Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

Abstract Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative Volume Element size related to the user-defined significance level and the discrepancy between bounds on the apparent properties. First, the tiling concept is employed to efficiently generate arbitrarily large, statistically consistent realizations of investigated microstructures. Second, benefiting from the regular structure inherent to the tiling concept, the Partition theorem, and statistical sampling, we construct confidence intervals of the apparent properties related to the size of a microstructure specimen. Based on the interval width and the upper and lower bounds on the apparent properties, we adaptively generate additional microstructure realizations in order to arrive at an RVE satisfying the prescribed tolerance. The methodology is illustrated with the homogenization of thermo-mechanical properties of three two-dimensional microstructure models: a microstructure with mono-disperse elliptic inclusions, foam, and sandstone.

[1]  Dominique Jeulin,et al.  Apparent and effective physical properties of heterogeneous materials: Representativity of samples of two materials from food industry , 2006 .

[2]  Jarkko Kari,et al.  An Aperiodic Set of Wang Cubes , 1996, J. Univers. Comput. Sci..

[3]  D. Fullwood,et al.  Microstructure reconstructions from 2-point statistics using phase-recovery algorithms , 2008 .

[4]  A Tudor,et al.  Three-dimensional reconstruction of statistically optimal unit cells of polydisperse particulate composites from microtomography. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  René de Borst,et al.  Computational homogenization for adhesive and cohesive failure in quasi‐brittle solids , 2010 .

[6]  N. Kikuchi,et al.  Simulation of the multi-scale convergence in computational homogenization approaches , 2000 .

[7]  Anton Alstes.  Wang Tiles for Image and Texture Generation , 2004 .

[8]  Determining the size of RVE for nonlinear random composites in an incremental computational homogenization framework , 2016 .

[9]  M. Utz,et al.  Efficient reconstruction of multiphase morphologies from correlation functions. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Dani Lischinski,et al.  Recursive Wang tiles for real-time blue noise , 2006, SIGGRAPH 2006.

[11]  Ares Lagae,et al.  An alternative for Wang tiles: colored edges versus colored corners , 2006, TOGS.

[12]  Peter Wriggers,et al.  Aspects of the computational testing of the mechanical properties of microheterogeneous material samples , 2001 .

[13]  Marcel Utz,et al.  Efficient reconstruction of multiphase morphologies from correlation functions. , 2001 .

[14]  Robert L. Berger.  The undecidability of the domino problem , 1966 .

[15]  Jan Novák,et al.  Aperiodic compression and reconstruction of real-world material systems based on Wang tiles. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  J. Hurtado,et al.  Determination of the critical size of a statistical representative volume element (SRVE) for carbon reinforced polymers , 2006 .

[17]  D. Jeulin,et al.  Virtual improvement of ice cream properties by computational homogenization of microstructures , 2011 .

[18]  Harm Askes,et al.  Quantification of stochastically stable representative volumes for random heterogeneous materials , 2006 .

[19]  Jan Havelka,et al.  Compression and reconstruction of random microstructures using accelerated lineal path function , 2016, 1601.04359.

[20]  Martin Ostoja-Starzewski,et al.  Random field models of heterogeneous materials , 1998 .

[21]  Marc Levoy,et al.  Fast texture synthesis using tree-structured vector quantization , 2000, SIGGRAPH.

[22]  J. Nov'ak,et al.  A jigsaw puzzle framework for homogenization of high porosity foams , 2015, 1507.08521.

[23]  Generalized boundary conditions on representative volume elements and their use in determining the effective material properties , 2013 .

[24]  Christian Huet,et al.  Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume , 1994 .

[25]  Javier Segurado,et al.  A numerical approximation to the elastic properties of sphere-reinforced composites , 2002 .

[26]  Wei Chen,et al.  Stochastic microstructure characterization and reconstruction via supervised learning , 2016 .

[27]  D. Fullwood,et al.  Optimized structure based representative volume element sets reflecting the ensemble-averaged 2-point statistics , 2010 .

[28]  Anna Kučerová,et al.  Compressing random microstructures via stochastic Wang tilings. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Jagan M. Padbidri,et al.  Minimal kinematic boundary conditions for simulations of disordered microstructures , 2005 .

[30]  Michaël Rao,et al.  An aperiodic set of 11 Wang tiles , 2015, ArXiv.

[31]  Jaroslav Kruis.  Domain Decomposition Methods for Distributed Computing , 2007 .

[32]  G. Povirk,et al.  Incorporation of microstructural information into models of two-phase materials , 1995 .

[33]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[34]  W. Curtin,et al.  Using microstructure reconstruction to model mechanical behavior in complex microstructures , 2006 .

[35]  Y. Monerie,et al.  Determination of the size of the representative volume element for random quasi-brittle composites , 2009 .

[36]  A. Gokhale,et al.  Representative volume element for non-uniform micro-structure , 2002 .

[37]  Karam Sab,et al.  Periodization of random media and representative volume element size for linear composites , 2005 .

[38]  Andrey L. Piatnitski,et al.  Approximations of Effective Coefficients in Stochastic Homogenization , 2002 .

[39]  Peter Wriggers,et al.  A method of substructuring large-scale computational micromechanical problems , 2001 .

[40]  M. Sahimi,et al.  Cross-correlation function for accurate reconstruction of heterogeneous media. , 2013, Physical review letters.

[41]  Daniel Balzani,et al.  Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations , 2014 .

[42]  Michel Fogli,et al.  Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior , 2012 .

[43]  L. J. Sluys,et al.  Numerical determination of representative volumes for granular materials , 2004 .

[44]  V. Kouznetsova,et al.  Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme , 2002 .

[45]  F. Larsson,et al.  On statistical strain and stress energy bounds from homogenization and virtual testing , 2015 .

[46]  J. Zeman,et al.  Microstructural enrichment functions based on stochastic Wang tilings , 2011, 1110.4183.

[47]  Xingchen Liu,et al.  Random heterogeneous materials via texture synthesis , 2015 .

[48]  Marc G. D. Geers,et al.  A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials , 2017, J. Comput. Phys..

[49]  D. Fullwood,et al.  Gradient-based microstructure reconstructions from distributions using fast Fourier transforms , 2008 .

[50]  Peter Wriggers,et al.  A domain decomposition method for bodies with heterogeneous microstructure basedon material regularization , 1999 .

[51]  Harm Askes,et al.  Representative volume: Existence and size determination , 2007 .

[52]  Christian Huet,et al.  Application of variational concepts to size effects in elastic heterogeneous bodies , 1990 .

[53]  Vinh Phu Nguyen,et al.  On the existence of representative volumes for softening quasi-brittle materials – A failure zone averaging scheme , 2010 .

[54]  Dominique Jeulin,et al.  Towards gigantic RVE sizes for 3D stochastic fibrous networks , 2014 .

[55]  V. G. Kouznetsova,et al.  Multi-scale computational homogenization: Trends and challenges , 2010, J. Comput. Appl. Math..

[56]  J. Larena,et al.  Power-law cosmic expansion in f(R) gravity models , 2009, 0906.3860.

[57]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[58]  J. Srinivas,et al.  Numerical evaluation of effective elastic properties of CNT-reinforced polymers for interphase effects , 2014 .

[59]  S. Torquato,et al.  Lineal-path function for random heterogeneous materials. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[60]  C. Miehe,et al.  Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy , 2003 .

[61]  J. Michel,et al.  Effective properties of composite materials with periodic microstructure : a computational approach , 1999 .

[62]  T. Zohdi.  Statistical ensemble error bounds for homogenized microheterogeneous solids , 2005 .

[63]  Daniel Balzani,et al.  Design of 3D statistically similar Representative Volume Elements based on Minkowski functionals , 2015 .

[64]  A. A. Gusev.  Representative volume element size for elastic composites: A numerical study , 1997 .

[65]  Dani Lischinski,et al.  Recursive Wang tiles for real-time blue noise , 2006, ACM Trans. Graph..

[66]  Adrian R. Russell,et al.  Microstructural pore changes and energy dissipation in Gosford sandstone during pre-failure loading using X-ray CT , 2013 .

[67]  Bernard Sonon,et al.  An advanced approach for the generation of complex cellular material representative volume elements using distance fields and level sets , 2015 .

[68]  Michal Šejnoha,et al.  From random microstructures to representative volume elements , 2007 .

[69]  D. Jeulin,et al.  Determination of the size of the representative volume element for random composites: statistical and numerical approach , 2003 .

[70]  M. Ostoja-Starzewski.  Material spatial randomness: From statistical to representative volume element☆ , 2006 .

[71]  S. Torquato,et al.  Random Heterogeneous Materials: Microstructure and Macroscopic Properties , 2005 .

[72]  D. Therriault,et al.  Assessment of existing and introduction of a new and robust efficient definition of the representative volume element , 2013 .

[73]  F. Stillinger,et al.  A superior descriptor of random textures and its predictive capacity , 2009, Proceedings of the National Academy of Sciences.

[74]  F. Stillinger,et al.  Modeling heterogeneous materials via two-point correlation functions: basic principles. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[75]  S. Torquato,et al.  Reconstructing random media , 1998 .

[76]  K. Sab.  On the homogenization and the simulation of random materials , 1992 .

[77]  Michal Šejnoha,et al.  Numerical evaluation of effective elastic properties of graphite fiber tow impregnated by polymer matrix , 2001 .

[78]  Hao Wang.  Proving theorems by pattern recognition — II , 1961 .

[79]  Michel Fogli,et al.  Various estimates of Representative Volume Element sizes based on a statistical analysis of the apparent behavior of random linear composites , 2012 .

[80]  Peter Wriggers,et al.  An Introduction to Computational Micromechanics , 2004 .

[81]  W. Drugan,et al.  A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites , 1996 .