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Optimal Analysis-Aware Parameterization of Computational Domain in Isogeometric Analysis

  • Gang Xu
  • Bernard Mourrain
  • Régis Duvigneau
  • André Galligo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)

Abstract

In isogeometric analysis (IGA for short) framework, computational domain is exactly described using the same representation as that employed in the CAD process. For a CAD object, we can construct various computational domain with same shape but with different parameterization. One basic requirement is that the resulting parameterization should have no self-intersections. In this paper, a linear and easy-to-check sufficient condition for injectivity of planar B-spline parameterization is proposed. By an example of 2D thermal conduction problem, we show that different parameterization of computational domain has different impact on the simulation result and efficiency in IGA. For problems with exact solutions, we propose a shape optimization method to obtain optimal parameterization of computational domain. The proposed injective condition is used to check the injectivity of initial parameterization constructed by discrete Coons method. Several examples and comparisons are presented to show the effectiveness of the proposed method. Compared with the initial parameterization during refinement, the optimal parameterization can achieve the same accuracy but with less degrees of freedom.

Keywords

isogeometric analysis analysis-aware parameterization of computational domain injectivity shape optimization steepest descent method 

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References

  1. 1.
    Aigner, M., Heinrich, C., Jüttler, B., Pilgerstorfer, E., Simeon, B., Vuong, A.-V.: Swept volume parametrization for isogeometric analysis. In: Hancock, E.R., Martin, R.R., Sabin, M.A. (eds.) MOS XIII 2009. LNCS, vol. 5654, pp. 19–44. Springer, Heidelberg (2009)Google Scholar
  2. 2.
    Auricchio, F., da Veiga, L.B., Buffa, A., Lovadina, C., Reali, A., Sangalli, G.: A fully locking-free isogeometric approach for plane linear elasticity problems: A stream function formulation. Computer Methods in Applied Mechanics and Engineering 197, 160–172 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bazilevs, Y., Beirao de Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for refined meshes. Mathematical Models and Methods in Applied Sciences 6, 1031–1090 (2006)CrossRefGoogle Scholar
  4. 4.
    Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid structure interaction: Theory, algorithms, and computations. Computational Mechanics 43, 3–37 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bazilevs, Y., Calo, V.M., Zhang, Y., Hughes, T.J.R.: Isogeometric fluid structure interaction analysis with applications to arterial blood flow. Computational Mechanics 38, 310–322 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bazilevs, Y., Hughes, T.J.R.: NURBS-based isogeometric analysis for the computation of flows about rotating components. Computational Mechanics 43, 143–150 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis using T-Splines. Computer Methods in Applied Mechanics and Engineering 199(5-8), 229–263 (2010)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cohen, E., Martin, T., Kirby, R.M., Lyche, T., Riesenfeld, R.F.: Analysis-aware Modeling: Understanding Quality Considerations in Modeling for Isogeometric Analysis. Computer Methods in Applied Mechanics and Engineering 199(5-8), 334–356 (2010)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cottrell, J.A., Hughes, T.J.R., Reali, A.: Studies of refinement and continuity in isogeometric analysis. Computer Methods in Applied Mechanics and Engineering 196, 4160–4183 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering 195, 5257–5296 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dokken, T., Skytt, V., Haenisch, J., Bengtsson, K.: Isogeometric representation and analysis–bridging the gap between CAD and analysis. In: 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, Florida, January 5-8 (2009)Google Scholar
  12. 12.
    Dörfel, M., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with T-splines. Computer Methods in Applied Mechanics and Engineering 199(5-8), 264–275 (2010)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Duvigneau, R.: An introduction to isogeometric analysis with application to thermal conduction. INRIA Research Report RR-6957 (June 2009)Google Scholar
  14. 14.
    Elguedj, T., Bazilevs, Y., Calo, V.M., Hughes, T.J.R.: \(\bar{B}\) and \(\bar{F}\) projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Computer methods in applied mechanics and engineering 197, 2732–2762 (2008)CrossRefGoogle Scholar
  15. 15.
    Evans, J.A., Bazilevs, Y., Babuka, I., Hughes, T.J.R.: n-Widths, supinfs, and optimality ratios for the k-version of the isogeometric finite element method. Computer Methods in Applied Mechanics and Engineering 198, 1726–1741 (2009)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Farin, G., Hansford, D.: Discrete coons patches. Computer Aided Geometric Design 16(7), 691–700 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gain, J.E., Dodgson, N.A.: Preventing self-Intersection under free-form deformation. IEEE Transactions on Visualization and Computer Graphics 7(4), 289–298 (2001)CrossRefGoogle Scholar
  18. 18.
    Gomez, H., Calo, V.M., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of the Cahn-Hilliard phase-field model. Computer Methods in Applied Mechanics and Engineering 197, 4333–4352 (2008)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194(39-41), 4135–4195 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hughes, T.J.R., Realli, A., Sangalli, G.: Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Computer methods in applied mechanics and engineering 197, 4104–4124 (2008)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hughes, T.J.R., Realli, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Computer Methods in Applied Mechanics and Engineering 199(5-8), 301–313 (2010)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Jüttler, B.: Shape-preserving least-squares approximation by polynomial parametric spline curves. Computer Aided Geometric Design 14, 731–747 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kestelman, H.: Mappings with non-vanishing Jacobian. Amer. Math. Monthly 78, 662–663 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kim, H.J., Seo, Y.D., Youn, S.K.: Isogeometric analysis for trimmed CAD surfaces. Computer Methods in Applied Mechanics and Engineering 198(37-40), 2982–2995 (2009)CrossRefGoogle Scholar
  25. 25.
    Martin, T., Cohen, E., Kirby, R.M.: Volumetric parameterization and trivariate B-spline fitting using harmonic functions. Computer Aided Geometric Design 26(6), 648–664 (2009)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Sevilla, R., Fernandes-Mendez, S., Huerta, A.: NURBS-enhanced finite element method for Euler equations. International Journal for Numerical Methods in Fluids 57, 1051–1069 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Wall, W.A., Frenzel, M.A., Cyron, C.: Isogeometric structural shape optimization. Computer Methods in Applied Mechanics and Engineering 197, 2976–2988 (2008)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Zhang, Y., Bazilevs, Y., Goswami, S., Bajaj, C., Hughes, T.J.R.: Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Computer Methods in Applied Mechanics and Engineering 196, 2943–2959 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gang Xu
    • 1
  • Bernard Mourrain
    • 1
  • Régis Duvigneau
    • 2
  • André Galligo
    • 3
  1. 1.GALAADINRIA Sophia-AntipolisCedexFrance
  2. 2.OPALEINRIA Sophia-AntipolisCedexFrance
  3. 3.University of Nice Sophia-AntipolisNice Cedex 02France

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