【作者】 李正光；

【导师】 吴柏生；

【作者基本信息】 吉林大学 ， 计算数学， 2004， 博士

【摘要】 在结构设计和优化过程中，有时需要对结构进行几十次甚至上百次的修改，而每作一次修改都执行一次完全的有限元分析其计算成本是巨大的。为了减少计算成本，以不直接求解结构修改后的隐式分析方程，而利用初始分析的相关信息设计高效算法来求解修改后结构响应的重分析方法，日益受到人们的重视并得到飞速发展。传统的结构修改主要侧重于参数修改。随着结构设计与优化技术的不断发展，人们发现结构布局的设计与优化比布局固定的设计与优化更具有节约潜力。 然而，由于在布局修改中，结构的有限元模型发生变化，其自由度数量的改变将导致分析方程组的结构和阶数发生变化。目前大部分重分析方法都是针对结构自由度没有变化的情形。 发展适用于结构布局修改的重分析方法是具有巨大挑战性和经济价值的。预条件技术是科学与工程计算中复杂问题高效求解器的最关键组成部分。 所谓预条件就是将一个难处理的问题转化成另一个解能够被迅速逼近的问题。 本文主要应用预条件技术解决结构布局修改的静力重分析问题。论文主要由五部分组成。第一章绪论，主要介绍了重分析问题的工程背景和国内外的研究状况。 第二章首先回顾了解结构修改后自由度未发生变化的重分析问题的组合逼近方法和预条件共轭梯度法，然后从计算复杂度和收敛精度等方面将二者做了比较，指出了预条件共轭梯度法的优越性。第三、四、五章是作者的主要研究工作(即摘要的1－3部分)，分别建立了求解结构自由度减少、增加和一般布局修改的静力重分析问题的预条件共轭梯度法。结点自由度减少情形的静力重分析方法给定一个初始结点自由度为的结构，相应的刚度矩阵记为，荷载向量记为，对应的位移可由求解如下平衡方程 <WP=93>(1)来获得。由初始分析，刚度矩阵可写成如下的Cholesky分解形式(2)式中是上三角矩阵。假设增加和删除一些单元并且从初始结构中删除一些结点后，结构有个自由度。记与分别是结构修改后阶刚度矩阵和荷载向量。则修改后的分析方程组可以写成如下形式：(3)式中是位移向量。我们选取(4)为预条件矩阵，再利用预条件共轭梯度法来求解。在方程(4)中，是阶投影矩阵，变换通过除去m-维向量中对应于被删除结点的分量而得到n-维向量；是将n-维向量转化为m-维向量，，在保留自由度上的分量与的对应分量相同，而在被删除自由度上的分量为零。本文建立了求解该问题的预条件共轭梯度法的有效辅助，研究了其计算复杂性。结点自由度增加情形的静力重分析方法在结点自由度为的初始设计基础上，假设增加和删除一些单元并增加一些新的结点，这使得结构的自由度数目增加。修改后的分析方程可以写为如下形式：(5) <WP=94>式中表示在修改设计中新增加结点的自由度数目，对应新增结点的子刚度矩阵，为联系初始结点和新增结点的子刚度矩阵。方程(5)可以分解为以下两个线性系统：(6)(7)由方程(7)解出，然后代入方程(6)可以得到如下方程：.(8)式中, , .(9)这样就把一个维的问题化为一个维的问题。本文提出的第一个方法是以为预条件矩阵应用预条件共轭梯度方法求解(8)。在求得后代入(7)式即可求得。本文提出的第二个方法是取(10)为预条件矩阵，再利用预条件共轭梯度方法来求解方程(5)。本文证明了作为预条件矩阵的必要性质——对称正定性，建立了求解该问题的两种预条件共轭梯度法的有效辅助并研究了它们的计算复杂度。 结构一般布局修改的静力重分析方法在结点自由度为的初始设计基础上，假设增加和删除一些单元，删除了个已有自由度并增加了个新的自由度，修改后结构的分析方程为(11) <WP=95>式中为阶刚度矩阵。修改后的刚度矩阵可以分块的形式写为：(12)下角标中小写字母表示子刚度矩阵的阶数, 其中。 本文构造(13)为预条件矩阵，用预条件共轭梯度方法求解方程(11)。在方程(13)中，是阶投影矩阵，变换通过除去m0-维向量中对应于被删结点的分量而得到m-维向量；将m-维向量变换为m0-维向量，中对应保留自由度上分量与向量的对应分量相同，而在被删除自由度上的分量为零。本文证明了作为预条件矩阵的必要性质——对称正定性，建立了求解该问题的预条件共轭梯度法的有效辅助并研究了该算法的计算复杂度。 在本文提出的预条件方法中，预条件算子的构造主要是基于修改前的刚度矩阵，这样，预条件算子的构造是廉价的且预条件后的系统很容易求解。预条件方法的主要优点是计算量小，需要的存储空间小，收敛速度快，算法容易辅助并能自适应地监视逼近解的精度，因而该方法能够大大地节省计算费用。更多还原

【Abstract】 In structural design or optimization, the procedures are generally iterative and require repeated analysis as the structure is progressively modified. For each trial design, the analysis equations must be solved and the corresponding analyses usually involve much computational effort. Reanalysis methods are intended to analyze efficiently structures that are modified due to changes in design. The object of reanalysis is to evaluate the structural response for successive modifications in the design without solving the set of the modified implicit equations so that the computational cost is significantly reduced. It has been recognized that the layout optimization of structures can greatly improve the design, and potential savings are generally more significant than those resulting from fixed-layout optimization. In layout modifications, nodes and elements may be added or deleted during the design process. Because of changes of the number of degrees of freedom (DOF) in the analysis model itself, more difficulties are involved in developing an effective reanalysis method for such cases. Most reanalysis methods are designed for the simple cases where the number of DOFs or the number of analysis equations is unchanged. Developing a reanalysis procedure for general layout modifications is most challenging, <WP=97>particularly in cases in which the number of DOF is modified and the structural behavior is significantly changed.It is widely recognized that preconditioning is the most critical ingredient in the development of efficient solvers for challenging problems in scientific and engineering computation. The preconditioned conjugate gradient (PCG) method is to convert the original linear system into a related system with a small condition number so that the total number of iterations required for solving the system to within some specified tolerances is decreased substantially. The present dissertation deals with PCG methods for reanalysis of structural layout modifications, using some particular effective pre-conditioners.This dissertation is consisted of five parts. In Chapter 1, we introduce the engineering back-ground of reanalysis problems and describe status all over the world in this research field. In Chapter 2, we review the combine approximation (CA) method and PCG method for the case of structural modifications where the number of DOF or the number of analysis equations is unchanged. We also compare CA method with PCG method both in efficiency and in accuracy in this chapter. It is shown that the PCG method is superior to CA method in both of the two aspects. The applicant’s major research work is composed of Chapters 3-5 which deal with static reanalysis of structures with the decreased DOF, increased DOF and general layout modifications, respectively.1 Reanalysis of structures with decreased degrees of freedomConsider an initial design with DOF’s (nodal displacements), the corresponding stiffness matrix and the load vector, the displacement vector can be obtained by solving the system of equilibrium equations.(1)The stiffness matrix is symmetric and positive definite and available from <WP=98>the initial analysis in a decomposed form(2)where is an upper triangular matrix.Assume the structure is modified by deletion and addition of elements, and deletion of some joints, and a new stable structure is obtained. The number of analysis equations is changed and the sizes of stiffness matrix and load vector are also decreased according to number of joints deleted from the structure. Let and be the stiffness matrix and the load vector, respectively, of the modified structure with the reduced number of DOF’s. The modified analysis equations can be written as.(3)where is now the reduced vector of modified displacements. The following matrix is chosen as the preconditioner for Eq. (3):(4)Here, ( matrix) is the projection operator from to , the action of is to reduce a -dimensional vector into the - dimensional vector where the displa更多还原