Structural Static And Dynamic Sensitivity Analysis Method Based On Epsilon Algorithm

In practical engineering problems,the dynamic design of structures plays a vital role.In order to achieve an optimal design,we have to modify the structural parameters and solve the generalized eigenvalue problem repeatedly.Engineering problems often involve many small modifications in the structural parameters,such as design modifications,manufacturing errors,material property variations,robustness analysis of control systems,random eigenvalue analysis,and so on.There are two computational problems in the structural dynamic modifications.One is the structural sensitivity analysis;the other is the reanalysis after the structure modified.Epsilon algorithm which very easy to calculate on the computer is based on the extrapolation processes of approximation theory.Epsilon algorithm has the character of fast convergence to the target value.This paper presents a new method for calculating static and dynamic sensitivity of structures based on Epsilon algorithm and Neumann series.In Chapter 2,the epsilon algorithm is introduced and expanded to vector and matrix series.The relationship between the epsilon algorithm and Pade approximation theory is discussed.Pade approximation theory is well developed and its convergence is proved.As a result of the direct relationship between the epsilon algorithm and Pade approximation,the rationality of the epsilon algorithm is accordingly proved.It is shown that its application scope of the epsilon algrithm does not depend on the convergence domain of series.Actually this result has been proved in Pade approximation theory.Another important result in algebra equations is that when applying the epsilon algorithm,the exact solution 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 sensitivity analysis.Two methods,the Neumann series and the perturbation are used to construct the vector basis.We apply the Epsilon algorithm to the partial sum of vector basis,and derive the accelerated convergence formula to obtain the static displacement sensitivity of structure.Engineering examples show that the excellent results are obtained for very large changes in the design.By comparing with the exact solutions,it is shown that the error of the present method is very small.2.The epsilon algorithm for eigenproblem sensitivity analysis.Using the Neumann series to construct the vector basis and derive the accelerated convergence formula.In fact,the perturbation method can be used as the vector basis of Epsilon algorithm according to the application of eigenproblem.An accumulating computation method is presented when the epsilon algorithm is invalid for the vector basis of Neumann series.In engineering examples,the precision and the computation effort of the Epsilon algorithm are compared with the approximation method of the extended Kirsch combination.