Epsilon Algorithm For Reanalysis Of Structural Modifications

In modern engineering problems, the dynamic design of structures isbecoming more important. In order to achieve an optimal design, we have tomodify the structural parameters and solve the generalized eigenvalue problemrepeatedly. Engineering problems often involve many small modifications in thestructural parameters, such as material property variations, manufacturing errors,iterative design of structural parameters, design sensitivity analysis, randomeigenvalue analysis, robustness analysis of control systems, and so on.There are two computational problems in the structural dynamicmodifications. One is the structural sensitivity analysis;the other is thereanalysis after the structure modified.The extrapolation processes in numerical analysis is used to develop a newreanalysis method by epsilon algorithm which is very easy to calculate on thecomputer.Firstly, the epsilon algorithm is introduced and expanded to vector andmatrix series. The relationship between the epsilon algorithm and Padeapproximation theory is discussed. Pade approximation theory is well developedand its convergence is proved. As a result of the direct relationship between theepsilon algorithm and Pade approximation, the rationality of the epsilonalgorithm is accordingly proved. It is shown that its application scope of theepsilon algrithm does not depend on the convergence domain of series. Actuallythis result has been proved in Pade approximation theory. Another importantresult in algebra equations is that when applying the epsilon algorithm, the exactsolution can be obtained after finite steps when a iterate solution is constructed,no matter whether convergent of the iterate series.The contents of this text include the following aspects:1. The epsilon algorithm for static displacement reanalysis. Two methods,the Neumann series and the perturbation are used to construct the vector basis.The solution step is straightward and it is easy to implement with the generalfinite element analysis system. The computational effort is much smaller thanthat of the full analysis of the modified structures. A numerical example of achassis structure is given to demonstrate the application of the present method.By comparing with the exact solutions and the kirsch combined approximatesolutions, it is shown that the excellent results are obtained for very largechanges in the design, and the computational efficiency of the present method ishigher than that of the kirsch combined approximation.2. The epsilon algorithm for eigenproblem reanalysis. Using the epsilonalgorithm table to obtain the approximate eigenvectors, the approximateeigenvalues are computed from the Rayleigh quotients. An accumulatingcomputation method is presented when the epsilon algorithm is invalid for thevector basis of Neumann series. In the examples, the precision and thecomputation effort of the epsilon algorithm are compared with theapproximation method of the extended Kirsch combination.3. In the topological optimization, the change of DOFs is a difficultproblem. When the DOF increases, the condensation method is used. A newdynamic reduction method, which is unrelated with eigenvalue is developed.With this new reduction transforming matrix, the terms of added DOF iscondensed to the DOF of the initial structure. After obtaining the eigenvaluesand eigenvectors on the DOF of the initial structure, we recove the eigenvectorson the appending DOF by new reduction transform matrix to constitute thecomplete eigenpair on the new DOF of the modified structure.4. In the optimization of structural analysis, the response and designsensitivity play an essential role. Significant works has been done in this area.On the detailed introduction of the modal method and its improved methods,two methods are discussed: â… . A hybrid method of modal method and epsilonalgorithm for response and sensitivity analysis is presented. â…¡. The expressionsof the modified structural response and sensitivity are developed by constructingvector basis using Neuman series and applying the Epsilon algorithm toaccelerating the convergence of the vectors basis.