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High Order Schemes on Three-Dimensional General Polyhedral Meshes — Application to the Level Set Method

Published online by Cambridge University Press:  20 August 2015

Thibault Pringuey  and
R. Stewart Cant
Thibault Pringuey*
Affiliation:
CFD Laboratory, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
R. Stewart Cant*
Affiliation:
CFD Laboratory, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

In this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

Keywords

65M0876-0476N99WENO schemethree-dimensionalunstructured meshmixed elementpolyhedral elementhyperbolic equationslevel set
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 1 - 41
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 144 (1994), 45–58.CrossRefGoogle Scholar
[2]Barth, T. and Frederickson, P., High order solution of the Euler equations on unstructured grids using quadratic reconstruction, Tech. Rep., 90-0013, American Institute of Aeronautics and Astronautics, 1990.Google Scholar
[3]Billett, S. and Toro, E., Numerical Methods for Wave Propagation, Chap: unsplit WAF-type schemes for three-dimensional hyperbolic conservation laws, Kluwer Academic Publishers, 1998, pp. 75–124.Google Scholar
[4]Chang, Y., Hou, T., Merriman, B. and Osher, S., A level-set formulation of Eulerian capturing methods for incompressible fluid flows, J. Comput. Phys., 124 (1996), 449–464.Google Scholar
[5]Couderc, F., Développement d’un code de calcul pour la simulation d’écoulements de fluides non miscibles, Application à la désintegration assistée d’un jet liquide par un courant gazeux, Ph.D. thesis, Ecole Nationale Supérieurede l’Aéronautique et de l’Espace- Toulouse, 2007.Google Scholar
[6]Desjardins, O., Moureau, V., Knudsen, E., Herrmann, M. and Pitsch, H., Conservative level set/ghost fluid method for simulating primary atomization, Proceedings of the 20th Annual Conference of the Institute for Liquid Atomization and Spray Systems-Americas.Google Scholar
[7]Dumbser, M. and Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221 (2007), 693–723.CrossRefGoogle Scholar
[8]Dumbser, M., Käser, M., Titarev, V. and Toro, E., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226 (2007), 204–243.Google Scholar
[9]Enright, D., Fedkiw, R., Ferziger, J. and Mitchell, I., A hybrid particle level set method for improved interface capturing, J. Comput. Phys., 183 (2002), 83–116.Google Scholar
[10]Forsythe, G., Malcolm, M. and Moler, C., Computer Methods for Mathematical Computations, NJ: Prentice-Hall, 1977.Google Scholar
[11]Forsythe, G. and Moler, C., Computer Solution of Linear Algebraic Systems, NJ: Prentice-Hall, 1967.Google Scholar
[12]Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144 (1998), 194–212.Google Scholar
[13]Fuster, D., Bague, A., Boeck, T., Le, L. Moyne, Leboissetier, A., Popinet, S., Ray, P., Scardovelli, R. and Zaleski, S., Simulation of primary atomization with an octree adaptive mesh refinement and VOF method, Int. J. Multiphase Flow, 35 (2009), 550–565.CrossRefGoogle Scholar
[14]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes iii, J. Comput. Phys., 71 (1987), 231–303.Google Scholar
[15]Harten, A. and Osher, S., Uniformly high order accurate non-oscillatory schemes i, SIAM J. Numer. Anal., 24 (1987), 279–309.Google Scholar
[16]Herrmann, M., A Eulerian level set/vortex sheet method for two-phase interface dynamics, J. Comput. Phys., 203 (2005), 539–571.CrossRefGoogle Scholar
[17]Hu, C. and Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), 97–127.Google Scholar
[18]Jiang, G.-S. and Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2126–2143.Google Scholar
[19]Jiang, G.-S. an Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202.Google Scholar
[20]Käser, M. and Iske, A., ADER schemes on adaptive triangular meshes for scalar conservation laws, J. Comput. Phys., 205 (2005), 486–508.Google Scholar
[21]Kim, D., Desjardins, O., Herrmann, M. and Moin, P., The primary breakup of a round liquid jet by a coaxial flow of gas, Proceedings of the 20th Annual Conference of the Institute for Liquid Atomization and Spray Systems-Americas.Google Scholar
[22]Krommer, A. and Ueberhuber, C., Computational Integration, Society for Industrial and Applied Mathematics, 1998.CrossRefGoogle Scholar
[23]LeVeque, R., High-resolution conservative algorithm for advection in incompressible flow, SIAM J. Numer. Anal., 33 (1996), 627–665.Google Scholar
[24]Liu, X.-D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200.Google Scholar
[25]Menard, T., Tanguy, S. and Berlemont, A., Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary break-up of a liquid jet, Int. J. Multiphase Flow, 33 (2007), 510–524.CrossRefGoogle Scholar
[26]Ollivier-Gooch, C. and Altena, M. van, A high-order-accurate unstructured mesh finite-volume scheme for the advectiondiffusion equation, J. Comput. Phys., 181 (2002), 729–752.Google Scholar
[27]Olsson, E. and Kreiss, G., A conservative level set method for two phase flow, J. Comput. Phys., 210 (2005), 225–246.Google Scholar
[28]Penrose, R. and Todd, A., On best approximate solutions of linear matrix equations, Math. Proc. Cambridge, 52 (1956), 17–19.Google Scholar
[29]Pilliod, J. and Puckett, E., Second-order accurate volume-of-fluid algorithms for tracking material interfaces, J. Comput. Phys., 199 (2004), 465–502.Google Scholar
[30]Pringuey, T. and Cant, R., High order schemes on 3D mixed-element unstructured meshes, Tech. Rep. CUED/A-AERO/TR29, University of Cambridge, Department of Engineering (Sept. 2010).Google Scholar
[31]Rider, W. and Kothe, D., Stretching and tearing interface tracking methods, AIAA Computational Fluid Dynamics Conference, 12th, San Diego CA, June 1995, Collection of Technical Papers (AIAA-1995-1717)(1995), 806–816.Google Scholar
[32]Shu, C. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. Comput. Phys., 83 (1989), 32–78.Google Scholar
[33]Stroud, A., Approximate Calculation of Multiple Integrals, NJ: Prentice-Hall, 1971.Google Scholar
[34]Tanguy, S., Développement d’une méthode de suivi d’interface, Applcations aux écoulements diphasiques, Ph.D. thesis, Université de Rouen, 2004.Google Scholar
[35]Titarev, V. and Drikakis, D., Uniformly high order schemes on arbitrary unstructured meshes for advection-diffusion equations, Comput. Fluids, 46 (2011), 467–471.Google Scholar
[36]Titarev, V., Tsoutsanis, P. and Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8 (2010), 585–609.Google Scholar
[37]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, Verlag, 1965.Google Scholar
[38]Tsoutsanis, P., Titarev, V. and Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230 (2011), 1585–1601.Google Scholar
[39]Ushakova, O., Conditions of nondegeneracy of three-dimensional cells: a formula of a volume of cells, SIAM J. Sci. Comput., 23 (2001), 1274–1290.Google Scholar
[40]Zalesak, S., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31 (1979), 335–362.CrossRefGoogle Scholar
[41]Zhang, Y.-T. and Chu, C.-W., High order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM J. Sci. Comput., 24 (2003), 1005–1030.Google Scholar
[42]Zhang, Y.-T. and Chu, C.-W., Third order WENO scheme on three dimensional tetrahedral meshes, Commun. Comput. Phys., 5 (2009), 836–848.Google Scholar