Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation

  • Denis DevaudEmail author
  • Gianluigi Rozza
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. We propose an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA), as well as reduced basis methods for parametric PDEs growing research themes in scientific computing and computational mechanics. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries. This work shows that it is also possible for some class of problems to deal with affine geometrical parametrization combined with a NURBS IGA formulation. In this work we show a certification of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation by introducing two numerical examples in heat transfer with different parametrization.


  1. 1.
    Y. Bazilevs, L. Beirao da Veiga, J.A. Cottrell, T.J.R. Hughes, G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    F. Chinesta, A. Huerta, G. Rozza, K. Willcox, Model Order Reduction. Encyclopedia of Computational Mechanics (Elsevier, Amsterdam, 2016)Google Scholar
  3. 3.
    J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric analysis: toward integration of CADand FEA (John Wiley & Sons, Chichester, 2009)CrossRefGoogle Scholar
  4. 4.
    M.G. Cox, The numerical evaluation of b-splines. IMA J. Appl. Math. 10(2), 134–149 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    C. De Boor, On calculating with B-splines. J. Approx. Theory 6(1), 50–62 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    C. De Falco, A. Reali, R. Vázquez, GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42(12), 1020–1034 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    J.S. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics (Springer, Berlin, 2015)Google Scholar
  8. 8.
    T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    D.B.P. Huynh, N.C. Nguyen, G. Rozza, A.T. Patera, rbMIT software: copyright MIT. Technology Licensing Office (2006/2007),
  10. 10.
    D.B.P. Huynh, G. Rozza, S. Sen, A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C.R. Math. 345(8), 473–478 (2007)Google Scholar
  11. 11.
    A. Manzoni, F. Salmoiraghi, L. Heltai, Reduced basis isogeometric methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils. Comput. Methods Appl. Mech. Eng. 284, 1147–1180 (2015)CrossRefMathSciNetGoogle Scholar
  12. 12.
    A.T. Patera, G. Rozza, Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT (2007),
  13. 13.
    A. Quarteroni, G. Rozza, A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1(1), 1–49 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    G. Rozza, Reduced basis approximation and error bounds for potential flows in parametrized geometries. Commun. Comput. Phys. 9, 1–48 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Meth. Eng. 15(3), 229–275 (2008)CrossRefzbMATHGoogle Scholar
  16. 16.
    F. Salmoiraghi, F. Ballarin, L. Heltai, G. Rozza, Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes. Adv. Model. Simul. Eng. Sci. 3, 21 (2016)CrossRefGoogle Scholar
  17. 17.
    The MathWorks Inc. Matlab. version (R2013a) (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.ETHZ SAM, Seminar for Applied MathematicsZurichSwitzerland
  2. 2.SISSA, International School for Advanced Studies, Mathematics Area, mathLabTriesteItaly

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