1. Natural Elements Method for Shallow Water Equations
M. Darbani, A. Ouahsine, P. Villon
University of Technology of Compiègne
Roberval laboratory UMR-CNRS 6253, France
MULTIPHYSICS-2009
09-11 December 2009, LILLE ,France
Ladies & jentlemen good afernoon, my name is mohsen darbani, I’m a phd student in advance mechanic at the university of tehnology of Compiègne in France. I am going to present you a part of our research developement on :Natural elements method for shallow water equations. Supervised by Pr Ouahsine and Pr Villon.

2. Contents Problem Formulation Shape Functions
Problem Shape Function SWE & Time discretizatio Nodal Integration Numerical test Appllication
Contents
Problem Formulation
Shape Functions
Shallow water Equations and Time discretization
Nodal Integration
Numerical tests and results application to Dam Break
During my presentation, I will speak about the following points:

3. Finite Elements method with meshing is not always convenient.
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Large deformation
Dam Break
The finit elements methode with meshing for modeling and soulving some problems in mechanic of fluide like dambreak or breaking waves with large deformation is not always convenient.We propose one kind of messhless methode called Natural Elements Method.
Breaking waves
Finite Elements method with meshing is not always convenient.

4. Why meshless method ? treating the problems is easier for large
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Why meshless method ?
In NEM method the possibility of
treating the problems is easier for large
deformations than in the finite elements method
Ability to insert or remove the nodes easier
Example: Domain enrichment
Shape functions depend only on the positions
of the nodes.

5. Voronoi Diagram and Delaunay triangles
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Voronoi Diagram and Delaunay triangles
Cloud of the nodes
Delaunay Triangulations
and Circumscribed circles
Voronoi diagram
Consider a cloude of nodes like this. With diagram of Voronoi we can divide the domain in to the subdomains which called the Voronoi diagram or Voronoi cells. The dual of the Voronoi diagram is the Delaunay Triangulation. For each triangle we have a Circumscribed circle.

6. Mathematical formulation
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Mathematical formulation
Cells 1st and 2d order
The first order and second order cells of the Voronoi diagram
are defined mathematically by :
At the following we need to two consepts, 1st and 2d order cells, I would like to present you another principal concept called "Natural neighbor".
Natural Neighbor
The Natural Neighbors of a node are the nodes associated with neighboring Voronoi cells. They are connected to the node by an edge of Delaunay triangle.

7. Example
In this example we have a diagram of Voronoi, we choose an arbitrary point like « P » in the domain , this polygon green is the cell first order for TP and in the next figure the polygon red shows the second order cell TP3

8. Sibsonian Shape Function
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Sibsonian Shape Function
Example
In the NEM we have two kinds of the shape function. The first one is the Sibsonian shape function which is defined with the following relation :

9. Non-Sibsonian Shape Function
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Non-Sibsonian Shape Function
di(x):
Euclidian distance between
points x and node ni
li(x):
length of the Voronoi edge
associated to x and node ni
The second one is Non-Sinsonian shape function which is defined by this equation :

10. Properties of NEM Shape Function
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Properties of NEM Shape Function
property of the Kronecker delta
Partition of unity
Local co-ordinate property
Linear consistency
I would like to point to some important properties of NEM shape functions.

11. Properties of NEM Shape Function
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Properties of NEM Shape Function
Remark :
There is an important remark about the properties of NEM shape functions. It’s necessary to consider that :
is C1 in every points of Circumscribed circles
is only continuous in every points of Circumscribed circles

12. Shallow Water Equations
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Shallow Water Equations
Governing and Continuty Equations for Incompressible Fluid
h(x,y,t)=η(x,y,t)+H0(x,y)
H(x,y,t)=h(x,y,t)+hb(x,y)
Here (u,v) is horizontal velocity & h is the depth of channel & η is the slope of the free surface & f0 is the coeffitiont of Coriolis f0=2*Ω*sin with Ω =7.3*10-3 rad/s rotation of earth and  =+-45° is latitude of the Earth & Tb is constraint at fond of the channel & cf is the coeffitiont of the friction of type Chezy.
Boundary Conditions
Stand for the Dirichlet
portion of the boundary

13. Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Here I consider only the temporal part of governing equations. The material derivative produce a terme non-linear. For avoiding of this terme we propose lagrangian formulation. For soulving the equation we use Gauss method, one kind of numerical integration method. But in the last equation the integration point ξ k is belong to present moment and  k belongs to previous time step.

14. Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Iterative process

15. The weak form of the Diffusion Term leads to the following integral :
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Diffusion Term
The weak form of the Diffusion Term leads to the following integral :
The above integral can be approximated by the following assumption with considering the method SCNI [Chen 2001] :
With method of Gauss for numerical integration we have:
Method of Stabilized Conforming Nodal Integration (SCNI)
n i:
External normal
Xg &ω i :
Integration point and weight of Gauss
Mes(ω i):
Area of ω i

16. The weak form of the Coriolis Term leads to the following integral:
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllicatio
Coriolis Term
The weak form of the Coriolis Term leads to the following integral:
By using we obtain :
and by using the second propriety we will obtain
Finally, the approximated Coriolis term will become :

17. Algebraic form of Sallow Water Equations
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Algebraic form of Sallow Water Equations
The global matricial form is :
α =0 : Euler Explicit
0< α <1 : Crank–Nicolson
α=1 : Euler Implicit

18. Numerical tests application of Dam Break
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Numerical tests application of Dam Break
We conseder a rectangular channel of water initially retained by a door that is instantaneously removed at time t=0. When the door is removed, water will flow under the action of gravity, consider as 9.81 m/s2.Density of water is 103 (thousand) kg/m3 and a viscosity of 0.1Ps.s was assumed. The matematical model was composed of nodes between 100 (hundred) and 1000 (thousand) .

19. Magnitude of elevation for transect 1
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Numerical tests application for Dam break
We have verified the manitude of elevation of water par time stepes. First we put a transect inside of the reservation of water. We can see the level of water decresed with time step.
Magnitude of elevation for transect 1

20. magnitude of elevation for transect 2
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Numerical tests application for Dam break
At the second time we put this transect in the empty side of the channel and we see the level of water increased prepare to time step.
magnitude of elevation for transect 2

21. 09-11 December 2009, LILLE ,France
MULTIPHYSICS-2009
09-11 December 2009, LILLE ,France
Thank you
for your attention
Any questions?

22. Natural Elements Method for Shallow Water Equations
UTC
Natural Elements Method for Shallow
Water Equations
M. Darbani, A. Ouahsine, P. Villon
Université de Technologie de Compiègne,
laboratoire Roberval UMR-CNRS 5263, France
MULTIPHYSICS-2009
09-11 December 2009, LILLE ,France

23. Numerical resolution :
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
5/5
Numerical resolution :
The global matricial form reads :
=0 : Euler Explicite
=1 : Euler Implicite

24. Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
1/3
Numerical resolution
Shallow water equations Integration over the total water depth

25. Why meshless method ? In Meshless Methods the possibility of treating
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Why meshless method ?
In Meshless Methods the possibility of treating
the problems is easier in large deformation than in the
finite element method
Ability to insert or remove the nodes easily
Example: Domain enrechissement
Shape functions depends only on the position of nodes

26. Shape Function of NEM (Sibsonian)
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Shape Function of NEM (Sibsonian)
Example
is at least C1 in every points but only
continuous at the nodal position

27. Mathematical formulation
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Mathematical formulation
Cells 1st and 2d order
The first order and second order cells of the Voronoï diagram
are defined mathematically by :
At the following we need to two consepts, 1st and 2d order cells, here you see their mathematical deffinitions
Natural Neighbor
The Natural Neighbors of a node are the nodes associated with neighboring Voronoi cell, or which are connected to the node by a edge of Delaunay triangle.

28. Iterative process Determine vertex of triangle at t (n-1) step :
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Iterative process
Determine vertex of triangle at t (n-1) step :
Finding velocity at the point with interpolation :
Adjusting the point position in the old triangle
Find at t (n-1) step with:

29. May be approximatted by
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
May be approximatted by
Thus, for any arbitrary vector b we can write

30. By taking into account of
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
By taking into account of

31. Method of Stabilized Conforming Nodal Integration (SCNI)
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Nodal Integration
Method of Stabilized Conforming Nodal Integration (SCNI)
Based on the substitution of the gradient term by an average gradient of each node in an area surrounding the representative node
Divergence theorem :

32. Shape Function of NEM (Sibsonian)
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Shape Function of NEM (Sibsonian)
Example

33. The pressure term leads to the following integral:
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Pressure Term
The pressure term leads to the following integral:
For example: Consider the nodes 1,2,3,,..7 as natural neighbors of the node i

34. Evolution of the particle nodes
Solution Stability
Initial
Final

35. Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

36. The weak form of pressure term leads to the following integral:
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Pressure Term
The weak form of pressure term leads to the following integral: