3D multiscale vectorial simulations of random models - Centre de ...

Introduction Implicit functions Random sets models Advantages of the method Conclusions

3D multiscale vectorial simulations of random models Matthieu Faessel, Dominique Jeulin Centre de Morphologie Mathématique, Mathématiques et Systèmes, MINES ParisTech

ICS13, October 19-23, 2011

Matthieu Faessel, Dominique Jeulin

3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

Table of Contents 1 2

3

Introduction Implicit functions Denition 2D Example Boolean operations Random sets models Point processes Boolean model Multiscale Boolean model

4

Advantages of the method

5

Conclusions Matthieu Faessel, Dominique Jeulin

3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

1

Introduction

2

Implicit functions

3

Random sets models

4

Advantages of the method

5

Conclusions

Matthieu Faessel, Dominique Jeulin

3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

Introduction Simulations of random models : Probabilistic approach to generate models and simulations of material microstructures Based on the theory of random sets Generally performed in the discrete space on a grid of points (2D or 3D images) New approach : 3D random models generated in the continuous space using implicit functions and level-sets. Matthieu Faessel, Dominique Jeulin

3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

1

Introduction

2

Implicit functions Denition 2D Example Boolean operations

3

Random sets models

4

Advantages of the method

5

Conclusions

Matthieu Faessel, Dominique Jeulin

Denition 2D Example Boolean operations

3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

Denition 2D Example Boolean operations

Denition Implicit functions are real valued functions dened in 3D space by ϕ(x , y , z ) = c

An implicit function representation denes a surface as a level set of a function ϕ, most commonly the set of points for which ϕ(~x ) = 0 The implicit function divides space into three regions : on the surface : ϕ(x , y , z ) = c outside of the surface : ϕ(x , y , z ) > c and inside the surface : ϕ(x , y , z ) < c Matthieu Faessel, Dominique Jeulin

3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

Denition 2D Example Boolean operations

2D Example Implicit function : ϕ(x , y ) = x 2 + y 2 − 1 Level set representations : ϕ = −0.75, ϕ = 0 and ϕ = 1.25 y φ����� φ�� φ������ O

Matthieu Faessel, Dominique Jeulin

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3D multiscale vectorial simulations of random models

Introduction Implicit functions Random sets models Advantages of the method Conclusions

Denition 2D Example Boolean operations

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φB��

φA��

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φA⋃B