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Euler characteristic
Introduction to the Euler characteristic is often made with the so-called Polyhedra Formula. Among its countless contributions, Euler found out that alternative the sum of the number of vertices (V ), edges (E) and faces (F) of a convex polyhedra is constant, no matter how it is constructed: V − E + F = 2. Now, P polyhedra at least glued together by one common face are considered. It occurs that, for a wild bunch of configurations, V − E + F − P = 1. This rather more general formula holds until the union of polyhedra form a hole in the resulting structure, leading to V − E + F − P = 0. Two holes inevitably lead to V −E+F −P = −1 no matter how many polyhedra are used and how they are constructed. Each hole reduces the alternative sum by 1. It is from this consideration that the topology field is born, creating a new descriptor for a solid subdivided in polyhedra, invariant under its geometrical properties. Named after its instigator, the Euler characteristic χ is defined for a solid subdivided into polyhedra as follows: χ = V − E + F − P. (1.27).
The contribution of Carl Friedrich Gauss to (classical) differential geometry has led to several useful concepts (Disquisitiones generales circa superficies curva, 1827), and among them, the Gaussian curvature. At a point on a surface, it can be defined as the product of the minimum and the maximum curvatures K = kminkmax. Things become interesting when focus is made on global values (integrated over the surface) of this curvatures. Known as one of the most elegant theorems of differential geometry, the Gauss- Bonnet theorem links this global geometrical value to the genus, a topological invariant (remains unchanged under homeomorphisms of the surface). If a compact surface (that closes on itself) in R3 is considered, its genus g is the number of its holes. It turns out that the integrated value of the Gaussian curvature follows the simple relationship Z ∂M KdS = 2πχ(∂M).
Gaussian Minkowski functionals
Minowski functionals are closely linked to LKCs and can be defined as follows for a subset A ⊂ RN: MN−j(A) = (N − j)! ωN−j Lj(A). (1.33) More popular than LKCs,Minkowski functionals are used in many fields in order to characterize morphologies. Especially in the astrophysics community (see the pioneer work of [Mecke et al., 1994] in which Steiner’s formula is presented with these functionals) leading to huge amount of literature on the topic [Mecke and Wagner, 1991,Winitzki and Kosowsky, 1997,Kerscher et al., 2001].
In contrast to LKCs, the Minkowski functionals are not intrinsic. Therefore, they depend on the used measure. This can be seen as a foretaste of how calculating geometrical properties of excursion sets is linked to the probability of a Gaussian RV to be in the socalled hitting set. Hence, the measure of a Gaussian distribution is of concern here. Let γk be such a measure in the Euclidean space Rk. If X = {Xi} is a standard Gaussian vector of size k in which Xi ∼ N (0, σ2), i = [1..k] are independent and A ⊂ Rk: γk(A) = P {X ∈ A} = 1 σk(2π)k/2 A e−kxk2/2σ2 dx.
Application to the Gaussian related χ-square distribution
It has previously been stated (EQ. (1.5)) that Gaussian related RFs can be defined by a transformation (noted S) of a Gaussian RF. It occurs that because these fields are a product of an underlying Gaussian distribution, the principle of Gaussian measurement stated just above can be applied. In order to retrieve the previous formulation, the transformation S can be taken into account for the hitting. Hence GMFs can still be used even if they are defined for a Gaussian measure.
This principle can be explained by the following statements. First, let gr be a Gaussian related RF defined by g, a vector valued Gaussian RF of size k and S, a transformation from Rk to R. The Gaussian related RF is then given by gr = S(g). Now, if Hs is the hitting set defined for gr, then attention has to be focused on the probability measure of gr to be in this hitting set. Starting from that point and using the decomposition of Gaussian related RF, the following equations can easily be yielded [Adler, 2008].
Validation of the numerical implementation
Excursion sets are represented by a discretized binary field defined by 1 if in the excursion and 0 if elsewhere. In order to validate the numerical implementation, morphological characteristics of actual excursion sets are compared to theoretical results.
Gaussian correlated RFs of zero means, standard deviation σ = 5 and Gaussian co- Meso-scale FE and morphological modeling of heterogeneous media variance of correlation length Lc = 10 are yielded in a three-dimensional cube of size a = 100. Attention is focused on the volume fraction Vf = L3/a3 and on the Euler characteristic χ = L0. It is recalled that the latter is calculated with the polyhedra formula EQ. (1.27). Results are plotted in FIG. (1.11) where mean values of both volume fraction and Euler characteristic are represented in terms of level set values κ from −25 to 25 (corresponding to the whole range of RF values). In order to collate relevant statistical information, calculations are made over 100 realizations. Analysis is made using the mean square error for Vf and χ over all level sets for each realization (results are given in TAB. (1.3)).
Table of contents :
Introduction
1 Morphological modeling: a generalized method based on excursion sets
1 Introduction
2 Review on correlated Random Fields
2.1 Basic definitions
2.2 Gaussian and Gaussian related distribution
2.3 Covariance functions
2.4 Numerical implementation
3 Excursion set theory
3.1 General principle
3.2 Measures of excursion set
3.3 Expectation formula
4 Application to cementitious materials at different scales
4.1 Meso-scale modeling of concrete
4.2 Micro-scale modeling of cement paste
5 Analytical model for size effect of brittle material
5.1 Correlation lengths as scale parameters
5.2 One-dimensional case
5.3 Validation, results and comments
6 Continuous percolation on finite size domains
6.1 Accounting for side effects
6.2 Representative Volume Element for percolation
7 Concluding remarks
2 Unified numerical implementation of quasi-brittle behavior for heterogeneous material through the Embedded Finite Element Method
1 Introduction
2 Kinematics of strain and displacement discontinuity
2.1 Jump in the strain field
2.2 Jump in the displacement field
2.3 Strong discontinuity analysis
3 Variational formulation
3.1 Three-field variational formulation
3.2 Assumed strain and double enhancement
3.3 Finite Element interpolation
4 Discrete model at the discontinuity
4.1 Localization
4.2 Traction-separation law
5 Resolution methodology
5.1 Integration and linearization with constant strain elements .
5.2 Solving the system
5.3 Application to spatial truss and spatial frame
5.4 Application to volume Finite Elements
6 Concluding remarks
3 Applications to cementitious materials modeling
1 Introduction
2 One-dimensional macroscopic loading paths
2.1 Analysis of the asymmetricmacroscopic response for traction and compression loading paths
2.2 Transversal strain
2.3 Dissipated energy
2.4 Induced anisotropy
2.5 Uniaxial cyclic compression loading
3 Representative Volume Element for elastic and failure properties
3.1 Experimental protocol
3.2 Young moduli analysis
3.3 Tensile and compressive strength
3.4 Comments
4 Application to the Delayed Ettringite Formation
4.1 Numerical simulation
4.2 Homogeneous mortar expansion
4.3 Residual Young modulus
5 Concluding remarks
Conclusions and perspectives
A Gaussian Minkowski Functionals
1 Volume of the unit ball
2 Probabilist Hermite polynomials
3 Gaussian volumes of spherical set in Rk
B Correlated Random Fields
1 Orthogonal decomposition of correlated random fields
2 Finite Element discretization of the Fredholm problem
3 The turning band method
Bibliography
