The Research Of Some Problems In Structural Static Reanalysis

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< 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.