Probabilistic learning and updating of a digital twin for composite material systems

Abstract

This paper presents an approach for characterizing and estimating statistical dependence between a large number of observables in a composite material system. Conditional regression is carried out using the estimated joint density function, permitting a systematic exploration of interdependence between fine scale and coarse observables that can be used for both prognosis and design of complex material systems. An example demonstrates the integration of experimental data with a computational database. The statistical approach is based on the probabilistic learning on manifolds recently developed by the authors. This approach leverages intrinsic structure detected through diffusion on graphs with projected stochastic differential equations to generate samples constrained to that structure.

Probabilistic learning on manifolds

“…In general, for ν sufficiently large (for instance, ν ∼ 10), the optimal value ε opt defined by Eqs. (14) and 15is such that ε opt 1 while, since ν ≤ N , Eq. (8) shows that s 2 /2 < 1.…”

“…For reasons of limitation of the paper length, we cannot reproduce the description of this application and we refer the reader to this reference. The reader can find further illustrations concerning the estimation of m opt and the preservation of the concentration of the probability measure in [1,11,13,14,15,16,32,33,35,36]. With respect to the notation introduced in Section 1.1, we have n w = 20, n q = 200, n = 220, and N = 200.…”

“…The exploration of the admissible solution space in these situations is thus hampered by available computational resources. The PLoM was successfully adapted to tackle these challenges for several related problems including nonconvex optimization under uncertainty [2,3] and the calculation of Sobol's indices [4].…”

“…For each value ε DM in this partition, the eigenvalues are computed as explained in Section 5.1 and the criterion defined by Eqs. (14) and (15) are examined. Note that this estimation has been used with success for many applications (see for instance, [11,13,15,16,18,32,33,35,36]).…”

Computational stochastic homogenization of heterogeneous media from an elasticity random field having an uncertain spectral measure

“…The homogenization of linear elastic materials with heterogeneous microstructures composed of several phases with well defined constituents (from a continuum mechanics point of view) and the calculation of the macroscopic properties (effective properties) have received considerable attention (see for instance [1,2,3,4,5,6,7,8,9,10,11]), for stochastic homogenization (see [12,13,14,15,16,17,18,19]), for computational multiresolution materials and multiscale method (see for instance [20,21,22,23,24,25,26]), and more recently, for data-driven and machine learning approaches applied to heterogeneous materials (see for instance [27,28,29,30,31,32,33]).…”