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A Finite Element Approach to Surface Reconstruction

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Computation of Curves and Surfaces

Part of the book series: NATO ASI Series ((ASIC,volume 307))

Abstract

This paper is devoted to the presentation of algorithms for interpolating with Simplicial Polynomial Finite Elements. We present first some general results on the construction of piecewise polynomial interpolation of class C k on a triangulated domain in ℝn. Then, we focus on the computation of surfaces interpolating scattered data of Lagrange or Hermite type. Finally, we investigate the evaluation of smoothing surfaces with a finite element minimization.

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References

  1. APPRATO, D., ARCANGELI, R., MANZANILLA, R., Sur la construction de Surfaces de classe C k à partir d’un grand nombre de données de Lagrange, Numer. Math., 1988.

    Google Scholar 

  2. ARCANGELI, R. and GOUT, J.L., Sur l’évaluation de l’erreur d’interpolation de Lagrange dans un ouvert de ℝn, RAIRO Analyse Numérique, 10 (1976), 5–27.

    Google Scholar 

  3. ATTEIA, M., Evaluation de l’erreur dans la méthode des éléments finis, Numer. Math., 28 (1977), 295–306.

    Article  MathSciNet  MATH  Google Scholar 

  4. ATTEIA, M., Fonctions splines et méthode d’éléments finis, RAIRO, R2, (1975), 13–40.

    MathSciNet  Google Scholar 

  5. BELL, K., A refined triangular plate bending element, Inst. J. Numer. Methods, 1, (1969), 101–122.

    Article  Google Scholar 

  6. BIERMANN, O., Uber naherungsweise Cubaturen, Monatsh. Math. Phys., 14 (1903), 211–225.

    Article  MathSciNet  MATH  Google Scholar 

  7. CHUNG, K.C. and YAO, T.H., On Lattice admitting unique Lagrange interpolation, SIAM J. Numer. Anal., 14 (1977), 735–743.

    Article  MathSciNet  MATH  Google Scholar 

  8. CHUI, C.K. and LAI, M.J., Vandermonde determinant and Lagrange interpolation in ℝn, CAT report 94, Texas A & M University, (1985).

    Google Scholar 

  9. CIARLET, P.G., Sur l’élément de Clough et Tocher, RAIRO, Anal. Num., 2 (1974), 19–27.

    MathSciNet  MATH  Google Scholar 

  10. CIARLET, P.G., The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, (1978).

    MATH  Google Scholar 

  11. CIARLET, P.G. and RAVIART, P.A.,General Lagrange and Hermite interpolation in &#x211Dn with applications to finite element methods, Arc. Rational Mech. Anal., 49 (1972), 177–199.

    MathSciNet  Google Scholar 

  12. COATMELEC, C., Approximation et interpolation des fonctions différentiables de plusieurs variables, Ann. Sci. Ecole Normale Sup., 83 (1966), 271–341.

    MathSciNet  Google Scholar 

  13. DAVIS, P.J., Construction of non-negative approximate quadratures, Math. Comp., 21 (1967), 578–582.

    Article  MathSciNet  MATH  Google Scholar 

  14. DUCATEAU, C.F., Etude de quelques problèmes d’interpolation, Thèse, Grenoble (1971).

    Google Scholar 

  15. DUCHON, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO Anal. Num. 10 (1976).

    Google Scholar 

  16. FERGUSON, D., The question of uniqueness for G.D. BIRKHOFF interpolation problems, J. Approx. Th. 2 (1969), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  17. FRANKE, R., Scattered Data Interpolation: Test of some methods, Math. Comp., 38 (1982).

    Google Scholar 

  18. GASCA, M. and MAETZU, J.I., On Lagrange and Hermite interpolation in ℝk, Numer. Math. 39 (1982), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  19. GLAESER, G., L’interpolation des fonctions de plusieurs variables, dans Proceeding Liverpool singularities, Symposium 2, Lecture Notes in Math., 209, Springer Verlag Berlin, Heidelberg, New York, (1972).

    Google Scholar 

  20. GOUT, J.L., Estimation de l’erreur d’interpolation d’Hermite dans ℝn, Numer. Math. 28 (1977), 407–429.

    Article  MathSciNet  MATH  Google Scholar 

  21. GUELFOND, A.O., Calcul des différences finies, Dunod Paris, (1963).

    MATH  Google Scholar 

  22. HAKOPIAN, H., Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type, J. Approx. Th. 34 (1982), 286–305.

    Article  MathSciNet  MATH  Google Scholar 

  23. JETTER, K., Some contribution to bivariate interpolation and cubature, Approximation Theory 4, C.K. Chui, L.L. Schumaker, J.D. Ward, eds Academic Press, New York, 1983, 533–538.

    Google Scholar 

  24. KERGIN, P., A natural interpolation of C k-functions, J. Approx. Th. 29 (1980), 278–293.

    Article  MathSciNet  MATH  Google Scholar 

  25. LAFRANCHE, Y., Représentation d’une surface de classe C n, Triangulation de Delaunay, Note d’étude LA N. 54. CCSA. CELAR (1982).

    Google Scholar 

  26. LAFRANCHE, Y., Thèse de 3e cycle, Université de Rennes (1984).

    Google Scholar 

  27. LAURENT, P.J., Approximation et optimisation, Hermann, Paris (1972).

    MATH  Google Scholar 

  28. LAWSON, C.L., Software for C 1 surface interpolation, Mathematical Software, ed. Rice. J.R. Academic Press. New York, 3, 1977.

    Google Scholar 

  29. LE MEHAUTE, A., Quelques méthodes explicites de prolongement de fonctions numé- riques de plusieurs variables, Thèse 3e cycle, Université de Rennes (1976).

    Google Scholar 

  30. LE MEHAUTE, A., Prolongement d’un champ taylorien connu sur les cotés d’un triangle, Numerical methods of approximation theory, INSM 52, Birkhauser verlag, Basel, 1980.

    Google Scholar 

  31. LE MEHAUTE, A., Taylor interpolation of order n at the vertices of a triangle, Applications for Hermite interpolation and finite elements, Approximation Theory and Applications, ed. S. Ziegler, Academic Press, 1981.

    Google Scholar 

  32. LE MEHAUTE, A., Explicit C n extensions of functions of two variables in a strip between two curves, or in a corner, in ℝn, Approximation Theory and Applications, ed. S. Ziegler, Academic Press, 1981.

    Google Scholar 

  33. LE MEHAUTE, A., Construction of surfaces of class C k on a domain Ω ⊂ ℝn after triangulation, Multivariate Approximation Theory 2, INSM 61, ed. W. Schempp and K. Zeller, Birkhauser Verlag, Bazel, 1982.

    Google Scholar 

  34. LE MEHAUTE, A., Représentation de surfaces par éléments finis, publication des Laboratoires d’Analyse Numérique et de Mécanique de l’INSA et de l’Université de Rennes, (1982).

    Google Scholar 

  35. LE MEHAUTE, A., On Hermite elements of class C k in ℝn, Approximation Theory 4, ed. C.K. Chui, L.L. Schumaker, J.D. Ward, Academic Press, New York, 1983.

    Google Scholar 

  36. LE MEHAUTE, A., Sur les éléments finis simpliciaux d’Hermite de classe C k, Publications des Séminaires de l’Université de Toulouse, (1983).

    Google Scholar 

  37. LE MEHAUTE, A., Approximation of derivatives in ℝn, Application: construction of surfaces in ℝn, in Approximation Theory and Applications, NATO.ASI, St John., Newfoundland, Canada, 1983.

    Google Scholar 

  38. LE MEHAUTE, A., Interpolation et approximation par des fonctions polynomiales par morceaux dans ℝn, Thèse de doctorat d’Etat, Université de Rennes, (1984).

    Google Scholar 

  39. LE MEHAUTE, A., Interpolation with minimizing triangular finite element in ℝn, Proceeding of the International Conference on methods of functional analysis in Approximation Theory, Bombay, (1985),.

    Google Scholar 

  40. LE MEHAUTE, A., An efficient algorithm for C k Simplicial finite element interpolation in ℝn, CAT report 111, Texas A & M University, (1986).

    Google Scholar 

  41. LORENTZ, G.G. and LORENTZ, R.A., Multivariate interpolation, Rational approximation and interpolation, P. Graves-Morris and al., eds, Lecture Notes in Math. No. 1105, Springer Verlag, Berlin, 1984, 136–144.

    Chapter  Google Scholar 

  42. MEINGUET, J., Sharp “a priori” error bounds for polynomial approximation in Sobo- lev Spaces, Multivariate Approximation Theory 2, INSM 61, ed. W. Schempp and K. Zeller, Birkhauser verlag, Basel, 1982.

    Google Scholar 

  43. MICCHELLI, C.A., A constructive approach to Kergin interpolation in ℝk: Multivariate B-splines and Lagrange interpolation, Rocky Mountains J. Math, 10, (1980), 485–497.

    Article  MathSciNet  MATH  Google Scholar 

  44. MORGAN, J. and SCOTT, R., A nodal basis for C 1 piecewise polynomials of degree ≥ 5, Math Comp. 29, (1975), 736–740.

    MathSciNet  MATH  Google Scholar 

  45. MORREY, C.B., Multiple integrals in the calculus of variations, Springer Verlag, New York, 1966.

    MATH  Google Scholar 

  46. NICOLAIDES, R.A., On a class of finite elements generated by Lagrange Interpolation 1, SIAM J. of Numer. Anal. 9, (1972), 435–445.

    Article  MathSciNet  MATH  Google Scholar 

  47. NICOLAIDES, R.A., On a class of finite elements generated by Lagrange Interpolation 2, SIAM J. of Numer. Anal. 10, (1973), 182–187.

    Article  MathSciNet  MATH  Google Scholar 

  48. POLYA, G., Bermerkug zur Interpolation und zur Nakerungstheorie der Balkenbiegung, Z. Angew Math Mech. 11, (1931), 445–449.

    Article  MATH  Google Scholar 

  49. SABLONNIERE, P., Bases de Bernstein et approximants splines,Thèse, Lille, (1982).

    Google Scholar 

  50. SCHOENBERG, I.J., On Hermite Birkhoff Interpolation, J. Math. Anal. Appl. 6, (1966), 538–543.

    Article  MathSciNet  Google Scholar 

  51. SCHUMAKER, L.L., Fitting surfaces to scattered data, Approximation Theory 2, ed. G.G. Lorentz, C.K. Chui, L.L. Schumaker, Academic Press, 1976.

    Google Scholar 

  52. SHARMA, A., Some poised and non-poised problems of interpolation, SIAM Review 14 N. 1, (1972).

    Google Scholar 

  53. STEFFENSEN, I.F., Interpolation, Chelsea, New York, 1950.

    MATH  Google Scholar 

  54. STRANG, G., Approximation in the finite element method, Numer. Math. 19, (1972), 81–98.

    Article  MathSciNet  MATH  Google Scholar 

  55. STRANG, G., Piecewise polynomials and the finite element method, Bull, of A.M.S., 79, N. 6, 1973.

    Google Scholar 

  56. STROUD, A.H., Approximate calculation of multiple integrals, Prentice Hall, Englewood Cliffs, New Jersey, 1971.

    MATH  Google Scholar 

  57. WALKER, R.J., Algebraic curves, Princeton University Press, Princeton N.J., 1950.

    MATH  Google Scholar 

  58. ZENISEK, A., Interpolation polynomials on the triangle, Numer. Math. 15, (1970), 283–286.

    Article  MathSciNet  MATH  Google Scholar 

  59. ZENISEK, A;, A general theorem on triangular C m elements, RAIRO. R2, (1974), 119–127.

    MathSciNet  Google Scholar 

  60. ZIENKIEWICS, O.C., The finite element method in Engineering Science, Mac Graw Hill, London, 1971.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’Analyse Numérique et d’Optimisation, Université de Lille 1 Flandres-Artois, 59655, Villeneuve d’Ascq Cedex, France

    A. Le Méhauté

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  1. A. Le Méhauté
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Editors and Affiliations

  1. Institut für Mathematik I, Freie Universität, Berlin, Germany

    Wolfgang Dahmen

  2. Department of Applied Mathematics, University of Zaragoza, Zaragoza, Spain

    Mariano Gasca

  3. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Charles A. Micchelli

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© 1990 Kluwer Academic Publishers

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Le Méhauté, A. (1990). A Finite Element Approach to Surface Reconstruction. In: Dahmen, W., Gasca, M., Micchelli, C.A. (eds) Computation of Curves and Surfaces. NATO ASI Series, vol 307. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2017-0_8

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  • DOI: https://doi.org/10.1007/978-94-009-2017-0_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7404-9

  • Online ISBN: 978-94-009-2017-0

  • eBook Packages: Springer Book Archive

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