Design and Analysis of Computer Experiments

In structural optimization, usually an approximation concept is introduced as interface between (FEM) structural analysis code and optimization algorithm. In some cases of optimm desigr. , a glcba! apprcxhaticl? ccncept car. be effectively applied. Then, apprmirnatior. 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 o p timization 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 defined model function is fitted 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 difficult to find 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 finally 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 effectiveness 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 effort 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 have an independent accuracy measure several additional computer experiments (test points) have to be performed. Main advantage of the approximations of Sacks et al., compared with response-surface models, is the flexibility to automatica! y adapt to the ca! cu! ated response data. The same sort, of flexibility also occurs for the moving least-squares variant of the least-squares parameter estimation. I t is recommended to investigate whether this method can add flexibility to, and solve the model function selection problems of, the response-surface strategy without an unmanageable amount of computations.