【作者】 张帆；

【导师】 陈飙松；

【作者基本信息】 大连理工大学 ， 计算力学， 2011， 硕士

【摘要】 面向大规模科学与工程计算的软件系统是进行科学研究与工程分析的重要工具,这类软件必须具有良好的开放性与通用性。然而,传统的软件设计开发方法已经不能满足科技进步对计算软件系统的要求。SiPESC (Software Integration Platform for Engineering and Scientific Computation)软件集成平台是采用面向对象设计方法与插件体系架构开发的面向大规模科学与工程计算的通用集成软件平台。本文基于SiPESC系统进行了有限元线性代数方程组求解相关算法的研究以及程序实现。本文实现了矩阵轮廓缩减的AD(Akhras-Dhatt)与RCM(Reverse Cuthill-Mckee)排序算法,并且引入了对有限元结构节点的坐标预优序处理。经排序后,一维变带宽格式存储的有限元总体刚度矩阵的数据量降低了约1个数量级；在此基础上,基于一维变带宽存储格式的LDLT分解求解器的计算效率提高了1-2个数量级。其中,坐标预优序明显提高了AD算法的计算效率,改善了其排序效果。为了发展内外存交换的线性代数方程组稀疏快速直接求解算法的求解器,也为当前的SiPESC.FEMS系统提供中小规模问题的求解方案,本文讨论并实现了在SiPESC架构上组集稀疏存储矩阵格式总体刚度矩阵的程序,并在此基础上集成了开放源代码的稀疏超节点Cholesky分解求解器CHOLMOD。算例表明,当问题规模允许CHOLMOD进行内存求解时,其计算速度较一维变带宽LDLT分解求解器有数量级上的提升。本文猜想矩阵轮廓对矩阵分解计算中产生的填入元数量可能具有约束作用。算例证明了这一猜想。通过算例对比,证明在规模较小的情况下轮廓缩减算法在填入元优化中的作用不逊于高效的填入元优化程序AMD(Approximate Minimum Degree)程序包。针对这一现象,本文作了初步的分析。更多还原

【Abstract】 As an important numerical simulation tool, massive computational software systems aiming at scientific and engineering computation should be open for functional extension and general purposes. However, the traditional method of software design no longer meets the requirements of advance computation software. By the object-orient software design method and the plugin software architecture, Software Integration Platform for Engineering and Scientific Computation (SiPESC) is developed as a generality integration platform for large-scale scientific and engineering computation. Based on SiPESC system, algorithm research on linear algebra equations and its implementation for finite element analysis are present in this thesis.This thesis realizes AD algorithm and RCM algorithm using for matrix profile reduction, and introduce the pre-ordering program for FE structure nodal coordinates. The size of FEA global stiffness matrix storage by skyline format has been reduced about 1 order of magnitude by nodal ordering, and the computation efficiency of LDLT factorization is also improved 1-2 orders of magnitude. It shows that the pre-ordering program improve the computation efficiency significantly, and modifies its ordering quality.In order to develop the out-of-core fast direct sparse solver of linear system, also to develop SiPESC.FEMS system for solution of medium scale problem, this thesis investigates and implements assembly program for global stiffness matrix by using compressed sparse column format. Furthermore, an open source sparse supernodal Cholesky factorization solver- CHOLMOD has been integrated. Numerical example show that the speed up of this solver over orders of magnitude than former LDLT solver if the problem could solve in-core.This thesis made a hypothesis that the matrix profile could be the constraint of fill-in of matrix factorization. Numerical example provided the evidence of this hypothesis. Preliminary compare have been made on the effects of profile reduction program and fill-in optimization program AMD(Approximate Minimum Degree) package. It is proved that profile reduction program reduces the fill-in of small-scale matrix effectively.更多还原