Three-scale concurrent topology optimization for porous structures

The natural world is replete with multiscale structures that exhibit exceptional functionality, particularly those possessing three-scale configurations. To address the challenge associated with the full-scale computational burden for such three-scale structures, this study proposes a three-scale concurrent topology optimization strategy based on the material-field series expansion (MFSE) method. In order to explore the design space of the three-scale structure, multiple microstructures are introduced at both the meso- and micro-scales. Microstructures at the microscale and the mesoscale are described using the MFSE-based multi-material interpolation approach, which significantly reduces the number of design variables. To address the geometric incompatibility at the interfaces of adjacent microstructures, a strategy called active shape connective (ASC) is proposed to ensure microstructural connectivity without imposing additional constraints in the optimization model. To avoid the sensitivity coupling phenomenon, the design sensitivity is derived based on the decoupled sensitivity analysis method. As a result, the topology optimization for a 2D porous structure with billions of full-scale elements can be efficiently realized within less than an hour on a desktop computer. Aiming at stiffness and buckling, several 2D and 3D three-scale concurrent examples featuring multiple connectable microstructures at the meso- and micro-scales are achieved, exemplifying the attainment of ultra-light porous structures in the real world.