Certification of projection-based reduced order modelling in computational homogenisation by the Constitutive Relation Error

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model.