Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

An extrapolative approach to integration over hypersurfaces in the level set framework


Authors: Catherine Kublik and Richard Tsai
Journal: Math. Comp.
MSC (2010): Primary 65D30, 65M06
DOI: https://doi.org/10.1090/mcom/3282
Published electronically: January 2, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g., curves in two dimensions that contain a finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.


References [Enhancements On Off] (What's this?)

  • [1] M. Burger, O.L. Elvetun, and M. Schlottbom, Analysis of the diffuse domain method for second order elliptic boundary value problems, Foundations of Computational Mathematics (2015), 1-48.
  • [2] C. Chen, C. Kublik, and R. Tsai, An implicit boundary integral method for interfaces evolving by Mullins-Sekerka dynamics, Submitted.
  • [3] C. Chen and R. Tsai, Implicit boundary integral methods for the Helmholtz equation in exterior domains, UCLA CAM Report 16-38.
  • [4] Li-Tien Cheng and Yen-Hsi Tsai, Redistancing by flow of time dependent eikonal equation, J. Comput. Phys. 227 (2008), no. 8, 4002-4017. MR 2403875, https://doi.org/10.1016/j.jcp.2007.12.018
  • [5] John Dolbow and Isaac Harari, An efficient finite element method for embedded interface problems, Internat. J. Numer. Methods Engrg. 78 (2009), no. 2, 229-252. MR 2510520, https://doi.org/10.1002/nme.2486
  • [6] Björn Engquist, Anna-Karin Tornberg, and Richard Tsai, Discretization of Dirac delta functions in level set methods, J. Comput. Phys. 207 (2005), no. 1, 28-51. MR 2143581, https://doi.org/10.1016/j.jcp.2004.09.018
  • [7] Catherine Kublik, Nicolay M. Tanushev, and Richard Tsai, An implicit interface boundary integral method for Poisson's equation on arbitrary domains, J. Comput. Phys. 247 (2013), 279-311. MR 3066175, https://doi.org/10.1016/j.jcp.2013.03.049
  • [8] Catherine Kublik and Richard Tsai, Integration over curves and surfaces defined by the closest point mapping, Res. Math. Sci. 3 (2016), Paper No. 3, 17. MR 3486053, https://doi.org/10.1186/s40687-016-0053-1
  • [9] W. Lorensen and H. Cline, Marching cubes: a high resolution 3d surface construction algorithm, Computer Graphics 21 (1987).
  • [10] Emmanuel Maitre and Fadil Santosa, Level set methods for optimization problems involving geometry and constraints. II. Optimization over a fixed surface, J. Comput. Phys. 227 (2008), no. 22, 9596-9611. MR 2467635, https://doi.org/10.1016/j.jcp.2008.07.011
  • [11] B. Müller, F. Kummer, and M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Internat. J. Numer. Methods Engrg. 96 (2013), no. 8, 512-528. MR 3130061, https://doi.org/10.1002/nme.4569
  • [12] Stanley Osher and Ronald Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 153, Springer-Verlag, New York, 2003. MR 1939127
  • [13] Stanley J. Osher and Fadil Santosa, Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum, J. Comput. Phys. 171 (2001), no. 1, 272-288. MR 1843648, https://doi.org/10.1006/jcph.2001.6789
  • [14] Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12-49. MR 965860, https://doi.org/10.1016/0021-9991(88)90002-2
  • [15] Giovanni Russo and Peter Smereka, A remark on computing distance functions, J. Comput. Phys. 163 (2000), no. 1, 51-67. MR 1777721, https://doi.org/10.1006/jcph.2000.6553
  • [16] Steven J. Ruuth and Barry Merriman, A simple embedding method for solving partial differential equations on surfaces, J. Comput. Phys. 227 (2008), no. 3, 1943-1961. MR 2450979, https://doi.org/10.1016/j.jcp.2007.10.009
  • [17] R. I. Saye, High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput. 37 (2015), no. 2, A993-A1019. MR 3338676, https://doi.org/10.1137/140966290
  • [18] P. Schwartz, D. Adalsteinsson, P. Collela, A. P. Arkin, and M. Onsum, Numerical computation of diffusion on a surface, Proc. Natl. Acad. Sci. USA 102 (2005), 11151-11156.
  • [19] J. A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), no. 4, 1591-1595. MR 1374010, https://doi.org/10.1073/pnas.93.4.1591
  • [20] J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 2nd ed., Cambridge Monographs on Applied and Computational Mathematics, vol. 3, Cambridge University Press, Cambridge, 1999. MR 1700751
  • [21] Peter Smereka, The numerical approximation of a delta function with application to level set methods, J. Comput. Phys. 211 (2006), no. 1, 77-90. MR 2168871, https://doi.org/10.1016/j.jcp.2005.05.005
  • [22] John D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007), no. 2, 915-931. MR 2284331, https://doi.org/10.1016/j.jcp.2006.05.037
  • [23] Yen-Hsi Richard Tsai, Li-Tien Cheng, Stanley Osher, and Hong-Kai Zhao, Fast sweeping algorithms for a class of Hamilton-Jacobi equations, SIAM J. Numer. Anal. 41 (2003), no. 2, 673-694. MR 2004194, https://doi.org/10.1137/S0036142901396533
  • [24] John N. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. Automat. Control 40 (1995), no. 9, 1528-1538. MR 1347833, https://doi.org/10.1109/9.412624
  • [25] Xin Wen, High order numerical quadratures to one dimensional delta function integrals, SIAM J. Sci. Comput. 30 (2008), no. 4, 1825-1846. MR 2407143, https://doi.org/10.1137/070682939
  • [26] Xin Wen, High order numerical methods to three dimensional delta function integrals in level set methods, SIAM J. Sci. Comput. 32 (2010), no. 3, 1288-1309. MR 2652078, https://doi.org/10.1137/090758295
  • [27] Xin Wen, High order numerical methods to two dimensional delta function integrals in level set methods, J. Comput. Phys. 228 (2009), no. 11, 4273-4290. MR 2524518, https://doi.org/10.1016/j.jcp.2009.03.004
  • [28] Sara Zahedi and Anna-Karin Tornberg, Delta function approximations in level set methods by distance function extension, J. Comput. Phys. 229 (2010), no. 6, 2199-2219. MR 2586244, https://doi.org/10.1016/j.jcp.2009.11.030

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65D30, 65M06

Retrieve articles in all journals with MSC (2010): 65D30, 65M06


Additional Information

Catherine Kublik
Affiliation: Department of Mathematics, University of Dayton, 300 College Park, Dayton, Ohio 45469
Email: ckublik1@udayton.edu

Richard Tsai
Affiliation: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden — and — Department of Mathematics and Institute for Computational Engineering and Sciences, University of Texas at Austin, 2515 Speedway, Austin, Texas 78712
Email: ytsai@ices.utexas.edu

DOI: https://doi.org/10.1090/mcom/3282
Received by editor(s): November 3, 2016
Received by editor(s) in revised form: March 6, 2017, and April 4, 2017
Published electronically: January 2, 2018
Additional Notes: The first author was supported by a University of Dayton Research Council Seed Grant.
The second author was supported partially by a National Science Foundation Grant DMS-1318975 and an ARO Grant no. W911NF-12-1-0519. He also thanks National Center for Theoretical Sciences, Taiwan, for hosting his stay at the center where part of the research for this paper was conducted.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society