一种求解三维Neumann边界条件线弹性问题二次有限元方程的并行高效预条件子
发布时间：2019-04-21 07:43
【摘要】：带纯Neumann边界条件三维线弹性模型是描述固体力学问题和计算材料力学问题的重要模型,有限元法是数值求解三维线弹性问题最常用的离散方法,并行代数多层网格(AMG)法是快速求解三维线弹性问题离散系统的最为有效的方法之一。本文针对一种满足适定性条件的纯Neumann边界的三维线弹性问题的二次有限元离散代数系统,研究其并行快速求解算法及解法器。首先,在文[33]的基础上,讨论了二次有限元离散化代数系统的适定性,数值实验表明二次有限元误差函数在L2(Ω)和H1(Ω)范数下均具有饱和误差阶。接着,针对二次有限元代数系统给出了两类求解算法,其中重点研究了两种基于AMG法和高斯赛德尔迭代法(GS)的组合型预条件子,并研制了相应的预条件GMRES(k)解法器(BAG-GMRES(k)和BvAG-GMRES(k))。数值实验验证了基于AMG的组合型预条件GMRES(k)法的迭代次数和求解时间均优于常用求解方法ILU(0)-GMRES(k)。相比较而言,BvAG-GMRES(k)解法器更稳定高效。最后,在上述串行预条件子算法解法器的基础上,设计了相应的具有极小化数据通信的并行AMG法和GS的组合型预条件算法,数值试验验证了该算法具有良好的扩展性。
[Abstract]:The three-dimensional linear elastic model with pure Neumann boundary conditions is an important model for describing the problems of solid mechanics and computational material mechanics. The finite element method is the most commonly used discrete method for numerical solution of three-dimensional linear elastic problems. The parallel algebraic multi-layer grid (AMG) method is one of the most effective methods for fast solving three-dimensional linear elastic problem discrete systems. In this paper, a quadratic finite element discrete algebraic system for three-dimensional linear elastic problem with pure Neumann boundary satisfying the well-posed condition is studied, and its parallel fast algorithm and solver are studied. Firstly, on the basis of [33], the well-posed quadratic finite element discrete algebraic system is discussed. Numerical experiments show that the quadratic finite element error function has saturation error order in L2 (惟) and H1 (惟) norm. Then, two kinds of solving algorithms are given for quadratic finite element algebraic system, in which two kinds of combinatorial preforms based on AMG method and Gao Si Seidel iterative method (GS) are studied emphatically. The corresponding preconditioned GMRES (k) solver (BAG-GMRES (k) and BvAG-GMRES (k).) is developed. Numerical experiments show that the iteration times and solution time of the combined preconditioned GMRES (k) method based on AMG are better than that of ILU (0)-GMRES (k). Method. In contrast, the BvAG-GMRES (k) solver is more stable and efficient. Finally, on the basis of the subalgorithm solver of serial prebar, the parallel AMG method with minimized data communication and the combined preconditioned algorithm of GS are designed. Numerical experiments show that the algorithm has good expansibility.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O343.2;O241.82
本文编号：2461984
[Abstract]:The three-dimensional linear elastic model with pure Neumann boundary conditions is an important model for describing the problems of solid mechanics and computational material mechanics. The finite element method is the most commonly used discrete method for numerical solution of three-dimensional linear elastic problems. The parallel algebraic multi-layer grid (AMG) method is one of the most effective methods for fast solving three-dimensional linear elastic problem discrete systems. In this paper, a quadratic finite element discrete algebraic system for three-dimensional linear elastic problem with pure Neumann boundary satisfying the well-posed condition is studied, and its parallel fast algorithm and solver are studied. Firstly, on the basis of [33], the well-posed quadratic finite element discrete algebraic system is discussed. Numerical experiments show that the quadratic finite element error function has saturation error order in L2 (惟) and H1 (惟) norm. Then, two kinds of solving algorithms are given for quadratic finite element algebraic system, in which two kinds of combinatorial preforms based on AMG method and Gao Si Seidel iterative method (GS) are studied emphatically. The corresponding preconditioned GMRES (k) solver (BAG-GMRES (k) and BvAG-GMRES (k).) is developed. Numerical experiments show that the iteration times and solution time of the combined preconditioned GMRES (k) method based on AMG are better than that of ILU (0)-GMRES (k). Method. In contrast, the BvAG-GMRES (k) solver is more stable and efficient. Finally, on the basis of the subalgorithm solver of serial prebar, the parallel AMG method with minimized data communication and the combined preconditioned algorithm of GS are designed. Numerical experiments show that the algorithm has good expansibility.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O343.2;O241.82
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