This work presents a parallel finite element solver of incompressible two-phase flow targeting large-scale simulations of three-dimensional dynamics in high-throughput microfluidic separation devices. The method relies on a conservative level set formulation for representing the fluid-fluid interface and uses adaptive mesh refinement on forests of octrees. An implicit time stepping with efficient block solvers for the incompressible Navier–Stokes equations discretized with Taylor–Hood and augmented Taylor–Hood finite elements is presented. A matrix-free implementation is used that reduces the solution time for the Navier–Stokes system by a factor of approximately three compared to the best matrix-based algorithms. Scalability of the chosen algorithms up to 32,768 cores and a billion degrees of freedom is shown.
AdamsMBrezinaMHuJ. (2003) Parallel multigrid smoothing: Polynomial versus Gauss–Seidel. Journal of Computational Physics188: 593–610.
2.
AdelsbergerJEsserPGriebelM. (2014) 3D incompressible two-phase flow benchmark computations for rising droplets. In: OñateEOliverJHuertaA (eds) Proceedings of the 11th world congress on computational mechanics (WCCM XI), Barcelona, Spain, 20–25 July 2014, pp. 5274–5285. Barcelona: CIMNE.
3.
ArnoldDBrezziFCockburnB. (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis39: 1749–1779.
4.
BangerthWBursteddeCHeisterT. (2011) Algorithms and data structures for massively parallel generic finite element codes. ACM Transactions on Mathematical Software38(2): 14:1–14:28.
5.
BangerthWHartmannRKanschatG (2007) deal.II – A general purpose object oriented finite element library. ACM Transactions on Mathematical Software33(4): 24:1–24:27.
6.
BangerthWHeisterTHeltaiL. (2016) The dealii library, version 8.3. The Archive of Numerical Software4(100): 1–11.
7.
BasiniPBlitzCBozdağE. (2012) SPECFEM 3D user manual. Technical Report, Computational Infrastructure for Geodynamics, Princeton University, University of PAU, CNRS, and INRIA.
8.
BenziMGolubGHLiesenJ (2005) Numerical solution of saddle point problems. Acta Numerica14: 1–137.
9.
BoffiDCavalliniNGardiniF. (2012) Local mass conservation of Stokes finite elements. Journal of Scientific Computing52(2): 383–400.
10.
BrackbillJUKotheDBZemachC (1992) A continuum method for modeling surface tension. Journal of Computational Physics100: 335–354.
11.
BrownJ (2010) Efficient nonlinear solvers for nodal high-order finite elements in 3D. Journal of Scientific Computing45(1-3): 48–63.
12.
BursteddeCWilcoxLCGhattasO (2011) p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM Journal on Scientific Computing33(3): 1103–1133.
13.
CahouetJChabardJP (1988) Some fast 3D finite element solvers for the generalized Stokes problem. International Journal for Numerical Methods in Fluids8(8): 869–895.
14.
CantwellCDSherwinSJKirbyRM. (2011) From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements. Computers and Fluids43: 23–28.
15.
ElmanHSilvesterDWathenA (2005) Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford: Oxford Science Publications.
16.
FriesTPBelytschkoT (2010) The extended/generalized finite element method: An overview of the method and its applications. International Journal for Numerical Methods in Engineering84(3): 253–304.
17.
GalusinskiCVigneauxP (2008) On stability condition for bifluid flows with surface tension: Application to microfluidicsJournal of Computational Physics227: 6140–6164.
18.
GeeMWSiefertCMHuJJ. (2006) ML 5.0 smoothed aggregation user’s guide. Technical Report 2006-2649, Sandia National Laboratories, USA.
19.
GiraultVRaviartPA (1986) Finite Element Methods for the Navier–Stokes Equations. New York: Springer–Verlag.
20.
GroßSReuskenA (2007) An extended pressure finite element space for two-phase incompressible flows with surface tension. Journal of Computational Physics224: 40–58.
21.
HagerGWelleinG (2011) Introduction to High Performance Computing for Scientists and Engineers. Boca Raton, FL: CRC Press.
22.
HerouxMABartlettRAHowleVE. (2005) An overview of the Trilinos project. ACM Transactions on Mathematical Software31: 397–423.
23.
HirtCWNicholsBD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics39: 201–225.
24.
HysingS (2006) A new implicit surface tension implementation for interfacial flows. International Journal for Numerical Methods in Fluids51: 659–672.
25.
JacqminD (2000) Contact-line dynamics of a diffuse fluid interface. Journal of Fluid Mechanics402: 57–88.
26.
KarniadakisGESherwinSJ (2005) Spectral/hp Element Methods for Computational Fluid Dynamics. 2nd ed. Oxford: Oxford University Press.
27.
KormannKKronbichlerM (2011) Parallel finite element operator application: Graph partitioning and coloring. In: LaureEHenningsonD (eds) Proceedings of the 7th IEEE international conference on eScience, Stockholm, 5–8 December 2011, pp.332–339. Los Alamitos, CA: IEEE.
28.
KronbichlerMHeisterTBangerthW (2012) High accuracy mantle convection simulation through modern numerical methods. Geophysical Journal International191: 12–29.
29.
KronbichlerMKormannK (2012) A generic interface for parallel finite element operator application. Computers and Fluids63: 135–147.
30.
MayDABrownJLe PourhietL (2014) pTatin3D: High-performance methods for long-term lithospheric dynamics. In: KunkelJMLudwigTMeuerHW (eds) Supercomputing (SC14), New Orleans, LA, 16–21 November 2014, pp.1–11. Los Alamitos, CA: IEEE.
31.
OlssonEKreissG (2005) A conservative level set method for two phase flow. Journal of Computational Physics210: 225–246.
32.
OlssonEKreissGZahediS (2007) A conservative level set method for two phase flow II. Journal of Computational Physics225: 785–807.
33.
OsherSSethianJA (1988) Fronts propagating with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics79: 12–49.
34.
PattersonDAHennessyJL (2009) Computer Organization and Design. 4th ed.
Burlington, MA: Morgan Kaufmann.
35.
PeskinC (1977) Numerical analysis of blood flow in the heart. Journal of Computational Physics25: 220–252.
36.
PeskinC (2002) The immersed boundary method. Acta Numerica11: 479–517.
37.
RasthoferUHenkeFWallWA. (2011) An extended residual-based variational multiscale method for two-phase flow including surface tension. Computer Methods in Applied Mechanics and Engineering200: 1866–1876.
38.
SaadY (2003) Iterative Methods for Sparse Linear Systems. 2nd ed.
Philadelphia, PA: SIAM.
39.
SayeRI (2015) High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles. SIAM Journal on Scientific Computing37(2): 993–1019.
40.
SethianJA (2000) Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge: Cambridge University Press.
41.
SundarHBirosGBursteddeC. (2012) Parallel geometric-algebraic multigrid on unstructured forests of octrees. In: HollingsworthJ (ed.) SC12: Proceedings of the international conference for high performance computing, networking, storage and analysis, Salt Lake City, UT, 10–16 November 2012, pp.1–11. Los Alamitos, CA: IEEE.
42.
SussmanMOhtaM (2009) A stable and efficient method for treating surface tension in incompressible two-phase flow. SIAM Journal on Scientific Computing31(4): 2447–2471.
43.
TadmorE (1984) Skew-self adjoint form for systems of conservation laws. Journal of Mathematical Analysis and Applications103: 428–442.
44.
TuminaroRTongC (2000) Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines. In: DonnelleyJ (ed.) Super computing 2000 proceedings, Dallas, TX, 4–10 November 2000, pp.1–21. Los Alamitos, CA: IEEE.
45.
TurekS (1999) Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach. Berlin: Springer.
46.
UnverdiSOTryggvasonG (1992) A front-tracking method for viscous, incompressible, multi-fluid flows. Journal of Computational Physics100: 25–37.
47.
WuZWillingBBjerketorpJ. (2009) Soft inertial microfluidics for high throughput separation of bacteria from human blood cells. Lab on a Chip9(9): 1193–1199.
48.
ZahediSKronbichlerMKreissG (2012) Spurious currents in finite element based level set methods for two-phase flow. International Journal for Numerical Methods in Fluids69: 1433–1456.
