A hybrid density/level set formulation for topology optimization of functionally graded lattice structures

Abstract

Additive manufacturing is capable of producing lightweight structures by introducing mesoscale lattice structures in the design without significant additional manufacturing costs. Nevertheless, designing efficient structures including complex lattices remains a challenging task. This paper presents a strategy for the design of additively manufactured structures that is able to simultaneously optimize the shape of the macroscale structure and the functionally graded lattices inside this design. Assuming a separation of scales between the lattice cell size and the macroscale structure, an effective material model based on numerical homogenization is adopted to model the lattice behavior in the analysis of the macroscale structure. While this material model is parametrized by volume densities to mimic functionally graded lattices, the shape of the macroscale structure is represented by a level set function and modeled by the extended finite element method inside the design domain. By adopting an explicit approach to the level set optimization the design problem can be formulated as a single nonlinear programming problem which can be solved with standard optimization algorithms for topology optimization. The proposed formulation avoids degenerate lattice members by excluding small densities in the optimization. Furthermore, the distinct representation of the macroscale structure enables to easily include geometric constraints on the macroscale level such as a minimum feature size or other constraints specific to the additive manufacturing process. The effectiveness of the method is demonstrated in a number of examples.