Sample-based synthesis of two-scale structures with anisotropy

Abstract A vast majority of natural or synthetic materials are characterized by their anisotropic properties, such as stiffness. Such anisotropy is effected by the spatial distribution of the fine-scale structure and/or anisotropy of the constituent phases at a finer scale. In design, proper control of the anisotropy may greatly enhance the efficiency and performance of synthesized structures. We propose a sample-based two-scale structure synthesis approach that explicitly controls anisotropic effective material properties of the structure on the coarse scale by orienting sampled material neighborhoods at the fine scale. We first characterize the non-uniform orientations distribution of the sample structure by showing that the principal axes of an orthotropic material may be determined by the eigenvalue decomposition of its effective stiffness tensor. Such effective stiffness tensors can be efficiently estimated based on the two-point correlation functions of the fine-scale structures. Then we synthesize the two-scale structure by rotating fine-scale structures from the sample to follow a given target orientation field. The effectiveness of the proposed approach is demonstrated through examples in both 2D and 3D.

[1]  W Chen,et al.  Characterization and reconstruction of 3D stochastic microstructures via supervised learning , 2016, Journal of microscopy.

[2]  W. J. Whitehouse.  The quantitative morphology of anisotropic trabecular bone , 1974, Journal of microscopy.

[3]  H. Wagner,et al.  Extended morphometric analysis of neuronal cells with Minkowski valuations , 2005, cond-mat/0507648.

[4]  Jerome J. Morgan.  New Method for the Determination of Potassium in Silicates. , 1921 .

[5]  Seeram Ramakrishna,et al.  Design strategies of tissue engineering scaffolds with controlled fiber orientation. , 2007, Tissue engineering.

[6]  Stephen C. Cowin,et al.  EIGENTENSORS OF LINEAR ANISOTROPIC ELASTIC MATERIALS , 1990 .

[7]  Pauli Pedersen,et al.  Simple transformations by proper contracted forms: Can we change the usual practice? , 1995 .

[8]  Rachid Harba,et al.  A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform , 2005, BMC Medical Imaging.

[9]  K. Mecke,et al.  Tensorial Minkowski functionals and anisotropy measures for planar patterns , 2010, Journal of microscopy.

[10]  T. Ryan,et al.  Quantification and visualization of anisotropy in trabecular bone , 2004, Journal of microscopy.

[11]  L. Gibson,et al.  Anisotropy of foams , 1988 .

[12]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[13]  Daniel Hug,et al.  Minkowski tensors of anisotropic spatial structure , 2010, 1009.2340.

[14]  S. Pietruszczak,et al.  Characterization of anisotropy in porous media by means of linear intercept measurements , 2003 .

[15]  Sylvain Lefebvre,et al.  Parallel controllable texture synthesis , 2005, ACM Trans. Graph..

[16]  Pauli Pedersen,et al.  On sufficiency conditions for optimal design based on extremum principles of mechanics , 1997 .

[17]  Michael Ashikhmin,et al.  Synthesizing natural textures , 2001, I3D '01.

[18]  Max Yi Ren,et al.  Deep Network-Based Feature Extraction and Reconstruction of Complex Material Microstructures , 2016, DAC 2016.

[19]  N. Kikuchi,et al.  A comparison of homogenization and standard mechanics analyses for periodic porous composites , 1992 .

[20]  N. Kikuchi,et al.  A homogenization method for shape and topology optimization , 1991 .

[21]  S. Kalidindi,et al.  Statistical construction of 3-D microstructures from 2-D exemplars collected on oblique sections , 2016 .

[22]  M. Wolcott.  Cellular solids: Structure and properties , 1990 .

[23]  P. Bickel,et al.  Texture synthesis and nonparametric resampling of random fields , 2006, math/0611258.

[24]  Ramin Farnood,et al.  Characterizing Anisotropy of the Deterministic Features in Paper Structure with Wavelet Transforms , 2007 .

[25]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[26]  Vadim Shapiro,et al.  Sample-Based Design of Functionally Graded Material Structures , 2016, DAC 2016.

[27]  Maria Cristina Pernice,et al.  Fabric Tensor-based Poroelastic Materials: a "Direct" Approach , 2008 .

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

[29]  Kun Zhou,et al.  Synthesis of progressively-variant textures on arbitrary surfaces , 2003, ACM Trans. Graph..

[30]  M. Ashby,et al.  Cellular solids: Structure & properties , 1988 .

[31]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[32]  H J Gundersen,et al.  Estimation of structural anisotropy based on volume orientation. A new concept , 1990, Journal of microscopy.

[33]  M. Ashby,et al.  The mechanics of three-dimensional cellular materials , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[34]  Aleš Linka,et al.  Analysis of Planar Anisotropy of Fibre Systems by Using 2D Fourier Transform , 2007 .

[35]  Masatoshi Saitou,et al.  Stripe pattern formation in ag–sb co-electrodeposition , 2005 .

[36]  Klaus Mecke,et al.  Additivity, Convexity, and Beyond: Applications of Minkowski Functionals in Statistical Physics , 2000 .

[37]  V. Sundararaghavan,et al.  A Markov random field approach for modeling spatio-temporal evolution of microstructures , 2016 .

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

[39]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[40]  Andres Tovar,et al.  An efficient 3D topology optimization code written in Matlab , 2014 .

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

[42]  William Thomson,et al.  XXI. Elements of a mathematical theory of elasticity , 1856, Philosophical Transactions of the Royal Society of London.

[43]  R. Hill.  A theory of the yielding and plastic flow of anisotropic metals , 1948, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[44]  R. Mann,et al.  Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor , 1984 .

[45]  V. Sundararaghavan.  Reconstruction of three-dimensional anisotropic microstructures from two-dimensional micrographs imaged on orthogonal planes , 2014, Integrating Materials and Manufacturing Innovation.

[46]  Claus Beisbart,et al.  Vector- und Tensor-Valued Descriptors for Spatial Patterns , 2002 .

[47]  William Thomson,et al.  I. Elements of a mathematical theory of elasticity , 1857, Proceedings of the Royal Society of London.

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

[49]  Hervé Moulinec,et al.  A numerical method for computing the overall response of nonlinear composites with complex microstructure , 1998, ArXiv.

[50]  Alexei A. Efros,et al.  Image quilting for texture synthesis and transfer , 2001, SIGGRAPH.

[51]  Anath Fischer,et al.  Volumetric texture synthesis of bone micro-structure as a base for scaffold design , 2009, 2009 IEEE International Conference on Shape Modeling and Applications.

[52]  Vadim Shapiro,et al.  Homogenization of material properties in additively manufactured structures , 2016, Comput. Aided Des..