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Proceedings of the 17th International Meshing Roundtable pp 159–176Cite as

Anisotropic Level Set Adaptation for Accurate Interface Capturing

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Summary

In fluid-structure interactions and fluid simulations, like incompressible two-phase flows involving high viscosity and density ratios, interface capturing and tracking is considered as a very challenging problem and has a strong impact on industrial applications. Usually, an adaptive grid is used to resolve the problem as well as to track the interface. In this paper, we describe an adaptive scheme based on the definition of an anisotropic metric tensor to control the generation of highly streched elements near an interface described with a level set function. In our approach, quasi-uniform anisotropic meshes are created with the objective of minimizing the interpolation errors by capturing the interface features using curvature-adapted anisotropic elements. The accuracy of the method is verified and numerical experiments are presented to show its efficiency.

Research supported by the INRIA ARC-LMN and a BQR grant from UPMC Scientific committee.

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References

  1. Abgrall, R.: Numerical discretisation of boundary conditions for first order Hamilton Jacobi equation n triangular meshes. Comm. Pure Appl. Maths. 49, 1339–1373 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. Journal of Computational Physics 118(2), 269–277 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alauzet, F., Frey, P.: Estimateur d’erreur géométrique et métriques anisotropes pour l’adaptation de maillage. Partie I: aspects théoriques, preprint INRIA, RR-4759 (2003)

    Google Scholar 

  4. Alauzet, F., Frey, P., George, P.L., Mohammadi, B.: 3D transient fixed-point mesh adaptation for time dependent problems: application to CFD simulations. J. Comp. Phys. 222, 592–623 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Apel, T.: Anisotropic finite elements: local estimates and applications, B.G. Teubner Stuttgart, Leipzig, Series of Advances in Numerical Mathematics (1999)

    Google Scholar 

  6. Apel, T., Berzins, M., Jimack, P.K., Kunert, G., Plaks, A., Tsukerman, I., Walkley, M.: Mesh shape and anisotropic elements: theory and practice. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications X, pp. 367–376. Elsevier, Oxford (2000)

    Chapter  Google Scholar 

  7. DAzevedo, E.F., Simpson, R.B.: On optimal triangular meshes for minimizing the gradient error. Numer. Math. 59, 321–348 (1991)

    Article  MathSciNet  Google Scholar 

  8. Babuška, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bui, T.T.C., Frey, P., Maury, B.: Méthode du second membre modifié pour la gestion de rapports de viscosité importants dans le problme de Stokes bifluide. C.R. Acad. Sci. Série I, 524–529 (2008)

    Google Scholar 

  10. Crandall, M., Lions, J.L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43, 1–19 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dervieux, A., Thomasset, F.: Multifluid incompressible flows by a finite element method. Lecture Notes in Physics 11, 158–163 (1981)

    Article  Google Scholar 

  12. Dompierre, J., Vallet, M.-G., Bourgault, Y., Fortin, M., Habashi, W.G.: Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part III: unstructured meshes. Int. J. Numer. Meth. Fluids 39, 675–702 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dijkstra, N.: A note on two problems in connexion with graphs. Numerische Mathematik 1(1), 269–271 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ducrot, V., Frey, P.: Contrôle de l’approximation géométrique d’une interface par une métrique anisotrope. C.R. Acad. Sci. 345, Série I, 537–542 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Enright, D., Fedkiw, R.: Robust treatment of interfaces for fluid flows and computer graphics. In: Hou, T., Tadmor, E. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 153–164. Springer, New York (2003)

    Google Scholar 

  16. Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89, 641–667 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Fortin, M.: Etude numérique d’estimations d’erreur a posteriori. REEF 9, 467–486 (2000)

    MATH  Google Scholar 

  18. Frey, P.J., George, P.L.: Mesh generation. Application to finite elements, 2nd edn. Wiley, Chichester (2008)

    MATH  Google Scholar 

  19. Frey, P., Alauzet, F.: Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Engrg. 194, 5068–5082 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Frey, P.J.: A differential approach to mesh generation. In: Ciarlet, P.G., Li, T. (eds.). Series in Contemporary Applied Mathematics. World Scientific, Singapore (2008)

    Google Scholar 

  21. Guèziec, A.: Meshsweeper: Dynamic Point-to-Polygonal-Mesh Distance and Applications. IEEE Trans. Visualization and Computer Graphics 7(1) (2001)

    Google Scholar 

  22. Hoch, P., Rascle, M.: Hamilton-Jacobi equations on a manifold and applications to grid generation or refinement. SIAM J. Sci. Comput. 23(6), 2055–2073 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huang, W.: Metric tensors for anisotropic mesh generation. J. Comput. Phys. 204, 633–665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiang, G.S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jiao, X., Colombi, A., Ni, X., Hart, J.C.: Anisotropic mesh adaptation for evolving triangulated surfaces. In: Proc. 15th Int. meshing Roundtable, pp. 173–190 (2006)

    Google Scholar 

  26. Kim, B., Liu, Y., Llamas, I., Rossignac, J.: Advections with significantly reduced dissipation and diffusion. IEEE Transactions on Visualization and Computer Graphics (2006)

    Google Scholar 

  27. Nagrath, S., Jansen, K.E., Lahey, R.T.: Computation of incompressible bubble dynamics with a stabilized finite element level set method. Comput. Methods Appl. Mech. Engrg. 194, 4565–4587 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys. 79, 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Osher, S., Shu, C.W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28, 907–922 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rudman, M.: Volume tracking methods for interfacial flow calculations. Int. Journal for Numerical Methods in Fluids 24(7), 671–691 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sethian, J.: Level set methods and fast marching methods. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  32. Smereka, P.: Semi-implicit level set method for curvature and surface diffusion motion. Journal of Scientific Computing 19, 439–456 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)

    Article  MATH  Google Scholar 

  34. Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. on Automatic Control 40(9), 1528–1538 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yamakawa, S., Shimada, K.: High quality anisotropic tetrahedral mesh generation via ellipsoidal bubble packing. In: Proc. 9th Int. Meshing Roundtable, pp. 263–273 (2000)

    Google Scholar 

  36. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. Journal of Computational Physics 31(3), 335–362 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zheng, X., Lowengrub, J., Anderson, A., Cristini, V.: Adaptive unstructured volume remshing: II: application to two and three dimensional level-set simulations of multiphase flow. J. Comput. Phys. 208, 626–650 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhu, J.Z., Zienkiewicz, O.C.: Superconvergence recovery technique and a posteriori error estimators. Inter. J. Numer. Meth. Engrg. 30, 1321–1339 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire J.L. Lions, UPMC Univ Paris 06, UMR 7598, F-75005, Paris, France

    V. Ducrot & P. Frey

  2. Centro de Modelamiento Matemático, Universidad de Chile, UMI 2807, Santiago, Chile

    P. Frey

Authors
  1. V. Ducrot
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  2. P. Frey
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Editors and Affiliations

  1. Sandia National Laboratories, Albuquerque, NM, USA

    Rao V. Garimella

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© 2008 Springer-Verlag Berlin Heidelberg

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Ducrot, V., Frey, P. (2008). Anisotropic Level Set Adaptation for Accurate Interface Capturing . In: Garimella, R.V. (eds) Proceedings of the 17th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87921-3_10

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  • DOI: https://doi.org/10.1007/978-3-540-87921-3_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-87920-6

  • Online ISBN: 978-3-540-87921-3

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