Second-order Sensitivity Of Static Displacement And Its Application To Interval Analysis

In a typical optimization process it is necessary to evaluate the response for numerous modified designs. For some of these designs, it is also necessary to evaluate the response derivatives. So, effective solution techniques for structural analysis and sensitivity are essential in the optimization of large-scale structures.In this paper, a unified approach for accurate approximations of displacements and displacement derivatives at various modified designs was presented. It was based on static displacement perturbation theory and Neumann Series. With the initial and modified stiffness, the explicit expression was constructed by Neumann expansion of the nodal displacements we needs. Unlike common approximations of the structural response, the approach presented is not based on calculation of derivatives. Rather, approximations of modified displacements are used to evaluate the modified displacement derivatives. Similar computational procedures were employed for evaluating displacements, first- and second-order derivatives of displacements. Numerical results show that accurate approximations of derivatives can be achieved with a small computational effort for designs obtained by large changes (about 10%) in the initial structures by considering second-order approximation. The approximations are based on results of a single exact analysis of initial design. The procedure is easy to implement, suitable for different types of design variables and structures and can be used with a general finite element system.Structural optimization and sensitivity analysis with FEM is based on accurate finite element model of structures and reasonable mathematical model of optimization problems. Modern commercial FEM software always involve sensitivity analysis module, which is used to find out the concerned variables quickly in the case of great number of variables. It is suitable to measure the weight of the variables to different estimation standard. However, the software usually calculates only the first-order derivatives which could not satisfy practical applications. The first- and second-order matrix computed by modified structures 1% in static analysis are used to replaced the first- and second-order derivatives in Taylor series in this paper. For without the tedious iterative and inverse process, it can be used in general software to fill the gas of second sensitivity.Furthermore, static reanalysis method discussed above can also be applied into interval function. In engineering design, the structure may have the errors or uncertainties such as manufacture errors and errors in observation. With the interval mathematics, interval finite element method can solve the problem of structural parameters with certain bounds. In this paper, applying the interval extension and static reanalysis method, four methods were given to solve the uncertainty problems of static response with interval parameter structure,and then the upper and lower bounds of the static displacements were obtained. Two Numerical examples were given to illustrate the validity of the present method.