Multiscale approximation of stochastic PDEs with fastly varying coefficients: certification of accuracy through error bounding

The goal of this research is to certify the accuracy of homogenisation schemes in the sense of engineering quantities of interest.

Homogenisation has been used for centuries to upscale some knowledge of the physics of heterogeneous materials to the engineering scale, where predictions are required. Homogenisation is typically a limit result, that delivers good predictions when the typical length-scale of the material heterogeneities is small compared to the engineering scale (i.e. the scales are well-separated). Broadly speaking, homogenisation fails in boundary regions that are dominated by stress concentrations (around sharp joints, holes, at the interface between different materials, ...).

We investigate here a methodology to quantify the error that is made when using homogenisation when scale separability is not satisfied. We started from the modelling error methodology developed at ICES Texas in the early 2000's. The approach proposed by this group is to bound the error that is made on engineering quantities of interest (QoI) when using an arbitrary homogenisation scheme as a approximation of the intractable, fine-scale heterogeneous problem. This was done by extending the equilibrated residual method (classically used to quantify discretisation errors) to the context of modelling error and combining it the adjoint methodology to convert error measures in energy norm into errors in QoI. The method was shown to deliver guaranteed error bounds, without requiring to solve the the underlying heterogeneous problem. However, the heterogeneous problem needs to be constructed (but not solved) in order to compute the bounds, which, in the case of large composite structures, remains a computational bottleneck.

We tackle this issue by representing the microscale heterogeneities as a random field. In addition to the fact that this is a realistic modelling approach, in the sense that we rarely know where the heterogeneities of composite materials are precisely located, we are able to completely alleviate the need for meshing and assembling the fine-scale heterogeneous problem. We therefore retrieve a numerical separation of scales for the computation of modelling error bounds.

We showed that this methodology could be applied to provide bounds for the stochastic homogenisation error made on both the first and second statistical moments of engineering QoI. These bounds can be implemented within a couple of hours in any finite element code. They can be interpreted as an extension of the classical Reuss-Voigt bounds, but without any a priori requirement in terms of scale separability.

All the numerical results have been obtained by Daniel Alves Paladim in the context of his PhD thesis. We have largely benefitted from the advices of Mathilde Chevreuil, University of Nantes, regarding the stochastic aspects of this work.