How can geometrical information and physical properties of a material be linked ?

How can geometrical information and physical properties of a material be linked ?

Research done in partnership with ICMCB laboratory (UPR 9048)

Specialised in pourous material study, and high resolution 3-dimensional pictures production.

Members participating to the project : Dominique Bernard

1. How calculations are usually done

Motivation : Fluid flow simulation

. An algorithm, based on the Navier-Stokes differential equations, is used to process 2d and 3d materials images.

. The algorithm stops when result stability is reached.

Main Problem :

Computation time can be very long (several days)

1. How calculations are usually done

. Calculations are not performed on the whole object, but on a sample of the object.

. The material can be seen as a tesselation of the sample, and the global flow can be seen as a tesselation of the local flow.

. This representation is equivalent to considering the opposite borders of the sample joined : the sample is embedded in a toric space.


2. Chosen method

Skeletonise the porous space of the sample in order to draw the main paths followed by the fluid flow through the material (curvilinear skeleton).

The different steps to complete :

.Threshold

.Remove the object’s grain (parts of the object which don’t touch the borders)

.Skeletonisation process [1].


in a toric space…

[1] G. Bertrand et M. Couprie, Transformations topologiques discrètes.

D. Coeurjolly, A. Montanvert et J.M. Chassery, Géométrie discrète et images numériques,

Hermès, 2007.

2. Chosen method

Main hypothesis:

The material (tesselation of the sample) has no grains

(connected component which does not touch « the borders » of the

image)

> We hardly imagine « floating pieces » in the full material.

> The grains are sources of 2d cycles in the skeleton of the porous space.

The skeleton of the porous space draws the main paths followed by the flow through the material.


3. Question

How can we identify, in the sample, the components of the object

which will create grains in the tesselation ?


The material, which is a tesselation of the sample,

still contains grains.

3. Question



It is not sufficient to filter the grains of the

sample in order to get rid of the grains in the

tesselation.


Contents

. Basic notions Toric spaces, neighbourhoods, loops, homotopy

. Characterizing a « grain » in a toric space

. Fundamental group of the torus Homotopy classes, wrapping vector

. Algorithm for detecting « toric loops » Example, elements of proof

Basic notions: Toric Space

Given d positive integer, we set Ζd = {0,…, d-1}.

The 1-dimensional toric space of size d is the group (Ζd , ) with

for all a,b Ζd , a b = (a + b) mod d

1. Toric space: definition

Let s = (s1, …, sn) Zn.

The n-dimensional toric space of size s is the group

(Tn, ) = (Ζs1, ) x … x (Ζsn , ) [2]

[2] J. Stillwell, Classical topology and combinatorial group theory, Springer.

Basic notions: Neighbourhood

In 2D, the 1- and 2-adjacency resp. correspond to the 4- and 8-neighbourhood.

In 3D, the 1- and 3-adjacency resp. correspond to the 6- and 26-neighbourhood.

Definition

Let (Tn, ) be an n-dimensional toric space.

. An m-step (0 < m ≤ n) is a vector v of {-1; 0; 1}n which has at most m non-null

coordinates.

. Two points a, b Tn are m-adjacent if there exists an m-step v such that

a v = b

Basic notions: Loops in toric space

Definition

Let (Tn, ) be an n-dimensional toric space, and p Tn.

We call m-loop of base point p a pair = ( p, V ), such that V = (v1, …, vk) is a

sequence of m-steps and (v1 … vk) = 0.

Example

We set p = (0,0)

v1 = (1,0)

v2 = (1,1)

v3 = (1,-1) B = ( p , (v2 , v1 , v3) ).

Let us consider the 2-loop B.

Basic notions: Loop homotopy

We started from the loop homotopy definition given in [3],

and small modifications were done in order to adapt it to toric spaces…


Two m-loops 1= ( p, V ) and 2= ( p, W ), with V = ( v1,…,vj ) and W = ( w1,…,wk ),

are directly homotopic if :

. V and W differ in one null vector, (Insertion / Deletion) or

. We have j=k, and there exists h [1 ; k-1], (Translation)

such that

. for all i [1; h-1] U [ h+2; k], vi = wi and

. vh + vh+1 = wh + wh+1 and

. (vh – wh) is an n-step.


Two loops β1 and β2 in Tn are homotopic if there exists a sequence (L1, … , Lk ) of

loops of Tn such that

. L1= β1, Lk= β2 and,

. for all i [1; k-1], Li and Li+1 are directly homotopic.

Characterizing grains in toric spaces


How can we identify, in a sample, the components of the object

which contains a loop non homotopic to a point ?

Thanks to the previous definitions, we can reformulate the problematic:

We call toric loop a loop non homotopic to a point.

How can we identify, in a sample, the components of the object



Wrapping vector of a loop

In (Tn, ), let β = (p, V) be a loop, with V = (v1, …, vk).

The wrapping vector of β is the vector

w = ∑ vii=1

k


Theorem

This theorem proves that there exists a value, easily computable for all loops, that

allows to know to which homotopy class a loop belongs to.

This value corresponds to the way the loop « wraps around » the toric space : it

completely describes the fundamental group of the torus (Zn) [4].

[4] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

Two loops are homotopic if and only if their wrapping vectors are equals.



which contain a loop with a non null wrapping vector ?


which contain a loop that is non homotopic to a point ?

Algorithm for detecting toric loops in an object

1. Main idea

. A connected component that contains a toric loop must be « broken », in

order to be embedded inside the « classical » space Zn.

. A connected component that does not contain any toric loop can be

embedded inside Zn without any distortion.

>> If an object of Tn can be embedded in Zn without distortion, then it does not

contain any toric loop.

How is it possible to know if an object of Tn

can be wrapped inside Zn without any distortion?

2. Algorithm for detecting toric loops

Data: image I (n-dimensional toric space), an m-connected object X of I, and a point p of X.

For all x X, do Coordinates [ x ] = false;

S = {p}; Coordinates [ p ] = 0n; (null n-dimensional vector)

While there exists x S do

S = S \ { x };

For all n-dimensional m-steps v, do

y = x v ; (y is a neighbour of x in the toric space)

If y X and Coordinates [ y ] == false, then

Coordinates [ y ] = Coordinates [ x ] + v ; S = S U { y };

Else if y X

If Coordinates [ y ] Coordinates [ x ] + v then return true;

return false; (no toric loop in X)

(there is a toric loop in X)


3. Example

. In (Z5 x Z5, ), we consider the 2-adjacency (8-neighbourhood) relation,

and a connected object.

The 2-dimensional 2-steps


3. Example

(1;-1)

(1;0)

(2;1) + (1;0) ≠ (-2;-1)

(1;0)

(0;0) + (1;0) = (1;0)

(-1;0)

(1;0) + (-1;0) = (0;0)S =

S = {p}; Coordinates [ p ] = (0;0);

While there exists x S do

S = S \ { x };

For all 2-steps v, do

y = x v ;

If y X and Coordinates [ y ] == false, then

Coordinates [ y ] = Coordinates [ x ] + v ;

S = S U { y };

Else if y X

If Coordinates [ y ] Coordinates [ x ] + v then

return false;

{ (0;0) }{ }{ (1;0) }{ (1;0),(-1;0),(-1;1),(0;1) }{ (-1;0),(-1;1),(0;1) }{ (-1;0),(-1;1),(0;1),(2;-1) }

return true;

{ (-1;1),(0;1),(2;-1),(-2;-1),(-2;0),(-2;1) }{ (-2;-1),(-2;0),(-2;1) }


4. Some results (1)

No toric loop in this component

This component containstoric loop


4. Some results (2)

Conclusion

Thanks to the definitions and theorems previously given (m-step, wrapping vector,…), the algorithm is proved.

Nowadays, the algorithm is used in order to detect and remove grains from object embedded in toric spaces.

A new version of the algorithm allows to compute the wrapping vector of all

loops contained in the object.

For a given dimension, the algorithm is linear ( O(n.m) ).

How can geometrical information and physical properties of a material be linked ?

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