A multi-material topology optimization approach to hybrid material structures with gradient lattices

In this paper, we present a novel approach for designing hybrid material structures that incorporate solid, void, and spatially-varying lattices. This approach extends the recursively defined power-law type multi-material interpolation model by incorporating additional gradient material phases. Unlike conventional models, the new extended model considers the secondary ``solid'' material phase as a type of lattice, enabling adjustment of lattice geometry and mechanical behavior with gradient control variables such as density or size. By means of numerical homogenization and interpolation, we explicitly express the effective mechanical behavior of the gradient lattice in relation to the gradient control variable. This approach enhances design freedom by allowing for flexible control of individual material volumes, and also the lower and upper bounds of gradient variables. By using implicit geometrical modeling techniques, the resulting gradient control variable field can be interpreted to create spatially-varying and smoothly-connected lattices at any arbitrary resolution. The effectiveness and flexibility of the proposed approach have been demonstrated through a series of benchmark design tests considering various lattice material types .