Building Efficient Response Surfaces of Aerodynamic Functions with Kriging and Cokriging

In this paper, we compare the global accuracy of different strategies to build response surfaces by varying sampling methods and modeling techniques. The final application of the response surfaces being aerodynamic shape optimization, the test cases are issued from CFD simulations of aerodynamic coefficients. For comparisons, a robust strategy for model fit using a new efficient initialization technique followed by a gradient optimization was applied. Firstly, a study of different sampling methods proves that including ’a posteriori’ information on the function to sample distribution can improve accuracy over classical space filling methods like Latin Hypercube Sampling. Secondly, comparing Kriging and gradient enhanced Kriging on two to six dimensional test cases shows that interpolating gradient vectors drastically improves response surface accuracy. Although direct and indirect Cokriging have equivalent formulations, the indirect Cokriging outperforms the direct approach. The slow linear phase of error convergence when increasing sample size is not avoided by Cokriging. Thus, the number of samples needed to have a globally accurate surface stays generally out of reach for problems considering more than four design variables. Nomenclature CV (x) Leave one out cross validation error D Domain of design variables dKdS(x) Leave one out sensitivity error f(x) Regression vector [np] F Regression matrix [N × np] L Model likelihood estimate N Order of the correlation matrix ndv Number of design variables np Dimension of regression vectors ns Number of samples p SCF power coefficients [ndv] R Correlation matrix [N × N ] Ph.D. Student, CFD, 42 Avenue Coriolis. Professor, Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, 4 place Jussieu.