Response surface methodology can be used to construct global and midrange approximations to functions in structural optimization. Since structural optimization requires expensive function evaluations, it is important to construct accurate function approximations so that rapid convergence may be achieved. In this paper techniques to ÿnd the region of interest containing the optimal design, and techniques for ÿnding more accurate approximations are reviewed and investigated. Aspects considered are experimental design techniques, the selection of the 'best' regression equation, intermediate response functions and the location and size of the region of interest. Standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size, while the selection of a larger number of points is not necessarily cost-e ective. In a further attempt to improve e ciency, di erent regression models were investigated. The results indicate that the use of the two methods investigated does not signiÿcantly improve the results. Finding an accurate global approximation is challenging, and su cient accuracy could only be achieved in the example problems by considering a smaller region of the design space. ? 1998 John Wiley & Sons, Ltd.

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SUMMARYFor the accurate prediction of the collapse behaviour of thin cylindrical shells, it is accepted that geometrical and other imperfections in material properties and loading have to be accounted for in the simulation. There are different methods of incorporating imperfections, depending on the availability of accurate imperfection data. The current paper uses a spectral decomposition of geometrical uncertainty (KarhunenLoève expansions). To specify the covariance of the required random field, two methods are used. First, available experimentally measured imperfection fields are used as input for a principal component analysis based on pattern recognition literature, thereby reducing the cost of the eigenanalysis. Second, the covariance function is specified analytically and the resulting Friedholm integral equation of the second kind is solved using a wavelet-Galerkin approach. Experimentally determined correlation lengths are used as input for the analytical covariance functions. The above procedure enables the generation of imperfection fields for applications where the geometry is slightly modified from the original measured geometry. For example, 100 shells are perturbed with the resulting random fields obtained from both methods, and the results in the form of temporal normal forces during buckling, as simulated using LS-DYNA ® , as well as the statistics of a Monte Carlo analysis of the 100 shells in each case are presented. Although numerically determined mean values of the limit load of the current and another numerical study differ from the experimental results due to the omission of imperfections other than geometrical, the coefficients of variation are shown to be in close agreement.

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SUMMARYInstabilities can introduce highly non-linear effects into structural problems. The instabilities, not clearly associated with a change in a parameter, result in a stochastic variation of the responses. This process variation can be distinguished from the effects of the parameter variation by mapping the response variation onto a predictable space and a residual space, where the predictable space contains the possible effects of the parameter variation, and the residual space contains the process effects. This study discusses the sources (mechanics) of the response variation in this class of problems, the use of response surfaces to distinguish between effects driven by design variable changes and bifurcations, and the visualization of unstable zones in the structure. Analytical problems, a headform impact problem, and an occupant safety study clarify the use of the proposed methodology.