【作者】 刘海峰；

【导师】 吴柏生；

【作者基本信息】 吉林大学 ， 工程数学， 2010， 博士

【摘要】 本文使用改进的Cholesky分解的秩一校正算法系统地研究了结构静力重分析问题。对于结构修改后自由度数目不变的情形,利用初始结构刚度矩阵的Cholesky分解和改进的Cholesky分解的秩一校正算法构造一个新的预处理矩阵,建立了预处理共轭梯度法,该方法属于逼近法。对于结构修改后自由度数目增加的情形,我们先对一个阶数相对较小的对称正定矩阵作一个Cholesky分解,再结合初始结构刚度矩阵的Cholesky分解,利用改进的Cholesky分解的秩一校正算法来计算结构修改后刚度矩阵的Cholesky分解,从而得到修改后的结构在给定荷载下的位移向量,该方法属于精确法。对于结构修改后边界支撑约束发生变化的情形,我们将其分成四种基本情形分别进行研究:增加的边界支撑约束方向与整体坐标系的某些坐标轴方向相同的情形,删除的边界支撑约束方向与整体坐标系的某些坐标轴方向相同的情形,增加斜边界支撑约束情形以及删除斜边界支撑约束情形。使用改进的Cholesky分解的秩一校正算法对以上四种情形逐一进行处理。利用初始结构刚度矩阵的Cholesky分解,计算结构修改后刚度矩阵的Cholesky分解,从而得到修改后的结构在给定荷载下的位移向量,该方法属于精确法。对于增加的边界支撑约束方向与整体坐标系的某些坐标轴方向相同的情形,我们还利用Cholesky分解的一个简单性质,提出了加行加列法,这种方法也属于精确法。更多还原

【Abstract】 With the social progress and the development of the productivity, people make demand of high standards for performance and quality of the engineering structures. In designs of aerospace engineering, civil engineering and mechanical engineering etc., the analysis of mechanical properties in many large and complex structures is required. At present, most of these analyses are usually completed by using computers and the finite element methods. However, for a large scale problem, the computational cost of each finite element analysis is quite expensive and the computational time is very unbearable. Furthermore, the design of an engineering structure usually needs repetitious modifications, each modification requires an analysis of mechanical properties. It takes a huge amount of computational cost and time to complete these repeated analyses, and the efficiency of structural design is greatly reduced. In order to improve the efficiency of structural design, save time and computational cost, one must develop efficient methods for the structural reanalysis problems.Structural reanalysis includes structural static and dynamic ones, respectively. The purpose of structural static reanalysis is to evaluate vector of displacement response under a given static load for successive modifications in the design without directly solving the set of the modified implicit equations so that the computational cost can be remarkably reduced. In this thesis, structural static reanalysis problems are systematically studied for the following cases of modification:1. Modify, delete and add some elements, change the geometric shape of the structure, but the number of degrees of freedom of the structure keeps unchanged.2. Modify, delete and add some elements, add some nodes on the original structure, change the geometric shape of the structure. The number of degrees of freedom of the structure increases.3. Modify, delete and add some boundary constraints. The number of nodes keeps unchanged and the number of degrees of freedom of the structure is likely to increase, decrease, or keep unchanged.1. Reanalysis method with unchanged number of degrees of freedomThe structural static reanalysis problem with unchanged number of degrees of freedom can be stated as follows. Assume an initial design variable vector Y0 has been given, the corresponding structural stiffness matrix is K 0, the number of degrees of freedom is m . The displacement vector x 0 can be obtained by solving the equilibrium equation: K 0 x0= R0 (1) where K 0∈R m×m is a symmetric positive definite (SPD) matrix, R 0 denotes the load vector. From the initial analysis, the Cholesky factorization of stiffness matrix has already been known: K 0 = L0D0LT0 (2) where L 0 is a unit lower-triangular matrix, LT0 represents the transpose of L 0, and D 0 is a diagonal matrix. Assume the modified design variables are Y = Y0 +?Y after structural modifications, the corresponding stiffness matrix is K = K0 +?K where ? K is the change of the stiffness matrix due to the changes in the design variables and is called incremental stiffness matrix. Thus the equilibrium equation after modifications is: Kx = ( K0 +?K)x =R (3) where R denotes the load vector after structural modifications, the stiffness matrix K∈R m×m is also a SPD. The purpose of structural static reanalysis is to solve the equilibrium equation (3) after modifications by utilizing the information of the original structure as much as possible so that the computational cost can be reduced.Preconditioned conjugate gradient (PCG) method is utilized to solve equation (3). The selection of preconditioner is the key step of PCG method. In this part, a new preconditioner is proposed by utilizing the Cholesky factorization of the original stiffness matrix and the improved algorithm for rank-one updating of the Cholesky factorization. Excellent results are achieved in the case of some large structural modifications by the proposed method.It can be proved that the structural stiffness matrix can be decomposed into a series of rank-one matrices. Thus, the incremental stiffness matrix ? K can also be decomposed into a series of rank-one matrices: where u i∈R m ( i=1, 2, , p), vi∈Rm ( i=1, 2, , q). Furthermore, ? K can be written as a sum of two parts: ?K 1 and ?K 2 so that ?K = ?K1 +?K2where we require p1 + q1<<p+q. That is that ? K1 represents the incremental stiffness matrix corresponding to a relative small number of elements with the significant changes in stiffness matrix and ? K2 denotes the incremental stiffness matrix related to the remaining elements. Then, K 0 + ?K1 is selected as the preconditioner of the PCG method for solving the equilibrium equation (3) after modifications. In order to ensure that K 0 + ?K1 should be a SPD matrix, it is necessary to require K 0 + ?K1 is the stiffness matrix of a real structure. In this way, the proposed preconditioned matrix approximates the modified stiffness matrix, and the structure corresponding to the preconditioner is close to the modified structure. For the purpose of solving system of equations ( K 0 + ?K1)y =z easily, the Cholesky factorization of K 0 + ?K1is calculated by utilizing the Cholesky factorization of the original stiffness matrix and the improved algorithm for rank-one updating of the Cholesky factorization. The cost of the proposed preconditioner is inexpensive due to the fewer number of corrected vectors. Once the Cholesky factorization of K 0 + ?K1 is obtained, the linear system ( K 0 + ?K1)y =LDLT y=z can be solved with little cost by the forward and back substitution. In the implementation of the proposed method, the decomposition of ? K2 into a series of rank-one matrices is not required.Numerical examples have shown the effectiveness of the proposed preconditioner.2. Reanalysis method with increased number of degrees of freedomThe structural static reanalysis problem with increased number of degrees of freedom can be stated as follows. Assume the corresponding structural stiffness matrix for an initial design is K 0∈R m×mwhere m is the number of degrees of freedom of the original structure. From the initial analysis, the Cholesky factorization of matrix K 0 has already been known, and is given by equation (2). Assume some new nodes and elements are added to the original structure for a modification, the equilibrium equation after modification is: where is a SPD stiffness matrix, K nn∈R n×n is the stiffness matrix corresponding to the newly added nodes, n denotes the number of degrees of freedom resulting from the newly added nodes, and n < m is usually satisfied, K nm∈R n×m, KTm n = Knm is the stiffness matrix expressing the connection of the original nodes with the new added nodes, R is the load vector after structural modifications, R m∈Rm denotes the load vector of the original nodes and R n∈Rn represents the additional load vector for the new nodes. The object of static reanalysis is to solve system of equilibrium equations (7) for the modified structure by utilizing the information of the original structure as much as possible so that the computational cost can be reduced.Assume K nn∈R n×n is the stiffness matrix corresponding to the newly added elements (all nodes shared with the original structure are fixed), thus it is a SPD matrix. The Cholesky factorization of K nn is firstly calculated: K∈R ( m+n)×( m+n) is also a SPD matrix due to the symmetric positive definiteness of K 0 and K nn. The Cholesky factorization of K is easily obtained by utilizing the Cholesky factorization of K 0 in equation (2) and the Cholesky factorization of K nn in equation (8):It can be proved that the matrix K ? K can be decomposed into a series of rank-one matrices, and the following relationship is satisfied: Thus, the Cholesky factorization of K can be obtained by utilizing the Cholesky factorization of K in equation (10) and the improved algorithm for rank-one updating of the Cholesky factorization. The displacement vector of the modified structure under a given load can be calculated by the forward and back substitution.3. Reanalysis method with changed boundary constraintsThe structural static reanalysis problem with changed boundary constraints can be stated as follows. Assume the corresponding structural stiffness matrix for an initial design is K 0∈R m×m where m is the number of degrees of freedom of the original structure. From the initial analysis, the Cholesky factorization of K 0 has already been known, and is given by equation (2). Assume boundary constraints have been changed after structural modification, the corresponding equilibrium equation is: Kx = R (12) where K∈R t×t is a SPD matrix, and its dimension t is likely to be greater than m (for examples, deletion of existing boundary constraints, or number of added new boundary constraints is smaller than that of deleted existing boundary constraints), be less than m (for examples, addition of new boundary constraints, or number of added new boundary constraints is greater than that of deleted existing boundary constraints), or equal m (number of added new boundary constraints is equal to that of deleted existing boundary constraints), R∈Rt denotes the load vector after structural modification.3.1 The method of adding rows and columnsA new method is proposed for dealing with the case of changed boundary constraints in this section. This method is only effective in the case of adding boundary constraints and the orientations of the added boundary constraints must be the same as some orientations of the axes of the global coordinate system, but it is invalid for the cases of adding skew boundary constraints and deleting boundary constraints. Assume k boundary constraints have been added after structural modification, and k << m is usually satisfied. The stiffness matrix K∈R ( m?k)×( m?k) corresponding to the modified structure is a SPD, it can be obtained by deleting k rows and k columns from K 0. The positions of deletion are determined by the corresponding numbers of nodes of adding boundary constraints. Suppose i1 , , ik rows and i1 , , ik columns are deleted, the following equation can be achieved: where L?∈R ( m?k)×m is obtained by deleting i1 , , ik rows from L 0. D 0 =diag ( d1 , , dm), and di > 0 ( i=1, , m). Let ( )D1 0 /2=diag d1 , , dm, 1/2L = L0D0, L~ = L?D10/2. Then, following equations are obtained:The equilibrium equation (12) after structural modifications can be calculated by solving the following two equations: L~ y =R (16) L~T x=y (17) Note that the coefficient matrices of two equations above are not square because L~∈R ( m?k)×m, y∈Rm, R∈R m?k, x∈R m?k. Now, L is written as the following block form: where L i∈Rm represents the transpose of the i th row of matrix L (i = 1, , m). Let the following equation can be obtained according to equation (16) and the definition of R in (19): where the ik th element of b i∈Rm ( i=1, 2, , k) is 1, the remainder are all 0. Pre-multiplying equation (20) by L? 1 yields: where The following equation is then satisfied: Combining equations (17), (21) and (24) together results in Pre-multiplying the two sides of equation (25) by L? Tyields Let One can then obtain:The following equation can be obtained by considering the i1 th, i2 th, , ik th component of the two sides of equation (28) only where u i j denotes the j th component ( j = i1, i2, , ik) of vector u i ( i=0, 1, 2, , k). The equation (29) is a linear system with k unknowns, and k is the number of the added boundary constraints. It can be proved that the coefficient matrix of equation (29) is a SPD one, thus the solution of equation (29) exists and is unique. Once the solution LT i y, LTiy, , LTiky12 has been calculated, substitution of them into equation (28) yields x . Then the solution x to equilibrium equation (12) after structural modifications can be easily obtained by the definition of x in equation (23).3.2 The method of dealing with changed boundary constraints by utilizing the improved algorithm for rank-one updating of the Cholesky factorizationIn this section, the augmented non-singular stiffness matrix is employed. The advantage of using the augmented non-singular stiffness matrix is that its dimension is only related to the number of the nodes of the structure, and unrelated to the number of the boundary constraints. Assume the stiffness matrix of the original structure is K 0∈R m×m, the augmented non-singular stiffness matrix is K~ 0∈R ( m+k1)×( m+k1), where k1 is the number of the boundary constraints of the original structure. The Cholesky factorization of K~ 0 can be easily obtained by utilizing the Cholesky factorization of K 0 in equation (2):Suppose the stiffness matrix of the modified structure is K∈R t×t, the augmented non-singular stiffness matrix is ( 2) ( 2)proved that the matrix K~ ? K~0 can be decomposed into a serious of rank-one matrices, thus,Based on equation (31), the Cholesky factorization of K~ can be calculated by utilizing the Cholesky factorization of K~ 0 in equation (30) and the improved algorithm for rank-one updating of the Cholesky factorization. Then the displacement vector of the modified structure under a given load can be obtained by using the forward and back substitution.更多还原