Multi-scale approach for reliability-based design optimization with metamodel upscaling

For multi-scale materials, the interplay of material and design uncertainties and reliability-based design optimization is complex and very dependent on the chosen modeling scale. Uncertainty quantification and management are often introduced at lower scales of the material, while a more macroscopic scale is the preferred design space at which optimization is performed. How the coupling between the different scales is handled strongly affects the efficiency of the overall model and optimization. This work proposes a new iterative methodology that combines a low-dimensional macroscopic design space with gradient information to perform accurate optimization and a high-dimensional lower-scale space where design variables uncertainties are modeled and upscaled. An inverse problem is solved at each iteration of the optimization process to identify the lower-scale configuration that meets the macroscopic properties in terms of some statistical description. This is only achievable thanks to efficient metamodel upscaling. The proposed approach is tested on the optimization of a composite plate subjected to buckling with uncertain ply angles. A particular orthonormal basis is constructed with Fourier chaos expansion for the metamodel upscaling, which provides a very efficient closed-form expression of the lamination parameters statistics. The results demonstrate a drastic improvement in the reliability compared to the deterministic optimized design and a significant computational gain compared to the approach of directly optimizing ply angles via a genetic algorithm.