Topology optimization of lightweight periodic lattices under stiffness and stability constraints

Abstract

This study advances the design of light, stiff lattice materials by taking into account stability considerations. The objective is to minimize a properly weighted relative density of the lattice, subject to the stiffness and stability constraints under axial and shear deformations. Timoshenko beam theory is employed to accurately evaluate the response when tangible shear deformations are expected. Taking advantage of the symmetry and through customized boundary conditions, stiffness and stability measures are evaluated only by two finite element and eigenvalue analyses on a reduced design domain mitigating the computational burden significantly. Two sets of design variables are defined; the physical variables representing elements’ cross-sectional properties and the existence variables that are used to both stabilize the optimization process and improve manufacturability. Analytical sensitivities are derived using the adjoint method to allow for the use of gradient-based optimizers. The utility of the proposed method is demonstrated through numerical examples for different scenarios of stiffness and stability constraints. It is shown that significant stability improvements for the specified stiffness ratios are achieved mainly through reconfiguration of elements in the optimized lattice in higher density regions with minimal change in weight. It is also shown that the designed lattices can dramatically outperform the classic lattice architectures such as Kagomé, in terms of stiffness and stability.