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A study of time integration schemes for the numerical modelling of ...

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A. MALIDI, S. DUFOUR AND D. N’DRI(a) (b) (c)Figure A2. Mass computation <strong>for</strong> each uid: (a) rst level; (b) second level; and (c) third level.ACKNOWLEDGEMENTSThis work was sponsored in part by <strong>the</strong> Auto21 Network <strong>of</strong> Centres <strong>of</strong> Excellence <strong>of</strong> Canada, <strong>the</strong>Natural Sciences and Engineering Research Council <strong>of</strong> Canada and <strong>the</strong> Fonds quebecois de la recherchesur la nature et les technologies.REFERENCES1. Floryan JM, Rasmussen H. Numerical methods <strong>for</strong> viscous ows with moving boundaries. In Applied MechanicsReview, Metzner AWK (ed.). ASME: New York, 1989; 323–341.2. Hirt CW, Nichols BD. Volume <strong>of</strong> uid (VOF) method <strong>for</strong> <strong>the</strong> dynamics <strong>of</strong> free boundaries. Journal <strong>of</strong>Computational Physics 1981; 39:201–225.3. Sethian JA. Level Set Methods. Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, andMaterials Science. Cambridge University Press: Cambridge, MA, 1999.4. Thompson E. Use <strong>of</strong> pseudo-concentration to follow creeping viscous ows during transient analysis.International Journal <strong>for</strong> Numerical Methods in Fluids 1986; 6:749–761.5. Dufour S, Malidi A. A free surface updating methodology <strong>for</strong> marker function-based Eulerian free surfacecapturing techniques on unstructured meshes. Communications in Numerical Methods in Engineering 2004;20:857–867.6. Dufour S, Pelletier D. Computations <strong>of</strong> multiphase ows with surface tension using an adaptive nite elementmethod. Numerical Heat Transfer, Part A 2001; 40:335–362.7. Johnson C. Numerical Solution <strong>of</strong> Partial Dierential Equations by <strong>the</strong> Finite Element Method. CambridgeUniversity Press: Cambridge, MA, 1987.8. Gao DM. A three-dimensional hybrid nite element-volume tracking model <strong>for</strong> mould lling in casting processes.International Journal <strong>for</strong> Numerical Methods in Fluids 1999; 29:877–895.9. Tezduyar ET, Aliabadi S. EDICT <strong>for</strong> 3D computation <strong>of</strong> two-uid interfaces. Computer Methods in AppliedMechanics and Engineering 2000; 190:403–410.10. Beliveau A, Fortin A, Demay Y. A two-dimensional <strong>numerical</strong> method <strong>for</strong> <strong>the</strong> de<strong>for</strong>mation <strong>of</strong> drops with surfacetension. International Journal <strong>of</strong> Computational Fluid Dynamics 1998; 10:225–240.11. Fortin M, Fortin A. A generalization <strong>of</strong> Uzawa’s algorithm <strong>for</strong> <strong>the</strong> solution <strong>of</strong> <strong>the</strong> Navier–Stokes equations.Communications in Applied Numerical Methods 1985; 1:205–208.12. Brackbill JU, Ko<strong>the</strong> DB, Zemach C. A continuum method <strong>for</strong> modeling surface tension. Journal <strong>of</strong>Computational Physics 1992; 100:335–354.13. Rudman M. A volume-tracking method <strong>for</strong> incompressible multiuid ows with large density variations.International Journal <strong>for</strong> Numerical Methods in Fluids 1998; 28:357–378.14. Williams MW, Ko<strong>the</strong> DB, Puckett EG. Accuracy and convergence <strong>of</strong> continuum surface-tension models. InFluid Dynamics at Interfaces, Shyy W, Narayanan R (eds). Cambridge University Press: Cambridge, MA,1999; 294–305.15. Popinet S, Zaleski S. A front-tracking algorithm <strong>for</strong> accurate representation <strong>of</strong> surface tension. InternationalJournal <strong>for</strong> Numerical Methods in Fluids 1999; 30:775–793.16. Dufour S, Pelletier D. A <strong>study</strong> <strong>of</strong> drop dynamics using an adaptive nite element method. 14th AIAAComputational Fluid Dynamics Conference, Norfolk, VA, AIAA Paper 99-3318, 1999; 12.Copyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)

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