Gradient based design optimization under uncertainty via stochastic expansion methods

We present a computational framework for robust and reliability based design optimization which combines stochastic expansion methods, namely polynomial chaos expansion, with design sensitivity analysis. It is well known that the statistical moments and their gradients with respect to design variables can be readily obtained from the polynomial chaos expansion. However, the evaluation of the failure probabilities of the cost and constraint functions and their gradients, requires integrations over failure regions. To simplify this we introduce an indicator function into the integrand, whereby the integration region becomes the known range of random variables and to alleviate the non-differentiable property of the indicator function, a smooth approximation is adopted to facilitate the sensitivity analysis. Both intrusive and non-intrusive polynomial chaos approaches for uncertainty propagation are employed in the design optimization of linear elastic structures. Guidelines to assess the computational costs associated with both polynomial chaos approaches are also presented.