Topology optimization approach for functionally graded metamaterial components based on homogenization of mechanical variables

The micron resolutions that 3D printers are reaching allow for a change of paradigm in the design of components, optimizing not only the topology of the component, but also the microstructure of the material, which can adopt different objectives at different locations in the component, reaching different mechanical properties and different optimal microstructures at those locations. This constitutes a flip in the design objectives to focus on tailored material work, aimed either at local objectives or at component-wide ones, instead of pursuing homogeneous material properties for global objectives. For this new concurrent structure-material design leading to optimized components made of functionally graded (FG) metamaterials, new topology optimization (TO) procedures are needed in which the objective is to obtain component-wise mechanical variables or distributions of these variables, rather than global component objectives as compliance, which is a typical objective of classical TO approaches. In this work we present a new TO approach aimed at FG metamaterial design pursuing homogeneous mechanical variables. We focus on homogenization of deformation level along the component by minimizing its standard deviation across the domain. The proposed formulation can be naturally generalized to anisotropy, nonlinear behavior, and other optimization objectives. Convergence to a suitable optimized solution, with a distribution of material properties resulting in a homogeneously deformed component, is obtained. These FG material properties may be obtained from local metamaterial design. Classical material/void configurations may also be obtained following our procedure, and results for such cases can be compared qualitatively to the Solid Isotropic Material with Penalization (SIMP) approach.