Microstructural enrichment functions based on stochastic Wang tilings

This paper presents an approach to constructing microstructural enrichment functions to local fields in non-periodic heterogeneous materials with applications in the partition of unity and hybrid finite element schemes. It is based on a concept of aperiodic tilings by the Wang tiles, designed to produce microstructures morphologically similar to original media and enrichment functions that satisfy the underlying governing equations. An appealing feature of this approach is that the enrichment functions are defined only on a small set of square tiles and extended to larger domains by an inexpensive stochastic tiling algorithm in a non-periodic manner. The feasibility of the proposed methodology is demonstrated on constructions of stress enrichment functions for two-dimensional mono-disperse particulate media.

[1]  A. Simone,et al.  A partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres , 2010 .

[2]  P. P. Castañeda,et al.  Effective-medium theory for infinite-contrast two-dimensionally periodic linear composites with strongly anisotropic matrix behavior : Dilute limit and crossover behavior , 2008, 0804.2817.

[3]  Erik D. Demaine,et al.  Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity , 2007, Graphs Comb..

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

[5]  Michal Šejnoha,et al.  Qualitative analysis of fiber composite microstructure: Influence of boundary conditions , 2006 .

[6]  L. J. Sluys,et al.  A partition of unity finite element method for simulating non‐linear debonding and matrix failure in thin fibre composites , 2011 .

[7]  S. Torquato.  Random Heterogeneous Materials , 2002 .

[8]  Ivo Babuška,et al.  An approach for constructing families of homogenized equations for periodic media. I: an integral representation and its consequences , 1991 .

[9]  Stefan Scheiner,et al.  From micron-sized needle-shaped hydrates to meter-sized shotcrete tunnel shells: micromechanical upscaling of stiffness and strength of hydrating shotcrete , 2008 .

[10]  Daniel Balzani,et al.  Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions , 2011 .

[11]  J. D. Eshelby,et al.  The elastic field outside an ellipsoidal inclusion , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

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

[13]  小山 毅,et al.  拡張有限要素法(XFEM)・一般化有限要素法(GFEM)を用いた材料モデリングのレビュー Ted Belytschko,Robert Gracie and Giulio Ventura:A Review of Extended/Generalized Finite Element Methods for Material Modeling [Modeling and Simulations in Materials Science and Engineering, Vol.17, 043001, June 2009](構造,文献抄録) , 2010 .

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

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

[16]  H. Moulinec,et al.  A fast numerical method for computing the linear and nonlinear mechanical properties of composites , 1994 .

[17]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

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

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

[20]  T. Belytschko,et al.  A review of extended/generalized finite element methods for material modeling , 2009 .

[21]  Jacob Fish,et al.  Multiscale enrichment based on partition of unity for nonperiodic fields and nonlinear problems , 2007 .

[22]  Ismael Herrera,et al.  Trefftz Method: A General Theory , 2000 .

[23]  Gennadi Vainikko,et al.  Periodic Integral and Pseudodifferential Equations with Numerical Approximation , 2001 .

[24]  Ares Lagae,et al.  A Comparison of Methods for Generating Poisson Disk Distributions , 2008, Comput. Graph. Forum.

[25]  David S. Ebert,et al.  Volume illustration using wang cubes , 2007, TOGS.

[26]  T. Belytschko,et al.  The extended/generalized finite element method: An overview of the method and its applications , 2010 .

[27]  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.

[28]  Kenji Yamamoto,et al.  Hierarchical adaptive nanostructured PVD coatings for extreme tribological applications: the quest for nonequilibrium states and emergent behavior , 2012, Science and technology of advanced materials.

[29]  Andrew S. Glassner.  Andrew Glassner's notebook , 2004, IEEE Computer Graphics and Applications.

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

[31]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[32]  Ivo Babuska,et al.  Generalized p-FEM in homogenization , 2000, Numerische Mathematik.

[33]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[34]  Jan Novák,et al.  Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients , 2010, J. Comput. Phys..

[35]  S. Dreyfus,et al.  Thermodynamical Approach to the Traveling Salesman Problem : An Efficient Simulation Algorithm , 2004 .

[36]  Robert Lipton,et al.  Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems , 2010, Multiscale Model. Simul..

[37]  Karel Culík,et al.  An aperiodic set of 13 Wang tiles , 1996, Discret. Math..

[38]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[39]  G. Milton.  The Theory of Composites , 2002 .

[40]  Chris J. Pearce,et al.  A micromechanics-enhanced finite element formulation for modelling heterogeneous materials , 2011, ArXiv.

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

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

[43]  Jacob Fish,et al.  Multiscale enrichment based on partition of unity , 2005 .

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

[45]  The jigsaw puzzles. , 2015, Work.

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

[47]  J. A. Freitas,et al.  Formulation of elastostatic hybrid-Trefftz stress elements , 1998 .

[48]  Chris J. Pearce,et al.  A corotational hybrid-Trefftz stress formulation for modelling cohesive cracks , 2009 .