  Understanding structure-property relations is essential to optimally design
materials for specific applications. Two-scale simulations are often employed
to analyze the effect of the microstructure on a component's macroscopic
properties. However, they are typically computationally expensive and
infeasible in multi-query contexts such as optimization and material design. To
make such analyses amenable, the microscopic simulations can be replaced by
surrogate models that must be able to handle a wide range of microstructural
parameters. This work focuses on extending the methodology of a previous work,
where an accurate surrogate model was constructed for microstructures under
varying loading and material parameters using proper orthogonal decomposition
and Gaussian process regression, to treat geometrical parameters. To this end,
a method that transforms different geometries onto a parent domain is
presented. We propose to solve an auxiliary problem based on linear elasticity
to obtain the geometrical transformations. Using these transformations,
combined with the nonlinear microscopic problem, we derive a fast-to-evaluate
surrogate model with the following key features: (1) the predictions of the
effective quantities are independent of the auxiliary problem, (2) the
predicted stress fields fulfill the microscopic balance laws and are periodic,
(3) the method is non-intrusive, (4) the stress field for all geometries can be
recovered, and (5) the sensitivities are available and can be readily used for
optimization and material design. The proposed methodology is tested on several
composite microstructures, where rotations and large variations in the shape of
inclusions are considered. Finally, a two-scale example is shown, where the
surrogate model achieves a high accuracy and significant speed up,
demonstrating its potential in two-scale shape optimization and material design
problems.
