Topology optimization and de-homogenization of graded lattice structures based on asymptotic homogenization

Abstract This paper presents a novel two-step homogenization-based topology optimization and de-homogenization method for the design of graded lattice structures. The lattice orientation and material layout are first optimized for square base cells in the macro scale. Then by introducing the lattice stretching design variables of micro base cells, which bridge the base cell distortion with lattice stretch, the error residual of mapping functions is integrated with compliance formulation to form a novel mixed optimization formulation, concurrently optimizing structural performance and mapping functions. The advantage of this formulation is two-fold. First, the micro design space is relaxed from square base cells to rectangular ones so that performance improvement is further expected. Second, an excellent agreement, in both shape and performance, between the projected single-scale lattice structures with the homogenization results is secured, as compared to the frequently adopted post-process procedure of constructing single-scale lattices, where performance deviation could arise for specific microstructural patterns. With the optimized mapping functions, de-homogenization procedure is carried out to construct single-scale spatially graded lattice structures, where a simple filter-projection operation is proposed to obtain fine-scale smoothed boundaries from coarse-scale homogenization results with zig-zag boundaries. Several numerical examples are presented and compared with conventional post-process treatment results to show the validity of the proposed method, and different kinds of lattice patterns adopted in this work show its versatility for a broad range of lattice patterns.