Research On The Adaptive Method Of Combined Approximations

With the rapid development of computer processing power and numerical methods, the subject of structural optimization, which involves mathematics, mechanics and computer science, has attracted substantial attention recently. Since design is an iterative process, the introduction of analysis in the design process demands repetitive analysis of the design concepts. Typically a sequence of design and redesign is carried out until certain features such as weight, natural frequency, mode shape or stress magnitudes attain suitable values. Therefore, the design optimization process becomes iterative and requires repeated analysis of structures obtained by progressive modification in design variable. This operation, which involves much computation effort, is one of the major obstacles when applying optimization methods to large systems. Because of the large number of analysis required, it is very important to develop efficient reanalysis techniques, especially for structures analyzed by finite element method, for only a small number of elements that are modified at each design step. Until now, the common approximations can be divided into the following classes:local approximations, global approximations and combined approximations which can be achieved by considering the terms of local approximations as basis vectors in a global expression. The Combined Approximations (CA) approach, which attracted substantial attention, has been proved useful not only in structural optimization but also in various analysis and design tasks.In the CA approach, it is assumed that the displacement vector of a new design can be approximated by a linear combination of s linearly independent basis vectors. The approximate displacement vector can be evaluated by solving the smaller s×s system instead of computing the exact solution by solving the large system, where s is assumed to be much smaller than the number of degrees of freedom. However, a question arises:how many basis vectors should be used to achieve a prescribed level of accuracy? In addition, as the number of basis vectors is increased the reduced system may become ill-conditioned, making it difficult to obtain an accurate solution in this case. As a result, the number of basis vectors should attain a critical value which the error will not decrease significantly. Considering that CA approach is used commonly in the reanalysis procedures, this paper presents a new adaptive CA method. By this technique, the number of basis vectors can be determined using a computation-inexpensive convergence criterion according modifications of structure at the very beginning of either static or eigenvalue reanalysis during arbitrary structural optimization process. If a large number of basis vectors are needed, the current system should be solved directly instead of reanalysis. As a result, a closer relationship between CA method and optimization is established.For this purpose, the mathematical relationship between basis vectors and the accuracy of results is discussed. After a lot of literature were searched and read, the standard of modifications and the equivalence between the CA approach and preconditioned conjugate gradient (PCG) method on Krylov subspace are presented. Finally, a new adaptive method which can estimate the number of basis vectors according the modifications is obtained based on previous study about preconditioning strategy. To improve the efficiency, the approach of estimating K-condition number is also studied. Four examples are given in the end to illustrate the validity of the proposed method.In the 1st chapter, the development of CA approach and its adaptive method is introduced. The significance and the status of research are showed by summing up the literatures. The research contents of this paper are presented here.In the 2nd chapter, solutions of static reanalysis and modal reanalysis are introduced. The standard of modifications is discussed by a classical example called 10-bar truss and the equivalence between CA approach and PCG method on Krylov subspace is proved.In the 3rd chapter, a new adaptive method which can estimate the number of basis vectors is proposed. The method of estimating PCG iteration number by K-condition number which has been proposed is illustrated after the relationship between K-condition number and spectral norm is proved. And then the convergence criterion is improved based on previous study about preconditioning strategy and the feature of CA approach. As a result, a new adaptive method which can estimate the number of basis vectors according the modifications is obtained. To improve the efficiency, the approach of estimating K-condition number is also studied. A 50-bar truss example is given to illustrate the validity of the proposed approach.In the 4th chapter, numerical examples such as 10-bar truss and 357-beam frame are used to validate the proposed adaptive method, and also a truck body structure is given to illustrate the efficiency.The final chapter is the summary and prospects for the whole thesis.