Polynomial chaos (PC) methods with Gauss-type quadrature formulae have been widely applied for robust design optimization. During the robust optimization, gradient-based optimization algorithms are commonly employed, where the sensitivities of the mean and variance of the output response with respect to design variables are calculated. For robust optimization with computationally expensive response functions, although the PC method can significantly reduce the computational cost, the direct application of the classical finite difference method for the analysis of the design sensitivity is impractical with a limited computational budget. Therefore, in this paper, a semi-analytical design sensitivity analysis method based on the PC method is proposed, in which the sensitivity is directly derived based on the Gauss-type quadrature formula without additional function evaluations. Comparative studies conducted on several mathematical examples and an aerodynamic robust optimization problem revealed that the proposed method can reduce the computational cost of robust optimization to a certain extent with comparable accuracy compared with the finite difference-based PC method.

具有高斯型正交公式的多项式混沌（PC）方法已广泛用于鲁棒性设计优化。在鲁棒优化过程中，通常采用基于梯度的优化算法，在该算法中，计算平均值和输出响应相对于设计变量的方差的敏感性。对于具有计算昂贵的响应函数的鲁棒优化，尽管PC方法可以显着降低计算成本，但是在有限的计算预算下，将经典的有限差分法直接用于设计灵敏度分析是不切实际的。因此，本文提出了一种基于PC方法的半分析设计灵敏度分析方法，其中灵敏度是根据高斯型正交公式直接得出的，而无需进行其他功能评估。在几个数学示例和一个空气动力学鲁棒优化问题上进行的比较研究表明，与基于有限差分的PC方法相比，该方法可以在一定程度上降低鲁棒优化的计算成本，并且具有相当的精度。