A reduced-order stochastic finite element approach for design optimization under uncertainty

This paper presents a computational framework for structural design optimization under uncertainty. The stochastic static response of linear elastic structures is predicted by a spectral stochastic finite element method (SSFEM) based on a polynomial chaos expansion (PCE). Traditional SSFEM approaches lead to large-scale numerical problems and computational costs rendering these methods impractical for design optimization purposes. To mitigate the computational burden, a Galerkin-based multi-point reduced-order model (ROM) is integrated into SSFEM. The ROM basis is spanned by displacements and derivatives of displacements with respect to design and random variables evaluated at adaptively selected calibration points in the design space. The design sensitivity equations are derived and a formal analysis of the computational costs in terms of floating-point operations is presented. The efficiency of the ROM based approach versus traditional SSFEM is studied with the shape optimization of a shell structure. For this example, the results show that the proposed reduced-order modeling scheme can lower the total turnaround time by more than a factor of 30.