代数多重网格算法研究及其在固体力学计算中的应用
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摘要
随着计算机日新月异的发展，科学计算特别是大规模、高性能科学计算越来越成为推动技术革命的强劲动力，计算方法是科学计算的核心，因此，计算机与计算方法的发展程度已成为科学计算能力提高的决定性因素。在物理、力学等应用领域里，有很多问题常常归结于偏微分方程(组)的求解，但由于实际问题的复杂性，根本无法得到它们的解析解，有限元数值求解已成为求解这类问题最为有效的途径和方法。有限元方法通常包括三个过程，即网格生成及优化、有限元离散代数系统的形成以及离散系统的代数求解，其中第一个和第三个过程是影响有限元分析整体求解效率的主要因素。多重网格法是求解偏微分方程(组)大规模离散化方程最为有效的方法，它一般可分为几何多重网格法和代数多重网格法。由于实际应用问题的错综复杂性，以及数值商业软件对“即插即用”型求解器的要求，使得几何多重网格法的应用变得越来越困难，而代数多重网格(AMG)法的“高效性”和“鲁棒性(robustness)”，使之成为了当今多重网格法的研究热点。
AMG方法的关键技术是建立生成各个粗网格层和相应的插值算子的代数方法。对大量的偏微分方程离散代数系统，其背景问题的许多重要特征仅仅通过总刚度矩阵是很难重构出来的。因此，借助于部分几何或分析信息，再通过代数途径来构造相应的高效AMG法是一种十分自然的想法，我们称这种AMG法为基于部分几何和分析信息的AMG法。目前，对标量椭圆型偏微分方程，AMG方法发展比较成熟，但对方程组情形，通常的AMG方法在求解效率上将变差，有时甚至失效。其主要原因是，对方程组情形，定义在每一个节点的自由度往往大于1，而且这些自由度又是相互耦合在一起的，总刚度矩阵所对应的代数网格图与相应的几何网格图根本不一致，利用通常的网格粗化技术，很难控制粗网格的规模及合理地设计插值(提升)算子，从而大大降低了其求解效率。特别是对三维情形，由于网格图的复杂度较二维情形有本质性的增加，相应的网格粗化技术在计算效率和鲁棒性等方面还不尽如人意。因此，我们需要发展新的网格粗化技术及插值算子的构造方法，以提高AMG方法的收敛速度，使其能应用于更多实际问题的快速求解。
本文主要针对固体力学计算中的几类应用问题，对其相应的AMG算法进行了深入的研究和探讨，借助于部分几何或分析信息，提出了几种有效的AMG算法，并进行了大量的数值实验与结果比较，得到了一些有意义的数值结果。这些研究进一步丰富和充实了AMG算法，拓宽了AMG方法在一些应用领域中的研究，具有重要的理论和工程应用价值。主要内容和结果包括：
(一) 针对一类(多尺度)离散应变原子模型的数值求解，将基于标量椭圆型偏微分方程的AMG方法推广应用于方程组情形，给出了相应的AMG算法，并详细介绍了其中的网格粗化算法及插值(提升)算子的构造。据我们所知，这是首次尝试设计快速方法求解离散应变原子模型。对二维问题作了大量的数值实验，并与工程计算中常用的数值方法进行了比较，结果表明，本文设计的AMG算法特别是AMG-CG方法对求解应变模型(包括多尺度耦合模型)是有效的，具有很好的“鲁棒性”。
(二) 针对含间断系数弹性结构力学问题的数值求解，建立了一类界面保持粗化多重网格方法，这样，只需构造简单的插值算子及选取点块Gauss-Seidel作磨光迭代，就可得到一类相当有效的多重网格方法。数值结果表明，这种界面保持粗化多重网格方法
With the rapid development of computer, scientific computing, especially large-scale , high-performance computing has become more and more the impetus to the advancing of science and technology. And computational methods are the essence of scientific computing. Hence, the development of computer and computational methods decide the enhancement of scientific computing capability and capacity. In the fields of physics and mechanics, we need to solve a partial differential equations (PDEs) or a system of PDEs whose exact solutions are in general difficult to be obtained and the finite element method is the most effective way for solving such problems. The finite element solutions usually involve the mesh generation and optimization, the assembly of discrete algebraic systems using the finite element basis, and the solution of these systems by some algebraic solvers. It is well-known that the performance of finite element analysis depends critically on both the first and the third processes. Multigrid methods are by far the most efficient methods for solving large scale algebraic systems arising from discretizations of PDEs or a system of PDEs. Generally speaking, there are two types of multigrid methods: geometric-based approach and algebraic approach. Since the complexities for practical application problems and the requirements for the "plug and play" solvers in numerical business softwares, it is difficult to construct a sequence of nested discretizations or meshes needed for geometric multigrid method. The algebraic multigrid (AMG) method has become the hotspot due to the high performance and robustness.
The key technique of AMG is to propose some algebraic ways for the coarse grid selection and the construction of prolongation operators. For many discrete systems arising from PDEs or a system of PDEs, it is difficult to reproduce the crucial properties of origin problems only by the global stiffness matrix. So it is a natural way to construct highly efficient AMG method such as AMGe and agglomeration methods, by using geometric and analytic information. This is a new direction in developing AMG algorithms currently. We call this method as geometric-based and analytic AMG method. AMG has been well developed for the scalar PDEs, namely for the case when d = 1, where d is the number of physical unknowns resides in each grid. However, the naive use of the scalar AMG does not lead to the robust and efficient solver, rather it deteriorates and oftentimes breaks down in the convergence for system cases in which d > 1 and those unknowns are coupled as well and the graph of global stiffness matrix is no longer the same as the corresponding graph of geometric meshes, especially for three dimensional problems. Therefore, it is a very significant work to develop new techniques and methodology for the coarse grid selection and the construction of prolongation operators to improve the AMG convergence rate, and so that the resulting AMG methods may be applied to more application problems.
In this paper, we make some in-depth studies for AMG algorithms for some types of important application problems. We propose some types of AMG methods by using geometric-based and analytic information. A number of numerical experiments have been performed, and some crucial numerical conclusions are obtained. These researches will make the AMG algorithms richer and apply AMG methods to more researching fields. The main contents and results are listed as follows:
AMG方法的关键技术是建立生成各个粗网格层和相应的插值算子的代数方法。对大量的偏微分方程离散代数系统，其背景问题的许多重要特征仅仅通过总刚度矩阵是很难重构出来的。因此，借助于部分几何或分析信息，再通过代数途径来构造相应的高效AMG法是一种十分自然的想法，我们称这种AMG法为基于部分几何和分析信息的AMG法。目前，对标量椭圆型偏微分方程，AMG方法发展比较成熟，但对方程组情形，通常的AMG方法在求解效率上将变差，有时甚至失效。其主要原因是，对方程组情形，定义在每一个节点的自由度往往大于1，而且这些自由度又是相互耦合在一起的，总刚度矩阵所对应的代数网格图与相应的几何网格图根本不一致，利用通常的网格粗化技术，很难控制粗网格的规模及合理地设计插值(提升)算子，从而大大降低了其求解效率。特别是对三维情形，由于网格图的复杂度较二维情形有本质性的增加，相应的网格粗化技术在计算效率和鲁棒性等方面还不尽如人意。因此，我们需要发展新的网格粗化技术及插值算子的构造方法，以提高AMG方法的收敛速度，使其能应用于更多实际问题的快速求解。
本文主要针对固体力学计算中的几类应用问题，对其相应的AMG算法进行了深入的研究和探讨，借助于部分几何或分析信息，提出了几种有效的AMG算法，并进行了大量的数值实验与结果比较，得到了一些有意义的数值结果。这些研究进一步丰富和充实了AMG算法，拓宽了AMG方法在一些应用领域中的研究，具有重要的理论和工程应用价值。主要内容和结果包括：
(一) 针对一类(多尺度)离散应变原子模型的数值求解，将基于标量椭圆型偏微分方程的AMG方法推广应用于方程组情形，给出了相应的AMG算法，并详细介绍了其中的网格粗化算法及插值(提升)算子的构造。据我们所知，这是首次尝试设计快速方法求解离散应变原子模型。对二维问题作了大量的数值实验，并与工程计算中常用的数值方法进行了比较，结果表明，本文设计的AMG算法特别是AMG-CG方法对求解应变模型(包括多尺度耦合模型)是有效的，具有很好的“鲁棒性”。
(二) 针对含间断系数弹性结构力学问题的数值求解，建立了一类界面保持粗化多重网格方法，这样，只需构造简单的插值算子及选取点块Gauss-Seidel作磨光迭代，就可得到一类相当有效的多重网格方法。数值结果表明，这种界面保持粗化多重网格方法
With the rapid development of computer, scientific computing, especially large-scale , high-performance computing has become more and more the impetus to the advancing of science and technology. And computational methods are the essence of scientific computing. Hence, the development of computer and computational methods decide the enhancement of scientific computing capability and capacity. In the fields of physics and mechanics, we need to solve a partial differential equations (PDEs) or a system of PDEs whose exact solutions are in general difficult to be obtained and the finite element method is the most effective way for solving such problems. The finite element solutions usually involve the mesh generation and optimization, the assembly of discrete algebraic systems using the finite element basis, and the solution of these systems by some algebraic solvers. It is well-known that the performance of finite element analysis depends critically on both the first and the third processes. Multigrid methods are by far the most efficient methods for solving large scale algebraic systems arising from discretizations of PDEs or a system of PDEs. Generally speaking, there are two types of multigrid methods: geometric-based approach and algebraic approach. Since the complexities for practical application problems and the requirements for the "plug and play" solvers in numerical business softwares, it is difficult to construct a sequence of nested discretizations or meshes needed for geometric multigrid method. The algebraic multigrid (AMG) method has become the hotspot due to the high performance and robustness.
The key technique of AMG is to propose some algebraic ways for the coarse grid selection and the construction of prolongation operators. For many discrete systems arising from PDEs or a system of PDEs, it is difficult to reproduce the crucial properties of origin problems only by the global stiffness matrix. So it is a natural way to construct highly efficient AMG method such as AMGe and agglomeration methods, by using geometric and analytic information. This is a new direction in developing AMG algorithms currently. We call this method as geometric-based and analytic AMG method. AMG has been well developed for the scalar PDEs, namely for the case when d = 1, where d is the number of physical unknowns resides in each grid. However, the naive use of the scalar AMG does not lead to the robust and efficient solver, rather it deteriorates and oftentimes breaks down in the convergence for system cases in which d > 1 and those unknowns are coupled as well and the graph of global stiffness matrix is no longer the same as the corresponding graph of geometric meshes, especially for three dimensional problems. Therefore, it is a very significant work to develop new techniques and methodology for the coarse grid selection and the construction of prolongation operators to improve the AMG convergence rate, and so that the resulting AMG methods may be applied to more application problems.
In this paper, we make some in-depth studies for AMG algorithms for some types of important application problems. We propose some types of AMG methods by using geometric-based and analytic information. A number of numerical experiments have been performed, and some crucial numerical conclusions are obtained. These researches will make the AMG algorithms richer and apply AMG methods to more researching fields. The main contents and results are listed as follows:
引文
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