Abstract
Although the isogeometric collocation (IGA-C) method has been successfully utilized in practical applications due to its simplicity and efficiency, only a little theoretical results have been established on the numerical analysis of the IGA-C method. In this paper, we deduce the convergence rate of the consistency of the IGA-C method. Moreover, based on the formula of the convergence rate, the necessary and sufficient condition for the consistency of the IGA-C method is developed. These results advance the numerical analysis of the IGA-C method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F Auricchio, L B Veiga, A Buffa, C Lovadina, A Reali, G Sangalli. A fully locking-free isogeometric approach for plane linear elasticity problems: a stream function formulation, Computer methods in applied mechanics and engineering, 2007, 197(1): 160–172.
F Auricchio, L B Veiga, T J R Hughes, A Reali, G Sangalli. Isogeometric collocation methods, Mathematical Models and Methods in Applied Sciences, 2010, 20(11): 2075–2107.
F Auricchio, L B Veiga, T J R Hughes, A Reali, G Sangalli. Isogeometric collocation for elastostatics and explicit dynamics, Computer methods in applied mechanics and engineering, 2012, 249: 2–14.
F Auricchio, L B Veiga, J Kiendl, C Lovadina, A Reali. Locking-free isogeometric collocation methods for spatial timoshenko rods, Computer Methods in Applied Mechanics and Engineering, 2013, 263(15): 113–126.
M Aigner, C Heinrich, B Jüttler, E Pilgerstorfer, B Simeon, A Vuong. Swept volume parameterization for isogeometric analysis, In: Mathematics of Surfaces XIII, 2009, 19–44.
C Anitescu, Y Jia, Y J Zhang, T Rabczuk. An isogeometric collocation method using superconvergent points, Computer Methods in Applied Mechanics and Engineering, 2015, 284: 1073–1097.
Y Bazilevs, L B Veiga, J A Cottrell, T J R Hughes, G Sangalli. Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Mathematical Models and Methods in Applied Sciences, 2006, 16(7): 1031–1090.
Y Bazilevs, V M Calo, T J R Hughes, Y Zhang. Isogeometric fluid-structure interaction: theory, algorithms, and computations, Computational Mechanics, 2008, 43(1): 3–37.
Y Bazilevs, V M Calo, Y Zhang, T J R Hughes. Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Computational Mechanics, 2006, 38(4): 310–322.
L B Veiga, C Lovadina, A Reali. Avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods, Computer Methods in Applied Mechanics and Engineering, 2012, 241: 38–51.
Y Bazilevs, J R Gohean, T J R Hughes, R D Moser, Y Zhang. Patient-specific isogeometric fluid-structure interaction analysis of thoracic aortic blood flow due to implantation of the jarvik 2000 left ventricular assist device, Computer Methods in Applied Mechanics and Engineering, 2009, 198(45): 3534–3550.
J A Cottrell, T J R Hughes, A Reali. Studies of refinement and continuity in isogeometric structural analysis, Computer methods in applied mechanics and engineering, 2007, 196(41): 4160–4183.
J A Cottrell, A Reali, Y Bazilevs, T J R Hughes. Isogeometric analysis of structural vibrations, Computer methods in applied mechanics and engineering, 2006, 195(41): 5257–5296.
C de Boor. A practical guide to splines, Springer Verlag, 1978, 27(149), DOI: https://doi.org/10.2307/2006241.
M Donatelli, C Garoni, C Manni, S Serra-Capizzano, H Speleers. Robust and optimal multiiterative techniques for IGA Galerkin linear systems, Computer Methods in Applied Mechanics and Engineering, 2015, 284: 230–264.
L D Lorenzis, J A Evans, T J R Hughes, A Reali. Isogeometric collocation: Neumann boundary conditions and contact, Computer Methods in Applied Mechanics and Engineering, 2015, 284: 21–54.
T Elguedj, Y Bazilevs, V M Calo, T J R Hughes. B and F projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order nurbs elements, Computer Methods in Applied Mechanics and Engineering, 2008, 197: 2732–2762.
T Elguedj, T J R Hughes. Isogeometric analysis of nearly incompressible large strain plasticity, Computer Methods in Applied Mechanics and Engineering, 2014, 268: 388–416.
J A Evans, R R Hiemstra, T J R Hughes, A Reali. Explicit higher-order accurate isogeometric collocation methods for structural dynamics, Computer Methods in Applied Mechanics and Engineering, 2018, 338: 208–240.
F Fahrendorf, L D Lorenzis, H Gomez. Reduced integration at superconvergent points in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2018, 328: 390–410.
H Gomez, L D Lorenzis. The variational collocation method, Computer Methods in Applied Mechanics and Engineering, 2016, 309: 152–181.
T J R Hughes, J A Cottrell, Y Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 2005, 194(39): 4135–4195.
T J R Hughes, A Reali, G Sangalli. Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS, Computer Methods in Applied Mechanics and Engineering, 2008, 197(49): 4104–4124.
T J R Hughes, A Reali, G Sangalli. Efficient quadrature for nurbs-based isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2010, 199(5): 301–313.
J Kiendl, F Auricchio, L B Veiga, C Lovadina, A Reali. Isogeometric collocation methods for the reissner-mindlin plate problem, Computer Methods in Applied Mechanics and Engineering, 2015, 284: 489–507.
J Kiendl, E Marino, L D Lorenzis. Isogeometric collocation for the reissner-mindlin shell problem, Computer Methods in Applied Mechanics and Engineering, 2017, 325: 645–665.
H Lin, Q Hu, Y Xiong. Consistency and convergence properties of the isogeometric collocation method, Computer Methods in Applied Mechanics and Engineering, 2013, 267: 471–486.
F Maurin, F Greco, L Coox, D Vandepitte, W Desmet. Isogeometric collocation for kirchhoff-love plates and shells, Computer Methods in Applied Mechanics and Engineering, 2018, 329: 396–420.
M Montardini, G Sangalli, L Tamellini. Optimal-order isogeometric collocation at galerkin superconvergent points, Computer Methods in Applied Mechanics and Engineering, 2017, 316: 741–757.
A Reali, H Gomez. An isogeometric collocation approach for bernoulli-euler beams and kirchhoff plates, Computer Methods in Applied Mechanics and Engineering, 2015, 284: 623–636.
D Schillinger, J A Evans, A Reali, M A Scott, T J R Hughes. Isogeometric collocation: Cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations, Computer Methods in Applied Mechanics and Engineering, 2013, 267: 170–232.
P Solin. Partial differential equations and the finite element method, Wiley-Interscience, 2006.
W A Wall, M A Frenzel, C Cyron. Isogeometric structural shape optimization, Computer Methods in Applied Mechanics and Engineering, 2008, 197(33): 2976–2988.
D Wang, D Qi, X Li. Superconvergent isogeometric collocation method with greville points, Computer Methods in Applied Mechanics and Engineering, 2021, 377: 113689.
O Weeger, S K Yeung, M L Dunn. Isogeometric collocation methods for cosserat rods and rod structures, Computer Methods in Applied Mechanics and Engineering, 2017, 316: 100–122.
G Xu, B Mourrain, R Duvigneau, A Galligo. Parameterization of computational domain in isogeometric analysis: methods and comparison, Computer Methods in Applied Mechanics and Engineering, 2011, 200(23): 2021–2031.
G Xu, B Mourrain, R Duvigneau, A Galligo. Optimal analysis-aware parameterization of computational domain in 3d isogeometric analysis, Computer-Aided Design, 2013, 45(4): 812–821.
Acknowledgement
We are sincerely grateful to the reviewers for their careful and helpful comments.
Funding
This work is supported by the National Natural Science Foundation of China(61872316), and the Natural Science Foundation of Zhejiang Province, China(LY19F020004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articles Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lin, Hw., Xiong, Yy., Hu, H. et al. The convergence rate and necessary-and-sufficient condition for the consistency of isogeometric collocation method. Appl. Math. J. Chin. Univ. 37, 272–289 (2022). https://doi.org/10.1007/s11766-022-4587-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-022-4587-2