  We present a two-scale topology optimization framework for the design of
macroscopic bodies with an optimized elastic response, which is achieved by
means of a spatially-variant cellular architecture on the microscale. The
chosen spinodoid topology for the cellular network on the microscale (which is
inspired by natural microstructures forming during spinodal decomposition)
admits a seamless spatial grading as well as tunable elastic anisotropy, and it
is parametrized by a small set of design parameters associated with the
underlying Gaussian random field. The macroscale boundary value problem is
discretized by finite elements, which in addition to the displacement field
continuously interpolate the microscale design parameters. By assuming a
separation of scales, the local constitutive behavior on the macroscale is
identified as the homogenized elastic response of the microstructure based on
the local design parameters. As a departure from classical FE$^2$-type
approaches, we replace the costly microscale homogenization by a data-driven
surrogate model, using deep neural networks, which accurately and efficiently
maps design parameters onto the effective elasticity tensor. The model is
trained on homogenized stiffness data obtained from numerical homogenization by
finite elements. As an added benefit, the machine learning setup admits
automatic differentiation, so that sensitivities (required for the optimization
problem) can be computed exactly and without the need for numerical derivatives
- a strategy that holds promise far beyond the elastic stiffness. Therefore,
this framework presents a new opportunity for multiscale topology optimization
based on data-driven surrogate models.
