时域有限差分和时域有限元电磁数值计算的研究
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摘要
论文主要针对时域有限差分(FDTD)方法和时域有限元(TDFEM)方法做了一系列的研究。
文中首先针对一种新型的可用于毫米波通信的带有磁化铁氧体球的微带环行器结构,从磁矩进动方程和麦克斯韦旋度方程出发,派生出计算包含铁氧体材料的电磁场的数学模型,用三维FDTD方法分析了这种带有铁氧体球的微带环行器结构。针对运用FDTD方法仿真分析该结构过程中时域波形出现的后期发散问题,文中分别运用两种时域外推技术:改良的矩阵束(Modified Matrix Pencil,MMP)方法与最小二乘的支持向量机(Least-Square Support Vector Machines,LSSVM)方法对获得的早期稳定时域波形进行外推,避免时域波形后期不稳定性。并详细的分析和阐述了这两种方法的数学原理和应用,且运用粒子群优化算法(PSO)对LS-SVM算法的参数进行优化选取,以减少人工干预,提高鲁棒性。
论文提出将短开路校准(SOC)技术与FDTD方法相结合来分析微带不连续性结构,SOC技术可以去除由电压源的近似以及微带线在传输信号过程中所引起的寄生误差,提高计算效率以及计算结果的准确度。运用此混合方法分析三维微带不连续性结构以及有限周期结构。与普通FDTD方法相比,运用该混合方法整个有限周期结构的散射参量可以仅通过计算一个周期单元得到。
传统FDTD方法时间步长的选取受到Courant-Friedrich-Levy(CFL)稳定性条件的约束,而无条件稳定的三维交替方向隐格式FDTD(ADI-FDTD)方法,随着时间步长的增大,其数值色散误差也增大,因此文中研究了一种三维无条件稳定的迭代ADI-FDTD(Iterative ADI-FDTD)方法。该方法不仅克服了传统FDTD方法时间步长的选取必须满足CFL稳定性条件的局限,并且随着时间步长的增大,可以消除ADI-FDTD方法所产生的分裂误差,能达到Crank-Nicolson FDTD(CN-FDTD)方法的计算精度,而不用像CN-FDTD方法一样每一个时间步都求解一次大型稀疏矩阵。迭代ADI-FDTD方法使得时间步长即使在取的很大的情况下也可以保持较高的精度。
针对FDTD方法分析复杂电磁问题时处理非规则边界的局限性,对TDFEM方法开展研究,实现对所研究对象的任意网格剖分,利用棱边基函数及其叠层(Hierarchical)矢量基函数,采用完全匹配层(PML)吸收边界条件,分析任意结构谐振腔、波导及微带等复杂结构。针对TDFEM方法中需要求解大型稀疏矩阵的问题,研究了Jacobi、SAI、SSOR、ILU0、SAI-SSOR等预条件Krylov子空间迭代算法(CG、GMRES),分析不同预条件迭代算法的收敛特性,同时将双步预条件技术与压缩矩阵带宽技术(reversing Cuthill-McKee RCM)相结合。v不仅仅为了解决处理不规则边界问题,同时为了避免求解TDFEM方法产生的大型稀疏矩阵,论文还研究了一种区域分解(DDM)TDFEM方法。该方法将整个计算区域分为多个互不重叠的子区域,在每个子域内基于二阶矢量波动方程来求解电场和磁场,电场和磁场基于相同的网格划分,在时间域上类似于FDTD方法的电场和磁场“蛙跳”格式交替求解。在时间步进前将每个子域的系统矩阵进行分解并存储,从而每个时间步计算时,各个子域场量的求解就可以利用预先分解好的矩阵有效的通过直接求解得到。与不区域分解的TDFEM方法相比节省了大量计算时间,可以分析较大结构。与保角映射(Conformal Mapping)后的CN-FDTD方法相比也有一定优势。
The research of this dissertation is focus on the Finite-Difference Time-Domain (FDTD) Method and Time-Domain Finite-Element Method (TDFEM).
A novel microstrip circulator with a magnetized ferrite sphere for millimeter wave communications is analyzed. The electromagnetic fields inside the ferrite junction are calculated using special updating equations derived from the equation of motion of the magnetization vector and Maxwell's curl equations in consistency. A three-dimensional FDTD method for the analysis of this ferrite sphere based microstrip circulator is presented.
The Modified Matrix Pencil (MMP) method and the Least-Squares Support Vector Machines (LS-SVM) technique are used in the FDTD method to eliminate the late time instability of time domain responses and extrapolate the time domain responses. The Particle Swarm Optimization (PSO) method is used to optimize the hyperparameterγ,σof the LS-SVM algorithm, which should be tried again and again randomly. By modeling the novel microstrip circulator, some of the instabilities that arise in late times in the time domain are eliminated.
The application of FDTD algorithm combined with the short-open calibration (SOC) technique to three-dimensional microstrip discontinuity is firstly studied. This SOC technique is directly accommodated in the FDTD algorithm. It is used to remove the unwanted parasitic errors brought by the approximation of the impressed voltage sources and the feed lines. This new method is used to analyze microstrip discontinuities and finite periodic structures. The scattering parameters of the whole periodic structure can be approximately obtained through analyzing only one cell of it.
The conventional FDTD method is limited by the Courant-Friedrich-Levy (CFL) condition while the unconditionally stable alternating-direction-implicit FDTD (ADI-FDTD) method has worse accuracy with the increase of the time step size. The iterative alternating-direction-implicit FDTD (Iterative ADI-FDTD) method is reseached here. This method is exactly the same as the original Crank-Nicolson (CN) method while recognizing the ADI-FDTD method as a special case of a more generalized iterative approach to solve the CN-FDTD method, which can reduce the splitting error of the ADI-FDTD method and no matrix need to be solved. Numerical results demonstrate that this 3D iterative ADI-FDTD method can improve the accuracy of the ADI-FDTD method by using the time step size greatly exceeding the CFL limit within several iterations.
The TDFEM method, which solves the second-order vector wave equations using Galerkin's method, is studied. Compared to FDTD, TDFEM can easily handle both complex geometry and inhomogeneous media by using tetrahedral edge elements. Edge basis function and its hierarchical vector basis functions are used while perfectly matched layers (PML) are used to terminate the waves when simulating different structures of cavity, waveguides and microstrips. Several preconditioning techniques, such as Jacobi、SSOR ILU0 and SAI-SSOR, are used to accelerate the convergence of iterative methods, such as CG and GMRES, which are used to solve the large system of linear equations resulted from TDFEM. Convergence properties and the time used of these conventional preconditioning techniques are compared and analyzed. Also the reversing Cuthill-McKee (RCM) ordering method is used to reorder the sparse matrices created by the hierarchical implicit TDFEM scheme in order to makes SAI-SSOR-CG method more efficient.
Not only to solve the starecasing limitation but also to avoid solving large sparse matrix, a new domain-decomposition TDFEM method (DDM TDFEM) is researched for numerical simulation of electromagnetic phenomena. The method divides the computation domain into several non-overlapping subdomains and computes both the electric and magnetic fields in each subdomain using the sparse direct solver solving second-order vector wave equations. Similar to FDTD method, a leapfrog-like scheme is employed in the time marching to update the alternating electric and magnetic fields. The system matrix for each subdomain is pre-factorized and stored before time marching, so the subdomain problems are solved efficiently using the local pre-factorized matrices at each time step. It could save much time compared to TDFEM method and could analyze big problems. It also has advantages compared to the comformal mapping CN-FDTD method.
文中首先针对一种新型的可用于毫米波通信的带有磁化铁氧体球的微带环行器结构,从磁矩进动方程和麦克斯韦旋度方程出发,派生出计算包含铁氧体材料的电磁场的数学模型,用三维FDTD方法分析了这种带有铁氧体球的微带环行器结构。针对运用FDTD方法仿真分析该结构过程中时域波形出现的后期发散问题,文中分别运用两种时域外推技术:改良的矩阵束(Modified Matrix Pencil,MMP)方法与最小二乘的支持向量机(Least-Square Support Vector Machines,LSSVM)方法对获得的早期稳定时域波形进行外推,避免时域波形后期不稳定性。并详细的分析和阐述了这两种方法的数学原理和应用,且运用粒子群优化算法(PSO)对LS-SVM算法的参数进行优化选取,以减少人工干预,提高鲁棒性。
论文提出将短开路校准(SOC)技术与FDTD方法相结合来分析微带不连续性结构,SOC技术可以去除由电压源的近似以及微带线在传输信号过程中所引起的寄生误差,提高计算效率以及计算结果的准确度。运用此混合方法分析三维微带不连续性结构以及有限周期结构。与普通FDTD方法相比,运用该混合方法整个有限周期结构的散射参量可以仅通过计算一个周期单元得到。
传统FDTD方法时间步长的选取受到Courant-Friedrich-Levy(CFL)稳定性条件的约束,而无条件稳定的三维交替方向隐格式FDTD(ADI-FDTD)方法,随着时间步长的增大,其数值色散误差也增大,因此文中研究了一种三维无条件稳定的迭代ADI-FDTD(Iterative ADI-FDTD)方法。该方法不仅克服了传统FDTD方法时间步长的选取必须满足CFL稳定性条件的局限,并且随着时间步长的增大,可以消除ADI-FDTD方法所产生的分裂误差,能达到Crank-Nicolson FDTD(CN-FDTD)方法的计算精度,而不用像CN-FDTD方法一样每一个时间步都求解一次大型稀疏矩阵。迭代ADI-FDTD方法使得时间步长即使在取的很大的情况下也可以保持较高的精度。
针对FDTD方法分析复杂电磁问题时处理非规则边界的局限性,对TDFEM方法开展研究,实现对所研究对象的任意网格剖分,利用棱边基函数及其叠层(Hierarchical)矢量基函数,采用完全匹配层(PML)吸收边界条件,分析任意结构谐振腔、波导及微带等复杂结构。针对TDFEM方法中需要求解大型稀疏矩阵的问题,研究了Jacobi、SAI、SSOR、ILU0、SAI-SSOR等预条件Krylov子空间迭代算法(CG、GMRES),分析不同预条件迭代算法的收敛特性,同时将双步预条件技术与压缩矩阵带宽技术(reversing Cuthill-McKee RCM)相结合。v不仅仅为了解决处理不规则边界问题,同时为了避免求解TDFEM方法产生的大型稀疏矩阵,论文还研究了一种区域分解(DDM)TDFEM方法。该方法将整个计算区域分为多个互不重叠的子区域,在每个子域内基于二阶矢量波动方程来求解电场和磁场,电场和磁场基于相同的网格划分,在时间域上类似于FDTD方法的电场和磁场“蛙跳”格式交替求解。在时间步进前将每个子域的系统矩阵进行分解并存储,从而每个时间步计算时,各个子域场量的求解就可以利用预先分解好的矩阵有效的通过直接求解得到。与不区域分解的TDFEM方法相比节省了大量计算时间,可以分析较大结构。与保角映射(Conformal Mapping)后的CN-FDTD方法相比也有一定优势。
The research of this dissertation is focus on the Finite-Difference Time-Domain (FDTD) Method and Time-Domain Finite-Element Method (TDFEM).
A novel microstrip circulator with a magnetized ferrite sphere for millimeter wave communications is analyzed. The electromagnetic fields inside the ferrite junction are calculated using special updating equations derived from the equation of motion of the magnetization vector and Maxwell's curl equations in consistency. A three-dimensional FDTD method for the analysis of this ferrite sphere based microstrip circulator is presented.
The Modified Matrix Pencil (MMP) method and the Least-Squares Support Vector Machines (LS-SVM) technique are used in the FDTD method to eliminate the late time instability of time domain responses and extrapolate the time domain responses. The Particle Swarm Optimization (PSO) method is used to optimize the hyperparameterγ,σof the LS-SVM algorithm, which should be tried again and again randomly. By modeling the novel microstrip circulator, some of the instabilities that arise in late times in the time domain are eliminated.
The application of FDTD algorithm combined with the short-open calibration (SOC) technique to three-dimensional microstrip discontinuity is firstly studied. This SOC technique is directly accommodated in the FDTD algorithm. It is used to remove the unwanted parasitic errors brought by the approximation of the impressed voltage sources and the feed lines. This new method is used to analyze microstrip discontinuities and finite periodic structures. The scattering parameters of the whole periodic structure can be approximately obtained through analyzing only one cell of it.
The conventional FDTD method is limited by the Courant-Friedrich-Levy (CFL) condition while the unconditionally stable alternating-direction-implicit FDTD (ADI-FDTD) method has worse accuracy with the increase of the time step size. The iterative alternating-direction-implicit FDTD (Iterative ADI-FDTD) method is reseached here. This method is exactly the same as the original Crank-Nicolson (CN) method while recognizing the ADI-FDTD method as a special case of a more generalized iterative approach to solve the CN-FDTD method, which can reduce the splitting error of the ADI-FDTD method and no matrix need to be solved. Numerical results demonstrate that this 3D iterative ADI-FDTD method can improve the accuracy of the ADI-FDTD method by using the time step size greatly exceeding the CFL limit within several iterations.
The TDFEM method, which solves the second-order vector wave equations using Galerkin's method, is studied. Compared to FDTD, TDFEM can easily handle both complex geometry and inhomogeneous media by using tetrahedral edge elements. Edge basis function and its hierarchical vector basis functions are used while perfectly matched layers (PML) are used to terminate the waves when simulating different structures of cavity, waveguides and microstrips. Several preconditioning techniques, such as Jacobi、SSOR ILU0 and SAI-SSOR, are used to accelerate the convergence of iterative methods, such as CG and GMRES, which are used to solve the large system of linear equations resulted from TDFEM. Convergence properties and the time used of these conventional preconditioning techniques are compared and analyzed. Also the reversing Cuthill-McKee (RCM) ordering method is used to reorder the sparse matrices created by the hierarchical implicit TDFEM scheme in order to makes SAI-SSOR-CG method more efficient.
Not only to solve the starecasing limitation but also to avoid solving large sparse matrix, a new domain-decomposition TDFEM method (DDM TDFEM) is researched for numerical simulation of electromagnetic phenomena. The method divides the computation domain into several non-overlapping subdomains and computes both the electric and magnetic fields in each subdomain using the sparse direct solver solving second-order vector wave equations. Similar to FDTD method, a leapfrog-like scheme is employed in the time marching to update the alternating electric and magnetic fields. The system matrix for each subdomain is pre-factorized and stored before time marching, so the subdomain problems are solved efficiently using the local pre-factorized matrices at each time step. It could save much time compared to TDFEM method and could analyze big problems. It also has advantages compared to the comformal mapping CN-FDTD method.
引文
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