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A Computational Comparison Between Isogeometric Analysis and Spectral Element Methods: Accuracy and Spectral Properties

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Abstract

In this paper, we carry out a systematic comparison between the theoretical properties of Spectral Element Methods and NURBS-based Isogeometric Analysis in its basic form, that is in the framework of the Galerkin method, for the approximation of the Poisson problem, which we select as a benchmark Partial Differential Equation. Our focus is on their convergence properties, the algebraic structure and the spectral properties of the corresponding discrete arrays (mass and stiffness matrices). We review the available theoretical results for these methods and verify them numerically by performing an error analysis on the solution of the Poisson problem. Where theory is lacking, we use numerical investigation of the results to draw conjectures on the behaviour of the corresponding theoretical laws in terms of the design parameters, such as the (mesh) element size, the local polynomial degree, the smoothness of the NURBS basis functions, the space dimension, and the total number of degrees of freedom involved in the computations.

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References

  1. Antolin, P., Buffa, A., Calabrò, 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)

    MathSciNet  MATH  Google Scholar 

  2. Auricchio, F., Calabrò, 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(252), 15–27 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G.: Isogeometric collocation methods. Math. Models Methods Appl. Sci. 20(11), 2075–2107 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G.: Isogeometric collocation for elastostatics and explicit dynamics. Comput. Methods Appl. Mech. Eng. 249(252), 2–14 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Bartezzaghi, A., Dedè, L., Quarteroni, A.: Isogeometric analysis for high order partial differential equations on surfaces. Comp. Methods Appl. Mech. Eng. 295, 446–469 (2015)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. Bartoň, M., Calo, V.M.: Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis. Comput. Aid. Des. 82, 57–67 (2017)

    MathSciNet  Google Scholar 

  8. Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16, 1–60 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Bernardi, C., Maday, Y.: Approximations Spectrales de Problèmes aux Limites Elliptiques. Springer, Paris (1992)

    MATH  Google Scholar 

  10. Bernardi, C., Maday, Y.: Spectral methods. In: Handbook of Numerical Analysis, vol. V, Handb. Numer. Anal., V, pp. 209–485. North-Holland, Amsterdam (1997)

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

    MathSciNet  Google Scholar 

  12. Canuto, C., Gervasio, P., Quarteroni, A.: Finite-Element Preconditioning of G-NI Spectral Methods. SIAM J. Sci. Comput., 31(6), 4422–4451 (2009/2010)

  13. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods, Fundamentals in Single Domains. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  14. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Heidelberg (2007)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for \(h\)-\(p\)-\(k\)-refinement in isogeometric analysis. Numer. Math. 118(2), 271–305 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Beirão da Veiga, L., Buffa, A., Sangalli, G., Vàzquez, R.: An introduction to the numerical analysis of isogeometric methods. In: Numerical Simulation in Physics and Engineering, vol. 9 of SEMA SIMAI Springer Ser., pp. 3–69. Springer, Hidelberg (2016)

  19. Beirão da Veiga, L., Lovadina, C., Reali, A.: Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods. Comput. Methods Appl. Mech. Eng. 241(244), 38–51 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Dedè, L., Quarteroni, A.: Isogeometric Analysis for second order partial differential equations on surfaces. Comput. Methods Appl. Mech. Eng. 284, 807–834 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Dedè, L., Santos, H.A.F.A.: B-spline goal-oriented error estimators for geometrically nonlinear rods. Comput. Mech. 49(1), 35–52 (2012)

    MathSciNet  MATH  Google Scholar 

  22. De Falco, C., Reali, A., Vázquez, R.: Geopdes: a research tool for isogeometric analysis of pdes. Adv. Eng. Softw. 42(12), 1020–1034 (2011)

    MATH  Google Scholar 

  23. Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Robust and optimal multi-iterative techniques for IgA collocation linear systems. Comput. Methods Appl. Mech. Eng. 284, 1120–1146 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Evans, J.A., Bazilevs, Y., Babuška, I., Hughes, T.J.R.: n-widths, sup-infs, and optimality ratios for the k-version of the Isogeometric finite element method. Comput. Methods Appl. Mech. Eng. 198, 1726–1741 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Evans, J.A., Hiemstra, R.R., Hughes, T.J.R., Reali, A.: Explicit higher-order accurate isogeometric collocation methods for structural dynamics. Comput. Methods Appl. Mech. Eng. 338, 208–240 (2018)

    MathSciNet  Google Scholar 

  26. Gahalaut, K., Tomar, S.: Condition number estimates for matrices arising in the isogeometric discretizations. Technical Report 2012-23, RICAM, (2012)

  27. Garoni, C., Manni, C., Pelosi, F., Serra Capizzano, S., Speleers, H.: On the spectrum of stiffness matrices arising from isogeometric analysis. Numer. Math. 127(4), 751–799 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Gervasio, P.: \(\text{CHQZ}\_\text{ lib }\): a MATLABlibrary for spectral element methods (2007). http://paola-gervasio.unibs.it/software

  29. Gomez, H., De Lorenzis, L.: The variational collocation method. Comput. Methods Appl. Mech. Eng. 309, 152–181 (2016)

    MathSciNet  Google Scholar 

  30. Gordon, W.J., Hall, C.A.: Construction of curvilinear co-ordinate systems and their application to mesh generation. Int. J. Numer. Meth. Eng. 7, 461–477 (1973)

    MATH  Google Scholar 

  31. Gordon, W.J., Hall, C.A.: Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Numer. Math. 21, 109–129 (1973)

    MathSciNet  MATH  Google Scholar 

  32. Gordon, W.J., Thiel, L.C.: Transfinite mappings and their application to grid generation. Appl. Math. Comput. 10, 171–233 (1982)

    MathSciNet  MATH  Google Scholar 

  33. Hughes, T.J.R., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5–8), 301–313 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Kiendl, J., Auricchio, F., Hughes, T.J.R., Reali, A.: Single-variable formulations and isogeometric discretizations for shear deformable beams. Comput. Methods Appl. Mech. Eng. 284, 988–1004 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Kiendl, J., Hsu, M.C., Wu, M.C.H., Reali, A.: Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials. Comput. Methods Appl. Mech. Eng. 291, 280–303 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Mantzaflaris, A., Jüttler, B., Khoromskij, B.N., Langer, U.: Low rank tensor methods in Galerkin-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 316, 1062–1085 (2017)

    MathSciNet  Google Scholar 

  37. Maurin, F., Dedè, L., Spadoni, A.: Isogeometric rotation-free analysis of planar extensible-elastica for static and dynamic applications. Nonlinear Dyn. 81(1–2), 77–96 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Melenk, J.M.: On condition numbers in \(hp\)-FEM with Gauss–Lobatto-based shape functions. J. Comput. Appl. Math. 139(1), 21–48 (2002)

    MathSciNet  MATH  Google Scholar 

  39. Montardini, M., Sangalli, G., Tamellini, L.: Optimal-order isogeometric collocation at Galerkin superconvergent points. Comput. Methods Appl. Mech. Eng. 316, 741–757 (2017)

    MathSciNet  Google Scholar 

  40. 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 (2017)

    MathSciNet  Google Scholar 

  41. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Heidelberg (1994)

    MATH  Google Scholar 

  42. Reali, A., Hughes, T.J.R.: An introduction to isogeometric collocation methods. In: CISM International Centre for Mechanical Sciences, vol. 561. Courses and Lectures. Springer, Vienna (2015)

  43. Sangalli, G., Tani, M.: Matrix-free weighted quadrature for a computationally efficient isogeometric \(k\)-method. Comput. Methods Appl. Mech. Engrg. 338, 117–133 (2018)

    MathSciNet  Google Scholar 

  44. Schillinger, D., Evans, J.A., Reali, A., Scott, M.A., Hughes, T.J.R.: Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput. Methods Appl. Mech. Eng. 267, 170–232 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Schillinger, D., Hossain, S.J., Hughes, T.J.R.: Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 277, 1–45 (2014)

    MATH  Google Scholar 

  46. Scholz, F., Mantzaflaris, A., Jüttler, B.: Partial tensor decomposition for decoupling isogeometric Galerkin discretizations. Comput. Methods Appl. Mech. Eng. 336, 485–506 (2018)

    MathSciNet  Google Scholar 

  47. Tagliabue, A., Dedè, L., Quarteroni, A.: Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Comput. Fluids 102, 277–303 (2014)

    MathSciNet  MATH  Google Scholar 

  48. van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  49. Vàzquez, R.: A new design for the implementation of isogeometric analysis in octave and matlab: Geopdes 3.0. Comput. Math. Appl. 72(3), 523–554 (2016)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. DICATAM, Università degli Studi di Brescia, via Branze 38, 25123, Brescia, Italy

    Paola Gervasio

  2. MOX, Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Luca Dedè & Alfio Quarteroni

  3. MNS, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    Ondine Chanon

  4. Institute of Mathematics, École Polytechnique Fédérale de Lausanne (EPFL), Station 8, 1015, Lausanne, Switzerland

    Alfio Quarteroni

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  1. Paola Gervasio
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Gervasio, P., Dedè, L., Chanon, O. et al. A Computational Comparison Between Isogeometric Analysis and Spectral Element Methods: Accuracy and Spectral Properties. J Sci Comput 83, 18 (2020). https://doi.org/10.1007/s10915-020-01204-1

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  • DOI: https://doi.org/10.1007/s10915-020-01204-1

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