Fitted front tracking methods for two-phase ncompressible Navier-Stokes flow: Eulerian and ALE finite element discretizations. (English) Zbl 1471.76084

Summary: We investigate novel fitted finite element approximations for two-phase Navier-Stokes flow. In particular, we consider both Eulerian and Arbitrary Lagrangian-Eulerian (ALE) finite element formulations. The moving interface is approximated with the help of parametric piecewise linear finite element functions. The bulk mesh is fitted to the interface approximation, so that standard bulk finite element spaces can be used throughout. The meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments, including convergence experiments and benchmark computations, for the introduced numerical methods, which demonstrate the accuracy and robustness of the proposed algorithms. We also compare the accuracy and efficiency of the Eulerian and ALE formulations.


76T99 Multiphase and multicomponent flows
76M10 Finite element methods applied to problems in fluid mechanics
35Q30 Navier-Stokes equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: arXiv Link


[1] H. Abels, H. Garcke and G. Gr¨un, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012) 1150013. · Zbl 1242.76342
[2] M. Agnese, Front tracking finite element methods for two-phase Navier-Stokes flow, Ph.D. thesis, Imperial College London, London (2017).
[3] M. Agnese and R. N¨urnberg, Fitted finite element discretization of two-phase Stokes flow, Internat. J. Numer. Methods Fluids, 82 (2016) 709-729.
[4] S. Aland and A. Voigt, Benchmark computations of diffuse interface models for twodimensional bubble dynamics, Internat. J. Numer. Methods Fluids, 69 (2012) 747-761.
[5] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in Annual review of fluid mechanics, Vol. 30, Annual Reviews, Palo Alto, CA (1998) pages 139-165. · Zbl 1398.76051
[6] ’L. Baˇnas and R. N¨urnberg, Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow, M2AN Math. Model. Numer. Anal., 51 (2017) 1089- 1117.
[7] E. B¨ansch, Finite element discretization of the Navier-Stokes equations with a free capillary surface, Numer. Math., 88 (2001) 203-235. · Zbl 0985.35060
[8] J. W. Barrett, H. Garcke and R. N¨urnberg, A parametric finite element method for fourth order geometric evolution equations, J. Comput. Phys., 222 (2007) 441-462. · Zbl 1112.65093
[9] J. W. Barrett, H. Garcke and R. N¨urnberg, On the parametric finite element approximation of evolving hypersurfaces inR3, J. Comput. Phys., 227 (2008) 4281-4307. · Zbl 1145.65068
[10] J. W. Barrett, H. Garcke and R. N¨urnberg, Eliminating spurious velocities with a stable approximation of viscous incompressible two-phase Stokes flow, Comput. Methods Appl. Mech. Engrg., 267 (2013) 511-530. · Zbl 1286.76040
[11] J. W. Barrett, H. Garcke and R. N¨urnberg, On the stable numerical approximation of twophase flow with insoluble surfactant, M2AN Math. Model. Numer. Anal., 49 (2015) 421-458. · Zbl 1315.35156
[12] J. W. Barrett, H. Garcke and R. N¨urnberg, A stable parametric finite element discretization of two-phase Navier-Stokes flow, J. Sci. Comp., 63 (2015) 78-117. · Zbl 1320.76059
[13] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Kl¨ofkorn, R. Kornhuber, M. Ohlberger and O. Sander, A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part II: Implementation and Tests in DUNE, Computing, 82 (2008) 121-138. · Zbl 1151.65088
[14] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Kl¨ofkorn, M. Ohlberger and O. Sander, A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework, Computing, 82 (2008) 103-119. · Zbl 1151.65089
[15] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Comput. & Fluids, 31 (2002) 41-68. · Zbl 1057.76060
[16] F. Boyer and S. Minjeaud, Hierarchy of consistentn-component Cahn-Hilliard systems, Math. Models Methods Appl. Sci., 24 (2014) 2885-2928. · Zbl 1308.35004
[17] T. Brochu and R. Bridson, Robust topological operations for dynamic explicit surfaces, SIAM J. Sci. Comput., 31 (2009) 2472-2493. · Zbl 1195.65017
[18] T. A. Davis, Algorithm 832:UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004) 196-199. · Zbl 1072.65037
[19] T. A. Davis, Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization, ACM Trans. Math. Software, 38 (2011) 1-22. · Zbl 1365.65122
[20] K. Deckelnick, G. Dziuk and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14 (2005) 139-232. · Zbl 1113.65097
[21] A. Dedner, R. Kl¨ofkorn, M. Nolte and M. Ohlberger, A Generic Interface for Parallel and Adaptive Scientific Computing: Abstraction Principles and the DUNE-FEM Module, Computing, 90 (2010) 165-196. · Zbl 1201.65178
[22] H. Ding, P. D. Spelt and C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226 (2007) 2078-2095. · Zbl 1388.76403
[23] J. Donea, Arbitrary Lagrangian Eulerian methods, Comput. Meth. Trans. Anal., 1 (1983) 474-516. · Zbl 0536.73062
[24] S. Dong, An efficient algorithm for incompressible N-phase flows, J. Comput. Phys., 276 (2014) 691-728. · Zbl 1349.76196
[25] S. Dong, Physical formulation and numerical algorithm for simulating N immiscible incompressible fluids involving general order parameters, J. Comput. Phys., 283 (2015) 98-128. · Zbl 1351.76234
[26] S. Elgeti and H. Sauerland, Deforming fluid domains within the finite element method: five mesh-based tracking methods in comparison, Arch. Comput. Methods Eng., 23 (2016) 323- 361. · Zbl 1348.76099
[27] X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44 (2006) 1049- 1072. · Zbl 1344.76052
[28] L. Formaggia and F. Nobile, Stability analysis of second-order time accurate schemes for ALE-FEM, Comput. Methods Appl. Mech. Engrg., 193 (2004) 4097-4116. · Zbl 1175.76091
[29] S. Ganesan, Finite element methods on moving meshes for free surface and interface flows, Ph.D. thesis, University Magdeburg, Magdeburg, Germany (2006).
[30] S. Ganesan, A. Hahn, K. Simon and L. Tobiska, ALE-FEM for two-phase and free surface flows with surfactants, in Transport processes at fluidic interfaces, Adv. Math. Fluid Mech., Birkh¨auser/Springer, Cham (2017) pages 5-31. · Zbl 1444.76043
[31] S. Ganesan and L. Tobiska, An accurate finite element scheme with moving meshes for computing 3D-axisymmetric interface flows, Internat. J. Numer. Methods Fluids, 57 (2008) 119- 138. · Zbl 1138.76045
[32] H. Garcke, M. Hinze and C. Kahle, A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Appl. Numer. Math., 99 (2016) 151-171. · Zbl 1329.76168
[33] C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009) 1309-1331. · Zbl 1176.74181
[34] S. Groß and A. Reusken, An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys., 224 (2007) 40-58. · Zbl 1261.76015
[35] S. Groß and A. Reusken, Numerical methods for two-phase incompressible flows, volume 40 ofSpringer Series in Computational Mathematics, Springer-Verlag, Berlin (2011). · Zbl 1222.76002
[36] G. Gr¨un and F. Klingbeil, Two-phase flow with mass density contrast: Stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model, J. Comput. Phys., 257 (2014) 708-725. · Zbl 1349.76210
[37] A. Hahn, K. Held and L. Tobiska, ALE-FEM for two-phase flows, PAMM. Proc. Appl. Math. Mech., 13 (2013) 319-320.
[38] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39 (1981) 201-225. · Zbl 0462.76020
[39] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977) 435-479.
[40] T. Hughes, W. Liu and T. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29 (1981) 329-349. · Zbl 0482.76039
[41] S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman, S. Ganesan and L. Tobiska, Quantitative benchmark computations of two-dimensional bubble dynamics, Internat. J. Numer. Methods Fluids, 60 (2009) 1259-1288. · Zbl 1273.76276
[42] D. Kay, V. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier- Stokes system, Interfaces Free Bound., 10 (2008) 15-43.
[43] R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997) 709-735. · Zbl 0879.76061
[44] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998) 2617-2654. · Zbl 0927.76007
[45] F. Nobile, Numerical Approximation Of Fluid-Structure Interaction Problems With Application To Haemodynamics, Ph.D. thesis, EPFL, Lausanne (2001).
[46] F. Nobile and L. Formaggia, A stability analysis for the arbitrary lagrangian: Eulerian formulation with finite elements, East-West J. Numer. Math., 7 (1999) 105-132. · Zbl 0942.65113
[47] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, volume 153 of Applied Mathematical Sciences, Springer-Verlag, New York (2003). · Zbl 1026.76001
[48] C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002) 479-517. · Zbl 1123.74309
[49] S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys., 228 (2009) 5838-5866. · Zbl 1280.76020
[50] Y. Renardy and M. Renardy, PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method, J. Comput. Phys., 183 (2002) 400-421. · Zbl 1057.76569
[51] A. Sacconi, Front-tracking finite element methods for a void electro-stress migration problem, Ph.D. thesis, Imperial College London, London, UK (2015).
[52] A. Schmidt and K. G. Siebert, Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, volume 42 ofLecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin (2005). · Zbl 1068.65138
[53] J. A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge (1999). · Zbl 0929.65066
[54] M. Sussman, P. Semereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994) 146-159. · Zbl 0808.76077
[55] P. Sv´aˇcek, On numerical approximation of incompresible fluid flow with free surface influenced by surface tension, EPJ Web Conf., 143 (2017) 02122.
[56] G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas and Y.-J. Jan, A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169 (2001) 708-759. · Zbl 1047.76574
[57] S. O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible multifluid flows, J. Comput. Phys., 100 (1992) 25-37. · Zbl 0758.76047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.