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Global MDS

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Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes

Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 22))

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Abstract

The main purpose of the current chapter is to define the global MDS based on the local analysis given in Sect. 3.4. One should try to “put together” local templates without contradictions. More precisely, the order of construction of the local MDS should be defined. It is important to remember that, even if local templates do not intersect geometrically, the construction of a local MDS is usually based on the assumption of some control points being already classified and the possibility to define dependencies on these control points. Therefore the order of defining the local MDS plays the principal role in the construction of the global MDS. We start with the quintic case where one gets a simple and elegant positive answer.

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References

  1. P. Alfeld, L.L. Schumaker, On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1. Numer. Math. 57, 651–661 (1990)

    MathSciNet  MATH  Google Scholar 

  2. P. Alfeld, B. Piper, L.L. Schumaker, An explicit basis for c 1 quartic bivariate splines. SIAM J. Numer. Anal. 24, 891–911 (1987)

    MathSciNet  MATH  Google Scholar 

  3. P. Alfeld, M. Neamtu, L.L. Schumaker, Bernstein-Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided Geom. Des. 13, 333–349 (1996)

    MATH  Google Scholar 

  4. L. Beirão da Veiga, A. Buffa, G. Sangalli, R. Vázquez, Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)

    MathSciNet  MATH  Google Scholar 

  5. M. Bercovier, O. Volpin, T. Matskewich, Globally G1 free form surfaces using \(\ll \) real \(\gg \) plate invariant energy methods, in Curves and Surfaces 1996, ed. by A. LeMehaute et al. (Vanderbilt U Press, Nashville, 1997), pp. 25–34

    MATH  Google Scholar 

  6. M. Bernadou, Finite Element Methods for Thin Shell Problems, first english language edition (Masson, Paris, 1996)

    Google Scholar 

  7. T.D. Blacker, M.B. Stephenson, Paving: a new approach to automated quadrilateral mesh generation. Int. J. Numer. Methods Eng. 32(4), 811–847 (1991)

    MATH  Google Scholar 

  8. F.K. Bogner, R.L. Fox, L.A. Schmit, The generation of inter element compatible stiffness and mass matrices by the use of interpolation formulas, in Proceedings of the Conference on Matrix Methods in Structural Mechanics (Wright Patterson A.F.B., Ohio, 1965), pp. 397–444

    Google Scholar 

  9. G.-P. Bonneau, S. Hahmann, Flexible G1 interpolation of quad meshes. Graph. Models 76(6), 669–681 (2014)

    Google Scholar 

  10. M.J. Borden, M.A. Scott, J.A. Evans, T.J. Hughes, Isogeometric finite element data structures based on Bézier extraction of nurbs. Int. J. Numer. Methods Eng. 87(1–5), 15–47 (2011)

    MATH  Google Scholar 

  11. P.G. Ciarlet, Numerical Analysis of the Finite Element Method (Les Presses de L’Universite de Montreal, Quebec, 1976)

    MATH  Google Scholar 

  12. P.G. Ciarlet, J.L. Lions, Handbook of Numerical Analysis: Finite Element Methods, vol. 1–2 (North-Holland, Amsterdam, 1991)

    Google Scholar 

  13. J.F. Ciavaldini, J.C. Nedelec, Sur l’element de Fraeijs de Veubeke et Sander. Revue Francaise d’automatique, Informatique, Recherche Opérationnelle R-2, 29–46 (1974)

    MATH  Google Scholar 

  14. R.W. Clough, J.L. Tocher, Finite element stiffness matrices for the analysis of plate bending, in Proceedings of the 1st Conference of Matrix Methods in Structural Mechanics (Wright-Patterson, Dayton, 1965), pp. 515–545

    Google Scholar 

  15. E. Cohen, R.F. Reisenfeld, G. Elber, Geometric Modeling with Splines - An Introduction (A. K. Peters, Natick, MA, 2001)

    Google Scholar 

  16. A. Collin, G. Sangalli, T. Takacs, Approximation properties of multi-patch c1 isogeometric spaces. Comput. Aided Geom. Des. 47, 93–113 (2016)

    MATH  Google Scholar 

  17. T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms, 2nd edn. (MIT Press, Cambridge, MA, 1994)

    MATH  Google Scholar 

  18. J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA (Wiley, Chichester, 2009)

    MATH  Google Scholar 

  19. C. De Boor. A Practical Guide to Splines (Springer, Heidelberg, 1978)

    MATH  Google Scholar 

  20. L. Demkovicz, P. Gatto, W. Qiu, A. Joplin, G1-interpolation and geometry reconstruction for higher order finite elements. Comput. Methods Appl. Mech. Eng. 198(13–14), 1198–1212 (2009)

    MATH  Google Scholar 

  21. D. Doo, M.A. Sabin, Behavior of recursive division surfaces near extraordinary points. Comput. Aided Geom. Des. 10, 356–360 (1978)

    Google Scholar 

  22. G. Elber, Guirit, A Graphics user interface to Irit. http://www.cs.technion.ac.il/~gershon/GuIrit/ (2008)

  23. A. Embar, J. Dolbow, I. Harari, Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. Int. J. Numer. Methods Eng. 83, 877–898 (2010)

    MathSciNet  MATH  Google Scholar 

  24. G.E. Farin, Curves and Surfaces for CAGD: A Practical Guide, 4th edn. (Academic Press, New York, 1997)

    MATH  Google Scholar 

  25. E. Fasshauer, L.L. Schumaker, Minimal energy surfaces using parametric splines. Comput. Aided Geom. Des. 13(1), 45–79 (1996)

    MathSciNet  MATH  Google Scholar 

  26. A.E. Frey, C.A. Hall, T.A. Porschin, Some results on the global inversion of bilinear and quadratic jacobian. isoparametric finite element transformations. Math. Comput. 32(143), 725–749 (1978)

    Google Scholar 

  27. G. Greiner, Surface construction based on variational principles, in Wavelets, Images, and Surface Fitting (Adams-Blake, Fair Oaks, 1994), pp. 277–286

    MATH  Google Scholar 

  28. G. Greiner, J. Loos, W. Wesselink, Data dependent thin plate energy and its use in interactive surface modeling. Comput. Graphics Forum (EUROGRAPHICS ’96 Proceedings) 15(3), C-175–C-185 (1996)

    Google Scholar 

  29. D. Groisser, J. Peters, Matched -constructions always yield -continuous isogeometric elements. Comput. Aided Geom. Des. 34, 67–72 (2015)

    MathSciNet  MATH  Google Scholar 

  30. K. Höllig, U. Reif, J. Wipper, Weighted extended b-spline approximation of dirichlet problems. SIAM J. Numer. Anal. 39(2), 442–462 (2001)

    MathSciNet  MATH  Google Scholar 

  31. D. Hong, Spaces of bivariate spline functions over triangulations. J. Approx. Theory Appl. 7, 56–75 (1991)

    MathSciNet  MATH  Google Scholar 

  32. J. Hoschek, D. Lasser, Fundamentals of Computer Aided Geometric Design (A. K. Peters, Wellesley, MA, 1993)

    MATH  Google Scholar 

  33. A. Ibrahim, L.L. Schumaker, Super spline spaces of smoothness r and degree \(d \geq 3r + 2\). Constr. Approx. 7, 401–423 (1991)

    MathSciNet  MATH  Google Scholar 

  34. R.E. Jones, Qmesh: a self-organizing mesh generation program. Technical Report SLA-73–1088 (Sandia Laboratories, Albuquerque, NM, 1974)

    Google Scholar 

  35. P. Kagan, A. Fischer, P. Bar-Yoseph, New B-spline finite element approach for geometrical design and numerical analysis. Int. J. Numer. Methods Eng. 41, 435–458 (1998)

    MATH  Google Scholar 

  36. M. Kapl, V. Vitrih, Space of C2-smooth geometrically continuous isogeometric functions on two-patch geometries. Comput. Math. Appl. 73(1), 37–59 (2017)

    MathSciNet  MATH  Google Scholar 

  37. M. Kapl, V. Vitrih, Space of C2-smooth geometrically continuous isogeometric func- tions on planar multi-patch geometries: dimension and numerical experiments. arXiv 1701.06753v1 (2017)

    Google Scholar 

  38. M. Kapl, V. Vitrih, B. Jüttler, K. Birner, Isogeometric analysis with geometrically continuous functions on two-patch geometries. Comput. Math. Appl. 70(7), 1518–1538 (2015)

    MathSciNet  MATH  Google Scholar 

  39. M. Kapl, F. Buchegger, M. Bercovier, B. Jüttler, Isogeometric analysis with geometrically continuous functions on multi-patch geometries. Comput. Methods Appl. Mech. Eng. 316, 209–234 (2017) [on line June 2016]

    Google Scholar 

  40. M. Kapl, G. Sangalli, T. Takacs, Dimension and basis construction for analysis-suitable G1 two-patch parameterizations. arXiv 1701.06442 (2017)

    Google Scholar 

  41. L. Kobbelt, Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Comput. Graphics Forum 15, 409–420 (1996)

    Google Scholar 

  42. M.J. Lai, Scattered data interpolation and approximation by using bivariate c 1 piecewise cubic polynomials. Comput. Aided Geom. Des. 13, 81–88 (1996)

    MATH  Google Scholar 

  43. M.J. Lai, L.L. Schumaker, On the approximation power of bivariate splines. Adv. Comput. Math. 9, 251–279 (1998)

    MathSciNet  MATH  Google Scholar 

  44. H. Lin, W. Chen, H. Bao, Adaptive patch-based mesh fitting for reverse engineering. Comput. Aided Des. 39, 1134–1142 (2007)

    Google Scholar 

  45. Q. Liu, T.C. Sun, G 1 interpolation of mesh curves. Comput. Aided Des. 26(4), 259–267 (1994)

    MATH  Google Scholar 

  46. T. Matskewich, Construction of C 1 surfaces by assembly of quadrilateral patches under arbitrary mesh topology. Dissertation, Hebrew University of Jerusalem, Computer Science Dpt. Givat Ram, Jerusalem, Israel (2001)

    Google Scholar 

  47. J. Morgan, R. Scott, A nodal basis for c 1 piecewise polynomials of degree \(n \geq 5\). Math. Comput. 29, 736–740 (1975)

    MathSciNet  MATH  Google Scholar 

  48. T. Nguyen, K. Karčiauskas, J. Peters, A comparative study of several classical, discrete differential and isogeometric methods for solving poisson’s equation on the disk. Axioms 3(2), 280–299 (2014)

    MATH  Google Scholar 

  49. G.M. Nielson, A method for interpolating scattered data based upon a minimum norm network. Math. Comput. 40(161), 253–271 (1983)

    MathSciNet  MATH  Google Scholar 

  50. M. Nielson, R. Ramaraj, Interpolation over a sphere based upon a minimum norm network. Comput. Aided Geom. Des. 4, 41–57 (1987)

    MathSciNet  MATH  Google Scholar 

  51. J. Peters, Smooth interpolation of a mesh of curves. Constr. Approx. 7, 221–246 (1991)

    MathSciNet  MATH  Google Scholar 

  52. J. Peters, Joining smooth patches around a vertex to form a ck surface. Comput. Aided Geom. Des. 9, 387–411 (1992)

    MATH  Google Scholar 

  53. L. Piegl, W. Tiller, The NURBS Book. Monographs in Visual Communication, 2nd edn. (Springer, Wellesley, MA, 1997)

    Google Scholar 

  54. B. Piper, Visually smooth interpolation with triangular bézier patches, in Geometric Modeling: Algorithms and New Trends, ed. by G. Farin (SIAM, Philadelphia, PA, 1987), pp. 221–234

    Google Scholar 

  55. H. Pottmann, Scattered data interpolation based upon generalized minimum norm networks. Constr. Approx. 7, 247–256 (1991)

    MathSciNet  MATH  Google Scholar 

  56. U. Reif, Biquadratic g-spline surfaces. Comput. Aided Geom. Des. 12, 193–205 (1995)

    MathSciNet  MATH  Google Scholar 

  57. R.F. Sarraga, g 1 interpolation of generally unrestricted cubic bézier curves. Comput. Aided Geom. Des. 4, 23–39 (1987)

    Google Scholar 

  58. R.F. Sarraga, Errata: g 1 interpolation of generally unrestricted cubic bézier curves. Comput. Aided Geom. Des. 6, 167–172 (1989)

    MathSciNet  MATH  Google Scholar 

  59. R.F. Sarraga, Recent methods for surface shape optimization. Comput. Aided Geom. Des. 15(5), 417–436 (1998)

    MathSciNet  MATH  Google Scholar 

  60. X. Shi, T. Wang, P. Yu, A practical construction of G1 smooth biquintic B-spline surfaces over arbitrary topology. Comput. Aided Des. 36(5), 413–424 (2004)

    MATH  Google Scholar 

  61. G. Strang, Piecewise polynomials and the finite element method. Bull. Am. Math. Soc. 79, 1128–1137 (1973)

    MathSciNet  MATH  Google Scholar 

  62. S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd edn. (McGraw-Hill, New York, 1970)

    MATH  Google Scholar 

  63. P. Wu, X. Shi, T. Wang, F. Liu, Reconstruction of convergent g1 smooth b-spline surfaces Comput. Aided Geom. Des. 21, 893–913 (2004)

    MATH  Google Scholar 

  64. O.C. Zienkiewicz, R.L. Taylor, D.D. Fox, Finite Element Method, 7th edn. (Butterworth Heinemann, London, 2013)

    Google Scholar 

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Authors and Affiliations

  1. Rachel and Benin School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem, Israel

    Michel Bercovier

  2. Microsoft Corporation, Redmond, Washington, USA

    Tanya Matskewich

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  1. Michel Bercovier
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  2. Tanya Matskewich
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Bercovier, M., Matskewich, T. (2017). Global MDS. In: Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes. Lecture Notes of the Unione Matematica Italiana, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-63841-6_4

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