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An Extended Discontinuous Galerkin Framework for Multiphase Flows

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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.

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References

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742 (1982)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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/S0036142901384162

    CrossRef  MathSciNet  MATH  Google Scholar 

  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-z

    MathSciNet  Google Scholar 

  7. Bluman, G., Cheviakov, A., Anco, S.: Applications of Symmetry Methods to Partial Differential Equations, vol. 168. Applied Mathematical Sciences. Springer, Berlin (2010)

    MATH  Google Scholar 

  8. Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized gaussian quadratures. SIAM J. Sci. Comput. 32(4), 1761 (2010). doi:10.1137/080737046

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2768

    MathSciNet  MATH  Google Scholar 

  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.2006.09.012

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2008.11.005

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. 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.2008.02.036

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Dziuk, G., Elliott, M.C.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Dziuk, G., Elliott, M.C.: Surface finite elements for parabolic equations. J. Comput. Math. 25, 385–407 (2007)

    MathSciNet  Google Scholar 

  16. Dziuk, G., Elliott, M.C.: Eulerian finite element method for parabolic PDEs on implicit surfaces. IMA J. Numer. Anal. 10, 119–138 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Dziuk, G., Elliott, C.M.: An Eulerian approach to transport and diffusion on evolving implicit surfaces. Comput. Vis. Sci. 13, 17–28 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2079

    CrossRef  MathSciNet  MATH  Google Scholar 

  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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Engwer, C.: An unfitted discontinuous Galerkin scheme for micro-scale simulations and numerical upscaling. Ph.D. thesis, Heidelberg (2009)

    Google Scholar 

  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)

    CrossRef  Google Scholar 

  22. Greer, J.B., Bertozzi, A., Sapiro, G.: Fourth order partial differential equations on general geometries. J. Comput. Phys. 216(1), 216–246 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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-y

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2005.06.020

    CrossRef  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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/27.1.87

    CrossRef  MATH  Google Scholar 

  27. Harper, J.F.: Stagnant-cap bubbles with both diffusion and adsorption rate-determining. J. Fluid Mech. 521, 115–123 (2004). doi:10.1017/S0022112004001843

    CrossRef  MathSciNet  MATH  Google Scholar 

  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/S1064827598337282

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Kallendorf, C.: An Eulerian discontinuous Galerkin method for the numerical simulation of interfacial transport. Ph.D. thesis, TU Darmstadt (2016)

    Google Scholar 

  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. 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. Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. 95(15), 8431–8435 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. Klein, B., Kummer, F., Oberlack, M.: A SIMPLE based discontinuous Galerkin solver for steady incompressible flows. J. Comput. Phys. 237, 235–250 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. Kummer, F.: Extended discontinuous Galerkin methods for two-phase flows: the spatial discretization. Int. J. Numer. Methods Eng. 109(2), 259–289 (2017)

    CrossRef  MathSciNet  Google Scholar 

  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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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/

    CrossRef  MathSciNet  Google Scholar 

  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.2012.06.001

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. 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.2005.213

    Google Scholar 

  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.2008.05.018

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2013.01.042

    CrossRef  MATH  Google Scholar 

  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. 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.2007.01.005

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2007.05.032

    Google Scholar 

  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.2008.07.021

    CrossRef  MathSciNet  MATH  Google Scholar 

  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)

    CrossRef  MATH  Google Scholar 

  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.2010.06.031

    CrossRef  MathSciNet  MATH  Google Scholar 

  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/s00466-010-0562-5

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. 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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.4569

    CrossRef  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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.2000.6636

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2013.04.036

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2013.04.010

    CrossRef  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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.2012.03.008

    CrossRef  Google Scholar 

  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. 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.2014.9.107

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2006.07.029

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2013.01.007

    MathSciNet  MATH  Google Scholar 

  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:1025328714359

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.1994.1155

    CrossRef  MATH  Google Scholar 

  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.4344

    CrossRef  Google Scholar 

  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/S0022112008005417

    Google Scholar 

  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/S0022112099005157

    CrossRef  MATH  Google Scholar 

  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/s10915-014-9856-7

    CrossRef  MathSciNet  MATH  Google Scholar 

  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.2009.10.027

    CrossRef  MathSciNet  MATH  Google Scholar 

  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:10.1137/090770291

    CrossRef  MathSciNet  MATH  Google Scholar 

  70. Zhao, H.: A fast sweeping method for Eikonal equations. Math. Comput. 74(250), 603–627 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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. 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

    CrossRef  MathSciNet  MATH  Google Scholar 

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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.

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Utz, T., Kallendorf, C., Kummer, F., Müller, B., Oberlack, M. (2017). An Extended Discontinuous Galerkin Framework for Multiphase Flows. In: Bothe, D., Reusken, A. (eds) Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-56602-3_3

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