Blending Function Interpolation: A Survey and Some New Results

  • Robert E. Barnhill
Part of the International Series of Numerical Mathematics book series (ISNM, volume 30)


Blending function methods permit the exact interpolation of data given along curves and/or surfaces. Appropriate discretisations yield finite dimensional schemes. These methods are useful for Finite Element Analysis and for Computer Aided Geometric Design.


Normal Derivative Boundary Condi Tions Curve Side Curve Element Smooth Interpolation 
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  1. 1.
    R E Barnhill. Smooth Interpolation over Triangles, Computer Aided Geometric Design, edited byGoogle Scholar
  2. R E Barnhill and R Riesenfeld, 45–70, Academic Press, New York, 1974.Google Scholar
  3. 2.
    R E Barnhill. Blending Function Finite Elements for Curved Boundaries, Proceedings of the Finite Element Conference at Brunel University, 1975, ed., J R Uhiteman, Academic Press (to appear).Google Scholar
  4. 3.
    R E Barnhill. A Survey of Blending Function Methods Proceedings of the Seminars in Finite Elements, ed., J R \hiteman, Brunel University. (to appear).Google Scholar
  5. 4.
    R E Barnhill. The Convergence of Complex Cubatures, SIAM J. Numer. Anal. 6, 82–89, 1969.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 5.
    R E Barnhill, G Birkhoff, and W J Gordon. Smooth Interpolatión in Triangles, J. Approx. Theory 8, 114–128, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 6.
    R E Barnhill and J H Brown. Nonconforming Finite Element for Curved Regions, Proceedings of the Numerical Analysis Conference at Dundee University 1975, ed., J Ll Morris, Springer Verlag (to appear).Google Scholar
  8. 7.
    R E Barnhill and J H Brown. Curved Nonconforming Elements for Plate Problems, University of Dundee Numerical Analysis Report No. 8, 1975.Google Scholar
  9. 8.
    R E Barnhill, 7i J Gordon, and D H Thomas. The Method of Successive Decomposition for Multivariate Integration, General Motors Publication GMR-1281, October, 1972.Google Scholar
  10. 9.
    R E Barnhill and J A Gregory. On Blending Function Interpolation and Finite Element Basis Functions, tale given at the Conference on the Numerical Solution of Differential Equations, Univ. of Dundee 1973.Google Scholar
  11. 10.
    R E Barnhill and J A Gregory. Compatible Smooth Interpolation in Triangles, J. Approx. Theory (to appear).Google Scholar
  12. 11.
    R E Barnhill and J A Gregory. Polynomial Interpolation to Boundary Data on Triangles, Math. Comp. (to appear).Google Scholar
  13. 12.
    R E Barnhill and J A Gregory. Sard Kernel Theorems on Triangular and Rectangular Domains with Extensions and Applications to Finite Element Error Bounds, Numer. Math. (to appear).Google Scholar
  14. 13.
    R E Barnhill and L Mansfield. Error Bounds for Smooth Interpolâtion on Triangles, J. Approx. Theory 11, 306–318, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 14.
    Biermann. Uber Näherungs weise Cubaturen, Monatsh. Math. Phys. 14, 211–225, 1903.Google Scholar
  16. 15.
    G Birkhoff and L Mansfield. Compatible Triangular Finite Elements, J. Math. Anal. Appl. 47, 531–553, 1974.Google Scholar
  17. 16.
    J H Brown. Conforming and Nonconforming Finite Element Methods for Curved Regions, Ph.D. thesis, University of Dundee (to appear).Google Scholar
  18. 17.
    S A Coons. Surfaces for Computer Aided Design, Design Division, Mech. Engin. Dept., MIT, 1964, revised, 1967.Google Scholar
  19. 18.
    S A Coons. Surface Patches and B-Spline Curves, see reference 1.Google Scholar
  20. 19.
    A R Forrest. On Coons and Other Methods for the Representation of Curved Surfaces, Computer Graphics and Image Processing 1, 341–359, 1972.MathSciNetCrossRefGoogle Scholar
  21. 20.
    W J Gordon. “Blending-function” Methods of Bivariate and Multivariate Interpolation_ and Approximation, SIAM J. Numer. Anal. 8, 158–177, 1971.Google Scholar
  22. 21.
    W J Gordon. Distributive Lattices and the Approxi-mation of Multivariate Functions, Proceedings of the Symposium on Approximation with Special Emphasis on Splines, ed., I J Schoenberg, Univ. of Wisconsin Press, Madison, Wisconsin, 1969.Google Scholar
  23. 22.
    W J Gordon and C A Hall. Transfinite Element Methods: Blending-Function Interpolation over Arbitrary Carved Element Domains, Numer. Math. 21, 109–129, 1973.zbMATHGoogle Scholar
  24. 23.
    W J Gordon and C A Hall. Geometric Aspects of the Finite Element Method, ed., A K Aziz, The Mathematical Foundations of the Finite Element Method and Application to Partial Differential Equations, Academic Press, New York, 1972.Google Scholar
  25. 24.
    J A Gregory. Symmetric Smooth Interpolation on Triangles, TR/34, Brunel University, 1973.Google Scholar
  26. 25.
    J A Gregory. Piecewise Interpolation Theory for Functions of TWo Variables, Ph.D. thesis, Brunel University, 1975.Google Scholar
  27. 26.
    J A Gregory. Smooth Interpolation without Twist Constraints, see reference 1.Google Scholar
  28. 27.
    J A Gregory. Blending Functions on Triangles, see reference 3.Google Scholar
  29. 28.
    B M Irons. A Conforming Quartic Triangular Element for Plate Bending, Int. J. Numer. Methods in Engin. 1, 29–46, 1969.CrossRefGoogle Scholar
  30. 29.
    L Mansfield. Higher Order Compatible Triangular Finite Elements, Numer. Math. 22, 89–97, 1974.MathSciNetzbMATHGoogle Scholar
  31. 30.
    L Mansfield. Interpolation to Boundary Data in Tetrahedra with Applications to Compatible Finite Elements, J. Math. Anal. Appl. (to appear).Google Scholar
  32. 31.
    L Mansfield. Approximation of the Boundary in the Finite Element Solution of Fourth Order Problems, manuscript, 1975.Google Scholar
  33. 32.
    J A Marshall and A R Mitchell. An Exact Boundary Technique for Improved Accuracy in the Finite Element Method, JIMA 12, 355–362, 1973.MathSciNetzbMATHGoogle Scholar
  34. 33.
    J A Marshall. Application of Blending Function Methods in the Finite Elément Method, Ph.D. thesis, University of Dundee, 1975.Google Scholar
  35. 34.
    R J McDermott. Graphical Representation of Surfaces over Triangles and Rectangles, see reference 1.Google Scholar
  36. 35.
    R McLeod and A R Mitchell. The Construction of Basis Functions for Curved Elements in the Finite Element Method, JIMA 10, 382–393, 1972.zbMATHGoogle Scholar
  37. 36.
    A R Mitchell. Basis Functions for Curved Elements in the Mathematical Theory of Finite Element, see reference 2.Google Scholar
  38. 37.
    A R Mitchell. Curved Elements and the Finite Element Method, see reference 3.Google Scholar
  39. 38.
    A R Mitchell and J A Marshall. Matching of Essential Boundary Conditions in the Finite Element Method“, Proceedings of Numerical Analysis Conference at Dublin, 1974.Google Scholar
  40. 39.
    C Poeppelmeier. A Boolean Sum Interpolation Scheme to Random Data for Computer Aided Geometric Design, M.S. thesis, University of Utah, 1975.Google Scholar
  41. 40.
    R F Piesenfeld. Aspects of Modelling in Computer Aided Geometric Design, Proc. of NCC, AFIPS Press, 1975 (to appear).Google Scholar
  42. 41.
    D Shepard. A Two Dimensional Interpolation Function for Irregularly Spaced Data, Proc. 23rd Nat. Conf. ACM, 517–523, 1965.Google Scholar
  43. 42.
    D D Stancu. The Remainder of Certain Linear Approximation Foriulas in Two Variables, SIAM J. Numer. Anal. 1, 137–163, 1964.MathSciNetGoogle Scholar
  44. 43.
    G Strang and G J Fix. An Analysis of the Finite Element Method, Prentice-Hall, 1973.Google Scholar
  45. 44.
    E L Wachspress. Algebraic Geometry Foundations for Finite Element Computation, Proceedings Conf. Numerical Solution of Differential Equations, Univ. of Dundee, Springer-Verlag, 1974.Google Scholar
  46. 45.
    D S Watkins. Blending Functions and Finite Elements, Ph.D. thesis, University of Calgary, 1974.Google Scholar
  47. 46.
    D S Watkins. Conforming Rectangular Plate Elements via Blending Functions, see reference 3.Google Scholar
  48. 47.
    C Zienkiewicz. The Finite Element Method in Engineering Science, 2nd ed., McGraw-Hill, New York, 1971.Google Scholar
  49. 48.
    M Zlami1. Curved Elements in the Finite Element Method I and II, SIAM J.-Numer. Anal. 10, 229–240, 1973 and 11, 347–362, 1974.Google Scholar
  50. 49.
    M Zlamal. The Finite Element Method in Domains with Curved Boundaries, Int. J. Numer. Methods in Engin. 5, 3677–373, 1973.Google Scholar
  51. 50.
    M Zlamal. Curved Elements and Questions of Numerical Integration, see reference 3.Google Scholar

Copyright information

© Springer Basel AG 1976

Authors and Affiliations

  • Robert E. Barnhill
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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