Design of thin micro-architectured panels with extension–bending coupling effects using topology optimization

We design the micro-architecture of thin elastic panels to control their macroscopic behavior, accounting simultaneously for in-plane stiffness, out-of-plane stiffness and the extension–bending coupling effects. Our topology optimization method combines inverse homogenization, the Hadamard shape derivative and a level set method in the diffuse interface context to systematically capture within the unit cell the optimal micro-architecture. The efficiency of the solution method is illustrated through four numerical examples where the designed shape yields an important extension–bending coupling. The deformation responses under tensile loading are assessed numerically both on the complete periodic panel and on its homogenized twin plate. The results demonstrate that the simultaneous control of the in-plane, out-of-plane and their coupled behavior enables to shift a flat panel into a dome or a saddle shaped structure. Moreover, the obtained unit cells are elementary blocks to create directly 3D printable objects with shape-morphing capabilities.