Nonlinear Dynamic Sensitivities of Structures Using Combined Approximations

Abstract

Calculation of design sensitivities involves much computational effort, particularly for nonlinear dynamic response of large structures. Modal analysis is often not effective for such problems, because eigenproblems must be solved many times. Approximation concepts, which are used to reduce the computational cost involved in structural analysis, are usually not sufficiently accurate for sensitivity analysis. In this study, approximate reanalysis concepts are used to improve the efficiency of nonlinear dynamic sensitivity analysis. Efficient evaluation of the derivatives, using modal analysis, finite-difference of eigenpairs and the recently developed combined approximations approach, is presented. Several approximate solution procedures are developed and compared in terms of the efficiency and the accuracy of the results. It is shown that some of the approximations presented reduce significantly the computational effort and provide sufficiently accurate results. I. Introduction Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives are used in the solution of various problems such as design optimization, generation of response approximations, including approximate reanalysis models, and explicit approximations of the constraint functions in terms of the structural parameters. Sensitivities are required also for assessing the effects of uncertainties in the structural properties on the system response. Calculation of design sensitivities involves much computational effort, particularly for nonlinear dynamic response of large structures. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Early and recent developments in methods for sensitivity analysis are discussed in various studies. 1-4 Methods of sensitivity analysis for discretized systems can be divided into the following classes: a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in large problems. The efficiency can be improved by using fast reanalysis techniques, but finite-difference approximations might have accuracy problems. b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems. c. Semi-analytical methods, which are based on a compromise between finite-difference methods and analytical methods. These methods are easy to implement but might provide inaccurate results. In general, the main factors considered in choosing a suitable sensitivity analysis method for a specific application include the accuracy, the computational effort, and the ease-of-implementation. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort. Dynamic sensitivity analysis has been demonstrated by several authors. In Ref. 5 the mode superposition approach has been considered for linear response. Assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion. In several studies