In this Note, we present a unified approach to the information-theoretic modeling and simulation of a class of elasticity random fields, for all physical symmetry classes. The new stochastic representation builds upon a Walpole tensor decomposition, which allows the maximum entropy constraints to be decoupled in accordance with the tensor (sub)algebras associated with the class under consideration. In contrast to previous works where the construction was carried out on the scalar-valued Walpole coordinates, the proposed strategy involves both matrix-valued and scalar-valued random fields. This enables, in particular, the construction of a generation algorithm based on a memoryless transformation, hence improving the computational efficiency of the framework. Two applications involving weak symmetries and sampling over spherical and cylindrical geometries are subsequently provided. These numerical experiments are relevant to the modeling of elastic interphases in nanocomposites, as well as to the simulation of spatially dependent wood properties for instance.
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@article{CRMECA_2017__345_6_399_0,
author = {Brian Staber and Johann Guilleminot},
title = {Stochastic modeling and generation of random fields of elasticity tensors: {A} unified information-theoretic approach},
journal = {Comptes Rendus. M\'ecanique},
pages = {399--416},
publisher = {Elsevier},
volume = {345},
number = {6},
year = {2017},
doi = {10.1016/j.crme.2017.05.001},
language = {en},
}
TY - JOUR AU - Brian Staber AU - Johann Guilleminot TI - Stochastic modeling and generation of random fields of elasticity tensors: A unified information-theoretic approach JO - Comptes Rendus. Mécanique PY - 2017 SP - 399 EP - 416 VL - 345 IS - 6 PB - Elsevier DO - 10.1016/j.crme.2017.05.001 LA - en ID - CRMECA_2017__345_6_399_0 ER -
%0 Journal Article %A Brian Staber %A Johann Guilleminot %T Stochastic modeling and generation of random fields of elasticity tensors: A unified information-theoretic approach %J Comptes Rendus. Mécanique %D 2017 %P 399-416 %V 345 %N 6 %I Elsevier %R 10.1016/j.crme.2017.05.001 %G en %F CRMECA_2017__345_6_399_0
Brian Staber; Johann Guilleminot. Stochastic modeling and generation of random fields of elasticity tensors: A unified information-theoretic approach. Comptes Rendus. Mécanique, Volume 345 (2017) no. 6, pp. 399-416. doi : 10.1016/j.crme.2017.05.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.05.001/
[1] Random field representations for stochastic elliptic boundary value problems and statistical inverse problems, Eur. J. Appl. Math., Volume 25 (2014) no. 03, pp. 339-373
[2] Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial derivative operators, Comput. Methods Appl. Mech. Eng., Volume 195 (2006), pp. 26-64
[3] Probabilistic models for stochastic elliptic partial differential equations, J. Comput. Phys., Volume 229 (2010), pp. 8406-8429
[4] Mathematical Elasticity, vol. I: Three-Dimensional Elasticity, Elsevier Science Publishers, 1988
[5] Microstructure models and material response by extreme value theory, SIAM/ASA J. Uncertain. Quantificat., Volume 4 (2016) no. 1, pp. 190-217
[6] Information theory and statistical mechanics I, Phys. Rev., Volume 106 (1957) no. 4, pp. 620-630
[7] Information theory and statistical mechanics II, Phys. Rev., Volume 108 (1957) no. 2, pp. 171-190
[8] A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures, Comput. Methods Appl. Mech. Eng., Volume 200 (2011), pp. 1637-1648
[9] On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties, J. Elast., Volume 111 (2013), pp. 109-130
[10] Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media, SIAM J. Multiscale Model. Simul., Volume 11 (2013), pp. 840-870
[11] Approximate solutions of Lagrange multipliers for information-theoretic random field models, SIAM/ASA J. Uncertain. Quantificat., Volume 3 (2015), pp. 599-621
[12] A random field formulation of Hooke's law in all elasticity classes, J. Elast., Volume 34 (2017) no. 2, pp. 269-302
[13] Microstructural Randomness and Scaling in Mechanics of Materials, Modern, Mechanics and Mathematics Series, Chapman & Hall/CRC/Taylor & Francis, 2008
[14] A nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probab. Eng. Mech., Volume 15 (2000) no. 3, pp. 277-294
[15] Stochastic modeling of anisotropy in multiscale analysis of heterogeneous materials: a comprehensive overview on random matrix approaches, Mech. Mater., Volume 44 (2012), pp. 35-46
[16] Eigentensors of linear anisotropic elastic materials, Q. J. Mech. Appl. Math., Volume 43 (1990) no. 1, pp. 15-41
[17] Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Comput. Methods Appl. Mech. Eng., Volume 194 (2005), pp. 1251-1294
[18] Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model, Int. J. Numer. Methods Biomed. Eng., Volume 90 (2012) no. 5, pp. 613-635
[19] Fourth-rank tensors on the thirty-two crystal classes: Multiplication tables, Proc. R. Soc. Lond. A, Volume 391 (1984), pp. 149-179
[20] Random matrices and information theory, Il Nuovo Cimento B, Volume 57 (1968) no. 1, pp. 183-193
[21] Convex Optimization, Cambridge University Press, 2004
[22] The size of compact subsets of Hilbert spaces and continuity of Gaussian processes, J. Funct. Anal., Volume 1 (1967), pp. 290-330
[23] Itô SDE–based generator for a class of non-Gaussian vector-valued random fields in uncertainty quantification, SIAM J. Sci. Comput., Volume 36 (2014), p. A2763-A2786
[24] An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc., Ser. B, Stat. Methodol., Volume 74 (2011) no. 4, pp. 423-498
[25] Exploring a new class of nonstationary spatial Gaussian random fields with varying local anisotropy, Stat. Sin., Volume 133 (2015), pp. 25-115
[26] Does non-stationary spatial data always require non-stationary random fields?, Spat. Stat., Volume 14 (2015), pp. 505-531
[27] Stochastic continuum modeling of random interphases from atomistic simulations. Application to a polymer nanocomposite, Comput. Methods Appl. Mech. Eng., Volume 303 (2016), pp. 430-449
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