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Journal of Scientific Computing

Erschienen in:

01.04.2020

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

verfasst von: Paola Gervasio, Luca Dedè, Ondine Chanon, Alfio Quarteroni

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2020

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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.

Vorheriger Artikel Optimal Energy Conserving and Energy Dissipative Local Discontinuous Galerkin Methods for the Benjamin–Bona–Mahony Equation

Nächster Artikel Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime

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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)MathSciNetMATH

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)MathSciNetMATH

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)MathSciNetMATH

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)MathSciNetMATH

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)MathSciNetMATH

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)MathSciNetMATH

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

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)MathSciNetMATH

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

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

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

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

15.

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

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

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)MathSciNetMATH

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)MathSciNetMATH

20.

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

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)MathSciNetMATH

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

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)MathSciNetMATH

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)MathSciNetMATH

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

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)MathSciNetMATH

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

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

31.

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

32.

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

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)MathSciNetMATH

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)MathSciNetMATH

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)MathSciNetMATH

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

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)MathSciNetMATH

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)MathSciNetMATH

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

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

41.

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

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

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)MathSciNetMATH

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

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

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)MathSciNetMATH

48.

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

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)MathSciNetMATH

Titel: A Computational Comparison Between Isogeometric Analysis and Spectral Element Methods: Accuracy and Spectral Properties
verfasst von: Paola Gervasio
Luca Dedè
Ondine Chanon
Alfio Quarteroni
Publikationsdatum: 01.04.2020
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2020
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01204-1

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