Design and Analysis of Computer Experiments: the Method of Sacks Et Al

In structural optimization, usually an approximation concept is introduced as interface between FEM structural analysis code and optimization algorithm. In some cases of optimum design, a global approximation concept can be eeectively applied. Then, approximation model functions are built of all objective and constraint functions, whose values, for a certain design point, follow from the structural analysis calculations. In this way, the original optimization problem is completely replaced by an explicitly known approximate optimization problem. Response-surface techniques are commonly applied to build global approximation models, especially when dealing with responses of physical experiments. For one response function, this means that a user deened model function is tted to the response data calculated at the design sites of some experimental design. Errors between model function and experimental response values are assumed to be randomly distributed. However, in the structural optimization case, an experiment is a computer analysis with a deterministic response as a result. This rules out the statistical assumptions response-surface model building is based upon. Furthermore, it is often diicult to nd model functions that approximate the true response behaviour within the desired accuracy. Sacks and coworkers proposed a new model building strategy, that is especially suited for deterministic computer responses. Their basic assumption is that computer responses can be modelled as a realization of a stochastic process. This nally leads to a response prediction that exactly describes all calculated computer responses. Additionally, the mean squared error of a prediction can be calculated, which serves as a measure of accuracy of the prediction. A review is given of the method of Sacks et al.. It has been implemented in MATLAB. Some analytical response functions and small design optimization problems have been used to test the model building capacities and the eeectiveness of the method when applied as global approximation concept in structural optimization. It is concluded that the strategy as proposed by Sacks and coworkers is not suited for implementation in a design optimization tool, mainly because of two reasons. Firstly, maximum likelihood parameter estimation is computationally expensive and not straightforward, while the quality of the parameter estimations is questionable. This computational eeort is disadvantageous if many constraint functions have to approximated, which often occurs in structural optimization. Secondly, the mean squared error can not be unconditionally trusted. To h a v e an independent accuracy measure several additional computer experiments test points have to be performed. Main advantage of …