Methods

Four methods are applied to the test case. They include three surrogate methods, i. e. gradient-enhanced radial basis functions (GERBF), gradient-enhanced Kriging (GEK) and gradient-enhanced point-collocation polynomial chaos (GEPC), and one direct integration method, i. e. quasi-Monte Carlo (QMC) quadrature. An introduc­tion of them is made in this section.

Since the gradients of the SRQ with respect to all the ten variables are computed by an adjoint solver at an additional cost of approximately one evaluation of the CFD model, to account for this additional cost we introduce the term elapsed time – penalized sample number M by making M = 2N for the three gradient-employing methods and M = N for QMC, with N the number of evaluations of the CFD model. Compared to the cost of evaluating the CFD model the computational overhead of constructing surrogates is negligible, so in the efficiency comparison we use M as the measure of computational cost.

In the aspect of design of experiment, the study in [23] shows surrogate models based on samples with relatively high degree of uniformity (using Latin Hyper-cube sampling) are more accurate than those based on samples of lower degree of uni­formity (using plain Monte-Carlo sampling). For all the four methods in this work we adopt the QMC sampling scheme [7] because it achieves even higher degree of sample uniformity than Latin Hyper-cube sampling.

We use the DLR-TAU code [10] to solve the CFD model. The geometry per­turbation is implemented by using a mesh deformation tool based on radial basis functions incorporated in the DLR-TAU code as described in [14].