Homogenization-Based Topology Optimization of Spatially Variant Structural Networks

Topology optimization of load-bearing components at low-weight constraints typically leads to cellular truss-, plate- and shells-based structures that provide optimal stiffness at minimal mass. While classical optimization techniques are well suited for the optimization of macroscale structures, they tend to reach their limits when dealing with large-scale problems involving extreme numbers of significantly smaller structural members – such as, e.g., in the popular area of architected metamaterials where typical dimensions on the macro- vs. microscales (i.e., overall problem size vs. underlying structural feature size) can differ by orders of magnitude. Here, the required spatial resolution incurs prohibitively high computational expenses, which is why multiscale approaches – based on a separation of scales – become the method of choice. In a two-scale setting, the macroscale problem deals with the optimization of a continuous body whose effective local mechanical response is obtained from the under­lying structural topology on the microscale. Decoupling these two scales is efficient and allows the use of con­ventional optimization techniques on the macroscale which find the locally optimal microstructural topology at each point of the macroscale body. Unfortunately, most existing approaches have resulted in technically sound but – from an eng­in­eering perspective – little useful results since strong spatial variations in the structural topology cannot easily be accommodated in practice (i.e., knowing the optimal structural topology at each point of a body does not automatically imply a compatible and manufacturable structure).

We here discuss a two-scale homogenization-based strategy to rigorously optimize spatially graded structural networks in a compatible fashion based on a continuum description of the underlying truss architecture in a macroscale topology optimization problem and the subsequent extraction of a manufacturable structure with a tailorable characteristic length scale. To this end, we parametrize the structural topology through a set of Bravais lattice vectors, and we continuously interpolate the latter on the macroscale, analogous to the finite element (FE) discretization of the deformation in a mechanical boundary value problem. Solving the coupled macroscale problem (optimizing over the field of topological parameters while solving for mech­anical equilibrium using a nonlinear homogenization approach for the effective mech­anical response) results in the sought functionally graded structural design. We demonstrate that, for simple truss topologies, we recover results obtained previously for simple 2D bench­marks, while our approach extends to more complex scenarios and, in particular, can generate structures having microscale features that are orders of magnitude smaller than the macroscale boundary value problem (thus amounting to millions and more of individual truss members). We outline the general theory and numerical realization of this homogenization-based topology optimization approach, and we present example applications.