Finite volumes and finite elements for the numerical simulation of wave breaking F. Golay

1. Finite volumes and finite elements for the numerical simulation of wave breaking F. Golay University of Toulon, France ANAM/MNC

2. Plan • Numerical simulation of wave breaking • Finite volume and finite element code • Mesh refinement

3. Numerical simulation of wave breaking • Mathematical model • Numerical model • Numerical results

4. Numerical simulation of wave breaking: Mathematical model ( ) = g j - re - g j p j p ( ) 1 ( ) ( ) 1 1 1 g + p ( p ) = j + j - ( 1 ) = c g j - g - g - ( ) 1 1 1 r w a g j p j g p g p ( ) ( ) w w a a = j + j - ( 1 ) g j - g - g - ( ) 1 1 1 w a P. Helluy, F. Golay:”Mathematical and Numerical aspects of Low Mach Number Flows”, Porquerolles 2004 where Sound velocity Equation Of State: stiffened gaz (Abgrall-Saurel, 1996)

5. Numerical simulation of wave breaking: Numerical model Ci Cj The system has the form of a system of conservation laws We solve it by a standard finite volume scheme • Second order extension:MUSCL • No pressure oscillation thanks to a special non-conservative discretisation of the fraction evolution.

6. Numerical simulation of wave breaking: Test case In the air sound velocity c=20m/s, p=105 Pa pa=-99636 Pa, ga=1.1 In the water sound velocity c=20m/s, p=105 Pa pw=263636 Pa, gw=1.1

7. Numerical simulation of wave breaking: Numerical results wave propagation Mesh: 2000x150

8. Numerical simulation of wave breaking: Numerical results wave breaking

9. Numerical simulation of wave breaking: Partial conclusion • Simple and efficient method: no interface tracking • The same code can be used for (un)compressible multifluid flows • Improvements: • Unstructured mesh, automatic mesh refinement • A posteriori error • Physical interaction • Mixed numerical method Integration in a finite element code

10. Finite volume in a finite element code • Finite element formulation • Finite volume formulation • Software architecture • Validation

11. Finite volume/element formulation Discontinuous finite element formulation Baumann, Oden (2000) Finite volume formulation Finite element formulation

12. FV & FE: Finite Volume formulation Geometrical node with no dof Centroid node with 5 dof N+1 5 2 1 4 3 4 2 3 1 Compute numerical flux exact Godunov scheme Helluy, Barberon, Rouy 2003 • Compute nodal load vector • Estimation of U with slope limiter • Display the result

13. FV & FE: Software architecture Object element Identifieur Object 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 Name Identifieur Template: Character array Real array Integer array … … 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 Object oriented finite element code: SIC (Systeme Interactif de Conception) Touzot, Aunay, Breitkopf 1985 • Exemple of object : • - a node • a element • a kinematic condition • a matrix • a vector • a command • a model • … • An object could be: • - created • duplicated • listed • modified • … http://sic.univ-tln.fr

14. FV & FE: Validation Test 2 Test 1 Stationnary choc

16. Mesh refinement / unrefinement / adaptation • Finite element mesh refinement • example: topologic optimization • Quadtree mesh refinement • Unrefinement

17. Mesh refinement: Finite elementmesh refinement e3 e4 e1 e3 e4 e2 e1 e2 e1 e3 e1 e1 e1 e1 e1 e1 e1 e2 e1 e1 e2 e2 e1 conformity e2 e1 e1 e3 e1 e1 e2 e3 e4 e2 Refinement

18. Mesh refinement: Mesh refinement Test 2 ò ] 1 Criterion 1: [ ò 2 2 2 h = + e å r R de h ( u ) D dl e ce e u 2 Î e ¶ face e e CriterionR. Verfürth (2000) 1 ì ü 2 2 £ h å error K í ý e î þ e Error P=0,2,4 P=4,6,8 P=8,10,12 P=12,14,16 Criterion 2: Verfürth Initial Mesh

19. Mesh refinement: Mesh refinement & topologic optimisation +1 W +1/2 y +1 ux=0 uy=0 +1 x ? +1 ux=0 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

20. time cpu improved • best precision • « static » front captured • but conformity! • local unrefinement is difficult

21. Mesh refinement: Quadtree mesh refinement Hierarchical approach on quadrilateral

22. Mesh refinement: Quadtree mesh unrefinement • Loop on volume to set a refinement criteria • 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

23. Mesh refinement: Wave breaking • To be continued ….. • New posteriori error criteria • Interface captured by the entropy jump

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