# A Locally Gradient-Preserving Reinitialization for Level Set Functions

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## Abstract

The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. The gradient of the level set function represents the stretching of the interface, which is of critical importance in many physical applications. The proposed locally gradient-preserving reinitialization (LGPR) method involves the solution of three PDEs of Hamilton–Jacobi type in succession; first the signed distance function is found using a traditional reinitialization technique, then the interface gradient is extended into the domain by a transport equation, and finally the new level set function is found by solving a generalized reinitialization equation. We prove the well-posedness of the Hamilton–Jacobi equations, with possibly discontinuous Hamiltonians, and propose numerical schemes for their solutions. A subcell resolution technique is used in the numerical solution of the transport equation to extend data away from the interface directly with high accuracy. The reinitialization technique is computationally inexpensive if the PDEs are solved only in a small band surrounding the interface. As an important application, we show how the LGPR procedure can be used to make possible the local level set approach to the Eulerian Immersed boundary method.

## Keywords

Level-set method Eulerian immersed boundary method Fluid–structure interaction Reinitialization## Notes

### Acknowledgments

X. Xu acknowledges support by NSF-DMS Grant 1159133.

## References

- 1.Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. J. Comput. Phys.
**118**, 269–277 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Aujol, J.F., Aubert, G.: Signed distance functions and viscosity solutions of discontinuous Hamilton–Jacobi equations. Technical Report RR-4507, INRIA (2002)Google Scholar
- 3.Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
- 4.Bottino, D.C.: Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method. J. Comput. Phys.
**147**, 86–113 (1998)CrossRefzbMATHGoogle Scholar - 5.Chang, Y.C., Hou, T.Y., Merriman, B., Osher, S.: A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys.
**124**, 449–464 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Choi, H.I., Choi, S.W., Moon, H.P.: Mathematical theory of medial axis transform. Pacific J. Math.
**181**(1), 57–88 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Chrispell, J.C., Cortez, R., Khismatullin, D.B., Fauci, L.J.: Shape oscillations of a droplet in an Oldroyd-B fluid. Phys. D
**240**(20), 1593–1601 (2011)CrossRefzbMATHGoogle Scholar - 8.Chrispell, J.C., Fauci, L.J., Shelley, M.: An actuated elastic sheet interacting with passive and active structures in a viscoelastic fluid. Phys. Fluids.
**25**(1), 013,103 (2013)CrossRefzbMATHGoogle Scholar - 9.Cottet, G.H., Maitre, E.: A level-set formulation of immersed boundary methods for fluid–structure interaction problems. C. R. Acad. Sci. Paris
**338**, 581–586 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Cottet, G.H., Maitre, E.: A level set method for fluid–structure interactions with immersed surfaces. Math. Models Methods Appl. Sci.
**16**, 415–438 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Cottet, G.H., Maitre, E.: Eulerian formulation and level set models for incompressible fluid–structure interaction. Math. Model Numer. Anal.
**42**, 471–492 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Crandall, M.G., Lions, P.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc.
**277**, 1–42 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Crandall, M.G., Lions, P.: Two approximations of solutions of Hamilton–Jacobi equations. Math. Comput.
**43**, 1–19 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Deckelnick, K., Elliott, C.M.: Uniqueness and error analysis for Hamilton–Jacobi equations with discontinuities. Interfaces Free Bound
**6**, 329–349 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence, RI (2010)Google Scholar
- 16.Festa, A., Falcone, M.: An approximation scheme for an eikonal equation with discontinuous coefficient. SIAM J. Numer. Anal.
**52**(1), 236–257 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Guy, R.D., Thomases, B.: Computational challenges for simulating strongly elastic flows in biology. In: Spagnolie, S.E. (ed.) Complex Fluids in Biological Systems, pp. 359–397. Springer, New York (2015)Google Scholar
- 18.Harten, A.: ENO schemes with subcell resolution. J. Comput. Phys.
**83**, 148–184 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Ishii, H.: Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Fac. Sci. Eng. Chuo Univ.
**28**, 33–77 (1985)MathSciNetzbMATHGoogle Scholar - 20.Ishii, H.: Existence and uniqueness of solutions of Hamilton–Jacobi equations. Funkc. Ekvacio
**29**, 167–188 (1986)MathSciNetzbMATHGoogle Scholar - 21.Ishii, H.: A simple, direct proof of uniqueness for solutions of the Hamilton–Jacobi equations of Eikonal type. Proc. Am. Math. Soc.
**100**, 247–251 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Jin, S., Liu, H.L., Osher, S., Tsai, R.: Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. J. Comput. Phys.
**210**, 497–518 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys.
**59**(2), 308–323 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Koike, S.: A Beginner’s Guide to the Theory of Viscosity Solutions. Mathematical Society of Japan, Tokyo (2004)Google Scholar
- 25.Lai, M.C., Peskin, C.S.: An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys.
**160**, 705–719 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Li, Z., Zhao, H., Gao, H.: A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid. J. Comput. Phys.
**201**, 281–304 (1999)CrossRefzbMATHGoogle Scholar - 27.Lieutier, A.: Any open bounded subset of \(R^n\) has the same homotopy type as its medial axis. Comput. Aided Des.
**36**(11), 1029–1046 (2004)CrossRefGoogle Scholar - 28.Lions, P.L.: Generalized Solutions of Hamilton–Jacobi Equations. Cambridge University Press, Cambridge (1992)Google Scholar
- 29.Malladi, R., Sethian, J.A.: Image processing via level set curvature flow. Proc. Natl. Acad. Sci.
**92**, 7046–7050 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Min, C.: On reinitializing level set functions. J. Comput. Phys.
**229**, 2764–2772 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Mori, Y., Peskin, C.S.: Implicit second-order immersed boundary methods with boundary mass. Comput. Methods Appl. Mech. Eng.
**197**, 2049–2067 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Mushenheim, P.C., Pendery, J.S., Weibel, D.B., Spagnolie, S.E., Abbott, N.L.: Straining soft colloids in aqueous nematic liquid crystals. Proc. Natl. Acad. Sci.
**113**, 5564–5569 (2016)CrossRefGoogle Scholar - 33.Newren, E.P., Fogelson, A.L., Guy, R.D., Kirby, R.M.: Unconditionally stable discretizations of the immersed boundary equations. J. Comput. Phys.
**222**, 702–719 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Osher, S., Fedkiw, R.: Level set methods: an overview and some recent results. J. Comput. Phys.
**169**, 463–502 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 35.Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys.
**79**, 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 36.Osher, S., Shu, C.: Higher-order essentially nonoscillatory schemes for Hamilton–Jacobi equations. SIAM J. Numer. Anal.
**28**, 907–922 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 37.Ostrov, D.N.: Extending viscosity solutions to Eikonal equations with discontinuous spatial dependence. Nonlinear Anal.
**42**, 709–736 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 38.Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys.
**155**, 410–438 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 39.Peskin, C.S.: The immersed boundary method. Acta Numer.
**11**, 479–517 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 40.Salac, D., Miksis, M.: A level set projection model of lipid vesicles in general flows. J. Comput. Phys.
**230**(22), 8192–8215 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 41.Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci.
**93**, 1591–1595 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 42.Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, vol. 3. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
- 43.Soravia, P.: Optimal control with discontinuous running cost: Eikonal equation and shape-from-shading. In: Proceedings of 39th IEEE Conference on Decision and Control, 2000, vol 1, pp. 79–84. IEEE (2000)Google Scholar
- 44.Soravia, P.: Boundary value problems for Hamilton–Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J.
**51**, 451–476 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 45.Soravia, P.: Degenerate Eikonal equations with discontinuous refraction index. ESAIM: COCV
**12**, 216–230 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 46.Strychalski, W., Copos, C.A., Lewis, O.L., Guy, R.D.: A poroelastic immersed boundary method with applications to cell biology. J. Comput. Phys.
**282**, 77–97 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 47.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)CrossRefzbMATHGoogle Scholar - 48.Teran, J., Fauci, L., Shelley, M.: Peristaltic pumping and irreversibility of a Stokesian viscoelastic fluid. Phys. Fluids
**20**(073), 101 (2008)zbMATHGoogle Scholar - 49.Teran, J., Fauci, L., Shelley, M.: Viscoelastic fluid response can increase the speed and efficiency of a free swimmer. Phys. Rev. Lett.
**104**(3), 038,101 (2010)CrossRefGoogle Scholar - 50.Thomases, B., Guy, R.D.: Mechanisms of elastic enhancement and hindrance for finite-length undulatory swimmers in viscoelastic fluids. Phys. Rev. Lett.
**113**(9), 098,102 (2014)CrossRefGoogle Scholar - 51.Tsai, Y.H.R., Cheng, L.T., Osher, S., Zhao, H.K.: Fast sweeping algorithms for a class of Hamilton–Jacobi equations. SIAM J. Numer. Anal.
**41**(2), 673–694 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 52.Zhao, H.K.: A fast sweeping method for Eikonal equations. Math. Comput.
**74**(250), 603–627 (2005)MathSciNetCrossRefzbMATHGoogle Scholar