Topology optimization of hierarchical lattice structures with substructuring

This work presents a generalized topology optimization approach for the design of hierarchical lattice structures with the development of an Approximation of Reduced Substructure with Penalization (ARSP) model. The structure is assumed to be composed of substructures with a common lattice geometry pattern. Unlike conventional homogenization-based designs assuming the separation of scales, this work considers two different yet connected scales. Each substructure is condensed first into a super-element with a reduced degrees of freedom and is associated with a density design variable indicating the material volume fraction. The density design variable is linked to a lattice geometry feature parameter. A surrogate model is particularly built with the aid of proper orthogonal decomposition and diffuse approximation, mapping the density to super-element stiffness matrix. The derivative of super-element matrix with respect to the associated density can therefore be evaluated efficiently and explicitly. The super-element matrix is further augmented with a penalized density to control the structural complexity. The optimality criteria method is used for the update of design variables. Numerical examples show that both the size and the lattice pattern of substructure have essential influences on the design, indicating the necessity of performing connected hierarchical modeling and design.