The Reanalysis Methods For Structures With Large Parametric Modification And Their Applications

The design process of an engineering structure is an iterative modification process. We have to modify the structure and resolve the finite element problems repeatedly in order to achieve an optimal design. The main obstacles of the reanalysis problems when the structural parameters are changed is the high computational cost involved in the solution of large-scale problems, so the approximation concepts in structural optimization are applied to reduce the computational costs. Reanalysis methods are intended to analyze efficiently structures due to changes of parameters without performing the full analysis and they usually use the mechanical characteristics of the original structure.From 1960, many researchers have done much work on various reanalysis methods for eigenproblem, static displacement and dynamic response of modified structures. When changes in design parameters are small, the 1st order and 2nd order perturbation method can give good approximate solutions. For large changes in the design, the accuracy of the 1st and 2nd perturbation solutions often deteriorates, and they may become meaningless. Thus, it is highly desirable to improve the perturbation method in its accuracy for the case of large modification of the structural parameters. Kirsch introduced combined approximation (CA) method into the reanalysis for static displacement of structures which improved both the efficiency of the calculations and the accuracy of the results. The CA method was also used to deal with the reanalysis of eigensolutions of the modified structures. Recently, a mathematical algorithm named epsilon-algorithm was introduced into the reanalysis for eigenproblem and static displacement in practical engineering applications. Using epsilon table to accelerate the convergence of the constructed Neumann series expansion, the accurate approximate results of eigenvalues and displacements can be obtained even the parameters have large modifications.Section 2 firstly introduces a reanalysis method for static displacement and eigenproblem based on CA method which can be used in structures with large parametric modification. Then this section extends CA method to the reanalysis of complex eigenproblem. The second order perturbations of the eigenvectors are constructed as the basis vectors which is used to form a small scale (3×3) complex eigenproblem. On condition that the modes of the structures do not have significant modification, the complex eigenvalues and the complex eigenvectors of the modified structures can be evaluated by solving this reduced problem. In the end, two numerical examples are presented to illustrate the validity and improvements of the proposed method and the results show that the new method is effective even if large modifications of structural parameters are made.Section 3 firstly introduces another reanalysis method for static displacement and eigenproblem which is based on epsilon-algorithm. Then this section proposes the epsilon-algorithm for dynamic response reanalysis of the modified structure. Based on the Newmark method, the approximate displacement response in each time step can be obtained by using the acceleration of the epsilon-algorithm table, whose vector sequence is constructed by the Neumann series expansion. Section 3 also presents an efficient computation for dynamic response of systems with time-varying mass, damping and stiffness matrices. The proposed method can avoid solving the inverse of time-varying equivalent stiffness matrix in each time step of Newmark method, thus reducing the computational efforts. In the end, the validity of the proposed methods is illustrated by numerical examples.In addition, the structural parameters are often uncertain in engineering situations, such as the inaccuracy of the measurement, errors in the manufacturing and assembly process, invalidity of some components and uncertainty in boundary conditions etc. The uncertainties of structural parameters may lead to large and unexpected excursion of responses that may lead to drastic reduction in accuracy and precision of the operation. Therefore, uncertainty plays an important role in the modern engineering structural analysis.Over the past decades, a number of methods such as the probabilistic model method, convex model method, fuzzy model method and interval analysis method have been developed that include uncertain model properties in the finite element analysis and aim at the quantification of the uncertainty on the analysis result. Since the mid-1960s, Moore and Alefeld have done the pioneering work on interval mathematics. Recently, Chen et al. developed an interval method for computing the upper and lower bounds of eigenvalues and static displacement by using the first-order Taylor series and the first-order perturbation approximation. But this method is only valid for the case when the uncertainties of parameters are small. If the parameter uncertainties are fairly large or a large number of uncertain parameters combine, the accuracy of the computational results will become unacceptable. Thus, it is highly desirable to present a more accurate method for computing the upper and lower bounds of responses of structures with fairly large uncertainties of interval parameters.Section 4 proposes an efficient method to estimate the interval eigenvalues and static displacements of structures for the case with fairly large uncertainties of parameters. The idea of the proposed method is that the eigenvalues or the static displacements are considered as functions of the structural parameters; using interval theory and the second-order Taylor series expansion, the finite element problems of structures with multi-parameter can be transformed into those of structures with single parameter; the reanalysis method mentioned in above sections is used to compute the upper and lower bounds of the eigenvalues or the static displacements of structures with single interval parameter, thus the interval eigenvalues or static displacements of the structures with multi-parameter can be obtained. Two engineering examples are given to illustrate the application of the proposed method. The results obtained by the proposed method are compared with those obtained by the exact solutions and the first-order approximation.In Section 5, the computer implementation of the above methods in real engineering structures is investigated based on I-DEAS Open Solution Technology.