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Block-Based Adaptive Mesh Refinement scheme based on numerical density of entropy production for conservation laws and applications. Mehmet Ersoy 1 , Fr ed eric Golay, Lyudmyla Yushchenko, Universit e de Toulon, IMATH, and Damien Sous,

  1. Mesh refinement indicator : principle & illustration Given w n k → compute w n +1 k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : 1 � ◮ S n S n k � α min S = ⇒ the cell is refined with S = k | Ω | Ω ◮ S n k � α max S = ⇒ the cell is coarsened ◮ Dynamic mesh refinement : ⋆ Non-structured grid : macro-cell ⋆ Dyadic tree (1D), Quadtree (2D), Octree (3D) ⋆ hierarchical numbering : basis 2,4,8 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 8 / 29

  2. Mesh refinement indicator : principle & illustration Given w n k → compute w n +1 k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : 1 � ◮ S n S n k � α min S = ⇒ the cell is refined with S = k | Ω | Ω ◮ S n k � α max S = ⇒ the cell is coarsened M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 8 / 29

  3. Mesh refinement indicator : principle & illustration Given w n k → compute w n +1 k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : 1 � ◮ S n S n k � α min S = ⇒ the cell is refined with S = k | Ω | Ω ◮ S n k � α max S = ⇒ the cell is coarsened M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 8 / 29

  4. Mesh refinement indicator : principle & illustration Given w n k → compute w n +1 k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : 1 � ◮ S n S n k � α min S = ⇒ the cell is refined with S = k | Ω | Ω ◮ S n k � α max S = ⇒ the cell is coarsened ◮ Simple approach but the scheme is locally non consistent [SO88, TW05] Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 8 / 29

  5. Mesh refinement indicator : principle & illustration Given w n k → compute w n +1 k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : 1 � ◮ S n S n k � α min S = ⇒ the cell is refined with S = k | Ω | Ω ◮ S n k � α max S = ⇒ the cell is coarsened ◮ Simple approach but the scheme is locally non consistent [SO88, TW05] ◮ Limit the mesh level of adjacent cells by 2 Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 8 / 29

  6. Mesh refinement indicator : principle & illustration Given w n k → compute w n +1 k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : 1 � ◮ S n S n k � α min S = ⇒ the cell is refined with S = k | Ω | Ω ◮ S n k � α max S = ⇒ the cell is coarsened ◮ Simple approach but the scheme is locally non consistent [SO88, TW05] ◮ Limit the mesh level of adjacent cells by 2 ◮ A correction can be obtained (work in progress) [AE15] Altazin T., Ersoy, M. Analyze of the inconsistency of adaptive scheme . Preprint (in progress) , 2015. Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 8 / 29

  7. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 9 / 29

  8. An example : the one-dimensional gas dynamics equations for ideal gas ρ ( t, x ) : density ∂t + ∂ρu ∂ρ ∂x = 0 u ( t, x ) : velocity p ( t, x ) : pressure ρu 2 + p � � ∂ρu ∂t + ∂ γ := 1 . 4 : ratio of the specific heats = 0 where ∂x E ( ε, u ) : total energy + ∂ ( ρE + p ) u ∂ρE ε : internal specific energy = 0 ∂t ∂x ε + u 2 = E p = ( γ − 1) ρε 2 Intel(R) Core(TM) i5-2500 CPU @ 3.30GHz M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 10 / 29

  9. An example : the one-dimensional gas dynamics equations for ideal gas ρ ( t, x ) : density ∂t + ∂ρu ∂ρ ∂x = 0 u ( t, x ) : velocity p ( t, x ) : pressure ρu 2 + p � � ∂ρu ∂t + ∂ γ := 1 . 4 : ratio of the specific heats = 0 where ∂x E ( ε, u ) : total energy + ∂ ( ρE + p ) u ∂ρE ε : internal specific energy = 0 ∂t ∂x ε + u 2 = E p = ( γ − 1) ρε 2 Conservative variables w = ( ρ, ρu, ρE ) t entropy � p � s ( w ) = − ρ ln of flux ψ ( w ) = u s ( w ) . ρ γ Intel(R) Core(TM) i5-2500 CPU @ 3.30GHz M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 10 / 29

  10. Sod’s shock tube problem Mesh refinement parameter α max : 0 . 01 , Mesh coarsening parameter α min : 0 . 001 , 1 Mesh refinement parameter ¯ � S n S : k b | Ω | k b CFL : 0 . 25 , Simulation time ( s ) : 0 . 4 , Initial number of cells : 200 , Maximum level of mesh refinement : L max . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 11 / 29

  11. Accuracy 1.2 1.4 5 0.1 ρ on adaptive mesh with L max = 4 level n ρ on uniform fixed mesh N = 681 S k 1.2 4.5 Numerical density of entropy production Numerical density of entropy production ρ ex | ρ - ρ ex | 1 0.08 n S k 1 4 Mesh refinement level 0.8 0.8 3.5 0.06 Density 0.6 3 0.6 0.04 0.4 2.5 0.4 0.2 2 0.02 0 1.5 0.2 1-0.2 1 1 0 -1 -0.5 0 0.5 -1 -0.5 0 0.5 x x (a) Density and numerical density of en- (b) Mesh refinement level, numerical tropy production. density of entropy production and local error. Figure: Sod’s shock tube problem : solution at time t = 0 . 4 s using the AB1M scheme on a dynamic grid with L max = 5 and the AB1 scheme on a uniform fixed grid of 681 cells. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 12 / 29

  12. Time restriction Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 13 / 29

  13. Time restriction, local time stepping approach Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k Local time stepping algorithm : ◮ Sort cells in groups w.r.t. to their level Muller S., Stiriba Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. SIAM J. Sci. Comput., 30(3) :493–531, 2007. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 13 / 29

  14. Time restriction, local time stepping approach & Aims Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k Local time stepping algorithm : ◮ Sort cells in groups w.r.t. to their level ◮ Update the cells following the local time stepping algorithm. Muller S., Stiriba Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. SIAM J. Sci. Comput., 30(3) :493–531, 2007. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 13 / 29

  15. Time restriction, local time stepping approach & Aims Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k Local time stepping algorithm : ◮ Sort cells in groups w.r.t. to their level ◮ Update the cells following the local time stepping algorithm. ◮ save the cpu-time keeping the accuracy. M. Ersoy, F. Golay, L. Yushchenko. Adaptive multi-scale scheme based on numerical entropy production for conservation laws. CEJM, Central European Journal of Mathematics, 11(8), pp 1392-1415, 2013. M. Ersoy, F. Golay, L. Yushchenko. Adaptive scheme based on entropy production : robustness through severe test cases for hyperbolic conservation laws. Preprint , https://hal.archives-ouvertes.fr/hal-00918773 , 2013. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 13 / 29

  16. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 14 / 29

  17. two and three dimensional case : BB-AMR Main difficulty : mesh and data structure. For fast computation, the following are required ◮ parallel treatment ◮ hierarchical grids M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 15 / 29

  18. two and three dimensional case : BB-AMR Main difficulty : mesh and data structure. Some interesting issues : ◮ 2D quad-tree [ZW11], ◮ 3D octree [LGF04], ◮ 2D/3D anisotropic AMR [HFCC13]. Zhang, M., and W.M. Wu. 2011. A two dimensional hydrodynamic and sediment transport model for dam break based on finite volume method with quadtree grid. Applied Ocean Research 33 (4) : 297 – 308. Losasso, F., F. Gibou, and R. Fedkiw. 2004. Simulating Water and Smoke with an Octree Data Structure. ACM Trans. Graph. 23 (3) : 457–462, 2004. Hachem, E., S. Feghali, R. Codina, and T. Coupez. Immersed stress method for fluid structure interaction using anisotropic mesh adaptation. International Journal for Numerical Methods in Engineering 94 (9) : 805–825, 2013. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 15 / 29

  19. two and three dimensional case : BB-AMR Main difficulty : mesh and data structure. The strategy adopted : M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 15 / 29

  20. two and three dimensional case : BB-AMR Main difficulty : mesh and data structure. The strategy adopted : 1 fixed domain= 1 fixed block=1 cpu :“failure” → synchronization depends on 1 the finest domain M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 15 / 29

  21. two and three dimensional case : BB-AMR Main difficulty : mesh and data structure. The strategy adopted : 1 fixed domain= 1 fixed block=1 cpu :“failure” → synchronization depends on 1 the finest domain 1 dynamic domain= n × static blocks = 1cpu :“good compromise” → each 2 domain has almost the same number number of cells → “better” synchronization = Block-Based Adaptive Mesh Refinement (BB-AMR) M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 15 / 29

  22. two and three dimensional case : BB-AMR Main difficulty : mesh and data structure. The strategy adopted : 1 fixed domain= 1 fixed block=1 cpu :“failure” → synchronization depends on 1 the finest domain 1 dynamic domain= n × static blocks = 1cpu :“good compromise” → each 2 domain has almost the same number number of cells → “better” synchronization = Block-Based Adaptive Mesh Refinement (BB-AMR) It certainly exists better strategy . . . 3 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 15 / 29

  23. BB-AMR How it works ? each domain has almost the same number of cells M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  24. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  25. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  26. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  27. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  28. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  29. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  30. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  31. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  32. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering more sophisticated numbering exists . . . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  33. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering more sophisticated numbering exists . . . re-numbering and re-meshing being expensive ◮ the mesh should be kept constant on a time interval M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  34. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering more sophisticated numbering exists . . . re-numbering and re-meshing being expensive ◮ the mesh should be kept constant on a time interval ◮ AMR time-step computed through the smallest block and not the smallest cell T n +1 − T n = ∆ T AMR is given by the CFL min k h block k ∆ T AMR � β max k � u block k � , 0 < β � 1 . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  35. BB-AMR How it works ? each domain has almost the same number of cells domain are defined using Cuthill-McKee numbering more sophisticated numbering exists . . . re-numbering and re-meshing being expensive ◮ the mesh should be kept constant on a time interval ◮ AMR time-step computed through the smallest block and not the smallest cell ◮ Gain is important and numerical stability is conserved ! Thomas Altazin, Mehmet Ersoy, Fr´ ed´ eric Golay, Damien Sous, and Lyudmyla Yushchenko. Numerical entropy production for multidimensional conservation laws using Block-Based Adaptive Mesh Refinement scheme. preprint, 2015. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 16 / 29

  36. Examples : A two dimensional example of BB-AMR with 3 domains and 9 blocks. (a) AMR T 0 (b) AMR T 1 (c) AMR T 2 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 17 / 29

  37. Examples : A two dimensional example of BB-AMR with 3 domains and 9 blocks. (f) AMR T 0 (g) AMR T 1 (h) AMR T 2 A three dimensional example of BB-AMR with 3 domains and 27 blocks. (i) Block-based (j) Domain mesh decomposition M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 17 / 29

  38. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 18 / 29

  39. Simulation of wave propagation, wave breaking and wave impacting Understanding of wave hydrodynamics is of primary interest for ocean and naval engineering applications : ◮ dynamics of ships and floating structures, ◮ stability of offshore structures, ◮ coastal erosion and submersion processes, . . . . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 19 / 29

  40. Simulation of wave propagation, wave breaking and wave impacting Understanding of wave hydrodynamics is of primary interest for ocean and naval engineering applications : It’s difficult to describe accurately wave dynamics and still a fairly open research field. breaking or impacting waves on rigid structures = violent transformations M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 19 / 29

  41. Simulation of wave propagation, wave breaking and wave impacting Understanding of wave hydrodynamics is of primary interest for ocean and naval engineering applications : It’s difficult to describe accurately wave dynamics and still a fairly open research field. involved physical processes, such as splash-ups or gas pockets entrapment, are quite complex and can hardly be characterized by field or laboratory experiments or analytical approaches : several models ! : M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 19 / 29

  42. Simulation of wave propagation, wave breaking and wave impacting Understanding of wave hydrodynamics is of primary interest for ocean and naval engineering applications : It’s difficult to describe accurately wave dynamics and still a fairly open research field. involved physical processes, such as splash-ups or gas pockets entrapment, are quite complex and can hardly be characterized by field or laboratory experiments or analytical approaches : several models ! : Therefore, numerical simulation of breaking and impacting waves is both ◮ an attractive research topic ◮ a challenging task for coastal and environmental engineering M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 19 / 29

  43. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 20 / 29

  44. The governing equations Assumptions : physics of impacting/breaking waves can be simplified ◮ mainly governed by pressure forces and overturning forces ◮ Mach number < 0.3 → fluid is slightly compressible M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  45. The governing equations Assumptions : physics of impacting/breaking waves can be simplified ◮ mainly governed by pressure forces and overturning forces ◮ Mach number < 0.3 → fluid is slightly compressible ◮ small-scale friction and dissipation process are neglected M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  46. The governing equations Assumptions : physics of impacting/breaking waves can be simplified ◮ mainly governed by pressure forces and overturning forces ◮ Mach number < 0.3 → fluid is slightly compressible ◮ small-scale friction and dissipation process are neglected ◮ two-phase flow Compressible Euler equations can be considered Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � ∂t + div = ρg p ( t, x ) : pressure ϕ : fluid’s fraction ∂ϕ ∂t + u · ∇ ϕ = 0 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  47. The governing equations Assumptions : physics of impacting/breaking waves can be simplified ◮ mainly governed by pressure forces and overturning forces ◮ Mach number < 0.3 → fluid is slightly compressible ◮ small-scale friction and dissipation process are neglected ◮ two-phase flow Compressible Euler equations can be considered ◮ An artificial linearized pressure law is used to compute low Mach flows [C67] Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � ∂t + div = ρg p ( t, x ) : pressure ϕ : fluid’s fraction ∂ϕ ∂t + u · ∇ ϕ = 0 with = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) p Chorin, A.J. . A Numerical Method for Solving Incompressible Viscous Flow Problems. Journal of Computational Physics 2 (1) : 12 – 26, 1967 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  48. The governing equations Assumptions Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � p ( t, x ) : pressure ∂t + div = ρg : fluid’s fraction ϕ ∂ϕ ∂t + u · ∇ ϕ = 0 with p = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) Equation of state with artificial sound speed → CFL less restrictive M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  49. The governing equations Assumptions Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � p ( t, x ) : pressure ∂t + div = ρg : fluid’s fraction ϕ ∂ϕ ∂t + u · ∇ ϕ = 0 with p = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) Equation of state with artificial sound speed → CFL less restrictive Explicit scheme → easy parallel implementation (MPI) M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  50. The governing equations Assumptions Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � p ( t, x ) : pressure ∂t + div = ρg : fluid’s fraction ϕ ∂ϕ ∂t + u · ∇ ϕ = 0 with p = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) Equation of state with artificial sound speed → CFL less restrictive Explicit scheme → easy parallel implementation (MPI) hyperbolic system Moreover, M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  51. The governing equations Assumptions Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � p ( t, x ) : pressure ∂t + div = ρg : fluid’s fraction ϕ ∂ϕ ∂t + u · ∇ ϕ = 0 with p = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) Equation of state with artificial sound speed → CFL less restrictive Explicit scheme → easy parallel implementation (MPI) hyperbolic system entropy available Moreover, M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  52. The governing equations Assumptions Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � p ( t, x ) : pressure ∂t + div = ρg : fluid’s fraction ϕ ∂ϕ ∂t + u · ∇ ϕ = 0 with p = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) Equation of state with artificial sound speed → CFL less restrictive Explicit scheme → easy parallel implementation (MPI) hyperbolic system entropy available Moreover, automatic mesh refinement M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  53. The governing equations Assumptions Model (2D and 3D) : low mach two phase ∂ρ ∂t + div ( ρu ) = 0 ρ ( t, x ) : density u ( t, x ) : velocity ∂ρu where ρu 2 + pI � � p ( t, x ) : pressure ∂t + div = ρg : fluid’s fraction ϕ ∂ϕ ∂t + u · ∇ ϕ = 0 with p = p 0 + c 0 ( ρ − ( ϕρ w + (1 − ϕ ) ρ a )) Equation of state with artificial sound speed → CFL less restrictive Explicit scheme → easy parallel implementation (MPI) hyperbolic system entropy available Moreover, automatic mesh refinement local time stepping M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 21 / 29

  54. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 22 / 29

  55. A two-dimensional dam-break problem [KTO95] capture the complex structure of the air-water interface after wave impact Koshizuka, S., H. Tamako, and Y. Oka. A particle method for incompressible viscous flow with fluid fragmentations. Computational Fluid Dynamics Journal, 4 (1) : 29–46, 1995. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  56. A two-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Experimental configuration M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  57. A two-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Numerical parameters : Mesh refinement parameter α max : 0 . 2 , Mesh coarsening parameter α min : 0 . 02 , Number of domain : 321 , Number of blocks : 321 , Number of processors : 120 , Maximum level of mesh refinement : L max = 5 , CFL : CFL = 0 . 8 , Simulation time : T = 1 . 5 , AMR time : AMR = 300 . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  58. A two-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Confrontation with experiments : T = 0 Figure: mesh (left), density with blue and red corresponding to air and water, respectively (center), mesh refinement level (1 to 5) per block (right) M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  59. A two-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Confrontation with experiments : T = 0 . 2 Figure: (a) Mesh ; (b) Density (air-blue, water-red) ; (c) Density of numerical entropy production (green-zero, blue-negative values) ; (d) Mesh refinement level per block (1 to 5) ; (e) Experiment ; (f) Mesh refinement criterion per block. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  60. A two-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Confrontation with experiments : T = 0 . 4 Figure: (a) Mesh ; (b) Density (air-blue, water-red) ; (c) Density of numerical entropy production (green-zero, blue-negative values) ; (d) Mesh refinement level per block (1 to 5) ; (e) Experiment ; (f) Mesh refinement criterion per block. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  61. A two-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Remarks : ◮ number of cells varies from 70 000 and 100 000 ◮ elapsed computing time about 5 hours ◮ 1 domain = 1 block → better results with BB-AMR. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 23 / 29

  62. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 24 / 29

  63. A three-dimensional dam-break problem [K05] capture the complex structure of the air-water interface after wave impact Kleefsman, K.M.T., G. Fekken, A.E.P. Veldman, B. Iwanowski, and B. Buchner. A Volume-of-Fluid based simulation method for wave impact problems. Journal of Computational Physics 206 (1) : 363 – 393, 2005. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  64. A three-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Experimental configuration Figure: domain geometry and sensors points from http://www.math.rug.nl/$\sim$veldman/comflow/dambreak.html M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  65. A three-dimensional dam-break problem 2 capture the complex structure of the air-water interface after wave impact Numerical parameters : Mesh refinement parameter α max : 0 . 2 , Mesh coarsening parameter α min : 0 . 02 , Number of domain : 48 , Number of blocks : 3628 , Number of processors : 48 , Maximum level of mesh refinement : L max = 4 , CFL : CFL = 0 . 8 , Simulation time : T = 4 . 8 , AMR time : AMR = 240 . 2. 48 Intel X5675 cores M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  66. A three-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Confrontation with experiments : Figure: Free surface computed by Kleefsman (left), the experimentation (center) and our (right) at t = 0.4, 0.6, 1, 1.8, 2, 4.8s M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  67. A three-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Confrontation with experiments : Lee, E.S., D. Violeau, R. Issa, and S. Ploix. Application of weakly compressible and truly incompressible SPH to 3-D water collapse in waterworks. Journal of Hydraulic Research 48 (sup1) : 50–60, 2010. Vincent, S., G. Balmig` ere, J.-P. Caltagirone, and E. Meillot. Eulerian-Lagrangian multiscale methods for solving scalar equations - Application to incompressible two-phase flows. Journal of Computational Physics 229 (1) : 73 – 106, 2010 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  68. A three-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Confrontation with experiments : Figure: Domains due to the BB-AMR scheme (left) and air-water interface (right) at time 0 . 4 s, 0 . 6 s, 1 . 0 s, 2 s . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  69. A three-dimensional dam-break problem capture the complex structure of the air-water interface after wave impact Remarks : ◮ number of cells varies from 800 000 cells up to about 1 500 000 cells ◮ elapsed computing time about 10 hours (instead of 24h [GH07]) Golay, F., and P. Helluy. Numerical schemes for low Mach wave breaking. International Journal of Computational Fluid Dynamics 21(2) : 69–86, 2007. YUSHCHENKO, L., GOLAY, F., ERSOY, M. Production d’entropie et raffinement de maillage. Application au d´ eferlement de vague. 21` eme Congr` es Francais de M´ ecanique, 26 au 30 aout 2013, Bordeaux, France (FR). Golay, F., Ersoy, M., Yushchenko, L., Sous, D. Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows. International Journal of Computational Fluid Dynamics 29.1, 67-81, 2015. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 25 / 29

  70. A three-dimensional dam-break problem [AEGDSL15] A“block”dam break problem with a confrontation of RK2 and AB2 Thomas Altazin, Mehmet Ersoy, Fr´ ed´ eric Golay, Damien Sous, and Lyudmyla Yushchenko. Numerical entropy production for multidimensional conservation laws using Block-Based Adaptive Mesh Refinement scheme. preprint, 2015. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 26 / 29

  71. A three-dimensional dam-break problem A“block”dam break problem with a confrontation of RK2 and AB2 Initial configuration � 1 2 × 1 2 × 1 � Figure: Unit cube 2 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 26 / 29

  72. A three-dimensional dam-break problem A“block”dam break problem with a confrontation of RK2 and AB2 Numerical parameters : Mesh refinement parameter α max : 0 . 2 , Mesh coarsening parameter α min : 0 . 02 , Number of domain : 1 , 2 , 4 , 8 , 32 , Number of blocks : 3375 , Number of processors : 40 , Maximum level of mesh refinement : L max = 4 , Simulation time : T = 2 . 5 , AMR time : AMR = 100 . M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 26 / 29

  73. A three-dimensional dam-break problem A“block”dam break problem with a confrontation of RK2 and AB2 Confrontation with experiments : (a) Speed up vs proc number (b) cpu time vs proc number Figure: AB2 vs RK2 M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 26 / 29

  74. A three-dimensional dam-break problem A“block”dam break problem with a confrontation of RK2 and AB2 Remarks : ◮ number of cells varies from 172215 cells up to about 587763 cells speed up ◮ The efficiency, i.e. number of processors , of the computation is roughly 85% for 8 domains and 60% for 32 domains. ◮ performance decrease after 20 processors → optimization is required to get more efficiency. M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 26 / 29

  75. Outline Outline 1 Principle of the method Generality 1d examples and local time stepping Data structure : BB-AMR 2 Applications The two phase low Mach model A two-dimensional dam-break problem A three-dimensional dam-break problem 3 Conclusions M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 27 / 29

  76. Conclusions Several numerical validation on Euler equations M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 28 / 29

  77. Conclusions Several numerical validation on Euler equations Several numerical validation (in progress) for shallow water equations Figure: (left) L and (right) Kleefsman test case (B. Cleirec) M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 28 / 29

  78. Conclusions & Perspectives Several numerical validation on Euler equations Several numerical validation (in progress) for shallow water equations local consistency error between two adjacent cells of different levels M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 28 / 29

  79. Conclusions & Perspectives Several numerical validation on Euler equations Several numerical validation (in progress) for shallow water equations local consistency error between two adjacent cells of different levels capture accurately rarefactions and contact discontinuities M. Ersoy (IMATH) BB-AMR 2015, 10-13 June, Porto, Portugal 28 / 29

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