Concurrent multi-scale optimization of macro- and micro-shapes of laminated porous shell structure

In this study, we propose a multi-scale shape optimization method that concurrently optimizes the shape of micropores and the curvature of the shell structure in a porous laminated shell structure. The homogenization method is used to bridge the macrostructure and the microporous structures. In the microporous structural optimization the pore shape of the unit cell is optimized in each sub-domain layer by layer. The thickness of each layer is assumed to be constant with respect to the shape variation. A squared error norm is minimized for controlling the displacements at arbitrary points of the laminated shell structure to the target values under the total volume constraint. The equilibrium equation of the macrostructure and the homogenization equations of the unit cells are also used as the constraints. The shape optimization problem is formulated as a distributed parameter optimization problem, and the theoretically derived shape gradient functions are applied to the H¹ gradient method to optimize the shape of macro and microporous structures. The validity of the proposed multi-scale method is confirmed by several numerical examples for designing the optimal shapes of the unit cells and the overall shape of the laminated shell structure concurrently. Macro–micro and micro–macro-hierarchical shape optimizations are also implemented for comparison. With the proposed method, arbitrary stiff and flexible porous laminated shell structures can be created by defining the target displacements suitably.