Numerical strategy for unbiased homogenization of random materials

Abstract

International audienceThis paper presents a numerical strategy that allows to lower the costs associated to the prediction of the value of homogenized tensors in elliptic problems. This is done by solving a coupled problem, in which the complex microstructure is confined to a small region and surrounded by a tentative homogenized medium. The characteristics of this homogenized medium are updated using a self-consistent approach and are shown to converge to the actual solution. The main feature of the coupling strategy is that it really couples the random microstructure with the deterministic homogenized model, and not one (deterministic) realization of the random medium with a homogenized model. The advantages of doing so are twofold: (a) the influence of the boundary conditions is significantly mitigated, and (b) the ergodicity of the random medium can be used in full through appropriate definition of the coupling operator. Both of these advantages imply that the resulting coupled problem is less expensive to solve, for a given bias, than the computation of homogenized tensor using classical approaches. Examples of 1D and 2D problems with continuous properties, as well as a 2D matrix-inclusion problem, illustrate the effectiveness and potential of the method

Fully scalable implementation of a volume coupling scheme for the modeling of multiscale materials

“…This allows the usage of each models' own solver and code implementation without changing the coupling algorithm. The deterministic / stochastic model pair application is especially important for the numerical homogenization of random materials [10,8,9]. Another interesting extensions of this algorithm consist on generalizing it for a parallel solver with time coupled models, or for reduced-order solvers, such as the PGD [28].…”

A Stochastic Multi-scale Approach for Numerical Modeling of Complex Materials—Application to Uniaxial Cyclic Response of Concrete

“…There are no external forces other thant applied on the ED and no dynamic effects are considered either. Then, the displacement vector field u, and the strain and stress tensor 5 fields, ε ε ε and σ σ σ , satisfy at any pseudo-time t ∈ [0, T ]:…”

“…With the improvement of computational ressources, stochastic homogenization of random heterogeneous media can now be achieved without introducing a mesoscale. In [5], an efficient numerical strategy is presented to obtain effective tensors of random materials by coupling random micro-structures to tentative effective models within the Arlequin framework for model superposition [1]. In [30], micro-structures composed of a medium with randomly distributed inclusions of random shapes are generated and their behaviors are simulated with the extended finite element method (XFEM); homogenized properties at macroscale are then derived through the computation of mean response using Monte Carlo simulations.…”

A coupling method for stochastic continuum models at different scales

“…Extending the upscaling process described above to the limit M → +∞, one obtains a deterministic homogeneous mechanical parameter k * . In general, this parameter k * is not equal to the parameter obtained from classical homogenization [56,57,58,59]. However, in the particular case of a micro-scale parameter with lognormal first-order marginal density in 2D, the homogenized tensor is equal to the geometric average of the field k * = k m /(1 + σ 2 m /k 2 m ) 1/2 .…”