Learning constitutive models from microstructural simulations via a non-intrusive reduced basis method: Extension to geometrical parameterizations

Two-scale simulations are often employed to analyze the effect of the microstructure on a component's macroscopic properties. Understanding these structure–property relations is essential in the optimal design of materials for specific applications. However, these two-scale simulations 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 inexpensive-to-evaluate surrogate models. Such surrogate models must be able to handle microstructure parameters in order to be used for material design. A previous work focused on the construction of an accurate surrogate model for microstructures under varying loading and material parameters by combining proper orthogonal decomposition and Gaussian process regression. However, that method works only for a fixed geometry, greatly limiting the design space. This work hence focuses on extending the methodology to treat geometrical parameters. To this end, a method that transforms different geometries onto a parent domain is presented, that then permits existing methodologies to be applied. We propose to solve an auxiliary problem based on linear elasticity to obtain the geometrical transformations. The method has a good reducibility and can therefore be quickly solved for many different geometries. 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 automatically 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, thus demonstrating its potential in two-scale shape optimization and material design problems.