Analytical robust design optimization based on a hybrid surrogate model by combining polynomial chaos expansion and Gaussian kernel

Robust design optimization (RDO) is one of the most popular methodologies in the presence of uncertainties, which aims to provide an insensitive design configuration. However, the assessment of the robustness index is computationally intensive for practical engineering. Surrogate assisted optimization is an effective way to reduce the computational expense. Various surrogate models have been used in RDO, such as support vector regression, Kriging, artificial neural network, radial basis function and so on. Recently, a new surrogate model named PC-GK-SBL is proposed, combining the polynomial chaos expansion (PCE) and Gaussian kernel (GK) in the sparse Bayesian learning (SBL) framework. In this paper, the PC-GK-SBL surrogate model is integrated into RDO, and the analytical formulae for the robustness index and its sensitivity are derived, which improves computational efficiency significantly. Furthermore, an adaptive active learning function, named robust local geometrical exploration (RLGE), is also proposed to select the new sample points for updating the surrogate model, in which the sigmoid function is combined with distance based geometrical exploration strategy to improve the accuracy of the RDO solution significantly. In RLGE, the sigmoid function is utilized to filter out the design spaces that have little influence on the robustness index, and the geometrical exploration is employed to avoid crowding of the added samples for the surrogate model. The RLGE active learning function is evaluated by a metaheuristic algorithm, the comprehensive learning Harris hawks-equilibrium optimization. Finally, the accuracy and efficiency of the proposed method were demonstrated through four numerical examples and one engineering example.