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Stochastic Upscaling via Linear Bayesian Updating
[chapter]

Sadiq M. Sarfaraz, Bojana V. Rosić, Hermann G. Matthies, Adnan Ibrahimbegović

2017
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Multiscale Modeling of Heterogeneous Structures
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In this work we present an upscaling technique for multi-scale computations based on a stochastic model calibration technique. We consider a coarse-scale continuum material model described in the framework of generalized standard materials. The model parameters are considered uncertain, and are determined in a Bayesian framework for the given fine scale data in a form of stored energy and dissipation potential. The proposed stochastic upscaling approach is independent w.r.t. the choice of
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... the choice of models on coarse and fine scales. Simple numerical examples are shown to demonstrate the ability of the proposed approach to calibrate coarse scale elastic and inelastic material parameters. scale features in a stochastic setting. Stefanou et al. (2015) employed computational homogenization and XFEM to study the effect of uncertainty in material properties and geometrical features on macro-scale. Clément et al. (2013) have proposed a strategy to construct stochastic energy functional from realizations of random micro-structure. Brady et al. (2006) have utilized a "Moving Window" approach to characterize micro-scale randomness, a similar idea is used to infer meso-scale random fields of material stiffness tensor from bi-phased micro scale by Demmie and Ostaja-Starzewski (2015) . Another way to achieve the coupling between scales with possibly completely different descriptions is to use concepts of machine learning as in Koutsourelakis (2007) , the theory of which is often, at least conceptually, grounded in Bayesian ideas. In this paper a Bayesian approach (Kaipio and Somersalo 2004 , Kennedy and O´Hagan 2001 , Hawkins-Daarud et al. 2013 ) is taken directly in its computationally cheaper Gauss-Markov-Kalman filter form, a generalisation of classical Kalman filtering that allows direct estimation of non-Gaussian distributions without sampling . The general set-up we propose here is as follows: on the macro scale a continuum material model is derived which not only covers the mean (i.e., homogenised) behavior, but also the possible deviations from it. As the micro-scale mechanical behavior, we have in mind involves both reversible (i.e., elastic) as well as irreversible (i.e., inelastic) behavior, this has to be reflected also in the constitutive models considered on the macro scale. Here the main goal is to show a proof-of-concept, so we will limit ourselves to a simple but sufficiently representative case of inelastic behavior (Liu et al. 2013) . For the sake of simplicity, we limit ourselves to isothermal conditions and we shall exclude strain-rate dependent behavior. Thus, for the inelastic or irreversible part we only consider ductile non-softening behavior, i.e., strain-rate independent plasticity and damage with hardening. However, one can consider choice of more complex structural/continuum models for upscaling e.g. (Do and Ibrahimbegović 2015 , Do et al. 2015 . As this is to be a model for possibly more complex behavior, we shall assume that the macroscale continuum model can be described as a generalised standard material model (Halpen and Nguyen 1974 , Halpen and Nguyen 1975 , Nguyen, 1977 . This has the advantage that these materials are completely characterized by the specification of two scalar functions, the stored energy resp. Helmholtz free energy, and the dissipation pseudo-potential. In this way the simple case chosen here can be generalized to very complex material behavior. In our view this description is also a nice and simple illustration for the connection with the micro-scale behavior. 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 be used as measurements in Bayesian inference to identify the macro-scale model parameters given micro-scale response energy. In some more detail, the identification of the macro-structure generalized standard material constitutive model proceeds as follows: the micro-structure is exposed to some external action resp. stimulus, here purely mechanical case this is chosen as large scale homogeneous deformation. The response is measured in the change of the two thermodynamic functions alluded to: the stored resp. Helmholtz free energy and the dissipation resp. entropy production. The main goal is to show that this idea is computationally feasible for identifying the macro-model material parameters. The outline of this paper is as follows: In Section 2 the problem is defined in an abstract sense

doi:10.1007/978-3-319-65463-8_9
fatcat:z6erzvl5yzb5finaet2oj7jyxu