Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

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

  • Chapter
  • First Online:
Numerical Methods for PDEs

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 15))

  • 1509 Accesses

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 C 1 quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M., Wajid, H.A.: Dispersive and dissipative behavior of the spectral element method. SIAM J. Numer. Anal. 47(5), 3910–3937 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ainsworth, M., Wajid, H.A.: Optimally blended spectral-finite element scheme for wave propagation and nonstandard reduced integration. SIAM J. Numer. Anal. 48(1), 346–371 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Akkerman, I., Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Hulshoff, S.: The role of continuity in residual-based variational multiscale modeling of turbulence. Comput. Mech. 41(3), 371–378 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antolin, P., Buffa, A., Calabro, F., Martinelli, M., Sangalli, G.: Efficient matrix computation for tensor-product isogeometric analysis: the use of sum factorization. Comput. Methods Appl. Mech. Eng. 285, 817–828 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Auricchio, F., Calabro, F., Hughes, T.J.R., Reali, A., Sangalli, G.: A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 249, 15–27 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Babuska, I.M., Sauter, S.A.: Is the pollution effect of the fem avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34(6), 2392–2423 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Banerjee, U.: A note on the effect of numerical quadrature in finite element eigenvalue approximation. Numer. Math. 61(1), 145–152 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Banerjee, U., Osborn, J.E.: Estimation of the effect of numerical integration in finite element eigenvalue approximation. Numer. Math. 56(8), 735–762 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Banerjee, U., Suri, M.: Analysis of numerical integration in p-version finite element eigenvalue approximation. Numer. Methods Partial Differ. Equ. 8(4), 381–394 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bartoň, M., Calo, V.M.: Gaussian quadrature for splines via homotopy continuation: rules for C2 cubic splines. J. Comput. Appl. Math. 296, 709–723 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bartoň, M., Calo, V.M.: Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 305, 217–240 (2016)

    Article  MathSciNet  Google Scholar 

  13. Bazilevs, Y., Calo, V.M., Cottrell, J., Hughes, T.J.R., Reali, A., Scovazzi, G.: Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 197(1), 173–201 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Calabrò, F., Sangalli, G., Tani, M.: Fast formation of isogeometric Galerkin matrices by weighted quadrature. Comput. Methods Appl. Mech. Eng. 316, 606–622 (2017)

    Article  MathSciNet  Google Scholar 

  15. Calo, V.M., Deng, Q., Puzyrev, V.: Dispersion optimized quadratures for isogeometric analysis. arXiv:1702.04540 (2017, preprint)

    Google Scholar 

  16. Collier, N., Pardo, D., Dalcin, L., Paszynski, M., Calo, V.M.: The cost of continuity: a study of the performance of isogeometric finite elements using direct solvers. Comput. Methods Appl. Mech. Eng. 213, 353–361 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Collier, N., Dalcin, L., Pardo, D., Calo, V.M.: The cost of continuity: performance of iterative solvers on isogeometric finite elements. SIAM J. Sci. Comput. 35(2), A767–A784 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Collier, N., Dalcin, L., Calo, V.M.: On the computational efficiency of isogeometric methods for smooth elliptic problems using direct solvers. Int. J. Numer. Methods Eng. 100(8), 620–632 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195(41), 5257–5296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cottrell, J., Hughes, T.J.R., Reali, A.: Studies of refinement and continuity in isogeometric structural analysis. Comput. Methods Appl. Mech. Eng. 196(41), 4160–4183 (2007)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  22. De Basabe, J.D., Sen, M.K.: Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations. Geophysics 72(6), T81–T95 (2007)

    Article  Google Scholar 

  23. De Basabe, J.D., Sen, M.K.: Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping. Geophys. J. Int. 181(1), 577–590 (2010)

    Article  Google Scholar 

  24. Dedè, L., Jäggli, C., Quarteroni, A.: Isogeometric numerical dispersion analysis for two-dimensional elastic wave propagation. Comput. Methods Appl. Mech. Eng. 284, 320–348 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Deng, Q., Bartoň, M., Puzyrev, V., Calo, V.M.: Dispersion-minimizing optimal quadrature rules for c 1 quadratic isogeometric analysis. Comput. Methods Appl. Mech. Eng. 328, 554–564 (2018)

    Article  Google Scholar 

  26. Elguedj, T., Bazilevs, Y., Calo, V.M., Hughes, T.J.R.: B-bar and F-bar projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput. Methods Appl. Mech. Eng. 197(33), 2732–2762 (2008)

    Article  MATH  Google Scholar 

  27. Esterhazy, S., Melenk, J.: An analysis of discretizations of the Helmholtz equation in L 2 and in negative norms. Comput. Math. Appl. 67(4), 830–853 (2014). https://doi.org/10.1016/j.camwa.2013.10.005

    Article  MathSciNet  MATH  Google Scholar 

  28. Ewing, R., Heinemann, R., et al.: Incorporation of mixed finite element methods in compositional simulation for reduction of numerical dispersion. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1983)

    Google Scholar 

  29. Fix, G.J.: Effect of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems. In: Aziz, A.K. (ed.) The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, pp. 525–556 (1972)

    Google Scholar 

  30. Gao, L., Calo, V.M.: Fast isogeometric solvers for explicit dynamics. Comput. Methods Appl. Mech. Eng. 274, 19–41 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Garcia, D., Pardo, D., Dalcin, L., Paszyski, M., Collier, N., Calo, V.M.: The value of continuity: refined isogeometric analysis and fast direct solvers. Comput. Methods Appl. Mech. Eng. 316, 586–605 (2016)

    Article  MathSciNet  Google Scholar 

  32. Garcia, D., Bartoň, M., Pardo, D.: Optimally refined isogeometric analysis. Proc. Comput. Sci. 108, 808–817 (2017)

    Article  Google Scholar 

  33. Gómez, H., Calo, V.M., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197(49), 4333–4352 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gomez, H., Hughes, T.J.R., Nogueira, X., Calo, V.M.: Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Comput. Methods Appl. Mech. Eng. 199(25), 1828–1840 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guddati, M.N., Yue, B.: Modified integration rules for reducing dispersion error in finite element methods. Comput. Methods Appl. Mech. Eng. 193(3), 275–287 (2004)

    Article  MATH  Google Scholar 

  36. Harari, I.: Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Eng. 140(1–2), 39–58 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Harari, I., Slavutin, M., Turkel, E.: Analytical and numerical studies of a finite element PML for the Helmholtz equation. J. Comput Acoust. 8(1), 121–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. He, Z., Cheng, A., Zhang, G., Zhong, Z., Liu, G.: Dispersion error reduction for acoustic problems using the edge-based smoothed finite element method (ES-FEM). Int. J. Numer. Methods Eng. 86(11), 1322–1338 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hiemstra, R.R., Calabrò, F., Schillinger, D., Hughes, T.J.R.: Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 316, 966–1004 (2016)

    Article  MathSciNet  Google Scholar 

  40. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hughes, T.J.R., Reali, 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. Comput. Methods Appl. Mech. Eng. 197(49), 4104–4124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hughes, T.J.R., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5), 301–313 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ihlenburg, F., Babuška, I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Methods Eng. 38(22), 3745–3774 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  45. Komatitsch, D., Tromp, J.: Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999)

    Article  Google Scholar 

  46. Komatitsch, D., Vilotte, J.P.: The spectral element method: an efficient tool to simulate the seismic response of 2d and 3d geological structures. Bull. Seismol. Soc. Am. 88(2), 368–392 (1998)

    MATH  Google Scholar 

  47. Lipton, S., Evans, J.A., Bazilevs, Y., Elguedj, T., Hughes, T.J.R.: Robustness of isogeometric structural discretizations under severe mesh distortion. Comput. Methods Appl. Mech. Eng. 199(5), 357–373 (2010)

    Article  MATH  Google Scholar 

  48. Liu, J., Dedè, L., Evans, J.A., Borden, M.J., Hughes, T.J.R.: Isogeometric analysis of the advective Cahn–Hilliard equation: spinodal decomposition under shear flow. J. Comput. Phys. 242, 321–350 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Marfurt, K.J.: Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics 49(5), 533–549 (1984)

    Article  Google Scholar 

  50. Motlagh, Y.G., Ahn, H.T., Hughes, T.J.R., Calo, V.M.: Simulation of laminar and turbulent concentric pipe flows with the isogeometric variational multiscale method. Comput. Fluids 71, 146–155 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. Nguyen, L.H., Schillinger, D.: A collocated isogeometric finite element method based on Gauss–Lobatto Lagrange extraction of splines. Comput. Methods Appl. Mech. Eng. 316, 720–740 (2016)

    Article  MathSciNet  Google Scholar 

  52. Pardo, D., Paszynski, M., Collier, N., Alvarez, J., Dalcin, L., Calo, V.M.: A survey on direct solvers for Galerkin methods. SeMA J. 57(1), 107–134 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Piegl, L., Tiller, W.: The NURBS Book. Springer, New York (1997)

    Book  MATH  Google Scholar 

  54. Puzyrev, V., Deng, Q., Calo, V.M.: Dispersion-optimized quadrature rules for isogeometric analysis: modified inner products, their dispersion properties, and optimally blended schemes. Comput. Methods Appl. Mech. Eng. 320, 421–443 (2017). http://dx.doi.org/10.1016/j.cma.2017.03.029. http://www.sciencedirect.com/science/article/pii/S004578251631920X

    Article  MathSciNet  Google Scholar 

  55. Reali, A.: An isogeometric analysis approach for the study of structural vibrations. Master’s Thesis, University of Pavia (2004)

    Google Scholar 

  56. Seriani, G., Oliveira, S.P.: Optimal blended spectral-element operators for acoustic wave modeling. Geophysics 72(5), SM95–SM106 (2007)

    Article  Google Scholar 

  57. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, vol. 12. Springer, New York (2013)

    MATH  Google Scholar 

  58. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method, vol. 212. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  59. Thompson, L.L., Pinsky, P.M.: Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech. 13(4), 255–275 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  60. Thompson, L.L., Pinsky, P.M.: A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation. Int. J. Numer. Methods Eng. 38(3), 371–397 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wang, D., Liu, W., Zhang, H.: Novel higher order mass matrices for isogeometric structural vibration analysis. Comput. Methods Appl. Mech. Eng. 260, 92–108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  62. Wang, D., Liu, W., Zhang, H.: Superconvergent isogeometric free vibration analysis of Euler–Bernoulli beams and Kirchhoff plates with new higher order mass matrices. Comput. Methods Appl. Mech. Eng. 286, 230–267 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. Yue, B., Guddati, M.N.: Dispersion-reducing finite elements for transient acoustics. J. Acoust. Soc. Am. 118(4), 2132–2141 (2005)

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain

    Michael Bartoň

  2. Department of Applied Geology, Western Australian School of Mines, Curtin University, Bentley, Perth, WA, Australia

    Victor Calo

  3. Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Kensington, Perth, WA, Australia

    Victor Calo

  4. Department of Applied Geology, Western Australian School of Mines, Curtin University, Bentley, Perth, WA, Australia

    Quanling Deng & Vladimir Puzyrev

Authors
  1. Michael Bartoň
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor Calo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Quanling Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vladimir Puzyrev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quanling Deng .

Editor information

Editors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, CNRS, Université Montpellier, Montpellier, France

    Daniele Antonio Di Pietro

  2. Universit Paris Est, CERMICS (ENPC), Marne la Vallée, France

    Alexandre Ern

  3. MOX - Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Luca Formaggia

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bartoň, M., Calo, V., Deng, Q., Puzyrev, V. (2018). Generalization of the Pythagorean Eigenvalue Error Theorem and Its Application to Isogeometric Analysis. In: Di Pietro, D., Ern, A., Formaggia, L. (eds) Numerical Methods for PDEs. SEMA SIMAI Springer Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-94676-4_6

Download citation

Publish with us

Policies and ethics

Search

Navigation