1. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 1/24 Finite volumes and finite elements for the numerical simulation of wave breaking F. Golay University of Toulon, France ANAM/MNC

2. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 2/24 Numerical simulation of wave breaking Finite volume and finite element code Mesh refinement Plan

3. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 3/24 Mathematical model Numerical model Numerical results Numerical simulation of wave breaking

4. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 4/24 Numerical simulation of wave breaking: Mathematical model where Equation Of State: stiffened gaz (Abgrall-Saurel, 1996) 1 )1( 11)( )()( 1 1 )1( 1 1 1)( 1 )()(1)(p a aa w ww aw )p( c Sound velocity P. Helluy, F. Golay: Mathematical and Numerical aspects of Low Mach Number Flows, Porquerolles 2004

5. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 5/24 The system has the form of a system of conservation laws We solve it by a standard finite volume scheme CiCi CjCj Second order extension:MUSCL No pressure oscillation thanks to a special non-conservative discretisation of the fraction evolution. Numerical simulation of wave breaking: Numerical model

6. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 6/24 Numerical simulation of wave breaking: Test case In the air sound velocity c=20m/s, p=10 5 Pa a =-99636 Pa, a =1.1 In the water sound velocity c=20m/s, p=10 5 Pa w =263636 Pa, w =1.1

7. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 7/24 Mesh: 2000x150 Numerical simulation of wave breaking: Numerical results wave propagation

8. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 8/24 Numerical simulation of wave breaking: Numerical results wave breaking

9. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 9/24 Simple and efficient method: no interface tracking The same code can be used for compressible multifluid flows Improvements: Unstructured mesh, automatic mesh refinement A posteriori error Physical interaction Mixed numerical method Numerical simulation of wave breaking: Partial conclusion Integration in a finite element code

10. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 10/24 Finite element formulation Finite volume formulation Software architecture Validation Finite volume in a finite element code

11. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 11/24 Finite volume/element formulation Finite element formulation Finite volume formulation Discontinuous finite element formulation Baumann, Oden (2000)

12. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 12/24 FV & FE: Finite Volume formulation Geometrical node with no dof Centroid node with 5 dof 1 2 4 3 Compute numerical flux exact Godunov scheme Helluy, Barberon, Rouy 2003 1 2 4 3 5 N+1 Compute nodal load vector Estimation of U with slope limiter Display the result

13. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 13/24 FV & FE: Software architecture Object oriented finite element code: SIC (Systeme Interactif de Conception) Touzot, Aunay, Breitkopf 1985 Object Name Identifieur Template: Character array Real array Integer array … An object could be: - created - duplicated - listed - modified - … Exemple of object : - a node - a element - a kinematic condition - a matrix - a vector - a command - a model - … Object element Identifieur model number zone Id material properties Id geometric properties Id element properties Id interpolation function Id save vector List of nodes List of load case Mesh refinement parameter edges number List of neighbour elements Object node Identifieur Id kinematic condition Id load case number X coordinate Y coordinate Z coordinate Degree of freedom Nodal properties Equation numbers List of elements http://sic.univ-tln.fr

14. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 14/24 FV & FE: Validation Stationnary choc Test 1 Test 2

15. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 15/24

16. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 16/24 Mesh refinement / unrefinement / adaptation Finite element mesh refinement example: topologic optimization Quadtree mesh refinement Unrefinement

17. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 17/24 Mesh refinement: Finite element mesh refinement e1 e2 e3 e4 e1 Refinement e1 e2 e3 e4 e1 e2 e1 e2 e3 e1 e2 e3 e1 e2 e3 e4 e1 conformity e1 e2 e1 e2

18. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 18/24 Mesh refinement: Mesh refinement Test P=0,2,4P=4,6,8P=8,10,12P=12,14,16 Criterion 1: Criterion 2: Verfürth Initial Mesh Error eface e 2 u dlD)u(h 2 1 e 2 ece deR 22 e r 2 1 e 2 e Kerror Criterion R. Verfürth (2000)

19. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 19/24 +1 ? u x =0 u y =0 x y u x =0 +1/2 +1 P=0,2,4 399 nodes 130 elements P=4,6,8 708 nodes 257 elements P=8,10,12 1016 nodes 389 elements P=12,14,16 1472 nodes 589 elements Mesh refinement: Mesh refinement & topologic optimisation

20. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 20/24 time cpu improved best precision « static » front captured but conformity! local unrefinement is difficult

21. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 21/24 Mesh refinement: Quadtree mesh refinement Hierarchical approach on quadrilateral

22. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 22/24 1)Loop on volume to set a refinement criteria 2)Loop on nodes to find patch to unrefine -4 volumes at same hierarchical level -4 edge at same hierarchical level Modification of the central node Destruction of the other central nodes Destruction of the central edge elements Modification of the peripheral edges Loop on the nodes to merge edges if necessary Mesh refinement: Quadtree mesh unrefinement

23. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 23/24 Mesh refinement: Wave breaking To be continued ….. New posteriori error criteria Interface captured by the entropy jump

24. Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 24/24 Conclusion Compressible bi-fluid model Finite volume formulation with exact Rieman solver (integration in FE code) Validation: simulation of wave breaking (confrontation with others models) Integration in a finite software architecture Quadtree mesh (un)refinement … 3D Parallel implementation A posteriori error Multiphysic simulation