  In order to optimally design materials, it is crucial to understand the
structure-property relations in the material by analyzing the effect of
microstructure parameters on the macroscopic properties. In computational
homogenization, the microstructure is thus explicitly modeled inside the
macrostructure, leading to a coupled two-scale formulation. Unfortunately, the
high computational costs of such multiscale simulations often render the
solution of design, optimization, or inverse problems infeasible. To address
this issue, we propose in this work a non-intrusive reduced basis method to
construct inexpensive surrogates for parametrized microscale problems; the
method is specifically well-suited for multiscale simulations since the coupled
simulation is decoupled into two independent problems: (1) solving the
microscopic problem for different (loading or material) parameters and learning
a surrogate model from the data; and (2) solving the macroscopic problem with
the learned material model. The proposed method has three key features. First,
the microscopic stress field can be fully recovered. Second, the method is able
to accurately predict the stress field for a wide range of material parameters;
furthermore, the derivatives of the effective stress with respect to the
material parameters are available and can be readily utilized in solving
optimization problems. Finally, it is more data efficient, i.e. requiring less
training data, as compared to directly performing a regression on the effective
stress. For the microstructures in the two test problems considered, the mean
approximation error of the effective stress is as low as 0.1% despite using a
relatively small training dataset. Embedded into the macroscopic problem, the
reduced order model leads to an online speed up of approximately three orders
of magnitude while maintaining a high accuracy as compared to the FE$^2$
solver.
