Morphological Models of Random Structures

Description

This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories. Preface Acknowledgements Symbols and Notations Contents 1 Introduction 1.1 Use of a probabilistic approach 1.2 Aims of probabilistic models 1.2.1 Descriptive aspect 1.2.2 Predictive aspect 1.3 Types of models 1.4 Model construction 1.4.1 Characterization of a random model 1.4.2 Choice of basic assumptions 1.4.3 Computation of the functional T (K) 1.5 Some general properties of the models 1.6 Organization of the book Part I Tools for Random Structures 2 Introduction to Random Closed Sets and to Semi-Continuous Random Functions 2.1 Introduction 2.2 Introduction to random closed sets (RACS) 2.2.1 Recall of basic definitions in probability theory 2.2.2 Definition of the random closed sets 2.2.3 The Choquet capacity 2.2.4 Measurability of a random closed set 2.2.5 Operations on random closed sets 2.2.6 Independent random sets 2.2.7 Spatial law 2.2.8 Geometrical interpretation of the Choquet capacity in the Euclidean case 2.3 Introduction to Random Functions 2.3.1 Semi continuous functions 2.3.2 Subgraph of a function 2.3.3 Choquet topology on Φ_f 2.3.4 The Borel σ algebra on Φ_f and Φ_g , and the Choquet capacity for RF 2.3.5 Multivariate Random Functions 2.3.6 Erosion and Dilation of functions 3 Quantitative Analysis of Random Structures 3.1 Introduction 3.2 Basic morphological transformations 3.2.1 Types of structures 3.2.2 Notion of structuring element 3.2.3 Dilations and Erosions 3.3 Basic morphological measurements 3.3.1 Definition of Minkowski functionals 3.3.2 Stereological aspects 3.3.3 Steiner formula 3.3.4 Specific measurements 3.3.5 Counting procedure and the Euler relations 3.3.6 Minkowski tensors 3.4 Size distributions 3.4.1 Axioms of size distributions; openings and closings 3.4.2 Linear size distributions 3.4.3 Two and three dimensional size distributions 3.4.4 Size distributions for functions 3.4.5 Stereological reconstruction of the size distribution of spheres 3.5 Morphological analysis of the spatial distribution 3.5.1 Random sets 3.5.2 Multi component random sets 3.5.3 Random functions 3.5.4 Covariance of orientations of vector fields 3.5.5 Definition of a Statistical Representative Volume Element (RVE) and problems of estimation 3.5.6 Distance functions 3.5.7 Random graphs 3.6 Geodesic criteria and connectivity 3.7 Robustness of morphological measurements to noise 3.8 Shape and texture classification and recognition 3.8.1 Automatic recognition of non metallic inclusions 3.8.2 3D Shape classification of particles with complex shapes 3.8.3 Texture analysis and classification 3.9 Conclusion 3.10 Exercises 3.10.1 Minkowski tensor of a cylinder 3.10.2 Variogram γ_1(h) 3.10.3 Correlation functions of two phase random composites 3.10.4 Specific measurements and derivatives of morphological moments 3.10.5 Covariance of a noisy binary image 3.10.6 Moments of inertia in R^3 Part II Models of Random Structures 4 Excursion Sets of Gaussian RF 4.1 Introduction 4.2 Random sets and truncated RF: the Gaussian case. 4.2.1 Construction of truncated Gaussian random sets 4.2.2 Second order central correlation function 4.2.3 Third order central correlation function 4.2.4 Order m central correlation function 4.3 Application to a food microstructure 4.4 Three Phase random model 4.4.1 Introduction 4.4.2 Three components models based on independent random sets 4.4.3 Three component model based on the Boolean model 4.4.4 Three component model based on truncated Gaussian random functions 4.5 Exercise 4.5.1 Combination of independent random sets 5 Stochastic Point Processes and Random Trees 5.1 Introduction 5.2 Tools to characterize point processes 5.3 The Poisson point process 5.3.1 Definition and properties of the Poisson point process 5.3.2 Distribution of points of a homogeneous Poisson point process 5.3.3 Simulation of a homogeneous Poisson point process 5.3.4 Distribution of points of a non stationary Poisson point process 5.4 Hard-core point processes 5.5 The Cox point process 5.5.1 Definition and main properties 5.5.2 Cox process with a random variable θ 5.5.3 Large scale behavior of the Cox process 5.6 Point processes on iteration of Boolean varieties 5.6.1 Two steps Boolean varieties 5.6.2 Three steps Boolean varieties in R^3 5.7 Gibbs point processes 5.8 Determinantal point processes 5.9 Introduction to models of random trees by branching processes 5.9.1 Discrete branching process 5.9.2 Continuous branching process 5.9.3 Discrete random trees generated by random functions 5.9.4 Biological applications 5.10 Exercises 5.10.1 Laplace transforms for sections of discs and of spheres 5.10.2 A model of random tree embedded in the Euclidean space 6 Boolean Random Sets 6.1 Introduction 6.2 Construction of the Boolean model 6.3 Choquet capacity 6.4 Some stereological properties 6.5 Connectivity numbers and counting 6.6 Connectivity and percolation of the Boolean model 6.7 IDRACS and semi Markovian RACS 6.8 Testing the Boolean model 6.8.1 Convexity of the primary grain A0 6.8.2 Dilation by convex sets 6.8.3 Infinite divisibility of the Boolean model 6.9 Identification of the Boolean model 6.10 Some examples of application 6.10.1 Boolean model of spheres 6.10.2 The Poisson Boolean model 6.10.3 Further applications of the Boolean model to materials 6.10.4 Multiphase textures 6.11 The Boolean model and the Poisson process 6.12 Cox Boolean model and multiscale models of RACS 6.12.1 Intersection of independent random sets 6.12.2 Introduction to the Cox Boolean model 6.12.3 Percolation of the Cox Boolean model 6.13 Boolean model and the Poisson varieties 6.13.1 Construction and properties of the linear Poisson varieties 6.13.2 Stable RACS 6.13.3 Boolean model on the linear Poisson varieties 6.13.4 Power laws variance scaling of the Boolean random varieties 6.13.5 Two steps iterated Poisson varieties 6.14 Multi component Boolean models 6.14.1 Construction of Boolean models with m components 6.14.2 Choquet capacity 6.14.3 Spatial law and covariances 6.14.4 Stability for the reunion 6.14.5 Multi component Boolean models on Poisson varieties 6.15 Exercises 6.15.1 Counting with the Euler relation 6.15.2 Hierarchical Boolean model on a mosaic 6.15.3 Fractal Boolean random set 6.15.4 A noise random set model 6.15.5 Simulation of percolating aggregates 6.15.6 Transverse and longitudinal covariances of a Boolean model of cylinders 6.15.7 Boolean varieties in R^2 and R^3 as limit case of Boolean models 7 Random Tessellations 7.1 Introduction 7.2 Reminder on random tessellations 7.3 Probability distributions of the classes 7.4 The Voronoi, Johnson-Mehl, and Laguerre tessellation 7.4.1 Voronoi tessellation 7.4.2 Johnson-Mehl and Laguerre random tessellations 7.5 Random tessellation generated by a geodesic distance 7.6 The Poisson tessellation 7.6.1 Definition and Choquet capacity of the Poisson hyperplanes 7.6.2 Stereological aspects of the Poisson hyperplanes 7.6.3 Characterization of the Poisson polyhedron containing the origin 7.6.4 Conditional invariance by erosion 7.6.5 Opening size distribution 7.7 Example of application to model concrete by multiscale Poisson polyhedra 7.8 The Cauwe tessellation 7.8.1 Construction of the Cauwe tessellation 7.8.2 Stereological properties 7.8.3 Choquet capacity on convex sets 7.8.4 Mixture of Poisson and Cauwe tessellations 7.9 Iteration of tessellations and the STIT model 7.10 Exercise 7.10.1 Superposition of random tessellations 8 The Mosaic Model 8.1 Introduction 8.2 Construction 8.3 Choquet capacity 8.4 Mosaic model in R^n 8.4.1 Calculation of P {N(K) = 1} 8.4.2 First order statistics 8.4.3 Second order statistics 8.4.4 Third order statistics 8.4.5 Higher order statistics 8.5 The Poisson mosaic 8.6 The STIT mosaic 8.7 The Mosaic random set 8.8 The multivariate mosaic model 8.9 Exercises 8.9.1 Multi component mosaic 8.9.2 Cross covariances of the multicomponent mosaic 8.9.3 Hierarchical mosaic model 9 Boolean Random Functions 9.1 Introduction 9.2 Construction of the Boolean random functions BRF 9.3 Choquet capacity of the BRF 9.4 Supremum stability and infinite divisibility 9.5 Characteristics of the primary functions 9.5.1 Transformation by anamorphosis 9.5.2 Moments of Z0∨(K) and mathematical expectation of theanamorphosed of Z0∨(K) 9.5.3 Geometrical covariogram of the primary function 9.6 Some stereological aspects of the BRF 9.7 BRF and counting 9.8 Identification of a BRF model 9.9 Test of the BRF 9.9.1 Convexity of AZ0t(z) 9.9.2 Change of support on convex sets 9.9.3 Supremum infinite divisibility 9.10 Examples of application to rough surfaces 9.10.1 Simulation of the evolution of surfaces and of stresses during shot peening 9.10.2 Simulation of the roughness transfer on steel sheets 9.11 Modeling three phase random media with some contact constraints 9.11.1 Microstructure of a current collector 9.11.2 Use of a Boolean random function 9.11.3 Use of truncated Gaussian random functions 9.12 Multiscale Boolean random functions 9.13 The Boolean varieties RF 9.14 The multivariate BRF and Varieties 9.14.1 Construction 9.14.2 Choquet capacity 9.14.3 Spatial law 9.14.4 Supremum stability 9.14.5 Distribution of maxima 9.14.6 Multivariate Boolean varieties RF 9.15 Exercises 9.15.1 BRF with cylinder primary random functions 9.15.2 A hierarchical BRF model 9.15.3 BRF with cones anamorphosis 10 Random Tessellations and Boolean Random Functions 10.1 Introduction 10.2 Extensions of the Voronoi tessellation 10.2.1 Random tessellations defined from local metrics 10.2.2 Calculation of the probability P (K) 10.3 Extension to Johnson-Mehl and to Laguerre random tessellations 10.4 Random tessellations and Boolean random functions 10.4.1 Connection between metric based random tessellations and some BRF 10.4.2 General random tessellations built from BRF 10.4.3 Random tessellations with thick boundaries 10.5 Some indications on model identification 10.6 Conclusion 11 Dead Leaves Models: from Space Tessellations to Random Functions 11.1 Introduction 11.2 Sequential random tessellations 11.2.1 Infinitesimal Boolean random set 11.2.2 Construction of the sequential random tessellations (Dead Leaves tessellation) 11.2.3 Probabilistic properties 11.2.4 Linear size distributions 11.2.5 Specific parameters 11.2.6 Higher order probabilistic properties 11.2.7 Statistics on intact grains and bias correction 11.2.8 Example of application to polycristalline salt 11.2.9 Iteration of Dead Leaves tessellations 11.2.10 Multi scale Cox DLRT 11.3 Random packings generated by the Dead Leaves model 11.3.1 Intact grains of the Dead Leaves model for the time homogeneous case 11.3.2 Intact grains of the Dead Leaves model for the general case 11.3.3 Examples of models of dense packings 11.3.4 Size distributions for dense packings 11.3.5 Pair correlation function of intact grains centers 11.3.6 Conclusion 11.4 Color Dead Leaves 11.4.1 Construction of the Color Dead Leaves 11.4.2 First order statistics 11.4.3 Second order statistics 11.4.4 Third order statistics 11.4.5 Correlation functions 11.4.6 An example of two phase medium 11.4.7 Specific connectivity numbers of the Color Dead Leaves 11.4.8 Some potential applications of the Color Dead Leaves 11.5 The Dead Leaves Random Functions (DLRF) 11.5.1 The infinitesimal Boolean Random Functions (IBRF) 11.5.2 Construction of sequential RF from IBRF 11.5.3 Probabilistic properties of the DLRF 11.5.4 Moment P (g, t) 11.5.5 Summits of the DLRF 11.5.6 Intact primary functions 11.5.7 Estimation of the parameters of the DLRF and tests 11.6 Multivariate DLRF and varieties 11.6.1 Construction of the Multivariate DLRF 11.6.2 Multivariate distribution in point x 11.6.3 Bivariate distributions 11.6.4 Cross covariances 11.6.5 Multivariate DLRF built on Poisson varieties 11.7 Dead Leaves of rank m 11.8 Transparent Dead Leaves 11.9 Application to the morphology of powders 11.9.1 Mixtures of Boolean models 11.9.2 Use of The DLRT 11.9.3 Use of The DLRF for grey level images 11.10 Sequential RF with Markovian jumps (MJF) 11.10.1 Construction of MJF 11.10.2 Properties of the domains A0(t) = ∪u≤tA00(u) 11.10.3 Probabilistic properties of the MJF 11.11 Elements of simulation 11.12 Conclusions 11.13 Exercises 11.13.1 Exponential distribution of intercepts (DLRT) 11.13.2 Correlation between contiguous intercepts (DLRT) 11.13.3 Linear size distributions (DLRT) 11.13.4 Construction of a random set from a DLRT 11.13.5 Composite grains (Color Dead Leaves) 11.13.6 Exponential covariance of a non Markov random mosaic (DLRF) 11.13.7 Statistics of intact grains 11.13.8 A two scale model of platelets 12 Sequential Cox Boolean and Conditional Dead Leaves Models 12.1 Introduction 12.2 Sequential Cox Boolean RACS 12.2.1 Notations and definition of the two components sequential Cox Boolean model 12.2.2 Probabilistic properties 12.2.3 Simplified expressions for the Choquet capacity 12.2.4 First variant of the sequential Cox Boolean model 12.2.5 Second variant of the sequential Cox Boolean model 12.2.6 Multi components version of the Cox sequential Boolean model 12.3 Sequential Cox Boolean RF 12.3.1 Notations and definition of the two components Sequential Cox BRF 12.3.2 Probabilistic properties 12.3.3 Simplified expressions for the Choquet capacity 12.3.4 Multi components Sequential Cox mosaic BRF 12.3.5 Extension of the Sequential Cox mosaic BRF to the continuous case 12.4 Conditional Dead Leaves tessellation 12.4.1 Construction of the model 12.4.2 Probabilistic properties 12.4.3 Simplified expressions of probabilistic properties 12.5 Conditional color Dead Leaves 12.5.1 Notations and definition of the model 12.5.2 Probabilistic properties 12.5.3 Simplified expressions of probabilistic properties 12.5.4 Simplified expression for the Choquet capacity of the set B0(t) 12.5.5 Probabilities Pi(t) of the conditional color Dead Leaves 12.5.6 Covariances of the conditional color Dead Leaves 12.5.7 Expressions for Pi(t) and for the covariances in some specific cases 12.6 Conclusion 13 Sequential Alternate Random Functions 13.1 Introduction 13.2 Sequential alternate RF 13.2.1 Construction of SARF 13.2.2 Construction of coupled SARF 13.2.3 Univariate distribution 13.2.4 Bivariate distribution 13.2.5 Apparent maxima and minima of the primary RF 13.3 Multiscale SARF 13.4 Example of application to modeling of the Electro Discharge Textures (EDT) 13.5 Multivariate SARF 13.6 SARF varieties 13.7 Elements of simulation 13.8 Exercise 14 Primary Grains and Primary Functions 14.1 Introduction 14.2 Random primary grains 14.2.1 Population of spheres and of ellipsoids in R^3 14.2.2 Random parallelepipeds 14.2.3 The Poisson polyhedra 14.2.4 Grains generated by the DLRT 14.2.5 Random aggregates 14.2.6 Cylinder primary RF 14.2.7 Restriction of a stationary RF to a random compact set 14.2.8 Some RF with spherical thresholds 14.2.9 Boolean RF with a compact support 14.2.10 Dead Leaves RF with a compact support 14.3 Exercises 14.3.1 Boolean aggregates of rectangles 15 Dilution Random Functions 15.1 Introduction 15.2 Construction of the multivariate Dilution random functions DRF 15.3 Convolution of a DRF 15.4 Characteristic functional of a multivariate DRF 15.5 Moments of a multivariate DRF 15.6 Example of application to TEM micrographs 15.6.1 A model for thick slices images 15.6.2 Covariance of the TEM images 15.6.3 Numerical method to estimate the transitive covariogram 15.7 Random tokens 15.8 Positive DRF 15.9 DRF of the number of grains covering x 15.9.1 Univariate distribution 15.9.2 Bivariate distribution 15.9.3 Trivariate distribution 15.9.4 Random set A_m 15.9.5 Random tessellation generated by the A_m 15.9.6 Excursion random sets of N(x) 15.10 Convergence of the DRF towards a Gaussian RF 15.11 Multivariate Dilution varieties RF 15.12 Exercises 15.12.1 The Dilution RF and the Cox process 15.12.2 Cox Dilution RF 15.12.3 Hybrid model of spheres and cylinders 15.12.4 Model of indentation 16 Reaction-Diffusion and Lattice Gas Models 16.1 Introduction 16.2 Reaction-Diffusion models 16.2.1 Reaction-Diffusion equations 16.2.2 Discrete Reaction-Diffusion models 16.3 Random Functions and the linearReaction-Diffusion model 16.4 Examples of simulations of non- linear Reaction-Diffusion Random Functions 16.4.1 Schlögl model 16.4.2 Turing structures 16.4.3 The complex Ginzburg-Landau model 16.4.4 Modifications of the Ginzburg-Landau model 16.5 Lattice gas models 16.5.1 Basic rules 16.5.2 Some indications on the evolution equations 16.5.3 Boundary conditions 16.5.4 Application to flow in porous media 16.5.5 Application to simulations of random media 16.5.6 Multi species lattice gas models 16.6 Conclusion 17 Texture Segmentation by Morphological Probabilistic Hierarchies 17.1 Introduction 17.2 Morphological texture descriptors 17.3 Texture classification 17.4 Probabilistic texture segmentation 17.4.1 Watershed texture segmentation 17.4.2 Probabilistic hierarchical segmentation 17.4.3 Higher order probabilistic segmentation 17.4.4 Probabilistic distances between sets 17.4.5 Use of random markers 17.4.6 Random markers and higher order fusion of regions 17.5 Conclusion Part III Random Structures and Change of Scale 18 Change of Scale in Physics of Random Media 18.1 Introduction 18.2 From microscopic to macroscopic 18.3 Homogeneous medium and heterogeneous medium 18.4 Practical interest of change of scale methods 18.5 Principle of calculation of effective properties 18.6 Exact results 18.6.1 Geometrical average estimate 18.6.2 Self-consistent and effective medium estimates 18.6.3 Composite spheres assemblage 18.7 Perturbation expansion in electrostatics 18.8 Formal expansion of the effective dielectric permittivity of random media 18.9 Perturbation approach in elasticity and calculation of the effective elastic tensor 18.10 Bounds of effective properties 18.10.1 Variational principles 18.10.2 Bounds of order 2N + 1 derived from the classical variational principle 18.11 Third order bounds of the dielectric permittivity 18.11.1 Reminder on third order bounds for RF 18.11.2 Third order bounds for the mosaic model 18.11.3 Third order bounds of the Dilution model 18.11.4 Transformation of basic models 18.11.5 Combination of the basic random functions models 18.11.6 A hierarchical model 18.12 Third order bounds of the real dielectric permittivity of random sets 18.13 Third order bounds of the complex dielectric permittivity and spectral measure of random sets 18.13.1 Bounds in the complex plane 18.13.2 Spectral measure of a random set 18.14 Third order bounds for elastic moduli of random sets 18.14.1 Third order bounds for two-phase composites 18.14.2 Bounds of the bulk modulus K 18.14.3 Bounds of the shear modulus G 18.14.4 Bounds of the Young’s modulus E 18.14.5 Bounds of the Poisson coefficient ν 18.14.6 Functions ζ_1(p) and η_1(p) 18.15 Third order bounds of some models of random sets 18.15.1 Bounds for the Boolean model 18.15.2 Bounds for the hard spheres model 18.15.3 Bounds for the mosaic model 18.15.4 Bounds for the Dead Leaves model 18.15.5 Bounds for excursion sets of Gaussian RF 18.15.6 Combination of the basic random sets models 18.16 Case of porous media 18.17 Optimal conductivity of two-components porous media 18.17.1 Optimization of two conductivities 18.17.2 Upper bounds of the optimal properties 18.17.3 Conclusion 18.18 Fields fluctuations 18.18.1 Second order statistics of fields 18.18.2 Distribution function of fields 18.19 Conclusion 18.20 Exercises 18.20.1 Properties of the operator Γ 18.20.2 Calculation of the second order perturbation term in the scalar isotropic case 18.20.3 Calculation of a third order term 18.20.4 Field averages 18.20.5 Field averages in components 18.20.6 Hill-Mandel condition 18.20.7 Geometrical average effective property in R^2 19 Digital Materials 19.1 Introduction to Digital Materials and to numerical approaches 19.2 Homogenization of random media by numerical simulations 19.3 Fluctuations of apparent properties, and statistical RVE 19.3.1 The integral range 19.3.2 Practical determination of the size of the RVE 19.4 A case study: the Voronoi mosaic 19.5 FE Computation on 3D confocal microscope micrographs 19.5.1 Experimental effective physical properties 19.5.2 Numerical estimation of apparent physical properties by FE 19.5.3 Determination of the integral ranges 19.5.4 Size of the RVE 19.5.5 Phase connectivity and effective properties 19.6 Improving elastic properties from Boolean models 19.7 Gigantic RVE: 3D Poisson fibres 19.8 Stochastic Finite Elements 19.9 Numerical solutions of the Lippmann-Schwinger equation by iterations of FFT 19.10 Fields fluctuations in dielectric random media 19.10.1 Fields in dielectric autodual RS 19.10.2 Fields in a 3D Boolean model of spheres 19.10.3 Optical properties of paints 19.11 Elastic and thermal response of heterogeneous media from 3D microtomography 19.11.1 Hotspots in a granular material 19.11.2 Stress localization in a mortar microstructure 19.11.3 Elastic and thermal properties of lightweight concretes 19.12 Elastic and viscoelastic properties of multiscale random media 19.12.1 Bulk modulus of the Boolean model of spheres 19.12.2 Linear elastic properties and conductivity of multiscale Cox Boolean models 19.12.3 Nonlinear elastic and conductivity of multiscale Cox Boolean models 19.12.4 Elastic and viscoelastic properties of rubber with carbon black filler 19.13 Elastic properties of fibrous materials 19.14 Diffusion and fluid flows in porous media 19.14.1 Diffusion in random porous media 19.14.2 Estimation of the fluid permeability of porous media 19.15 RVE of acoustic properties of fibrous media 19.15.1 Thermoacoustic equations and homogenization of acoustic properties of porous media 19.15.2 Homogenization of acoustic properties of a periodic fibrous network 19.15.3 Acoustic fields simulated on random unit cells 19.15.4 Statistical RVE and integral ranges 19.16 Further examples of application 19.17 Conclusion 20 Probabilistic Models for Fracture Statistics 20.1 Introduction 20.2 Choice of a fracture criterion 20.2.1 Local fracture criteria 20.2.2 Global fracture criteria 20.3 Brittle fracture and weakest link 20.3.1 Introduction 20.3.2 Examples of stress fields used for the weakest link model 20.3.3 The Boolean random varieties and the weakest link model 20.3.4 Randomization of Boolean varieties 20.3.5 Weakest link model and iterated Boolean Varieties 20.3.6 The Dead Leaves varieties and the weakest link model 20.3.7 Competition between fracture mechanisms 20.3.8 Multicriteria and multiscale weakest link models 20.4 Fracture statistics models with a damage threshold 20.5 Fracture statistics model with a crack arrest criterion 20.5.1 Crack propagation and the Griffith’s criterion fort wo-dimensional random media 20.5.2 Types of probability distributions obtained from the models 20.5.3 Probability of fracture and scale effects for the Poisson Mosaic 20.5.4 Probability of fracture and scale effects for the Boolean Mosaic 20.5.5 Conclusions 20.6 Models of random damage 20.6.1 Introduction 20.6.2 Basic assumptions 20.6.3 Random damage under a homogeneous load 20.6.4 Random damage under a non homogeneous load 20.6.5 Conclusion 20.7 Elements of practical use of fracture statistics models in numerical simulations 20.8 Conclusion 20.9 Exercises 20.9.1 Crack propagation in a random polycrystal with grain boundary fracture energy 20.9.2 3D crack propagation in a random medium 21 Crack Paths in Random Media 21.1 Introduction 21.2 Probabilistic fracture of 3D random tessellations 21.2.1 Basic assumptions 21.2.2 Fracture of a random tessellation and percolation 21.3 Phase field model for crack initiation and propagation in random media with anisotropic fracture energy 21.3.1 Reminder on phase field models for fracture of homogeneous isotropic media 21.3.2 A phase field model for heterogeneous anisotropic fracture energy 21.3.3 Introduction to full fields estimation by FFT 21.3.4 Estimation of an effective toughness 21.4 Conclusion References Index