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Generalization of the Pythagorean Eigenvalue Error Theorem and Its Application to Isogeometric Analysis

  • Michael Bartoň
  • Victor Calo
  • Quanling Deng
  • Vladimir Puzyrev
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 15)

Abstract

This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propagation and structural vibration problems. The dispersion error of the isogeometric elements is minimized by optimally blending two standard Gauss-type quadrature rules. These blending rules approximate the inner products and increase the convergence rate by two extra orders when compared to those with fully-integrated inner products. To quantify the approximation errors, we generalize the Pythagorean eigenvalue error theorem of Strang and Fix. To reduce the computational cost, we further propose a two-point rule for C1 quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules.

Notes

Acknowledgements

This publication was made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided by the National Priorities Research Program grant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation), by the European Union’s Horizon 2020 Research and Innovation Program of the Marie Skłodowska-Curie grant agreement No. 644202, and by the Projects of the Spanish Ministry of Economy and Competitiveness MTM2016-76329-R (AEI/FEDER, EU). The Spring 2016 Trimester on “Numerical methods for PDEs”, organized with the collaboration of the Centre Emile Borel at the Institut Henri Poincare in Paris supported VMC’s visit to IHP in October, 2016.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Michael Bartoň
    • 1
  • Victor Calo
    • 2
    • 3
  • Quanling Deng
    • 4
  • Vladimir Puzyrev
    • 4
  1. 1.Basque Center for Applied MathematicsBilbaoSpain
  2. 2.Department of Applied Geology, Western Australian School of MinesCurtin UniversityBentley, PerthAustralia
  3. 3.Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)Kensington, PerthAustralia
  4. 4.Department of Applied Geology, Western Australian School of MinesCurtin UniversityBentley, PerthAustralia

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