用非结构网格离散, 对Maxwell方程采用伽辽金加权法得到弱解形式, 单元之间通过数值通量联系交换数据.该算法的计算复杂度介于FDTD和FETD之间.本文首先介绍近年来在FDTD和FETD无条件稳定方面的新进展, 随后介绍DGTD方法方面的进展.
1 时域无条件稳定方法的研究进展
近年来, 众多学者在克服传统FDTD方法和FETD方法在稳定性条件的限制方面做了很多贡献, 发展了很多改进的FDTD方法和FETD方法以适应新的计算需求, 提高计算效率.这些使时间步长和空间步长无关的FDTD方法或FETD方法称为无条件稳定时域有限差分(unconditionally stable finite-difference time-domain, US-FDTD)方法或无条件稳定时域有限元(unconditionally stable finite element time-domain, US-FETD)方法.下面将分别介绍这两种无条件稳定算法的研究进展.
1.1 US-FDTD方法的研究进展
目前, 主要的US-FDTD方法包括:交替方向隐式时域有限差分(alternating direction implicit finite-difference time-domain, ADI-FDTD)法、局部一维时域有限差分(locally one-dimensional finite-difference time-domain, LOD-FDTD)法、Crank-Nicolson时域有限差分(Crank-Nicolson finite-difference time-domain, CN-FDTD)法、加权拉盖尔多项式时域有限差分(weighted Laguerre polynomials finite-difference time-domain, WLP-FDTD)法、消除空间不稳定模的显式无条件稳定FDTD法和连带埃尔米特时域有限差分(associated Hermite finite-difference time-domain, AH-FDTD)法，以及混合显隐式差分时域有限差分(hybrid implicit explicit-finite difference time-domain, HIE-FDTD)法等, 如图 1所示.
1999年, Takefumi Namiki提出了ADI-FDTD方法, 并将该方法应用于二维横向电磁(transverse electromagnetic, TE)波问题中[1-2].同年, Fenghua Zheng将ADI-FDTD方法应用在三维电磁场求解问题中[3]. ADI-FDTD既有显式差分法计算简单的优点, 又有隐式差分法无条件稳定的特性, 时间步长能突破传统FDTD方法的CFL(Courant-Friedrichs-Lewy)条件的限制.但该方法是传统FDTD方法的近似, 计算精度不高, 且随着时间步长选取的增大计算误差随之增大.
2005年, J. Shibayama等人[4], V. E. do Nascimento等人[5]提出了LOD-FDTD方法. 2006年, Valtemir E. do Nascimento等人将分裂场完全匹配层(perfectly matched layer, PML)应用在了LOD-FDTD方法中[6]. 2007年, Iftikhar Ahmed等人将卷积完全匹配层(convolutional perfectly matched layer, CPML)应用在LOD-FDTD中[7], Erping Li等人对LOD-FDTD方法进行了改进并分析其数值色散性[8], Eng Leong Tan将LOD-FDTD应用于三维maxwell方程中[9]. 2008年, Iftikhar Ahmed等人发展了三步迭代的三维LOD-FDTD方法[10]. 2009年, Qi-Feng Liu等人提出了任意阶的LOD-FDTD方法, 并对其稳定性和数值色散进行了研究[11]. 2010年, Iftikhar Ahmed等人研究了CPML吸收边界在三维LOD-FDTD方法中的应用[12], 并分析了三维LOD-FDTD的数值色散[13]. 2017年, Eng Leong Tan等人研究了非均匀时间步长的LOD-FDTD方法, 并分析了该方法的稳定性[14].
CN-FDTD算法通过将Crank-Nicolson方案和传统FDTD算法相结合, 用n时刻和n+1时刻电磁场量的几何平均值近似代替n+1/2时刻的场量, 得到电场分量和磁场分量的隐式迭代方程.相比于ADI-FDTD算法和LOD-FDTD算法, CN-FDTD算法的数值精度高, 然而, 采用CN-FDTD算法得到的电场分量和磁场分量的最终迭代方程会形成大型的稀疏矩阵, 直接求解需要消耗大量的计算资源, 这在一定程度上制约了CN-FDTD算法的发展. 2003年, G. Sun等人将Douglas-Cunn应用到Crank-Nicolson (CN)方法中, 用来解决二维问题, 极大提高了CN-FDTD算法的计算效率[15]. 2004年, G. Sun等人又将CN方法和近似因子分解(approximate-factorization-splitting, AFS)相结合用来解决三维问题, 该方法被称为CNAFS[16]. 2010年, K. Xu等人用通用处理器(general processing unit, GPU)对CN-FDTD方法进行加速[17]. 2011年, Jianbao Wang等人用CN-FDTD解决斜入射平面波与周期结构的相互作用问题[18]. 2017年, Seyed-Mojtaba Sadrpour等人用CN-FDTD处理波动方程[19], Xiao-Kun Wei等人将区域分解法和CN-FDTD结合用来解决色散金属光栅问题[20].
WLP-FDTD和AH-FDTD属于时域正交展开法, 这两种方法通过采用不同的正交基函数将Maxwell方程中的时间相关项展开, 实现按阶步进. 2003年, Young-Seek Chung等人将加权拉盖尔多项式(weighted Laguerre polynomials, WLP)作为时间基函数来处理二维TE波[21]. 2007年, Yun Yi等人提出了WLP-FDTD方法中总场/散射场边界(total-field/scattered-field, TF/SF)以及PML吸收边界的实现方法[22]. 2009年, Yan-Tao Duan等人提出了一种分解分裂方案, 提高WLP-FDTD方法在二维问题中的计算效率[23]. 2011年, Yan-Tao Duan等人又将分解分裂方法扩展到三维问题中[24]. 2013年, Zheng Chen等人采用了迭代方法来降低分裂误差, 并简化了WLP-FDTD的迭代方程, 提高了计算精度[25]. AH-FDTD方法属于最新的US-FDTD法, 该方法最早由黄正宇等人于2014年提出[26], 目前关于该方法的论文及书籍均由黄正宇研究团队所发表. 2015年, 黄正宇等人提出了AH-FDTD中总场/散射场的处理以及各向异性完全匹配层(uniaxial anisotropic perfectly matched layer, UPML)吸收边界的处理[27]; 采用特征值变换技术实现了对矩阵方程的按阶并行处理, 提高了计算效率[28]; 并对一般色散问题进行了分析[29-30]; 2016年, 采用了平面波注入法引入AH-FDTD方法中的平面波源; 2017年, 推导了二维周期AH-FDTD方法并实现了对光子带隙结构的分析[31], 实现了柱坐标系下的AH-FDTD方法[32], 在按阶并行方法基础上, 通过引入“交替方向”迭代技术实现了二维AH-FDTD的高效算法[33].石立华等人将FDTD方法中高阶算法的思想引入到AH-FDTD方法中, 有效降低了色散误差[34]. 2018年, 黄正宇团队出版专著《连带Hermite基无条件稳定时域有限差分方法》[35].
消除空间不稳定模方法最早是由Dan Jiao团队于2013年引入到FDTD方法中[36], 该方法通过将FDTD法中空间离散所产生的不稳定高频波滤除使其达到无条件稳定, 该团队将该方法称为显式无条件稳定FDTD法. 2014年, Dan Jiao团队的M. Gaffer等人又对该方法进行了详细描述[37]. 2015年, M. Gaffer等人用该方法对有耗问题进行了分析[38], 提出了更高效的去除不稳定模的方法[39], 并求解了三维有耗问题[40], 同年, Zhizhang (David) Chen研究团队的范为也对该方法进行了研究, 分析了谐振腔的谐振频率[41]. 2016年, Jin Yan等人对系统矩阵的特征值进行了研究, 阐述了不稳定模所对应的特征值均为细网格及其细网格周围产生的网格, 提出了高效求解系统矩阵不稳定模所对应的特征值的方法[42], 周孟桥等人用亚网格算法模拟了有两个光电管(photoelectric cell, PEC)鳍片的鳍状波导[43]. 2017年, Jin Yan等人又对该方法进行了详细描述和分析[44], 并提出了对称半正定的亚网格算法[45]. 2018年, Jin Yan等人提出了无条件稳定的非对称亚网格算法[46], Yang Wu等人提出了在空间上离散求解而在时间上连续求解的方法, 达到了无条件稳定[47].
2003年, Huang等人提出了二维HIE-FDTD方法[48], 给出了该方法的基本迭代公式, 讨论了该方法的时间稳定性条件和色散误差, 但其分析只限于2D TE波. 2010年, Chen将该方法推广至三维坐标[49], 详细给出了该算法的基本迭代公式[50].文献[51-53]描述了HIE-FDTD方法的时间稳定性条件.文献[53]给出了HIE-FDTD方法的时间稳定性条件和色散误差模型.为了克服HIE-FDTD方法的色散误差, Dong在2016年提出了四阶HIE-FDTD方法[54].该方法主要是采用Taylor级数对空间求导进行近似, 使得HIE-FDTD方法在空间域具有四阶精度, 从而大大降低了HIE-FDTD方法的色散误差, 但是, 该方法会导致非常复杂的迭代方程, 并且会在一定程度上降低计算效率.为了分析HIE-FDTD方法的计算精度, 文献将HIE-FDTD方法看做FDTD方法的近似方法, 对两者的理论误差进行了推导, 分析结果表明, HIE-FDTD方法的计算精度与时间步长和计算空间的场的变化率密切相关[50]:时间步长越大或者空间场变化越剧烈, HIE-FDTD方法的计算精度越低; 但是, 与ADI-FDTD方法相比, 同样时间步长下, HIE-FDTD方法的计算精度明显高于ADI-FDTD方法. HIE-FDTD方法和ADI-FDTD方法计算精度的详细比较可参考文献[50, 55].文献[56]详细讨论了HIE-FDTD中连接边界的设置.文献[57-60]分析了CPML在HIE-FDTD方法中的应用.为了对具有周期结构的电磁目标进行模拟, 文献[61]将HIE技术和SFDTD方法相结合, 提出了HIE-SFDTD方法, 该方法对任意角度入射的电磁波均可以进行周期边界设置, 但存在要求解非三角矩阵的问题.为了克服这个问题, 文献[62]利用局部一维(locally one-dimensional, LOD)技术将迭代方程分裂为两步进行求解, 因此, 该方法只要求解一个隐式方程, 但其只适用于周期边界沿精细结构方向的问题. Lei等人对该方法进一步进行了改进, 使得该方法无论精细结构沿哪个方向, 都只需求解一个三角矩阵方程[63].但是, 这些方法的时间步长与电磁波入射角度有关, 2D坐标下其时间稳定性条件为: Δt≤cos θ Δx/c, 此处, θ表示波的入射角度.显然, 当入射角度接近90°时, 时间步长会明显减小, 这大大降低了算法的计算效率.在应用方面, 2007年, Chen首先将HIE-FDTD方法用于计算开有细缝的屏蔽腔[64], 计算结果表明, 在相同计算精度下, HIE-FDTD方法的计算效率是FDTD方法的6倍. 2016年, Chen利用HIE-FDTD计算了无限大石墨烯层的透射系数[65], 计算结果表明, HIE-FDTD方法的计算时间仅为FDTD方法的1/16 500.在此基础上, 许多基于石墨烯的电磁器件, 例如:极化转换器[65]、屏蔽板[66-67]、吸波体[67-70]、耦合器[71]和频率选择表面(frequency selection surface, FSS)[72]等均采用了HIE-FDTD方法进行分析.除此之外, HIE-FDTD方法在其他方面也都得到广泛运用, 例如, 具有精细结构的天线[73]、线性/非线性集总元件和导电媒质[74]、包含线性网络的混合系统[75-77]等等. HIE-FDTD方法近年来取得了很大进展, 主要集中在几个方面:1)复杂坐标系和色散媒质中HIE-FDTD方法. 2008年, HIE-FDTD方法被推广至旋转对称坐标系统[78].该方法在旋转对称坐标系下, 不用求解三角矩阵方程, 简便易于实现, 且计算效率远远高于FDTD方法.为了模拟具有圆柱结构的电磁目标, 如圆柱谐振腔, HIE-FDTD方法也被拓展至圆柱坐标系[79].此外, 为了分析色散媒质, 研究者们对频率相关的HIE-FDTD方法也进行了大量研究[80-84]. 2) one-step leapfrog HIE-FDTD方法.为了进一步降低空间网格长度对时间步长的限制, 2014年, Wang等人提出了one-step leapfrog HIE-FDTD方法[85].为了克服上述方法数值色散误差较大的问题, Dong等人发展了一种四阶one-step leapfrog HIE-FDTD方法[86].此外, Zhai等人[87]对该方法的PML吸收边界进行了讨论, Gao等人[88]和Zhu等人[89]分别对有耗媒质和旋转对称坐标系下的该方法进行了分析.除了one-step leapfrog HIE-FDTD方法, 研究者们针对如何降低HIE-FDTD方法的时间稳定性条件问题, 也提出了许多其他改进方法, 具体可参看文献[90-94]. 3) HIE-PSTD方法. HIE-PSTD方法是将混合显-隐式差分和伪谱技术相结合的一种方法[95-98].伪谱技术采用傅里叶正变换和逆变换代替空间求导, 这样就使得空间网格长度与波长的关系只用满足奈奎斯特抽样定理, 即Δ≤λ/2.因此, HIE-PSTD方法非常适用于沿一个方向具有精细结构, 沿另一个方向具有电大尺寸的电磁目标的模拟, 如:细长缝隙[96]、微结构金属光栅[97]、电大薄层[98]等.数值结果表明, 相对于FDTD方法, HIE-PSTD方法在模拟同时具有精细结构和电大尺寸的电磁目标时, 计算效率大大提高, 内存需求也明显减小. 4)HIE/C-FDTD方法. HIE/C-FDTD方法是将HIE-FDTD方法和共形时域有限差(conformal finite-difference time-domain, C-FDTD)方法相结合的一种方法[99-100]. C-FDTD方法利用共行技术模拟曲面边界, 所以HIE/C-FDTD方法非常适合于模拟复杂布线的印刷电路板.数值算例结果表明, 当用于模拟印刷电路板时, HIE/C-FDTD方法的计算速度是C-FDTD方法的486倍[99].
1.2 US-FETD方法的研究进展
FETD是以Maxwell一阶微分方程或其导出的二阶波动方程为基础方程, 空间上采用有限元网格离散, 时间上采用差分方法近似, 利用加权余量或变分的途径分析推导出有限元方程, 通过求解方程获得电磁问题数值解的一种方法[101-102].目前, US-FETD可分为两类: 1)基于二阶波动方程的US-FETD, 该方法主要有:时域正交展开方法[103-104]、Newmark-beta方法[105]和后向差分方法、消除空间不稳定模方法[106-107]; 2)基于Maxwell一阶旋度方程US-FETD的主流方法有:交替方向隐式时域有限元(alternating direction implicit finite element time-domain, ADI-FETD)方法[108]和Crank-Nicolson时域有限元(Crank-Nicolson finite element time-domain, CN-FETD)方法[109], 消除空间不稳定模方法[110], 如图 2所示.
2003年, Y. S. Chung等人提出了基于加权Laguerre多项式的US-FETD[103], 类似于无条件稳定的Laguerre-FDTD方法, 属于时域正交展开方法.通过正交基函数在时域将Maxwell方程时间相关项展开, 使用伽辽金方法, 消除时间变量, 得到US-FETD的隐式求解方案.由于该方法的内存需求会随着Laguerre多项式的阶数呈线性增长, 2012年, 何国强等人提出了一种高效的Laguerre-FETD递推公式[104], 降低了该方法对内存的需求, 新方法的内存降为由未知量数目决定的一个定值.
1995年, S. D. Gedney等人提出了基于二阶矢量波动方程Newmark-beta方法[105], 以三维谐振腔为例, 研究了Newmark-beta参数对求解误差的影响.在此基础上给出了参数的最优选择. 2007年, M. Movahhedi等人将交替方向隐式(alternating direction implicit, ADI)方法直接应用于Maxwell一阶旋度方程[108], 得到无条件稳定的FETD, 并给出了三维ADI-FETD方法的数值计算公式.研究表明, 与ADI-FDTD不同, 该方法一般不会形成三对角方程组.与此同时, M. Movahhedi等人[108]又将Crank-Nicolson引入基于Maxwell一阶旋度方程FETD, 得到了无条件稳定的CN-FETD. 2009年, Chen等人提出了新的CN-FETD实现方法[109].该方法利用Whitney I-型和Whitney II-型基函数的关系, 使电场的更新需要求解稀疏线性矩阵方程; 而磁场是显式获得的, 即每一时间步只求解一个稀疏矩阵方程, 大大节省了内存和计算时间.上述Newmark-beta方法、CN方法以及后向差分方法分别是不同的有限差分方法, 通过对方程左侧的矩阵施加不同的权重, 使数值系统的放大因子1而达到无条件稳定.由于上述差分方案都需要求解质量矩阵和刚度矩阵之和, 被认为是隐式求解方案; ADI-FETD将一个时间步分解为两个子步, 在每个子步使用不同的差分方案进行迭代, 本质上, ADI-FETD通过执行两次CN方法来保证数值结果的稳定性.上述隐式方案需要求解的矩阵复杂度较高, 计算量较大.且研究发现, 当时间步长较大时, Newmark-beta方法会出现较大的数值误差, 甚至会出现时域仿真后期不稳定现象[106].
消除空间不稳定模的方法是一种新的无条件稳定方法, 不稳定性的消除可通过两种方案完成. Dan Jiao等通过定量分析显式FETD和FDTD不稳定的根本原因, 提出了消除空间不稳定模的基于二阶矢量波动方程的US-FETD和US-FDTD的显式求解方案[37, 111].他们采用的方法必须通过传统FETD或者FDTD以较小的时间步迭代来获取计算系统所有的稳定模式(稳定模式是计算系统所有模式中的一部分)进行预处理, 再在稳定模式所构成的新的向量空间中求解问题.此时由于向量空间维度的降低, 在后续以较大的时间步迭代求解过程中可以大大节省计算资源.显然, 这一算法的预处理不能脱离原有的FETD或者FDTD, 甚至某些问题需要消耗大量时间进行预处理, 造成计算资源的浪费.
2015年, Dan Jiao团队又提出了可以替代原有的基于二阶波动方程的无条件稳定显式求解方案的另外一种新方案[112], 并将其成功应用于FDTD方法中.该方法通过求解系统矩阵的特征值问题得到不稳定模式, 利用不稳定模式组合的矩阵修改原始FETD的矩阵方程, 从而使显式时域迭代达到无条件稳定.这种替代方案无需传统FETD进行预处理, 只需对传统FETD方程进行微小的修改即可消除不稳定性, 便于实现.由于系统的不稳定模式对应着较大的特征值, 可以通过只求解少量相关的较大特征值来获取不稳定模式而无需计算系统矩阵的所有特征值. 2018年, Dan Jiao团队将基于二阶波动方程的无条件稳定显式求解的新方案应用于有耗介质的计算[113].
2018年, 针对基于Maxwell一阶旋度方程的FETD, Tadatoshi Sekine等人给出了消除空间不稳定模的无条件稳定显式求解方案[114].该方法利用传统FETD方法进行预处理来获取稳定模式的解空间, 使在稳定模式的解空间中的时域计算达到无条件稳定.同样, 由于无法脱离传统FETD, 对于某些问题预处理时间过长, 计算效率不高.
综上所述, 截至目前, 为了突破传统FDTD和FETD中时间步长和最小空间离散尺寸相关这一限制, 学者们发展了众多的无条件稳定算法.在这些无条件稳定算法中, ADI、CN、时域正交展开、消除不稳定空间模等方法既可用于FDTD中, 又可用于FETD中. ADI、CN、时域正交展开等方法已在传统FDTD和传统FETD中发展较为成熟, 消除空间不稳定模的FDTD和FETD方法是近几年由Dan Jiao团队发展起来的, 主要用于处理高速电路中的电磁兼容问题.混合显隐式差分FDTD方法非常适合于模拟沿一个方向具有精细结构的电磁问题.在上述方法中, 基于消除空间不稳定模的US-FDTD和US-FETD的应用范围还很窄, 在理论、技术等方面还具有广阔的研究前景.一方面, 要完善消除空间不稳定模的理论, 研究US-FDTD和US-FETD的快速算法, 另一方面, 要扩大算法的应用范围, 比如空中、陆地或海上目标的雷达回波模拟, 天线阵列, 载体共形天线等的电磁仿真和集成化电路引发的电磁兼容、信号完整性等多尺度问题.
2 DGTD方法的研究进展
非连续伽辽金(discontinuous Galerkin, DG)方法最初用于求解偏微分方程, 随后该思想被引入到流体力学当中以及时域有限体积法(finite volume time-domain, FVTD)中.结合FETD和FVTD思想, 人们提出了DGTD算法.该算法在各向异性、色散等复杂介质、波导不连续问题以及大规模高效计算等方面都有广泛的应用前景, 是近年来学者的研究热点.
DGTD采用非结构网格离散计算域, 对复杂结构目标外形拟合好, 单元之间采用数值通量来交换数据使得时间步迭代时只需求解局部小型矩阵方程, 从全域来看则为显式迭代, 这不但能提高计算精度而且便于高效并行计算, 计算内存和资源消耗随着未知量的增加呈线性增大. DGTD方法的发展脉络基本如下:
1973年美国Los Alamos国家重点实验室W. H. Reed和T. R. Hill在求解线性中子方程时最先提出DG方法[115].美国Brown大学应用数学系的J. S. Hethaven教授等人于2002年提出了基于四面体网格的结点DG方法[110], 并将其应用到Maxwell方程的求解当中, 此为DGTD在计算电磁领域完整理论的首篇文献. 2005年, Loula Fezoui等人用数学推理的方式分析了DGTD局域基函数、中心通量和二阶蛙跳时间离散方案的稳定性优势[116]. 2006年G. Cohen等人提出一种局部时间步方案用于六面体、四面体网格的混合求解, 结合高阶基函数求解Maxwell方程组, 有效提高了计算效率[117]. 2008年, E. Montesny等人在数值通量公式中加入了耗散补偿项[118], 在G. Cohen等局部时间步方案基础上提出了多级局域时间步方案. Salvador G. Garcia等人提出了FDTD与DGTD的混合计算方法, 在目标区局部复杂几何结构处使用DGTD方法, 在其他主要区域应用FDTD算法, 因此能够缩短计算时间和获得更高的计算精度[119]. 2010年F. Hassan和D. Victorita等人研究了非结构网格显隐式混合DGTD方法求解Maxwell方程组[120-121].针对多尺度层状结构, 结合区域分解方法, 2011年J. F. Chen和Q. H. Liu等人提出一种显隐式时间步方案, 即每层都被作为一个子域处理并且是独立的网格离散, 并采用DGTD方法显式求解了多极子域问题[122-123].
2012年Stylianos Dosopoulos博士论文中提出一种内部补偿(inner penalized, IP)的DGTD方法, 讨论了时间离散化和稳定性条件; 为了解决多规模应用中非常小的时间步问题, 采用了本地时间步长(local time step, LTS)策略以缩短时间; 研究了IP通量情形下的共形完全匹配层方法, 并将IP-DGTD方法应用到集成无源集总元件电磁分析当中[124]. 2014年西班牙格拉纳达大学Jesus Alvarez Gonzalez博士论文中给出常规框架下半离散方程[125], 半离散方程将不同通量评估成功应用到DG方法中; 讨论了三维情形中DGTD方法电磁特性; 研究了一阶SM-ABC, 以及UPML, 然后将时间积分方案应用到半离散DG方程中; 给出了蛙跳式DG算法和局域时间步策略; 研究了空间半离散方案和LFDT算法, 分析了数值色散和耗散、奇异性、稳定性各向异性误差和收敛性误差. 2015年杜克大学Qiang Ren博士论文研究了DGTD区域分解技术[126], 文中指出子域级DG方法将原始的大型全局系统划分为一些更容易解决的小型系统, 并且还提供了并行化的可能性方案; 文中使用单元混合来减少系统的总自由度等级, 低阶四面体用于捕获几何精细部分, 高阶六面体用于离散均匀和几何粗糙部分; 非共形网格不仅允许不同种类的单元, 而且还允许单元尺寸的急剧变化, 因此可以进一步减少自由度等级.
在吸收边界和复杂介质处理方面:20世纪90年代Shankar和Hall等人提出FVTD中最简单吸收边界条件(Silver-Muller absorbing boundary condition, SM-ABC)[127-128], 2004年他们将SM-ABC边界用在DGTD计算天线散射问题中[129]. 2004年北卡罗来纳大学的Wei Cai和北京大学Tiao Lu等人在DGTD计算中使用UPML层截断计算域[130]. 2010年Xiao等人将三维坐标伸缩PML技术应用于DGTD[131]. 2010年李金发等人在三维DGTD算法中采用共形UPML, 实现了共形截断面对电磁波的吸收[132]. 2004年开始Tiao Lu和Wei Cai等人的小组对二维DGTD情形复杂介质中电磁特性进行了持续的研究, 计算了线性色散介质Debye模型方柱的点源散射[130]; 研究了二维情形下色散介质Drude模型和银纳米线的耦合问题等[133]. 2012年美国肯塔基大学(University of Kentucky) S. D. Gedney和J. C. Young等人研究了三维常见三种色散介质模型(Drude模型、Debye和Lorentz模型)中电磁波散射, 文中采用辅助微分方程作为中间变量, 时间步迭代采用龙格库塔方法[134]. 2013年内华达大学Jia Jia Waters博士论文从数学的角度用DGTD方法研究了超材料和色散介质中Maxwell方程的解和收敛性等问题[135].
国内DGTD研究主要集中在几所高校, 相关的研究成果目前还较少.西安电子科技大学王俊和电子科技大学关瑜分别对一维和二维DGTD方法的基本理论和关键技术进行了研究[136-137].南京航空航天大学刘梅林的博士论文采用结点DGTD方法研究了电磁场谐振腔、电磁波传播、目标散射等问题[138].国防科技大学寇龙泽研究了曲边六面体的DGTD方法, 结合现有网格离散工具, 给出了非结构化六面体离散网格生成策略[139].邓聪博士论文给出了一种FVTD和DGTD的混合算法[140].彭达博士论文系统研究了基于leapfrog和Verlet方法的两种形式局部时间步策略[141].龚俊儒等人对现有DGTD方法进行了总结, 给出DGTD方法核心思想, 提出一种基于标准六面体单元的改进DGTD算法[142].南京大学王斯乐从数学角度讨论了求解一维变系数双曲方程时采用的基于偏迎风通量的龙格库塔DGTD方法[143].
西安电子科技大学魏兵小组系统研究了二维和三维DGTD算法:从最基础的数值通量和激励源的引入开始到一阶吸收边界条件[144-145], 到不同理论形式的PML吸收层[146]; 从非共形到共形PML吸收边界的实现[147-148]; 从平面波的引入到外推RCS的单、双站计算[149]; 从高阶标准基函数和叠层型基函数的展开到大规模并行计算等都进行了详细的研究.该小组提出移位算子时域非连续伽辽金(shift operator DGTD, SO-DGTD)算法来处理各向同性和各向异性色散介质问题[150-152], 此方法对于不同形式的色散介质模型可以采用统一的形式处理.葛德彪、魏兵的专著《电磁场时域非连续伽辽金法》2019年在科学出版社出版[153].
3 结论
本文介绍了US-FDTD法、US-FETD法以及DGTD法的研究新进展. FDTD法、FETD法发展时间较早, 其理论研究日益完善, 应用范围较为广泛.但是这两种方法需要满足稳定性条件, 在处理多尺度及精细结构问题时计算效率较低.目前, 无条件稳定是这两种方法的新发展方向. DGTD方法是近些年来发展的新的时域算法, 该方法在众多学者的努力下逐渐发展成熟, 应用范围也越来越广.但该方法在求解时也需要满足稳定性条件, 目前无条件稳定的DGTD方法还未有较多的报道.
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