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A numerical method for solving partial differential equations on highly irregular evolving grids

Nature volume 376, pages 655–660 (1995)Cite this article

Abstract

An efficient numerical method is described for solving partial differential equations in problems where traditional eulerian and lagrangian techniques fail. The approach makes use of the geometrical concept of 'natural neighbours', the properties of which make it suitable for solving problems involving large deformation and solid–fluid interactions on a deforming mesh, without the need for regridding. The approach can also be applied to high-order partial differential equations (such as the Navier-Stokes equation), even in cases where the evolving mesh is highly irregular.

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References

  1. Roache, P. J. Computational Fluid Dynamics (Hermosa, Albuquerque, NM, 1982).

    MATH  Google Scholar 

  2. Zienkiewicz, O. C. The Finite Element Method 3rd edn (McGraw-Hill, Maidenhead, 1977).

    MATH  Google Scholar 

  3. Zienkiewicz, O. C. & Phillips, D. V. Int. J. num. Meth. Engng 8, 519–528 (1971).

    Article  Google Scholar 

  4. Bathe, K.-J., Ramm, E. & Wilson, E. L. Int. J. num. Meth. Engng 9, 353–386 (1975).

    Article  Google Scholar 

  5. Sibson, R. Math. Proc. Camb. phil. Soc. 87, 151–155 (1980).

    Article  MathSciNet  Google Scholar 

  6. Voronoi, M. G. J. reine angew. Math. 134, 198–287 (1908).

    MathSciNet  Google Scholar 

  7. Delaunay, B. N. Bull. Acad. Sci. USSR: Class. Sci. Math. 7, 793–800 (1934).

    Google Scholar 

  8. Fortune, S. in Computing in Euclidean Geometry (eds Du, D. Z. & Hwang, F.) (World Scientific, Singapore, 1992).

    Google Scholar 

  9. Barber, B., Dobkin, D. P. & Huhdanpaa, H. Technical Rep. GCG53 (The Geometry Centre, Univ. Minesota, Minneapolis, 1993).

    Google Scholar 

  10. Fortune, S. Algorithmica 2, 153–174 (1987).

    Article  MathSciNet  Google Scholar 

  11. Braun, J. & Sambridge, M. Earth planet Sci. Lett. 124, 211–220 (1994).

    Article  ADS  Google Scholar 

  12. Sibson, R. in Interpreting Multivariate Data (ed. Barnet, V.) 21–36 (Wiley, Chichester, 1981).

    Google Scholar 

  13. Watson, D. F. Contouring: a Guide to the Analysis and Display of Spatial Data (Pergamon, Oxford, 1992).

    Google Scholar 

  14. Finlayson, B. A. The Method of Weighted Residuals and Variational Principles (Academic, New York, 1972).

    MATH  Google Scholar 

  15. Sambridge, M., Braun, J. & McQueen, H. Geophys. J. int. (In the press).

  16. Lasserre, J. B. J. Opt. Theory Applic. 39, 363–377 (1983).

    Article  Google Scholar 

  17. Bathe, K.-J. Finite Element Procedures in Engineering Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1982).

    Google Scholar 

  18. Van Norton, R. in Mathematical Methods for Digital Computers Vol. 1 (eds Ralston, A. &Wilf, H. S.) (Wiley, New York, 1960).

    Google Scholar 

  19. Hughes, T. J. R., Winget, J., Levit, T. & Tedzuyar, T. E. in Computer Methods for Non-linear Solids and Structure (eds Atluri, S. N. & Perrone, N.) 75–109 (Am. Soc. Mech. Eng., New York, 1983).

    Google Scholar 

  20. Batchetor, G. K. An Introduction to Fluid Dynamics (Cambridge Univ. Press, 1967).

    Google Scholar 

  21. Fung, Y. C. Foundations of Solid Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1965).

    Google Scholar 

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Authors and Affiliations

  1. Research School of Earth Sciences, Institute of Advanced Studies,

    Jean Braun & Malcolm Sambridge

  2. Centre for Information Science Research, Australian National University, Canberra, ACT, 0200, Australia

    Malcolm Sambridge

Authors
  1. Jean Braun
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  2. Malcolm Sambridge
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Braun, J., Sambridge, M. A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 376, 655–660 (1995). https://doi.org/10.1038/376655a0

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