辛时域多分辨率算法理论与应用研究
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摘要
近年来,随着计算机性能的飞速发展和计算数学、计算物理中各种新型算法的出现,计算电磁学呈现出空前繁荣的局面。经典的时域有限差分法(FDTD)以概念清晰、操作简单、通用性强、便于并行等优点,其在宽带分析、瞬态分析、全波分析中有着广泛的应用。然而,FDTD数值稳定性较低,数值色散误差较大,为了确保精度,需要细网格划分,空间步长至少取十分之一波长,同时,受数值稳定性的限制,时间步长也不能太大,使得在计算电大尺寸时,将耗费大量的内存和时间,效率极低。为了消除FDTD方法的缺陷,许多学者提出了改善方法,其中,1996年,Krumpholz提出的时域多分辨分析(MRTD)方法,把小波引入到时域电磁计算中,虽然很大的改善了色散误差,但是对稳定性要求更高。Zhizhang Chen提出ADI-MRTD来摆脱CFL稳定性条件的束缚,但ADI-MRTD方法的数值色散性较FDTD法差。曹群生把龙格库塔引入到MRTD计算中,提出了龙格库塔时域多分辨(RK-MRTD)方法,RK-MRTD在时间上有高阶精度和高稳定度的特点,但该算法有幅度误差,且耗费大量的计算机内存。随着人们对问题的物理本质认识的深入,意识到在追求算法高精度的同时,还应力求保持原系统的内在性质。由于电磁场方程可以转化为一无穷维Hamilton系统,而Hamilton系统具有一系列的内在性质,因而在对Hamilton系统的数值求解时,保持其内在性质就显得尤为重要。辛算法正是保持Hamilton系统内在性质的一种新型数值方法,该算法在长时间的数值计算中,具有常见数值方法无可比拟的计算优势。本文将辛算法引入到电磁计算中,结合传统MRTD的空间高阶特性,构造新型的具有高稳定度特性的高效的电磁计算方法—辛时域多分辨率算法。
本文对辛时域多分辨率算法进行了系统研究。时间方向上,采用高阶辛积分,在长期仿真中保持麦克斯韦方程的辛结构；空间方向上,对电磁场分量用小波尺度函数展开,减小数值色散,提高数值精度；网格剖分上,基于多区域分解技术,结合等效电磁参数概念,解决材质不连续性问题,通过上述相互匹配的算法和技术,来建立快速度、低内存、高精度的时域算法。
针对“辛时域多分辨率算法的理论和应用研究”这一课题,本文主要研究工作与贡献如下：
(1)对基于Daubechies尺度函数的MRTD方法进行理论研究,详细推导了相应的电磁场计算的迭代公式。
(2)探讨了自由空间麦克斯韦方程的辛性质,证明了其时间演化矩阵是辛矩阵,且该矩阵保持了电磁场的能量守恒性。将辛算法高稳定度特性和MRTD的空间高阶特性结合起来,构造了新型的高效的辛时域多分辨率算法的理论框架和迭代公式。
(3)对比了各种时域高阶算法的数值稳定性和数值色散性,证明了辛时域多分辨率算法在长时间仿真、能量守恒、数值精度等方面的优势。
(4)探讨了将辛时域多分辨率算法具体应用到时域电磁仿真中所必需的三大关键技术：平面波源引入技术、吸收边界条件、近远场变换技术。
(5)基于辛时域多分辨率算法多区域分解技术,结合等效电磁参数概念,提出了针对介质目标的共形辛时域多分辨率算法,介质目标的电磁散射计算的数值结果表明这该方法能有效解决材质不连续问题以及Yee氏蛙跳式网格划分造成的阶梯近似误差问题,可以显著提高计算的效率和精度。
In recent years, with the rapid development of the computer performance and the emergence of various new algorithm in computational mathematics and computational physics, computational electromagnetics have shown a situation of unprecedented prosperity.The traditional FDTD method has been widely applied to broad-band, transient, and full-wave analyses owing to its simplicity, generality, and facility for parallel computing.Due to the constraints of the numerical dispersion and the numerical stability,the method is needed to consume much memory and computing time in the calculation of electrically large size of the electromagnetic problems.As a result, the FDTD method is restricted the computation of the electrically large problems.Many efforts have been made in relaxing or removing the above two constraints in order to reduce the computational expenditures. Among them,the multiresolution time-domain (MRTD) method,proposed in1996,has achieved low numerical dispersion with numerical grid resolutions as low as two points per wavelength. However,the CFL stability condition still remains. In order to further explore efficient methods for optimum electromagnetic simulation, other improved time strategies are proposed. For example, the high-order Runge-Kutta (R-K) approach was introduced in. However, the approach is dissipative and needs large amount of memory. Another alternative method is the alternating direction implicit (ADI) algorithm. Although it saves CPU time owing to unconditional stability, undesirable numerical precision and dispersion will happen once the high CFL number is adopted.
Along with the people to the understanding of the physical nature of the problem, to realize in the pursuit of the algorithm with high precision, also should strive to maintain the original system intrinsic properties.Due to the electromagnetic field equations can be transformed into an infinite dimensional Hamilton system, Hamilton system has the intrinsic properties of a series of, so in the numerical solution of the Hamilton system, it is particularly important to maintain its intrinsic properties.Symplectic method is a new numerical method to maintain the intrinsic nature of Hamilton system, the algorithm of numerical calculation in long time, has a computational advantage there is nothing comparable to this common numerical methods.In this paper, the symplectic algorithm is introduced to the electromagnetic calculations and combined with higher-order spatial characteristics of traditional MRTD.As a result, symplectic Multi-Resolution Time Domain scheme,a high efficient computational electromagnetic method.is developed.
This dissertation systematically studies the symplectic Multi-Resolution Time Domain scheme. For temporal direction, the high-order symplectic integration scheme is used to keep the global symplectic structure of Maxwell's equations for long-term simulation. For spatial direction, the equations were evaluate with multi-resolution approximations to reduce numerical dispersion and improve numerical precision. For grid generation, the multi-region decomposition technology and the concept of effective dielectric constant are proposed to model material discontinuities. With the help of these matched schemes and techniques, we can build a fast, low-consumed, and accurate time-domain solver.
Focusing on the "theoretical and applied study of symplectic Multi-Resolution Time Domain scheme", The main researches and contributions are made as follows:
(1) The MRTD schemes based on Daubechies scaling functions" are studied theoretically.The iterative equations of the electromagnetic fields are derived in detail.
(2) The symplectiness of Maxwell's equations in free space is discussed. It is verified that the time evolution matrix of Maxwell's equations is symplectic matrix and conserves the total energy of electromagnetic field. Then, the symplectic algorithm is combined with higher-order spatial characteristics of traditional MRTD.As a result, symplectic Multi-Resolution Time Domain scheme,a high efficient computational electromagnetic method,is developed. The iterative equations of the electromagnetic fields are derived in detail.
(3) Numerical dispersion and stability are compared for a variety of high-order time-domain schemes. In particular, through numerical experiments on long-term simulation, energy conservation, and numerical precision, the advantages of the symplectic Multi-Resolution Time Domain scheme are demonstrated.
(4) The key technology of symplectic multi-resolution time-domain algorithm in electromagnetic simulation is investigated,which includes:is introduction of plane wave source, absorbing boundary conditions, the near to far field transformation technique,etc.
(5) According to the multi-region decomposition technology and the concept of effective dielectric constant,a conformal symplectic multi-resolution time-domain scheme based on Daubechies scaling functions used to dielectric targets is proposed.The numerical results of the electromagnetic scattering computation of dielectric targets show that out scheme can solve the discontinuous surface in dielectric case and the staircase error of Yee's leapfrog meshing,and also can improve computational efficiency and accuracy obviously.
本文对辛时域多分辨率算法进行了系统研究。时间方向上,采用高阶辛积分,在长期仿真中保持麦克斯韦方程的辛结构；空间方向上,对电磁场分量用小波尺度函数展开,减小数值色散,提高数值精度；网格剖分上,基于多区域分解技术,结合等效电磁参数概念,解决材质不连续性问题,通过上述相互匹配的算法和技术,来建立快速度、低内存、高精度的时域算法。
针对“辛时域多分辨率算法的理论和应用研究”这一课题,本文主要研究工作与贡献如下：
(1)对基于Daubechies尺度函数的MRTD方法进行理论研究,详细推导了相应的电磁场计算的迭代公式。
(2)探讨了自由空间麦克斯韦方程的辛性质,证明了其时间演化矩阵是辛矩阵,且该矩阵保持了电磁场的能量守恒性。将辛算法高稳定度特性和MRTD的空间高阶特性结合起来,构造了新型的高效的辛时域多分辨率算法的理论框架和迭代公式。
(3)对比了各种时域高阶算法的数值稳定性和数值色散性,证明了辛时域多分辨率算法在长时间仿真、能量守恒、数值精度等方面的优势。
(4)探讨了将辛时域多分辨率算法具体应用到时域电磁仿真中所必需的三大关键技术：平面波源引入技术、吸收边界条件、近远场变换技术。
(5)基于辛时域多分辨率算法多区域分解技术,结合等效电磁参数概念,提出了针对介质目标的共形辛时域多分辨率算法,介质目标的电磁散射计算的数值结果表明这该方法能有效解决材质不连续问题以及Yee氏蛙跳式网格划分造成的阶梯近似误差问题,可以显著提高计算的效率和精度。
In recent years, with the rapid development of the computer performance and the emergence of various new algorithm in computational mathematics and computational physics, computational electromagnetics have shown a situation of unprecedented prosperity.The traditional FDTD method has been widely applied to broad-band, transient, and full-wave analyses owing to its simplicity, generality, and facility for parallel computing.Due to the constraints of the numerical dispersion and the numerical stability,the method is needed to consume much memory and computing time in the calculation of electrically large size of the electromagnetic problems.As a result, the FDTD method is restricted the computation of the electrically large problems.Many efforts have been made in relaxing or removing the above two constraints in order to reduce the computational expenditures. Among them,the multiresolution time-domain (MRTD) method,proposed in1996,has achieved low numerical dispersion with numerical grid resolutions as low as two points per wavelength. However,the CFL stability condition still remains. In order to further explore efficient methods for optimum electromagnetic simulation, other improved time strategies are proposed. For example, the high-order Runge-Kutta (R-K) approach was introduced in. However, the approach is dissipative and needs large amount of memory. Another alternative method is the alternating direction implicit (ADI) algorithm. Although it saves CPU time owing to unconditional stability, undesirable numerical precision and dispersion will happen once the high CFL number is adopted.
Along with the people to the understanding of the physical nature of the problem, to realize in the pursuit of the algorithm with high precision, also should strive to maintain the original system intrinsic properties.Due to the electromagnetic field equations can be transformed into an infinite dimensional Hamilton system, Hamilton system has the intrinsic properties of a series of, so in the numerical solution of the Hamilton system, it is particularly important to maintain its intrinsic properties.Symplectic method is a new numerical method to maintain the intrinsic nature of Hamilton system, the algorithm of numerical calculation in long time, has a computational advantage there is nothing comparable to this common numerical methods.In this paper, the symplectic algorithm is introduced to the electromagnetic calculations and combined with higher-order spatial characteristics of traditional MRTD.As a result, symplectic Multi-Resolution Time Domain scheme,a high efficient computational electromagnetic method.is developed.
This dissertation systematically studies the symplectic Multi-Resolution Time Domain scheme. For temporal direction, the high-order symplectic integration scheme is used to keep the global symplectic structure of Maxwell's equations for long-term simulation. For spatial direction, the equations were evaluate with multi-resolution approximations to reduce numerical dispersion and improve numerical precision. For grid generation, the multi-region decomposition technology and the concept of effective dielectric constant are proposed to model material discontinuities. With the help of these matched schemes and techniques, we can build a fast, low-consumed, and accurate time-domain solver.
Focusing on the "theoretical and applied study of symplectic Multi-Resolution Time Domain scheme", The main researches and contributions are made as follows:
(1) The MRTD schemes based on Daubechies scaling functions" are studied theoretically.The iterative equations of the electromagnetic fields are derived in detail.
(2) The symplectiness of Maxwell's equations in free space is discussed. It is verified that the time evolution matrix of Maxwell's equations is symplectic matrix and conserves the total energy of electromagnetic field. Then, the symplectic algorithm is combined with higher-order spatial characteristics of traditional MRTD.As a result, symplectic Multi-Resolution Time Domain scheme,a high efficient computational electromagnetic method,is developed. The iterative equations of the electromagnetic fields are derived in detail.
(3) Numerical dispersion and stability are compared for a variety of high-order time-domain schemes. In particular, through numerical experiments on long-term simulation, energy conservation, and numerical precision, the advantages of the symplectic Multi-Resolution Time Domain scheme are demonstrated.
(4) The key technology of symplectic multi-resolution time-domain algorithm in electromagnetic simulation is investigated,which includes:is introduction of plane wave source, absorbing boundary conditions, the near to far field transformation technique,etc.
(5) According to the multi-region decomposition technology and the concept of effective dielectric constant,a conformal symplectic multi-resolution time-domain scheme based on Daubechies scaling functions used to dielectric targets is proposed.The numerical results of the electromagnetic scattering computation of dielectric targets show that out scheme can solve the discontinuous surface in dielectric case and the staircase error of Yee's leapfrog meshing,and also can improve computational efficiency and accuracy obviously.
引文
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