Design of graded cellular structures with data-driven inverse homogenization for microstructures with cubic symmetry

We reconsider the Inverse Homogenization Generative Adversarial Network (IH-GAN) model, a recently introduced data-driven method for the inverse homogenization of cellular structures, and propose a number of extensions and improvements. In particular, we discuss the application of inverse homogenization to functionally graded cellular structures and show that the optimization procedure presented in the original introduction of IH-GAN contains an implicit isotropy assumption on the homogenized stiffness tensors, which only exhibit cubic symmetry for the triply periodic minimal surface (TPMS) structures in the original design space. We therefore propose an extension of IH-GAN that includes the axial shear modulus as an additional input parameter and show by numerical experiments that if the optimization algorithm is adjusted accordingly, the optimization results can be significantly improved. Furthermore, we apply a similar extension to the previously introduced LatticeOptDiff model, which utilizes a diffusion architecture instead of a generative adversarial network, and show that the resulting inverse homogenization model—called CellularOptDiff—outperforms even the improved IH-GAN model for functionally graded cellular structure optimization tasks. Our numerical experiments are based on classical finite-element simulations and include compliance minimization problems for different geometries as well as a target deformation task. In the former, CellularOptDiff achieves a reduction of up to 30.1% in strain energy and up to 19.4% with respect to the maximum displacement, while in the latter, the mean squared deviation from the target deformation is reduced by 57.8% compared to IH-GAN.