Concurrent design for structures and material microstructures under hybrid uncertainties

Highlights: • A novel robust optimization method for concurrent designs is presented. • Smooth evolutionary method is adopted to optimize the two-scale topologies. • Both load and material uncertainties are systematically investigated. • Conclusions are universal for concurrent designs with multiple microstructures. This study presents a novel topology optimization method for the robust design of structures and material microstructures. Uncertainties are usually ubiquitous and of different sources, and especially hybrid uncertainties widely exist in structural designs including external loads and material properties. Firstly, an orthogonal decomposition and uniform sampling (ODUS) method will be proposed to avoid the time-consuming double loops, in terms of load uncertainties described by upper and lower bounds. Secondly, a non-intrusive polynomial chaos expansion (NIPCE) is implicitly implemented, in terms of base material uncertainties subjected to Gaussian distributions. In the optimization formulation, the robust objective function is defined according to both the expectation and standard variation of structural compliance, and the sensitivity information with respect to the two-scale design variables are given in detail. Finally, an effective evolutionary method is employed to iteratively find the optimal topologies of the design. In addition, this study also defines a dimensionless index to evaluate the robustness of deterministic and robust designs. Three numerical examples are provided to demonstrate the efficiency of the proposed method, and 3D design results are fabricated by using appropriate additive manufacturing techniques.