Skip to main content
×
×
Home

Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures

  • Meng Wu (a1) (a2), Bernard Mourrain (a1), André Galligo (a3) and Boniface Nkonga (a3)
Abstract
Abstract

Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the L 2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

Copyright
Corresponding author
*Corresponding author. Email addresses: meng.wu@hfut.edu.cn, wumeng@mail.ustc.edu.cn (M. Wu), Bernard.Mourrain@inria.fr (B. Mourrain), Andre.Galligo@unice.fr (A. Galligo), boniface.nkonga@unice.fr (B. Nkonga)
References
Hide All
[1] Czarny O. and Huysmans G.. Bézier surfaces and finite elements for MHD simulations. Journal of Computational Physics, vol. 227, p. 74237445, 2008.
[2] Huysmans G. T. A., Goedbloed J. P.. “Isoparametric Bicubic Hermite Elements for Solution of the Grad-Shafranov Equation”. International Journal of Modern Physics C, p. 371376, 1991.
[3] Huebner K. H., Thornton E. A.. The Finite ElementMethod for Engineers. John Wiley & Sons, ISBN 0-471-09159-6, 1982.
[4] Höig K.. Finite Element Methods with B-Splines. SIAM, 2003.
[5] Forsey D. R. and Bartels R. H.. Hierarchical B-spline refinement. In Proceedings of the 15th annual conference on Computer graphics and interactive techniques, SIGGRAPH ’88, pages 205212, New York, NY, USA, 1988. ACM.
[6] Kraft R., Adaptive and linearly independent multilevel B-splines. In Surface Fitting and Multiresolution Methods, Méhauté A. L., Rabut C., and Schumaker L. L., Eds., vol. 2. Vanderbilt University Press, p. 209216.
[7] Giannelli C., Jüttler B., and Speleers H.. THB-splines: The truncated basis for hierarchical splines. Computer Aided Geometric Design, 29(7):485498, 2012.
[8] Dokken T., Lyche T., and Pettersen K.. Polynomial splines over locally refined box-partitions. Computer Aided Geometric Design, 30(3), p. 331356, 2013.
[9] Sederberg T. W., Zheng J., Bakenov A., Nasri A.. T-splines and T-NURCCs. ACM Trans. Graph., vol. 22, p. 161172, 2003.
[10] Ergatoudis I., Irons B., and Zienkiewicz O.. Curved isoparametric “quadrilateral” elements for finite element analysis. Int. J. Solids Structures, vol. 4, p. 3142, 1968.
[11] Hughes T. J. R., Cottrell J. A. and Bazilevs Y.Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement”. Computer Methods in Applied Mechanics and Engineering, vol. 194, Issues. 39-41, p. 41354195, 2005.
[12] Gu X., He Y., Qin H., Manifold splines. Graphical Models, vol. 68, p. 237254, 2006.
[13] He Y., Wang K., Wang H., Gu X., Qin H., Manifold T-spline. Geometric Modeling and Processing-GMP 2006, Lecture Notes in Computer Science, vol. 4077, p. 409422, 2006.
[14] Wang H., He Y., Li X., Gu X., Qin H., Polycube Splines. Computer-Aided Design, vol. 40, p. 721733, 2008.
[15] Ying L., Zorin D., A simple manifold-based construction of surfaces of arbitrary smoothness. ACM Trans. Graph., vol. 23, p. 271275, 2004.
[16] Peter J., Reif U., Subdivision Surfaces. Geometry and Computing, Springer-Verlag, ISBN 878-3-540-76406-9, 2008.
[17] Gregory J. A. Smooth interpolation without twist constraints. Waltham, Massachusetts: AcademicPress, p. 7188, 1974.
[18] DeRose T., Loop C.. The S-patch: a new multisided patch scheme. ACM TransGraph, vol. 8, p. 204233, 1989.
[19] Piper B. R.. Visually smooth interpolation with triangular Bézier patches. In: Farin G., editor. Geometric modeling. Philadelphia: SIAM, p. 221233, 1987.
[20] Peter J.. Smooth free-form surfaces over irregular meshes generalizing quadratic splines. Computer Aided Geometric Design, vol. 10, p. 347361, 1993.
[21] Peter J.. C 1-surface splines. SIAM J. Numer Anal, vol. 32, p. 645666, 1995.
[22] Catmull E., Clark J.. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Geometric Design, vol. 10, p. 350355, 1978.
[23] Reif U.. Biquadratic G-spline surfaces. Computer Aided Geometric Design, vol. 12, p. 193205, 1995.
[24] Reif U.. A refineable space of smooth spline surfaces of arbitrary topological genus. Journal of Approximation Theory, vol. 90, p. 174199, 1997.
[25] Deng J., Chen F., and Feng Y.. Dimensions of spline spaces over T-meshes. Journal of Computational and Applied Mathematics, vol. 194, p. 267283, 2006.
[26] Schumaker L. L. and Wang L.. Approximation power of polynomial splines on T-meshes Computer Aided Geometric Design 29, p. 599612, 2012
[27] Takacs T., Jüttler B.. “Existence of Stiffness Matrix Integrals for Singularly Parameterized Domains in Isogeometric Analysis”. Computer Methods in Applied Mechanics and Engineering, vol. 200, p. 35683582, 2011.
[28] Quarteroni A., Valli A.. Numerical Approximation of Partial Differential Equations. Springer-Verlag, ISBN 3-540-57111-6, 1997.
[29] Schumaker L. L., Wang L.. On Hermite interpolation with polynomial splines on T-meshes. Journal of Computational and Applied Mathematics, vol. 240, p. 4250, 2013.
[30] Pataki A., Cerfon A. J., Freidberg J. P., Greengard L. and Neil M. O.. A fast, high-order solver for the Grad-Shafranov equation. J. Comput. Phys., vol. 243, p. 2845, 2013.
[31] Mourrain B.. On the dimension of spline spaces on planar T-meshes. Mathematics of Computation, 83, p. 847871, 2014.
[32] Burden R. L., Faires J. D.. Numerical Analysis. Prindle, Weber and Schnidt, Boston, MA 1985.
[33] Mourrain B., Vidunas R. and Villamizar N. Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology. Computer Aided Geometric Design, in press, 2016.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 31 *
Loading metrics...

Abstract views

Total abstract views: 151 *
Loading metrics...

* Views captured on Cambridge Core between 7th February 2017 - 20th January 2018. This data will be updated every 24 hours.