Skip to main content
Springer Nature Link
Log in

Sixth-Order Accurate Schemes for Reinitialization and Extrapolation in the Level Set Framework

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The level set method is a common approach for handling moving boundary problems, which allows a moving, irregular surface to be described implicitly on a Cartesian grid. This approach often requires reinitialization of the level set function and extrapolation of fields defined only on the interface. Because many applications in physics and engineering involve calculation of second derivatives of the interface curvature and fourth order derivatives of surface fields, accurate simulations of these problems require high-order methods for reinitialization and extrapolation. Here we build off WENO schemes for Hamilton–Jacobi equations to develop novel sixth-order accurate methods for reinitialization and extrapolation. We present numerical results in three dimensional spaces demonstrating fourth-order accuracy of the interfacial curvature and sixth-order accuracy for the extrapolated surface fields. We then show that the extrapolation scheme can be integrated into the closest point method for surface PDEs and present an example of computing geodesic curves on surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Adalsteinsson, D., Sethian, J.A.: Transport and diffusion of material quantities on propagating interfaces via level set methods. J. Comput. Phys. 185(1), 271–288 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bardi, M., Osher, S.: The nonconvex multidimensional Riemann problem for Hamilton–Jacobi equations. SIAM J. Math. Anal. 22(2), 344–351 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, L.-T., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level-set approach. J. Comput. Phys. 175(2), 604–644 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chopp, D., Sethian, J.A.: Motion by intrinsic Laplacian of curvature. Interfaces Free Bound. 1(1), 107–123 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Coquerelle, M., Glockner, S.: A fourth-order accurate curvature computation in a level set framework for two-phase flows subjected to surface tension forces. J. Comput. Phys. 305, 838–876 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crane, K., Weischedel, C., Wardetzky, M.: Geodesics in heat: a new approach to computing distance based on heat flow. ACM Trans. Graph. (TOG) 32(5), 152 (2013)

    Article  Google Scholar 

  7. du Chéné, A., Min, C., Gibou, F.: Second-order accurate computation of curvatures in a level set framework using novel high-order reinitialization schemes. J. Sci. Comput. 35(2–3), 114–131 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB, vol. 6. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  9. Goldman, R.: Curvature formulas for implicit curves and surfaces. Comput. Aided Geom. Des. 22(7), 632–658 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Greer, J.B., Bertozzi, A.L., Sapiro, G.: Fourth order partial differential equations on general geometries. J. Comput. Phys. 216(1), 216–246 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guckenberger, A., Schraml, M.P., Chen, P.G., Leonetti, M., Gekle, S.: On the bending algorithms for soft objects in flows. Comput. Phys. Commun. 207, 1–23 (2016)

    Article  MATH  Google Scholar 

  12. Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes. I. SIAM J. Numer. Anal. 24(2), 279–309 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Helfrich, W.: Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift für Naturforschung C 28(11–12), 693–703 (1973)

    Article  Google Scholar 

  14. Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Laadhari, A., Saramito, P., Misbah, C., Székely, G.: Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods. J. Comput. Phys. 343, 271–299 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lipnikov, K., Morgan, N.: A high-order discontinuous Galerkin method for level set problems on polygonal meshes. J. Comput. Phys. 397, 108834 (2019)

    Article  MathSciNet  Google Scholar 

  18. Liu, X.-D., Osher, S., Chan, T., et al.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Macdonald, C.B., Ruuth, S.J.: Level set equations on surfaces via the closest point method. J. Sci. Comput. 35(2–3), 219–240 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Min, C.: On reinitializing level set functions. J. Comput. Phys. 229(8), 2764–2772 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155(2), 410–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163(1), 51–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Salac, D., Miksis, M.: A level set projection model of lipid vesicles in general flows. J. Comput. Phys. 230(22), 8192–8215 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sezgin, E., Levental, I., Mayor, S., Eggeling, C.: The mystery of membrane organization: composition, regulation and roles of lipid rafts. Nat. Rev. Cell Biol. 18(1), 361–374 (2017)

    Article  Google Scholar 

  26. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Smit, J., van Sint Annaland, M., Kuipers, J.A.M.: Grid adaptation with weno schemes for non-uniform grids to solve convection-dominated partial differential equations. Chem. Eng. Sci. 60(10), 2609–2619 (2005)

    Article  Google Scholar 

  28. Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114(1), 146–159 (1994)

    Article  MATH  Google Scholar 

  29. The Stanford 3D Scanning Repository. http://graphics.stanford.edu/data/3Dscanrep/

  30. Towers, J.D.: Two methods for discretizing a delta function supported on a level set. J. Comput. Phys. 220(2), 915–931 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, J.-J., Zhao, H.-K.: An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 19(1–3), 573–594 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhao, H.-K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhong-can, O.-Y., Helfrich, W.: Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39, 5280–5288 (1989)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Arizona, Tucson, AZ, 85721, USA

    Tiankui Zhang & Charles W. Wolgemuth

  2. Department of Molecular and Cellular Biology, University of Arizona, Tucson, AZ, 85721, USA

    Charles W. Wolgemuth

Authors
  1. Tiankui Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Charles W. Wolgemuth
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Charles W. Wolgemuth.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by National Institute of Health (NIH) Grant U54CA210172.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, T., Wolgemuth, C.W. Sixth-Order Accurate Schemes for Reinitialization and Extrapolation in the Level Set Framework. J Sci Comput 83, 26 (2020). https://doi.org/10.1007/s10915-020-01210-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01210-3

Keywords

Mathematics Subject Classification