Skip to main content

Towards multi-phase flow simulations in the PDE framework Peano

Complex geometries, thermohydraulics, and two-phase flow

Abstract

In this work, we present recent enhancements and new functionalities of our flow solver in the partial differential equation framework Peano. We start with an introduction including an overview of the Peano development and a short description of the basic concepts of Peano and the flow solver in Peano concerning the underlying structured but adaptive Cartesian grids, the data structure and data access optimisation, and spatial and time discretisation of the flow solver. The new features cover geometry interfaces and additional application functionalities. The two geometry interfaces, a triangulation-based description supported by the tool preCICE and a built-in geometry using geometry primitives such as cubes, spheres, or tetrahedra allow for the efficient treatment of complex and changing geometries, an essential ingredient for most application scenarios. The new application functionality concerns a coupled heat-flow problem and two-phase flows. We present numerical examples, performance and validation results for these new functionalities.

This is a preview of subscription content, access via your institution.

References

  1. Akkerman I, Bazilevs Y, Kees CE, Farthing MW (2011) Isogeometric analysis of free-surface flow. J Comput Phys 230(11): 4137–4152

    Article  MATH  Google Scholar 

  2. Aoki T (1997) Interpolated differential operator (IDO) scheme for solving partial differential equations. Comput Phys Commun 102: 132–146

    Article  Google Scholar 

  3. Atanasov A, Bungartz H-J, Frisch J, Mehl M, Mundani R-P, Rank E, van Treek C (2010) Computational steering of complex flow simulations. In: Siegfried W, Arndt B, Gerhard W (eds) High performance computing in science and engineering, Garching 2009. Springer, Berlin (to appear)

  4. Aulisa E, Manservisi S, Scardovelli R (2006) A novel representation of the surface tension force for two-phase flow with reduced spurious currents. Comput Methods Appl Mech Eng 195(44–47): 6239–6257

    Article  MATH  Google Scholar 

  5. Badalassi VE, Ceniceros HD, Banerjee S (2003) Computation of multiphase systems with phase field models. J Comput Phys 190(2): 371–397

    Article  MathSciNet  MATH  Google Scholar 

  6. Balay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, Curfman McInnes L, Smith BF, Zhang H (2004) PETSc users manual. Technical Report ANL-95/11-Revision 2.1.5, Argonne National Laboratory

  7. Blanke C (2004) Kontinuitätserhaltende Finite-Element-Diskretisierung der Navier–Stokes–Gleichungen. Diploma Thesis, Technische Universität München

  8. Blum EK (1962) A modification of the Runge–Kutta fourth-order method. Math Comput

  9. Brachet ME, Mininni PD, Rosenberg DL, Pouquet A (2008) High-order low-storage explicit Runge-Kutta schemes for equations with quadratic nonlinearities. Technical Report, CERN, August 2008 (arXiv:0808.1883)

  10. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface-tension. J Comput Phys 100(2): 335–354

    Article  MathSciNet  MATH  Google Scholar 

  11. Brázdová V, Bowler DR (2008) Automatic data distribution and load balancing with space-filling curves: implementation in conquest. J Phys Condens Matter 20

  12. Brenk M, Bungartz H-J, Daubner K, Mehl M, Muntean IL, Neckel T (2008) An Eulerian approach for partitioned fluid-structure simulations on Cartesian grids. Comput Mech 43(1): 115–124

    Article  MATH  Google Scholar 

  13. Brenk M, Bungartz H-J, Mehl M, Muntean IL, Neckel T, Weinzierl T (2008) Numerical simulation of particle transport in a drift ratchet. SIAM J Sci Comput 30(6): 2777–2798

    Article  MathSciNet  MATH  Google Scholar 

  14. Bungartz H-J, Gatzhammer B, Mehl M, Neckel T (2010) Partitioned simulation of fluid–structure interaction on cartesian grids. In: Bungartz H-J, Mehl M, Schäfer M (eds) Fluid–structure interaction—modelling, simulation, optimisation, part II. LNCSE. Springer, Berlin, pp 255–284

    Google Scholar 

  15. Bungartz H-J, Mehl M, Neckel T, Weinzierl T (2010) The PDE framework Peano applied to fluid dynamics: an efficient implementation of a parallel multiscale fluid dynamics solver on octree-like adaptive Cartesian grids. Comput Mech 46(1):103–114 (published online)

    Google Scholar 

  16. Caboussat A (2005) Numerical simulation of two-phase free surface flows. Arch Comput Methods Eng 12(2): 165–224

    Article  MathSciNet  MATH  Google Scholar 

  17. Ceniceros HD, Roma AM, Silveira-Neto A, Villar MM (2010) A robust, fully adaptive hybrid level-set/front-tracking method for two-phase flows with an accurate surface tension computation. Commun Comput Phys 8(1): 51–94

    MathSciNet  Google Scholar 

  18. Dullien FAL (1992) Porous media: fluid transport and pore structure, 2nd edn. Academic Press, London

    Google Scholar 

  19. Eckhardt W, Weinzierl T (2010) A blocking strategy on multicore architectures for dynamically adaptive PDE solvers. In: Wyrzykowski R, Dongarra J, Karczewski K, Wasniewski J (eds) Parallel processing and applied mathematics, PPAM 2009. Lecture notes in computer science, vol 6068. Springer, Berlin, pp 567–575

    Google Scholar 

  20. Farthing MW, Kees CE, Akkerman I, Bazilevs Y (2011) A conservative level set method suitable for variable-order approximations and unstructured meshes. J Comput Phys 230(12): 4536–4558

    Article  MATH  Google Scholar 

  21. Gresho PM, Sani RL (1998) Incompressible flow and the finite element method. Wiley, London

    MATH  Google Scholar 

  22. Griebel M, Dornseifer Th, Neunhoeffer T (1998) Numerical simulation in fluid dynamics, a practical introduction. SIAM

  23. Griebel M, Zumbusch G (1999) Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves. Parallel Comput 25(7): 827–843

    Article  MathSciNet  MATH  Google Scholar 

  24. Günther F, Krahnke A, Langlotz M, Mehl M, Pögl M, Zenger Ch (2004) On the parallelization of a cache-optimal iterative solver for PDEs based on hierarchical data structures and space-filling curves. In: Proceedings of the recent advances in parallel virtual machine and message passing interface: 11th European PVM/MPI Users Group Meeting Budapest, Hungary, September 19–22, 2004. Lecture notes in computer science, vol 3241. Springer, Heidelberg

  25. Günther F, Mehl M, Pögl M, Zenger C (2006) A cache-aware algorithm for PDEs on hierarchical data structures based on space-filling curves. SIAM J Sci Comput 28(5): 1634–1650

    Article  MathSciNet  MATH  Google Scholar 

  26. Hungershöfer J, Wierum JM (2002) On the quality of partitions based on space-filling curves. In: ICCS’02: proceedings of the international conference on computational science—part III. Springer, pp 36–45

  27. Imai Y, Aoki T, Takizawa K (2008) Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics. J Comput Phys 227: 2263–2285

    MATH  Google Scholar 

  28. Jacqmin D (1999) Calculation of two-phase Navier–Stokes flows using phase-field modeling. J Comput Phys 155(1): 96–127

    Article  MathSciNet  MATH  Google Scholar 

  29. Langlotz M, Mehl M, Weinzierl T, Zenger C (2005) Skvg: cache-optimal parallel solution of PDEs on high performance computers using space-trees and space-filling curves. In: Bode A, Durst F (eds) High performance computing in science and engineering, Garching 2004. Springer, Berlin, pp 71–82

    Chapter  Google Scholar 

  30. Lieb M (2008) A full multigrid implementation on staggered adaptive Cartesian grids for the pressure poisson equation in computational fluid dynamics. Master’s thesis, Institut für Informatik, Technische Universität München

  31. Lubachevsky BD, Stillinger F (1990) Geometric properties of random disk packings. J Stat Phys 60(5,6): 561–583

    Article  MathSciNet  MATH  Google Scholar 

  32. Mehl M, Neckel T, Neumann P (2011) Navier–stokes and lattice–boltzmann on octree-like grids in the Peano framework. Int J Numer Methods Fluids 65(1):67–86. doi:10.1002/fld.2469 (published online)

    Google Scholar 

  33. Mehl M, Weinzierl T, Zenger C (2006) A cache-oblivious self-adaptive full multigrid method. Numer Linear Algebra Appl 13(2–3): 275–291

    Article  MathSciNet  MATH  Google Scholar 

  34. Meidner D, Vexler B (2007) Adaptive space–time finite element methods for parabolic optimization problems. SIAM J Control Optim 46(1): 116–142

    Article  MathSciNet  MATH  Google Scholar 

  35. Muntean IL, Mehl M, Neckel T, Weinzierl T (2008) Concepts for efficient flow solvers based on adaptive cartesian grids. In: Wagner S, Steinmetz M, Bode A, Brehm M (eds) High performance computing in science and engineering, Garching 2007. Springer, Berlin

    Google Scholar 

  36. Neckel T (2009) The PDE framework Peano: an environment for efficient flow simulations. Verlag Dr. Hut, Munich

    Google Scholar 

  37. Neckel T, Mehl M, Bungartz H-J, Aoki T (2008) CFD simulations using an AMR-like approach in the PDE framework Peano. In: CFD2008. CFD22, December 2008

  38. Neckel T, Mehl M, Zenger C (2010) Enhanced divergence-free elements for efficient incompressible flow simulations in the PDE framework Peano. In: Proceedings of the fifth European conference on computational fluid dynamics, ECCOMAS CFD 2010, 14–17 June 2010, Lissabon

  39. Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2): 463–502

    Article  MathSciNet  MATH  Google Scholar 

  40. Roma AM, Peskin CS, Berger MJ (1999) An adaptive version of the immersed boundary method. J Comput Phys 153(2): 509–534

    Article  MathSciNet  MATH  Google Scholar 

  41. Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Annu Rev Fluid Mech 31: 567–603

    Article  MathSciNet  Google Scholar 

  42. Scheidegger AE (1974) The physics of flow through porous media. University of Toronto Press, Toronto

    Google Scholar 

  43. Sethian JA, Smereka P (2003) Level set methods for fluid interfaces. Annu Rev Fluid Mech 35(1): 341–372

    Article  MathSciNet  Google Scholar 

  44. Shin S, Abdel-Khalik SI, Daru V, Juric D (2005) Accurate representation of surface tension using the level contour reconstruction method. J Comput Phys 203(2): 493–516

    Article  MATH  Google Scholar 

  45. Shin S, Juric D (2002) Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity. J Comput Phys 180(2): 427–470

    Article  MATH  Google Scholar 

  46. Steensland J, Handra SC, Parashar M (2002) An application-centric characterization of domain-based SFC partitioners for parallel SAMR. IEEE Trans Parallel Distrib Syst 13(12): 1275–1289

    Article  Google Scholar 

  47. Sussman M, Fatemi E, Smereka P, Osher S (1997) An improved level set method for incompressible two-phase flows. Comput Fluids 27: 663–680

    Article  Google Scholar 

  48. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible 2-phase flow. J Comput Phys 114(1): 146–159

    Article  MATH  Google Scholar 

  49. Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Space–time finite element computation of complex fluid–structure interactions. Int J Numer Methods Fluids 64: 1201–1218

    Article  MATH  Google Scholar 

  50. Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Taubner W, Han J, Nas S, Jan YJ (2001) A front-tracking method for the computations of multiphase flow. J Comput Phys 169(2): 708–759

    Article  MATH  Google Scholar 

  51. Unverdi SO, Tryggvason G (1992) A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys 100(1): 25–37

    Article  MATH  Google Scholar 

  52. Wand CM, Tay ZY (2010) Hydroelastic analysis and response of pontoon-type very large floating structures. In: Bungartz H-J, Mehl M, Schäfer M (eds) Fluid–structure interaction—modelling, simulation, optimisation, part II. LNCSE. Springer, Berlin, pp 103–130

    Google Scholar 

  53. Weinzierl T (2005) Eine cache-optimale Implementierung eines Navier–Stokes Lösers unter besonderer Berücksichtigung physikalischer Erhaltungssätze. Diploma Thesis, Institut für Informatik, Technische Universität München

  54. Weinzierl T (2009) A framework for parallel PDE solvers on multiscale adaptive Cartesian grids. Verlag Dr. Hut, Munich

    Google Scholar 

  55. Weinzierl T, Mehl M (2011) Peano—a traversal and storage scheme for octree-like adaptive cartesian multiscale grids. SIAM J Sci Comput (accepted)

  56. Zumbusch GW (2003) Parallel multilevel methods advances in numerical mathematics. Teubner Verlag, Stuttgart

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Miriam Mehl.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bungartz, HJ., Gatzhammer, B., Lieb, M. et al. Towards multi-phase flow simulations in the PDE framework Peano. Comput Mech 48, 365–376 (2011). https://doi.org/10.1007/s00466-011-0626-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-011-0626-1

Keywords

  • PDE framework
  • Octree-like Cartesian grids
  • Computational fluid dynamics
  • Thermohydraulics
  • Two-phase flow