Skip to main content

Part of the book series: Topics in Biomedical Engineering ((TOBE))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

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

    Article  MathSciNet  Google Scholar 

  2. D. Adalsteinsson and J.A. Sethian. The fast construction of extension velocities in level set methods. Journal of Computational Physics, 48(l):2–22, 1999.

    MathSciNet  Google Scholar 

  3. S. Chen, B. Merriman, M. Kang, R. E. Caflisch, C. Ratsch, L. T. Cheng, M. Gyure, R. P. Fedkiw, C. Anderson, and S. Osher. A level set method for thin film epitaxial growth. J. Comput. Phys., 167:475–500, 2001.

    Article  Google Scholar 

  4. D. L. Chopp. Computing minimal surfaces via level set curvature flow. Journal of Computational Physics, 106(1):77–91, May 1993.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. L. Chopp. Numerical computation of self-similar solutions for mean curvature flow. Journal of Experimental Mathematics, 3(1):1–15, 1994.

    MathSciNet  MATH  Google Scholar 

  6. D. L. Chopp. A level-set method for simulating island coarsening. Journal of Computational Physics, 162:104–122, 2000.

    Article  MATH  Google Scholar 

  7. D. L. Chopp. Some improvements of the fast marching method. SIAM J. Sci. Comp., 23(1):230–244, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. L. Chopp, L. C. Evans, and H. Ishii. Waiting time effects for Gauss curvature flows. Indiana U. Math Journal, 48(1):311–334, 1999.

    MathSciNet  Google Scholar 

  9. D. L. Chopp and J. A. Sethian. Flow under curvature: Singularity formation, minimal surfaces, and geodesics. Journal of Experimental Mathematics, 2(4):235–255, 1993.

    MathSciNet  Google Scholar 

  10. D. L. Chopp and J. A. Sethian. Motion by intrinsic Laplacian of curvature. Interfaces and Free Boundaries, 1:107–123, 1999.

    MathSciNet  Google Scholar 

  11. D. L. Chopp and N. Sukumar. Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. SIAM Journal of Scientific Computing, 2001. submitted, preprint available at http://www.esam.northwestern.edu/chopp/pubs.html.

  12. D. L. Chopp, A. Tongen, and J. A. Sethian. Fast approximations of surface diffusion, preprint, 2000.

    Google Scholar 

  13. D.L. Chopp. Computing minimal surfaces via level set curvature flow. Journal of Computational Physics, 106:77–91, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. H. Chung. A level set approach for computing solutions to inviscid compressible flow with moving solid boundary using fixed cartesian grids. Int. J. Numer. Methods Fluids, 36:373–389, 2001.

    Article  MATH  Google Scholar 

  15. C. Daux, N. Moës, J. Dolbow, N. Sukumar, and T. Belytschko. Arbitrary cracks and holes with the extended finite element method. International Journal for Numerical Methods in Engineering, 48(12):1741–1760, 2000.

    Article  Google Scholar 

  16. J. W. Deng and H. T. Tsui. A fast level set method for segmentation of low contrast noisy biomedical images. Pattern Recognit. Lett., 23:161–169, 2002.

    Article  Google Scholar 

  17. T. Deschamps and L. D. Cohen. Fast extraction of minimal paths in 3d images and applications to virtual endoscopy. Med. Image Anal., 5:281–299, 2001.

    Article  Google Scholar 

  18. J. Dolbow, N. Moës, and T. Belytschko. Discontinuous enrichment in finite elements with a partition of unity method. Finite Elements in Analysis and Design, 36:235–260, 2000.

    Article  Google Scholar 

  19. Q. Du, D. Z. Li, Y. Y. Li, R. Li, and P. W. Zhang. Simulating a double casting technique using level set method. Comput. Mater. Sci., 22:200–212, 2001.

    Article  Google Scholar 

  20. T. Elperin and A. Vikhansky. Variational model of granular flow in a three-dimensional rotating container. Physica A, 303:48–56, 2002.

    Article  MathSciNet  Google Scholar 

  21. M. Gage. An isoperimetric inequalitywith aplications to curve shortening. Duke Math Journal, 50(4):1225–1229, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Gage. Curve shortening makes convex curves circular. Inventiones Mathematica, 76(2):357–364, 1984.

    MathSciNet  MATH  Google Scholar 

  23. M. Gage and R. S. Hamilton. The equation shrinking convex planes curves. Journal of Differential Geometry, 23:69–96, 1986.

    MathSciNet  Google Scholar 

  24. A. Gravouil, N. Moës, and T. Belytschko. Non-planar 3d crack growth by the extended finite element and the level sets. Part II: Level set update and discretization techniques. 2001. preprint.

    Google Scholar 

  25. M. Grayson. The heat equation shrinks embedded plane curves to round points. Journal of Differential Geometry, 26(285):285–314, 1987.

    MathSciNet  MATH  Google Scholar 

  26. A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy. Uniformly high-order accurate essentially nonoscillatory schemes. Journal of Computational Physics, 71(2):231–303, 1987.

    Article  MathSciNet  Google Scholar 

  27. G. Huisken. Asymptotic behavior for singularities of the mean curvature flow. Journal of Differential Geometry, 31:285–299, 1991.

    MathSciNet  Google Scholar 

  28. H. Ji, D. Chopp, and J. E. Dolbow. A hybrid extended finite element/level set method for modeling phase transformations. International Journal of Numerical Methods for Engineering, 2001. to appear.

    Google Scholar 

  29. M. Khenner, A. Averbuch, M. Israeli, and M. Nathan. Numerical simulation of grain-boundary grooving by level set method. J. Comput. Phys., 170:764–784, 2001.

    Article  Google Scholar 

  30. S. Kodambaka, V. Petrova, A. Vailionis, P. Desjardins, I. Petrov, and J. E. Greene. Time sequence images of annealing TiN(002) at 800°. Technical report, University of Illinois Urbana-Champaign, Dept. of Materials Science and Engineering and Matierals Research Laboratory, 1999. To be published.

    Google Scholar 

  31. R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhäuser Verlag, 1992.

    Google Scholar 

  32. R. Malladi, J.A. Sethian, and B.C. Vemuri. Evolutionary fronts for topology-independent shape modeling and recovery. In Proceedings of Third European Conference on Computer Vision, Stockholm, Sweden, Lecture Notes in Computer Science, volume 800, pages 3–13, 1994.

    Google Scholar 

  33. J. M. Melenk and I. Babuska. The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139:289–314, 1996.

    Article  MathSciNet  Google Scholar 

  34. B. Merriman, J. Bence, and S.J. Osher. Motion of multiple junctions: A level set approach. Journal of Computational Physics, 112:334–363, 1994.

    Article  MathSciNet  Google Scholar 

  35. N. Moës, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46(1):131–150, 1999.

    Google Scholar 

  36. N. Moës, A. Gravouil, and T. Belytschko. Non-planar 3d crack growth by the extended finite element and the level sets. Part I: Crack representation and stress intensity computation. 2001. preprint.

    Google Scholar 

  37. S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79(1):12–49, November 1988.

    Article  MathSciNet  Google Scholar 

  38. S. J. Osher and F. Santosa. Level set methods for optimization problems involving geometry and constraints i. frequencies of a two-density inhomogeneous drum. J. Comput. Phys., 171:272–288, 2001.

    Article  MathSciNet  Google Scholar 

  39. S. B. Pillapakkam and P. Singh. A level-set method for computing solutions to viscoelastic twophase flow. J. Comput. Phys., 174:552–578 2001.

    Article  Google Scholar 

  40. J. A. Sethian. A review of recent numerical algorithms for hypersurfaces moving with curvature-dependent speed. Journal of Differential Geometry, 31:131–161, 1989.

    MathSciNet  Google Scholar 

  41. J. A. Sethian and J. Strain. Crystal growth and dendrite solidification. Journal of Computational Physics, 98(2):231–253, 1992.

    Article  MathSciNet  Google Scholar 

  42. J.A. Sethian. Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science. Cambridge University Press, 1996.

    Google Scholar 

  43. J.A. Sethian. A marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Sciences, 93(4):1591–1595, 1996.

    MathSciNet  MATH  Google Scholar 

  44. J.A. Sethian. Fast marching methods. SIAM Review, 41(2):199–235, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  45. K. A. Smith, F. J. Solis, and D. L. Chopp. A projection method for motion of triple junctions by level sets. Interfaces and Free Boundaries, 2001. to appear.

    Google Scholar 

  46. K.A. Smith, F.J. Solis, L. Tao, K. Thornton, and M.O. de la Cruz. Domain growth in ternary fluids: A level set approach. Phys. Rev. Lett., 84(1):91–94, 2000.

    Article  Google Scholar 

  47. G. H. Son. Numerical study on a sliding bubble during nucleate boiling. KSME Int. J., 15:931–940, 2001.

    Google Scholar 

  48. M. Stolarska, D. L. Chopp, N. Moës, and T. Belytschko. Modeling crack growth by level sets and the extended finite element method. International Journal for Numerical Methods in Engineering, 51(8):943–960, 2001.

    Article  Google Scholar 

  49. N. Sukumar, D. L. Chopp, N. Moes, and T. Belytschko. Modeling holes and inclusions by level sets in the extended finite-element method. Comput. Meth. Appl. Mech. Eng., 190:6183–6200, 2001.

    Article  MathSciNet  Google Scholar 

  50. N. Sukumar, D. L. Chopp, N. Moës, and T. Belytschko. Modeling holes and inclusions by level sets in the extended finite element method. Computer Methods in Applied Mechanics and Engineering, 190(46–47):6183–6200, 2001.

    MathSciNet  Google Scholar 

  51. N. Sukumar, D. L. Chopp, N. Moës, and T. Belytschko. Modeling holes and inclusions by level sets in the extended finite element method. Computer Methods in Applied Mechanics and Engineering, 190(46–47):6183–6200, 2001.

    MathSciNet  Google Scholar 

  52. N. Sukumar, D. L. Chopp, and B. Moran. Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. 2001. submitted.

    Google Scholar 

  53. N. Sukumar, D. L. Chopp, and B. Moran. Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Engineering Fracture Mechanics, 2001. to appear.

    Google Scholar 

  54. N. Sukumar, N. Moës, B. Moran, and T. Belytschko. Extended finite element method for three-dimensional crack modeling. International Journal for Numerical Methods in Engineering, 48(11):1549–1570, 2000.

    Article  Google Scholar 

  55. Y. Sumi, C. Yang, and Z. Wang. Morphological aspects of fatigue crack propagation, part ii — effects of stress biaxiality and welding residual stresses. Technical report, Department of Naval Architecture and Ocean Engineering, Yokohama National University, Japan, 1995.

    Google Scholar 

  56. M. Sussman, P. Smereka, and S.J. Osher. A level set method for computing solutions to incompressible two-phase flow. Journal of Computational Physics, 114:146–159, 1994.

    Article  Google Scholar 

  57. S. Y. Yeung, H. T. Tsui, and A. Yim. Global shape from shading for an endoscope image. Lect Note Comput Sci, 1679:318–327, 1999.

    Google Scholar 

  58. K. Yokoi and F. Xiao. Mechanism of structure formation in circular hydraulic jumps: numerical studies of strongly deformed free-surface shallow flows. Physica D, 161:202–219, 2002.

    Article  MathSciNet  Google Scholar 

  59. H.K. Zhao, T. Chan, B. Merriman, and S. Osher. A variational level set approach to multiphase motion. Journal of Computational Physics, 127:179–195, 1996.

    Article  MathSciNet  Google Scholar 

  60. J. Zhu and J. A. Sethian. Projection methods coupled to level set interface techniques. Journal of Computational Physics, 102(1):128–138, 1992.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Kluwer Academic Publishers

About this chapter

Cite this chapter

Chopp, D. (2002). Level Set Extentions, Flows and Crack Propagation. In: Suri, J.S., Laxminarayan, S. (eds) PDE and Level Sets: Algorithmic Approaches to Static and Motion Imagery. Topics in Biomedical Engineering. Springer, Boston, MA. https://doi.org/10.1007/0-306-47930-3_2

Download citation

  • DOI: https://doi.org/10.1007/0-306-47930-3_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-306-47353-1

  • Online ISBN: 978-0-306-47930-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics