Abstract
In this paper, we propose a numerical method for computing the signed distance function to a discrete domain, on an arbitrary triangular background mesh. It mainly relies on the use of some theoretical properties of the unsteady Eikonal equation. Then we present a way of adapting the mesh on which computations are held to enhance the accuracy for both the approximation of the signed distance function and the approximation of the initial discrete contour by the induced piecewise affine reconstruction, which is crucial when using this signed distance function in a context of level set methods. Several examples are presented to assess our analysis, in two or three dimensions.
Similar content being viewed by others
References
Alauzet, F., Frey, P.: Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Eng. 194, 5068–5082 (2005)
Alauzet, F., Frey, P.: Estimateur d’ erreur géometrique et métriques anisotropes pour l’ adaptation de maillage. Partie I: aspects théoriques. INRIA, Technical Report, p. 4759 (2003)
Anglada, M.V., Garcia, N.P., Crosa, P.B.: Directional adaptive surface triangulation. Comput. Aided Geom. Des. 16, 107–126 (1999)
Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Series of Advances in Numerical Mathematics. B.G. Teubner, Stuttgart, Leipzig (1999)
Aujol, J.F., Auber, G.: Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi equations. INRIA, Technical Report, p. 4507 (2002)
Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves and Surfaces. Graduate Texts in Mathematics. Springer, Berlin (1987)
Bui, T.T.C., Frey, P., Maury, B.: A coupling strategy for solving two-fluid flows. Int. J. Numer. Methods Fluids (2010). doi:10.1002/fld.2730
Cheng, L.-T., Tsai, Y.-T.: Redistancing by flow of time dependent eikonal equation. J. Comput. Phys. 227, 4002–4017 (2008)
Chopp, D.: Computing minimal surfaces via level-set curvature flow. J. Comput. Phys. 106, 77–91 (1993)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
Claisse, A., Frey, P.: Level set driven smooth curve approximation from unorganized or noisy point set. In: Proceedings ESAIM (2008)
Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)
D’Azevedo, E.F., Simpson, R.B.: On optimal triangular meshes for minimizing the gradient error. Numer. Math. 59, 321–348 (1991)
Delfour, M.C., Zolesio, J.-P.: Shapes and Geometries. SIAM, Philadelphia (2001)
Ducrot, V., Frey, P.: Contrôle de l’approximation géometrique d une interface par une métrique anisotrope. C. R. Acad. Sci., Ser. 1 Math. 345, 537–542 (2007)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1997)
Frey, P.J., George, P.L.: Mesh Generation: Application to Finite Elements, 2nd edn. Wiley, New York (2008)
Dobrzynski, C., Frey, P.: Anisotropic Delaunay mesh adaptation for unsteady simulations. In: Proc. of the 17th IMR, pp. 177–194 (2008)
Fuhrmann, A., Sobottka, G., Gross, C.: Abstract distance fields for rapid collision detection in physically based modeling. In: Proceedings of International Conference Graphicon (2003)
Gomes, A.J.P., Voiculescu, I., Jorge, J., Wyvill, B., Gallsbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. Springer, Berlin (2009)
Guigue, P., Devillers, O.: Fast and robust triangle-triangle overlap test using orientation predicates. J. Graph. GPU Game Tools 8, 25–42 (2003)
Vallet, M.G., Hecht, F., Mantel, B.: Anisotropic control of mesh generation based upon a Voronoi type method. In: Numerical Grid Generation in Computational Fluid Dynamics and Related Fields (1991)
Huang, W.: Metric tensors for anisotropic mesh generation. J. Comput. Phys. 204, 633–665 (2005)
Ishii, H.: Existence and uniqueness of solutions of Hamilton-Jacobi equations. Funkc. Ekvacioj 29, 167–188 (1986)
Jones, M.W.: 3D distance from a point to a triangle. Department of Computer Sciences, University of Wales Swansea, technical report (1995)
Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998)
Mantegazza, C., Mennucci, A.C.: Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47, 1–25 (2003)
Marchandise, E., Remacle, J.-F., Chevaugeon, N.: A quadrature free discontinuous Galerkin method for the level set equation. J. Comput. Phys. 212, 338–357 (2005)
Mut, F., Buscaglia, G.C., Dari, E.A.: A new mass-conserving algorithm for level-set redistancing on unstructured meshes. Mec. Comput. 23, 1659–1678 (2004)
Page, D.L., Sun, Y., Koschan, A.F., Paik, J., Abidi, M.A.: Normal vector voting: crease detection and curvature estimation on large, noisy meshes. Graph. Models 64, 199–229 (2004)
Qian, J., Zhang, Y.T., Zhao, H.: Fast sweeping methods for Eikonal equations on triangular meshes. SIAM J. Numer. Anal. 45, 83–107 (2007)
Osher, S.J., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2002)
Osher, S.J., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)
Memoli, F., Sapiro, G.: Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces. J. Comput. Phys. 173, 730–764 (2001)
Sethian, J.A.: Fast marching methods. SIAM Rev. 41, 199–235 (1999)
Sethian, J.A.: A fast marching method for monotonically advancing fronts. Proc. Natl. Acad. Sci. USA 93, 1591–1595 (1996)
Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)
Osher, S.J., Smereka, P., Sussman, M.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)
Strain, J.: Fast tree-based redistancing for level set computations. J. Comput. Phys. 152, 664–686 (1999)
Sussman, M., Fatemi, E.: An efficient, interface-preserving level set redistancing algorithm and its applications to interfacial incompressible fluid flow. SIAM J. Sci. Comput. 20, 1165–1191 (1999)
Sussman, M., Fatemi, E., Smereka, P., Osher, S.: An improved level-set method for incompressible two-phase flows. Comput. Fluids 27, 663–680 (1997)
Zhao, H.: A fast sweeping method for eikonal equations. Math. Comput. 74, 603–627 (2005)
Zhao, H., Osher, S.J., Fedkiw, R.: Fast surface reconstruction using the level set method. In: Proceedings of IEEE Workshop on Variational and Level Set Methods in Computer Vision (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is partially supported by the Chair “Mathematical modelling and numerical simulation, F-EADS, Ecole Polytechnique, INRIA”.
Rights and permissions
About this article
Cite this article
Dapogny, C., Frey, P. Computation of the signed distance function to a discrete contour on adapted triangulation. Calcolo 49, 193–219 (2012). https://doi.org/10.1007/s10092-011-0051-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10092-011-0051-z
Keywords
- Signed distance function
- Eikonal equation
- Level set method
- Anisotropic mesh adaptation
- ℙ1-finite elements interpolation