Two-scale minimum compliance optimization of spatially graded plates with variable-height parallelogram microstructures

This work investigates the two-scale design optimization of graded plates with variable-height parallelogram microstructures based on asymptotic homogenization. First, updated unit cell problems, with an emphasis on rotational degrees of freedom, are proposed to evaluate shell-structured microstructures, so that a pointwise effective stiffness calculation can be efficiently conducted with shell elements, which is otherwise computationally prohibitive with solid elements. Then, a two-scale optimization framework seeking minimum compliance is established, where the design variables controlling both in-plane parallelogram shape and out-of-plane stiffener height are introduced to enlarge the design space. In addition, mapping functions are integrated into the optimization formulation and concurrently optimized, enabling the production of well-defined dehomogenized plates. Finally, numerical examples are presented to validate the effectiveness and correctness of the proposed method by comparing the deflections of the graded plates with those of the homogenized plates.