| 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, 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 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...
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