节点间断伽辽金有限元方法及其在计算电磁学中的应用研究
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摘要
从上个世纪六十年代以来,计算电磁学得到了迅猛的发展,并得到了广泛的应用。经过四十多年众多计算电磁学和计算数学领域科学工作者的共同努力,开发了多种有效的数值计算方法求解各种电磁场问题。这些方法主要有时域有限差分方法、有限元方法、有限体积方法和矩量法等。这些方法在各种电磁场问题中得到了广泛的应用。然而这些方法又各自有自己的缺点,比如:时域有限差分方法并不适合计算几何结构复杂的问题;有限元方法和矩量法由于单元之间互相依赖,因此这两种方法并不适合并行计算;有限体积方法不适合使用高阶基函数。为了利用这些方法的优点,同时又尽量回避这些传统方法的缺点,本论文研究了一类较新的数值计算方法,这类方法是间断伽辽金有限元方法,并实现了其中一种基于节点的间断伽辽金有限元方法。间断伽辽金有限元方法是一种基于有限体积方法和有限元方法而发展出来的一种方法。我们研究了如何使用节点间断伽辽金有限元方法求解电磁场谐振腔问题、电磁波传播问题、电磁目标散射问题、不确定形状电磁目标随机散射等问题。
电磁场谐振腔问题是电磁场领域的基本问题之一。在研究运用一维到三维的节点间断伽辽金有限元方法求解Maxwell方程组时,我们首先用该方法求解了电磁场谐振腔问题。
电磁场传播问题也是电磁场领域的基本问题之一。我们首先研究了如何使用节点间断伽辽金有限元方法求解电磁波在自由空间的传播,然后研究了使用该方法求解电磁波在有耗介质中的传播问题。
接着,我们重点研究了使用节点间断伽辽金有限元方法求解电磁场散射问题。电磁场散射问题在电磁场领域的研究具有非常重要的意义。自从第二次世界大战雷达发明以来,不确定形状目标散射的计算一直是电磁场研究领域的热点问题,其对于军事目标的隐身和反隐身具有不言而喻的重要意义。首先,我们研究了二维金属圆柱的散射问题,然后研究了二维金属方柱的散射问题。最后,我们在已有开源网格生成程序基础上开发了计算不确定形状目标散射问题所需要的复杂动态网格生成程序,并结合稀疏网格积分方法和随机配置方法研究了不确定形状目标散射问题。稀疏网格积分方法(Sparse Grid Method)是在求解高维积分问题时大大减少计算量的一种有效方法。随机配置方法(Stochastic Collocation Method)则为求解随机问题提供了一种和蒙特-卡洛方法(Monte-Carlo Method)一样容易实现,但是其比蒙特-卡洛方法收敛更快。为了提高计算速度,我们提出了基于Richardson外插方法的自适应稀疏网络积分方法。其核心思想是在稀疏网格积分方法中从第三层起使用Richardson外插方法,由当前层和上一层的计算结果来预测下一层的结果。如果预测的结果与当前层的结果之间的误差在设计的误差范围之内,则停止计算,反之,则继续计算下一层,从而加速求解不确定目标随机散射问题。当所求解问题的维数越高,节省的计算时间就越多。
随着现代科学研究的不断发展,科学研究和工程要解决的问题规模越来越大。科学计算和仿真是理论研究和实验研究之外的第三种重要的方法。与此同时,人们对求解大规模的电磁场问题需求也日夜增长。单台个人计算机或工作站的计算能力通常满足不了计算需求。因此,并行计算是求解大规模电磁场问题的唯一途径。间断伽辽金有限元方法由于单元之间通过数值通量进行耦合,只有单元边界上的值与相邻单元有联系,所以特别容易实现并行计算。我们研究了在超级计算机环境下三维的间断伽辽金有限元方法实现。在实现中,我们采用了开源的基于消息传递接口的并行编程函数库,以及使用了ParMETIS快速并行网格划分函数库进行负载均衡。
间断伽辽金有限元方法在空间离散时通常采用高阶的多项式基函数。因此,为了使整个计算格式保持高阶精度,间断伽辽金有限元方法中采取的时间离散方法通常也采用高阶的常微分方程求解方法,比如:两阶的蛙跳格式,两到四阶的龙格-库塔方法,多步的预测器与校正器方法等。但是这些方法都是基于解析或半解析方法推导的。因此,这些方法的精度是固定的。要想推导同样的更高精度格式,非常困难乃至不可能。本文研究并实现了一种完全基于数值方法构造的时间离散格式。这种新格式,在形式上和传统的预测器与校正器方法一样。但是,不同的是,新格式不是采用多项式函数近似常微分方程的解,而是采用指数函数来近似常微分方程的解。
所有的这些研究为进一步推广间断伽辽金有限元方法在计算电磁学领域的应用提供了较好的参考价值。
Computational Electromagnetics (CEM) has been made a great leap and been extensively applied to electromagnetic (EM) industry and academic research since the subject emerged in 1960s. CEM community has developed a few popular and effective numerical methods to solve all kinds of EM problems in the past four decades. These methods mainly include Finite-Difference Time-Domain (FDTD) method, Finite Element Method (FEM), Finite Volume Method (FVM), and Method of Moment (MoM), etc. All these methods are extensively employed to solve almost all kinds of different EM problems. However, they have different disadavantages in these methods. For example, FDTD method is not suitable for handling structures with complex geotry; FEM and MOM methods are difficult to be parallelized because of the coupling relationship between on element to all other elements; FVM can not use high-order basis functions. To overcome the shortcomings of these traditional numerical methods, we studied a class of relatively new and novel numerical method– Discontinuous Galerkin Finite Element Method (DG-FEM) and implemented nodal based DG-FEM. We studied how to resolve EM resonator problems, EM wave propogation problems, EM scattering problems and stochastic EM wave scattering problems for object with uncertain shape, based on nodal DG-FEM.
EM resonator problem is one of the fundemantal problems in electromagnetics. We solved EM resonator problems from one dimension to three dimensions using nodal DG-FEM.
EM wave propagation is another of essential problems in electromagnetics. First of all, we solved the EM wave propagation in free space using nodal DG-FEM. Then we utilized nodal DG-FEM to deal with EM wave propagation in a lossy dielectric medium.
Next, we focused on how to deal with EM wave scattering problem using nodal DG-FEM. EM wave scattering problem is a very hot research topic without any doubt in CEM community. In particularly, how to quantify the scattering characteristic of an object with uncertain shape is very important for the stealth and anti-stealth of military target since RADAR was invented in World War II. First, we solved the 2 dimensional (2-D) metallic cylinder scattering problem using nodal DG-FEM. Second, we solved a 2-D square cylinder scattering problem. Finally, we solved object with uncertain shape stochastic scattering problem by combining with the Sparse Grid integration method as well as the Stochastic Collocation method. Sparse Grid method is a very effective method to reduce computation cost for high dimensional integration problems. And stochastic collocation method is an easy-programing method as Monte-Carlo method for stochastic problem, but it has higher convergence rate than the Monte-Carlo method does. To speed up the computation, we proposed an adaptive sparse grid method based on the Richardson extrapolation method. The key idea is to predict the result of the next level through those of the current level and the previous level. The program will be terminated when the error between the predicted result and that of the current level is below a given tolerance; otherwise, the program will continue to compute the next level. The adaptive sparse grid will save more CPU time when the problem to be solved has higher dimensionality.
Scientific computing and simulation is one of three methods in scientific research. Simulations are indispensable for theoretical and experimental research. There are gradually increasing interests to simulate enormous problem in EM research and industry societies as the science and technology evolve. One single personal computer (PC) or workstation is not powerful enough to satisfy the computational resource requirements. Therefore parallel computing is the only way for conquering huge EM problem. It is very easy to implement a parallel version of DG-FEM because DG-FEM exchanges information between elements through numerical flux and one element only relates with its neighbours. We implemented a 3-D parallel nodal DG-FEM EM solver under supercomputer environment. Our implementation bases on the Open Message-Passing Interface (OpenMPI) library and the ParMETIS parallel graph partition library for load balancing.
DG-FEM usually employs high-order polynomials in space discretization. To match the high-order accuracy in space discretization in DG-FEM, it also adopts high-order time discretization schemes, for instance, second order frog leap scheme, second to fourth order Runge-Kutta method, predicator-corrector method in multi-step methods, etc. However, these methods are derived via analytical or semi-analytical methods and their accuracies are fixed. It is very difficult or even impossiable to derive more accuracy similar methods. A new highly accuracy time discretization scheme is implemented in this work. The new scheme is fully numerically constructured. And the form of the new schem is same as the traditional predictor-corretor method. But the new method approximates the solution of ordinary differential equations (ODEs) using exponential functions, not polynomials functions in traditional-corretor method.
All of our contributions have laid a good foundation for further development using DG-FEM for EM analysis and application.
电磁场谐振腔问题是电磁场领域的基本问题之一。在研究运用一维到三维的节点间断伽辽金有限元方法求解Maxwell方程组时,我们首先用该方法求解了电磁场谐振腔问题。
电磁场传播问题也是电磁场领域的基本问题之一。我们首先研究了如何使用节点间断伽辽金有限元方法求解电磁波在自由空间的传播,然后研究了使用该方法求解电磁波在有耗介质中的传播问题。
接着,我们重点研究了使用节点间断伽辽金有限元方法求解电磁场散射问题。电磁场散射问题在电磁场领域的研究具有非常重要的意义。自从第二次世界大战雷达发明以来,不确定形状目标散射的计算一直是电磁场研究领域的热点问题,其对于军事目标的隐身和反隐身具有不言而喻的重要意义。首先,我们研究了二维金属圆柱的散射问题,然后研究了二维金属方柱的散射问题。最后,我们在已有开源网格生成程序基础上开发了计算不确定形状目标散射问题所需要的复杂动态网格生成程序,并结合稀疏网格积分方法和随机配置方法研究了不确定形状目标散射问题。稀疏网格积分方法(Sparse Grid Method)是在求解高维积分问题时大大减少计算量的一种有效方法。随机配置方法(Stochastic Collocation Method)则为求解随机问题提供了一种和蒙特-卡洛方法(Monte-Carlo Method)一样容易实现,但是其比蒙特-卡洛方法收敛更快。为了提高计算速度,我们提出了基于Richardson外插方法的自适应稀疏网络积分方法。其核心思想是在稀疏网格积分方法中从第三层起使用Richardson外插方法,由当前层和上一层的计算结果来预测下一层的结果。如果预测的结果与当前层的结果之间的误差在设计的误差范围之内,则停止计算,反之,则继续计算下一层,从而加速求解不确定目标随机散射问题。当所求解问题的维数越高,节省的计算时间就越多。
随着现代科学研究的不断发展,科学研究和工程要解决的问题规模越来越大。科学计算和仿真是理论研究和实验研究之外的第三种重要的方法。与此同时,人们对求解大规模的电磁场问题需求也日夜增长。单台个人计算机或工作站的计算能力通常满足不了计算需求。因此,并行计算是求解大规模电磁场问题的唯一途径。间断伽辽金有限元方法由于单元之间通过数值通量进行耦合,只有单元边界上的值与相邻单元有联系,所以特别容易实现并行计算。我们研究了在超级计算机环境下三维的间断伽辽金有限元方法实现。在实现中,我们采用了开源的基于消息传递接口的并行编程函数库,以及使用了ParMETIS快速并行网格划分函数库进行负载均衡。
间断伽辽金有限元方法在空间离散时通常采用高阶的多项式基函数。因此,为了使整个计算格式保持高阶精度,间断伽辽金有限元方法中采取的时间离散方法通常也采用高阶的常微分方程求解方法,比如:两阶的蛙跳格式,两到四阶的龙格-库塔方法,多步的预测器与校正器方法等。但是这些方法都是基于解析或半解析方法推导的。因此,这些方法的精度是固定的。要想推导同样的更高精度格式,非常困难乃至不可能。本文研究并实现了一种完全基于数值方法构造的时间离散格式。这种新格式,在形式上和传统的预测器与校正器方法一样。但是,不同的是,新格式不是采用多项式函数近似常微分方程的解,而是采用指数函数来近似常微分方程的解。
所有的这些研究为进一步推广间断伽辽金有限元方法在计算电磁学领域的应用提供了较好的参考价值。
Computational Electromagnetics (CEM) has been made a great leap and been extensively applied to electromagnetic (EM) industry and academic research since the subject emerged in 1960s. CEM community has developed a few popular and effective numerical methods to solve all kinds of EM problems in the past four decades. These methods mainly include Finite-Difference Time-Domain (FDTD) method, Finite Element Method (FEM), Finite Volume Method (FVM), and Method of Moment (MoM), etc. All these methods are extensively employed to solve almost all kinds of different EM problems. However, they have different disadavantages in these methods. For example, FDTD method is not suitable for handling structures with complex geotry; FEM and MOM methods are difficult to be parallelized because of the coupling relationship between on element to all other elements; FVM can not use high-order basis functions. To overcome the shortcomings of these traditional numerical methods, we studied a class of relatively new and novel numerical method– Discontinuous Galerkin Finite Element Method (DG-FEM) and implemented nodal based DG-FEM. We studied how to resolve EM resonator problems, EM wave propogation problems, EM scattering problems and stochastic EM wave scattering problems for object with uncertain shape, based on nodal DG-FEM.
EM resonator problem is one of the fundemantal problems in electromagnetics. We solved EM resonator problems from one dimension to three dimensions using nodal DG-FEM.
EM wave propagation is another of essential problems in electromagnetics. First of all, we solved the EM wave propagation in free space using nodal DG-FEM. Then we utilized nodal DG-FEM to deal with EM wave propagation in a lossy dielectric medium.
Next, we focused on how to deal with EM wave scattering problem using nodal DG-FEM. EM wave scattering problem is a very hot research topic without any doubt in CEM community. In particularly, how to quantify the scattering characteristic of an object with uncertain shape is very important for the stealth and anti-stealth of military target since RADAR was invented in World War II. First, we solved the 2 dimensional (2-D) metallic cylinder scattering problem using nodal DG-FEM. Second, we solved a 2-D square cylinder scattering problem. Finally, we solved object with uncertain shape stochastic scattering problem by combining with the Sparse Grid integration method as well as the Stochastic Collocation method. Sparse Grid method is a very effective method to reduce computation cost for high dimensional integration problems. And stochastic collocation method is an easy-programing method as Monte-Carlo method for stochastic problem, but it has higher convergence rate than the Monte-Carlo method does. To speed up the computation, we proposed an adaptive sparse grid method based on the Richardson extrapolation method. The key idea is to predict the result of the next level through those of the current level and the previous level. The program will be terminated when the error between the predicted result and that of the current level is below a given tolerance; otherwise, the program will continue to compute the next level. The adaptive sparse grid will save more CPU time when the problem to be solved has higher dimensionality.
Scientific computing and simulation is one of three methods in scientific research. Simulations are indispensable for theoretical and experimental research. There are gradually increasing interests to simulate enormous problem in EM research and industry societies as the science and technology evolve. One single personal computer (PC) or workstation is not powerful enough to satisfy the computational resource requirements. Therefore parallel computing is the only way for conquering huge EM problem. It is very easy to implement a parallel version of DG-FEM because DG-FEM exchanges information between elements through numerical flux and one element only relates with its neighbours. We implemented a 3-D parallel nodal DG-FEM EM solver under supercomputer environment. Our implementation bases on the Open Message-Passing Interface (OpenMPI) library and the ParMETIS parallel graph partition library for load balancing.
DG-FEM usually employs high-order polynomials in space discretization. To match the high-order accuracy in space discretization in DG-FEM, it also adopts high-order time discretization schemes, for instance, second order frog leap scheme, second to fourth order Runge-Kutta method, predicator-corrector method in multi-step methods, etc. However, these methods are derived via analytical or semi-analytical methods and their accuracies are fixed. It is very difficult or even impossiable to derive more accuracy similar methods. A new highly accuracy time discretization scheme is implemented in this work. The new scheme is fully numerically constructured. And the form of the new schem is same as the traditional predictor-corretor method. But the new method approximates the solution of ordinary differential equations (ODEs) using exponential functions, not polynomials functions in traditional-corretor method.
All of our contributions have laid a good foundation for further development using DG-FEM for EM analysis and application.
引文
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[88] https://computing.llnl.gov/tutorials/parallel_comp/
[89] http://www.scala-lang.org/
[90] W. Gropp, E. Lusk and A. Skjellum. Using MPI, 2nd ed. Massachusetts: The MIT Press, 1999.
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[92] http://www.mpi-forum.org/
[93] http://www.open-mpi.org/
[94] http://glaros.dtc.umn.edu/gkhome/metis/parmetis/overview
[95] http://www.cs.sandia.gov/zoltan/
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[97] http://hpc.kaust.edu.sa/
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