Keywords: metamodelling, moving least squares method, design of experiments, nested design of experiments, cross-validation, metamodel benchmarking.

The use of metamodels in lieu of a direct link between an optimizer and a finite element or a computational fluid dynamics (CFD) simulation software is becoming commonplace in applications of design optimization to computationally intensive industrial problems. In order to build a metamodel, the response of the system is to be evaluated by running the simulation model at a series of sets of parameters defining a design of experiments (DoE) in the range of variation of these parameters.

This paper focuses on a relatively new metamodelling technique, the moving least squares method (MLSM), its tuning and benchmarking, aiming to provide practical recommendations for its use in industrial applications. MLSM as a metamodel building technique has been suggested for the use in the meshless form of the finite element method but only recently proposed for the applications to design optimization. It is a generalization of a traditional weighted least squares model building where weights do not remain constant but are functions of the Euclidian distance from a sampling point to a point where the metamodel is evaluated (evaluation point). The weight, associated with a particular sampling point, decays as an evaluation point moves away from the sampling point. It is possible to control the "closeness of fit" of the metamodel into the sampling data set by changing a parameter in a weight decay function that defines the rate of weight decay or the radius of a sphere beyond which the weight is assumed to be zero. This feature of MLSM allows it to efficiently handle the issue of numerical noise in the response by adjusting the "closeness of fit" setting it to a close fit in a noiseless situation or changing it to a loose fit when the response exhibits a considerable amount of numerical (or experimental) noise.

Several strategies have been investigated for the automatic determination of the optimal value of the closeness of fit parameter in the MLSM-built metamodels, these include the use of nested DoEs (model building and model validation DoEs) with the final rebuilding of the metamodel on the merged set, leave-one-out cross-validation using the PRESS criterion and k-fold cross-validation. Exhaustive testing on a benchmarking problem has been performed resulting in a set of practical recommendations for the use of MLSM in computationally intensive design optimization problems.

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