An Extended Discontinuous Galerkin Framework for Multiphase Flows

  • Thomas Utz
  • Christina Kallendorf
  • Florian Kummer
  • Björn Müller
  • Martin Oberlack
Chapter
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

Abstract

We present a framework for handling cut cells in a high order discontinuous Galerkin (DG) context. To describe the boundary between fluid phases, we use a level-set formulation. When the interface cuts a computational cell, we discretize the resulting sub-cells with the same DG method as used on standard cells. This requires a suitable quadrature procedure. Within this framework, we present a solver for the two-phase Navier-Stokes equation, a reinitialization procedure for the level-set and a solver for transport-processes on the surface.

Notes

Acknowledgements

This work is supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt. The work of T. Utz and C. Kallendorf is supported by the German Science Foundation (DFG) within the Priority Program (SPP) 1506 “Transport Processes at Fluidic Interfaces”. The work of F. Kummer is supported by the German DFG through Research Fellowship KU 2719/1-1. The work of B. Müller is supported by the German Research Foundation (DFG) through Research Grant WA 2610/2-1.

References

  1. 1.
    Anco, S., Bluman, G.: Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566 (2002)MATHGoogle Scholar
  2. 2.
    Anco, S., Bluman, G.: Direct construction method for conservation laws of partial differential equations. Part II: general treatment. Eur. J. Appl. Math. 13, 567–585 (2002)MATHGoogle Scholar
  3. 3.
    Anco, S.C., Bluman, G.W., Cheviakov, A.F.: Construction of conservation laws: how the direct method generalizes Noether’s theorem. In: Proceedings of 4th Workshop “Group Analysis of Differential Equations & Integrability”, vol. 1, pp. 1–23 (2009)Google Scholar
  4. 4.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742 (1982)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002). doi:10.1137/S0036142901384162MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Basting, C., Kuzmin, D.: A minimization-based finite element formulation for interface-preserving level set reinitialization. Computing 95(1), 13–25 (2012). doi:10.1007/s00607-012-0259-zMathSciNetGoogle Scholar
  7. 7.
    Bluman, G., Cheviakov, A., Anco, S.: Applications of Symmetry Methods to Partial Differential Equations, vol. 168. Applied Mathematical Sciences. Springer, Berlin (2010)MATHGoogle Scholar
  8. 8.
    Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized gaussian quadratures. SIAM J. Sci. Comput. 32(4), 1761 (2010). doi:10.1137/080737046MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cheng, K.W., Fries, T.P.: Higher-order XFEM for curved strong and weak discontinuities. Int. J. Numer. Methods Eng. 82(5), 564–590 (2010). doi:10.1002/nme.2768MathSciNetMATHGoogle Scholar
  10. 10.
    Cheng, Y., Shu, C.W.: A discontinuous Galerkin finite element method for directly solving the Hamilton–Jacobi equations. J. Comput. Phys. 223(1), 398–415 (2007). doi:10.1016/j.jcp.2006.09.012MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Desjardins, O., Pitsch, H.: A spectrally refined interface approach for simulating multiphase flows. J. Comput. Phys. 228(5), 1658–1677 (2009). doi:10.1016/j.jcp.2008.11.005MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69. Springer, Berlin (2011). http://books.google.de/books?id=ak-qQvWGA5oC
  13. 13.
    Düster, A., Parvizian, J., Yang, Z., Rank, E.: The finite cell method for three-dimensional problems of solid mechanics. Comput. Methods Appl. Mech. Eng. 197(45–48), 3768–3782 (2008). doi:10.1016/j.cma.2008.02.036MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dziuk, G., Elliott, M.C.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dziuk, G., Elliott, M.C.: Surface finite elements for parabolic equations. J. Comput. Math. 25, 385–407 (2007)MathSciNetGoogle Scholar
  16. 16.
    Dziuk, G., Elliott, M.C.: Eulerian finite element method for parabolic PDEs on implicit surfaces. IMA J. Numer. Anal. 10, 119–138 (2008)MathSciNetMATHGoogle Scholar
  17. 17.
    Dziuk, G., Elliott, C.M.: An Eulerian approach to transport and diffusion on evolving implicit surfaces. Comput. Vis. Sci. 13, 17–28 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Elias, R.N., Martins, M.A.D., Coutinho, A.L.G.A.: Simple finite element-based computation of distance functions in unstructured grids. Int. J. Numer. Methods Eng. 72(9), 1095–1110 (2007). doi:10.1002/nme.2079MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Elliott, M.C., Eilks, C.: Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J. Comput. Phys. 227, 9727–9741 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Engwer, C.: An unfitted discontinuous Galerkin scheme for micro-scale simulations and numerical upscaling. Ph.D. thesis, Heidelberg (2009)Google Scholar
  21. 21.
    Fröhlcke, A., Gjonaj, E., Weiland, T.: A boundary conformal DG approach for electro-quasistatics problems. In: Michielsen, B., Poirier, J.R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry, vol. 16, pp. 153–161. Springer, Berlin (2012)CrossRefGoogle Scholar
  22. 22.
    Greer, J.B., Bertozzi, A., Sapiro, G.: Fourth order partial differential equations on general geometries. J. Comput. Phys. 216(1), 216–246 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Groß, S., Reichelt, V., Reusken, A.: A finite element based level set method for two-phase incompressible flows. Comput. Vis. Sci. 9(4), 239–257 (2006). doi:10.1007/s00791-006-0024-yMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Grooss, J., Hesthaven, J.S.: A level set discontinuous Galerkin method for free surface flows. Comput. Methods Appl. Mech. Eng. 195(25–28), 3406–3429 (2006). doi:10.1016/j.cma.2005.06.020MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hadamard, J.: Mouvement permanent lent d’ une sphere liquide et visqueuse dans un liquide visqueux. C. R. Acad. Sci. Paris 152, 1735–1738 (1911)MATHGoogle Scholar
  26. 26.
    Harper, J.F.: On spherical bubbles rising steadily in dilute surfactant solutions. Q. J. Mech. Appl. Math. 27(1), 87–100 (1974). doi:10.1093/qjmam/27.1.87CrossRefMATHGoogle Scholar
  27. 27.
    Harper, J.F.: Stagnant-cap bubbles with both diffusion and adsorption rate-determining. J. Fluid Mech. 521, 115–123 (2004). doi:10.1017/S0022112004001843MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Hu, C., Shu, C.: A Discontinuous Galerkin finite element method for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(2), 666–690 (1999). doi:10.1137/S1064827598337282MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kallendorf, C.: An Eulerian discontinuous Galerkin method for the numerical simulation of interfacial transport. Ph.D. thesis, TU Darmstadt (2016)Google Scholar
  30. 30.
    Kallendorf, C., Cheviakov, A.F., Oberlack, M., Wang, Y.: Conservation laws of surfactant transport equations. Phys. Fluids 24(10), 102105 (2012). doi:http://dx.doi.org/10.1063/1.4758184. http://scitation.aip.org/content/aip/journal/pof2/24/10/10.1063/1.4758184
  31. 31.
    Kallendorf, C., Fath, A., Oberlack, M., Wang, Y.: Exact solutions to the interfacial surfactant transport equation on a droplet in a stokes flow regime. Phys. Fluids 27(8), 082104 (2015). doi:http://dx.doi.org/10.1063/1.4928547. http://scitation.aip.org/content/aip/journal/pof2/27/8/10.1063/1.4928547
  32. 32.
    Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. 95(15), 8431–8435 (1998)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Klein, B., Kummer, F., Oberlack, M.: A SIMPLE based discontinuous Galerkin solver for steady incompressible flows. J. Comput. Phys. 237, 235–250 (2013)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kummer, F.: Extended discontinuous Galerkin methods for two-phase flows: the spatial discretization. Int. J. Numer. Methods Eng. 109(2), 259–289 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kummer, F., Oberlack, M.: An extension of the discontinuous Galerkin method for the singular Poisson equation. SIAM J. Sci. Comput. 35(2), A603–A622 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kummer, F., Warburton, T.: Patch-recovery filters for curvature in discontinuous Galerkin-based level-set methods. Commun. Comput. Phys. 19(02), 329–353 (2016). http://tubiblio.ulb.tu-darmstadt.de/80852/ MathSciNetCrossRefGoogle Scholar
  37. 37.
    Legrain, G., Chevaugeon, N., Dréau, K.: High order x-FEM and levelsets for complex microstructures: uncoupling geometry and approximation. Comput. Methods Appl. Mech. Eng. 241–244, 172–189 (2012). doi:10.1016/j.cma.2012.06.001MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lenz, M., Nemadjieu, S., Rumpf, M.: Finite volume method on moving surfaces. In: Eymard, R., Hérald, J.M. (eds.) Finite Volumes for Complex Applications V, pp. 561–576. Wiley, New York (2008)Google Scholar
  39. 39.
    Li, C., Xu, C., Gui, C., Fox, M.: Level set evolution without re-initialization: a new variational formulation. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005. CVPR 2005, vol. 1, pp. 430–436 (2005). doi:10.1109/CVPR.2005.213Google Scholar
  40. 40.
    Li, F., Shu, C.W., Zhang, Y.T., Zhao, H.: A second order discontinuous Galerkin fast sweeping method for Eikonal equations. J. Comput. Phys. 227(17), 8191–8208 (2008). doi:10.1016/j.jcp.2008.05.018MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Luo, S.: A uniformly second order fast sweeping method for Eikonal equations. J. Comput. Phys. 241, 104–117 (2013). doi:10.1016/j.jcp.2013.01.042CrossRefMATHGoogle Scholar
  42. 42.
    Marchandise, E.: Simulation of three-dimensional two-phase flows: coupling of a stabilized finite element method with a discontinuous level set approach. Ph.D. thesis, Université Catholique de Louvain (2006)Google Scholar
  43. 43.
    Marchandise, E., Geuzaine, P., Chevaugeon, N., Remacle, J.F.: A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics. J. Comput. Phys. 225(1), 949–974 (2007). doi:10.1016/j.jcp.2007.01.005MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Min, C., Gibou, F.: Geometric integration over irregular domains with application to level-set methods. J. Comput. Phys. 226(2), 1432–1443 (2007). doi:16/j.jcp.2007.05.032Google Scholar
  45. 45.
    Min, C., Gibou, F.: Robust second-order accurate discretizations of the multi-dimensional heaviside and dirac delta functions. J. Comput. Phys. 227(22), 9686–9695 (2008). doi:10.1016/j.jcp.2008.07.021MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)CrossRefMATHGoogle Scholar
  47. 47.
    Mousavi, S.E., Sukumar, N.: Generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method. Comput. Methods Appl. Mech. Eng. 199(49–52), 3237–3249 (2010). doi:10.1016/j.cma.2010.06.031MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Mousavi, S.E., Sukumar, N.: Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput. Mech. 47(5), 535–554 (2011). doi:10.1007/s00466-010-0562-5MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Müller, B., Krämer-Eis, S., Kummer, F., Oberlack, M.: A high-order discontinuous Galerkin method for compressible flows with immersed boundaries. Int. J. Numer. Methods Eng. pp. n/a–n/a (2016). doi:10.1002/nme.5343. http://dx.doi.org/10.1002/nme.5343. Nme.5343
  50. 50.
    Müller, B., Kummer, F., Oberlack, M., Wang, Y.: Simple multidimensional integration of discontinuous functions with application to level set methods. Int. J. Numer. Methods Eng. 92(7), 637–651 (2012)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Müller, B., Kummer, F., Oberlack, M.: Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int. J. Numer. Methods Eng. 96(8), 512–528 (2013). doi:10.1002/nme.4569MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions, 1st edn. Cambridge University Press, New York, NY (2010)MATHGoogle Scholar
  53. 53.
    Osher, S., Fedkiw, R.P.: Level set methods: an overview and some recent results. J. Comput. Phys. 169(2), 463–502 (2001). doi:10.1006/jcph.2000.6636MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Owkes, M., Desjardins, O.: A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows. J. Comput. Phys. 249, 275–302 (2013). doi:10.1016/j.jcp.2013.04.036MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Pochet, F., Hillewaert, K., Geuzaine, P., Remacle, J.F., Marchandise, M.: A 3d strongly coupled implicit discontinuous Galerkin level set-based method for modeling two-phase flows. Comput. Fluids 87, 144–155 (2013). doi:10.1016/j.compfluid.2013.04.010MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)MATHGoogle Scholar
  57. 57.
    Qin, R., Krivodonova, L.: A discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. J. Comput. Sci. 4(1–2), 24–35 (2013). doi:10.1016/j.jocs.2012.03.008CrossRefGoogle Scholar
  58. 58.
    Rybczynski, W.: Über die fortschreitende bewegung einer flüssigen kugel in einem zähen medium. Bull. Acad. Sci. de Cracovie A 40–46 (1911)Google Scholar
  59. 59.
    Saye, R.: High-order methods for computing distances to implicitly defined surfaces. Commun. Appl. Math. Comput. Sci. 9(1), 107–141 (2014). doi:10.2140/camcos.2014.9.107MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Shahbazi, K., Fischer, P., Ethier, R.C.: A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations. J. Comput. Phys. 222(1), 391–407 (2007). doi:10.1016/j.jcp.2006.07.029MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Sudhakar, Y., Wall, W.A.: Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods. Comput. Methods Appl. Mech. Eng. (2013). doi:10.1016/j.cma.2013.01.007MathSciNetMATHGoogle Scholar
  62. 62.
    Sussman, M., Hussaini, M.Y.: A discontinuous spectral element method for the level set equation. J. Sci. Comput. 19(1–3), 479–500 (2003). doi:10.1023/A:1025328714359MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114(1), 146–159 (1994). doi:10.1006/jcph.1994.1155CrossRefMATHGoogle Scholar
  64. 64.
    Utz, T., Kummer, F., Oberlack, M.: Interface-preserving level-set reinitialization for DG-FEM. Int. J. Numer. Meth. Fluids 84(4), 183–198 (2017). doi:10.1002/fld.4344CrossRefGoogle Scholar
  65. 65.
    Vlahovska, P.M., Blawzdziewicz, J., Loewenberg, M.: Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624 (2009). doi:10.1017/S0022112008005417Google Scholar
  66. 66.
    Wang, Y., Papageorgiu, D.T., Maldarelli, C.: Increased mobility of a surfactant-retarded bubble at high bulk concentrations. J. Fluid Mech. 390, 251–270 (1999). doi:10.1017/S0022112099005157CrossRefMATHGoogle Scholar
  67. 67.
    Wu, L., Zhang, Y.T.: A Third order fast sweeping method with linear computational complexity for Eikonal equations. J. Sci. Comput. 62(1), 198–229 (2014). doi:10.1007/s10915-014-9856-7MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Xiao, H., Gimbutas, Z.: A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions. Comput. Math. Appl. 59(2), 663–676 (2010). doi:10.1016/j.camwa.2009.10.027MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Zhang, Y., Chen, S., Li, F., Zhao, H., Shu, C.: Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations. SIAM J. Sci. Comput. 33(4), 1873–1896 (2011). doi:10.1137/090770291MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    Zhao, H.: A fast sweeping method for Eikonal equations. Math. Comput. 74(250), 603–627 (2005)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996). doi:http://dx.doi.org/10.1006/jcph.1996.0167. http://www.sciencedirect.com/science/article/pii/S0021999196901679
  72. 72.
    Zienkiewicz, O., Zhu, J.: The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Eng. 101(1–3), 207–224 (1992). doi:10.1016/0045-7825(92)90023-D. http://linkinghub.elsevier.com/retrieve/pii/004578259290023D MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Thomas Utz
    • 1
  • Christina Kallendorf
    • 1
  • Florian Kummer
    • 1
  • Björn Müller
    • 1
  • Martin Oberlack
    • 1
  1. 1.Institute of Fluid DynamicsTechnische Universität DarmstadtDarmstadtGermany

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