PAMM · Proc. Appl. Math. Mech. 17, 869 – 870 (2017) / DOI 10.1002/pamm.201710401 Stochastic upscaling of random microstructures Bojana Rosi´ c 1, * , Muhammad Sadiq Sarfaraz 1 , Hermann G. Matthies 1 , and Adnan Ibrahimbegovi´ c 2 1 Institut für Wissenschaftliches Rechnen, Mühlenpfordstraße 23, 38106 Braunschweig, Germany 2 Lab. de Mécanique Roberval, Centre de Recherche Royallieu, 60200 Compiégne, France In this work we present an upscaling technique for multi-scale computations based on random microstructures modelled as realisations of lognormally distributed random fields, or described by randomly distributed inclusions in a homogeneous matrix. Their corresponding coarse-scale model parameters are considered as uncertain, and are approximated by random variables, the distributions of which are obtained via polynomial chaos based Bayesian procedures in which the fine-scale energy is used as an observation. c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The spatial scales on which concrete microstructure heterogeneities occur are orders of magnitudes smaller than the ones for which one would like to do response predictions. To resolve the small scales (micro-scale) on the predictive scale (macro- scale), here we consider the stochastic upscaling framework as described in [1]. Thus, the coupling between scales with possibly completely different descriptions (e.g. discrete fine vs. continuum coarse scale models) is here achieved by using Bayesian concepts. The approach as described in [3] is extended to encounter not only spatial variation but also the random nature of microstructure. The macro-scale continuum model is described here as a generalised standard material model completely characterised by the specification of two scalar functions, the stored energy resp. Helmholtz free energy, and the dissipation pseudo-potential. In our view this description is also a key for the connection with the micro-scale behaviour. No matter how the physical and mathematical/computational description on the micro-scale has been chosen, in all cases where the description is based on physical principles it will be possible to define the stored (Helmholtz free) energy and the dissipation (entropy production). These two thermodynamic functions will thus serve as observation data in Bayesian estimation. 2 Bayesian upscaling of random microstructures and numerical example Let the fine-scale energy Ψ m (ξ (ω)) representing random microstructures be defined on the probability space (Ω ξ , B ξ , P ξ ) in which Ω ξ defines the set of elementary events, B ξ is the σ-algebra and P ξ is the probability measure. The goal is to use infor- mation on Ψ m (ξ (ω)) in order to calibrate (upscale) the set of material parameters q of the standard generalised coarse-scale model. To achieve this, q is assumed to be uncertain (unknown) and further modelled a priori as random variables q(θ(ω))— prior—belonging to the probability space (Ω θ , B θ , P θ ). Hence, a priori prediction of the coarse scale energy Ψ M (θ(ω)) is obtained by propagating the uncertainty in q(θ(ω)) through the standard generalised model, i.e. by solving the stochastic forward problem as described in [2]. Following discussion in [3], in order to incorporate the micro-scale information on the coarse-scale level one may use an approximated Bayes formula q a (ξ,θ)= q f (θ(ω)) + ϕ(Ψ m (ξ (ω))) - ϕ(Ψ M (θ(ω)) (1) which assimilates the prior (forecast) random variable q f with the measurement data Ψ s . As a result the posterior (assimilated) random variable q a is the function of both ξ (ω) and θ(ω). The former random variable describes randomness of the structure, whereas the latter expresses our knowledge. In addition, the accuracy of the estimate q a depends on the computationally acceptable choice for mapping ϕ ranging all measurable maps from the measurement to the parameter space. In this paper we assume ϕ to represent the linear mapping described by the Kalman gain K, i.e. q a (ξ,θ)= q f (θ(ω)) + K(Ψ s (ξ (ω)) - Ψ S (θ(ω)), K = cov qΨ cov -1 Ψ , (2) which finally results in sub-optimal estimates but smaller computational cost. As the updating is performed sequentually in time by taking small time increments (see [3]), the solution q a is assumed to be near optimal. For computational purposes the formula in Eq. 2 is further discretised by the generalised polynomial chaos approximation, for more details please see [2,3]. However, the main issue in Eq. 2 is the fact that the random variable Ψ m (ξ (ω)) is not ∗ Corresponding author: e-mail bojana.rosic@tu-bs.de, phone +00 49 531 391-3016, fax +00 49 531 391-3002 c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim