  The data-driven approach is emerging as a promising method for the
topological design of multiscale structures with greater efficiency. However,
existing data-driven methods mostly focus on a single class of microstructures
without considering multiple classes to accommodate spatially varying desired
properties. The key challenge is the lack of an inherent ordering or distance
measure between different classes of microstructures in meeting a range of
properties. To overcome this hurdle, we extend the newly developed
latent-variable Gaussian process (LVGP) models to create multi-response LVGP
(MR-LVGP) models for the microstructure libraries of metamaterials, taking both
qualitative microstructure concepts and quantitative microstructure design
variables as mixed-variable inputs. The MR-LVGP model embeds the mixed
variables into a continuous design space based on their collective effects on
the responses, providing substantial insights into the interplay between
different geometrical classes and material parameters of microstructures. With
this model, we can easily obtain a continuous and differentiable transition
between different microstructure concepts that can render gradient information
for multiscale topology optimization. We demonstrate its benefits through
multiscale topology optimization with aperiodic microstructures. Design
examples reveal that considering multiclass microstructures can lead to
improved performance due to the consistent load-transfer paths for micro- and
macro-structures.
