M. Darbani, A. Ouahsine, P. Villon University of Technology of Compiègne

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

2. Problem Shape Function SWE & Time discretizatioNodal 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

3. Problem Shape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Large deformation Dam Break Breaking waves Finite Elements method with meshing is not always convenient.

4. Problem Shape Function SWE&Time discretizatioNodal 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. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Voronoi Diagram and Delaunay triangles Cloud of the nodes DelaunayTriangulations and Circumscribed circles Voronoidiagram

6. ProblemShape Function SWE&Time discretizatioNodal 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 : 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

8. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication SibsonianShape Function Example

9. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Non-SibsonianShape Function

10. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Properties of NEM Shape Function

11. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Properties of NEM Shape Function Remark :

12. ProblemShape FunctionSWE&Time discretizatio Nodal Integration Numerical test Appllication Shallow Water Equations Governing and Continuty Equations for IncompressibleFluid h(x,y,t)=η(x,y,t)+H0(x,y) H(x,y,t)=h(x,y,t)+hb(x,y) Boundary Conditions Stand for the Dirichlet portion of the boundary

13. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Time discretization

14. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Iterative process

15. ProblemShape 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:

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

17. ProblemShape Function SWE&Time discretizatio Nodal IntegrationNumerical 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. ProblemShape Function SWE&Time discretizatioNodal IntegrationNumerical test Appllication Numerical tests application of Dam Break

19. ProblemShape Function SWE&Time discretizatioNodal IntegrationNumerical test Appllication Numerical tests application for Dam break Magnitude of elevation for transect 1

20. ProblemShape Function SWE&Time discretizatioNodal IntegrationNumerical test Appllication Numerical tests application for Dam break magnitude of elevation for transect 2

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

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

24. ProblemShape FunctionSWE&Time discretizatio Nodal Integration Numerical test Appllication 1/3 Numerical resolution Shallow water equations Integration over the total water depth

25. Problem Shape Function SWE&Time discretizatioNodal 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. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Shape Function of NEM (Sibsonian) Example is at leastC1 in every points but only continuous at the nodal position

27. ProblemShape Function SWE&Time discretizatioNodal 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 : 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. Finding velocity at the point with interpolation : Find at t (n-1) stepwith: ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Iterative process Determine vertex of triangle at t (n-1) step : Adjusting the point position in the old triangle

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

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

31. ProblemShape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication Nodal Integration Method ofStabilized 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. ProblemShape Function SWE&Time discretizatioNodal Integration Numerical test Appllication Shape Function of NEM (Sibsonian) Example

33. ProblemShape 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 SolutionStability Initial Final

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

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