An isogeometric collocation method using superconvergent points

Cosmin Anitescu; Yue Jia; Yongjie Jessica Zhang; Timon Rabczuk

doi:10.1016/j.cma.2014.11.038

An isogeometric collocation method using superconvergent points

Cosmin Anitescu
, Yue Jia
, Yongjie Jessica Zhang
, Timon Rabczuk^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

123 Citations (Scopus)

Abstract

We develop an IGA collocation method modified by collocating at points other than the standard Greville abscissae. The method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory, therefore we refer to this method as "super-collocation" (IGA-SC). By carefully choosing the collocation points, it can be seen that the IGA-SC converges in the first derivative (energy) norms at rates similar to that of the Galerkin solution. This is different from the collocation at Greville abscissae (IGA-C), where the convergence in energy norm for odd polynomial degrees is typically suboptimal. The method is tested on 1D, 2D and 3D numerical examples, in which it is compared to IGA-C and Galerkin's method (IGA-G). The comparison includes a detailed cost vs. accuracy analysis, which shows an improved efficiency of the proposed method in particular for odd polynomial degrees.

Original language	English
Pages (from-to)	1073-1097
Number of pages	25
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	284
DOIs	https://doi.org/10.1016/j.cma.2014.11.038
Publication status	Published - 2015 Feb 1

Bibliographical note

Funding Information:
The authors would like to thank the support by the European Union through the FP7-grant ITN (Marie Curie Initial Training Networks) INSIST (Integrating Numerical Simulation and Geometric Design Technology) PITN-GA-2011-289361 . Y. Zhang was supported in part by the PECASE Award N00014-14-1-0234 and NSF CAREER Award OCI-1149591 .

Publisher Copyright:
© 2014 Elsevier B.V.

Keywords

Galerkin method
Greville abscissae
IGA collocation
Least-squares
Orthogonal collocation
Superconvergence

ASJC Scopus subject areas

Computational Mechanics
Mechanics of Materials
Mechanical Engineering
General Physics and Astronomy
Computer Science Applications

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10.1016/j.cma.2014.11.038

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title = "An isogeometric collocation method using superconvergent points",

abstract = "We develop an IGA collocation method modified by collocating at points other than the standard Greville abscissae. The method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory, therefore we refer to this method as {"}super-collocation{"} (IGA-SC). By carefully choosing the collocation points, it can be seen that the IGA-SC converges in the first derivative (energy) norms at rates similar to that of the Galerkin solution. This is different from the collocation at Greville abscissae (IGA-C), where the convergence in energy norm for odd polynomial degrees is typically suboptimal. The method is tested on 1D, 2D and 3D numerical examples, in which it is compared to IGA-C and Galerkin's method (IGA-G). The comparison includes a detailed cost vs. accuracy analysis, which shows an improved efficiency of the proposed method in particular for odd polynomial degrees.",

keywords = "Galerkin method, Greville abscissae, IGA collocation, Least-squares, Orthogonal collocation, Superconvergence",

author = "Cosmin Anitescu and Yue Jia and Zhang, \{Yongjie Jessica\} and Timon Rabczuk",

note = "Funding Information: The authors would like to thank the support by the European Union through the FP7-grant ITN (Marie Curie Initial Training Networks) INSIST (Integrating Numerical Simulation and Geometric Design Technology) PITN-GA-2011-289361 . Y. Zhang was supported in part by the PECASE Award N00014-14-1-0234 and NSF CAREER Award OCI-1149591 . Publisher Copyright: {\textcopyright} 2014 Elsevier B.V.",

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AU - Anitescu, Cosmin

AU - Jia, Yue

AU - Zhang, Yongjie Jessica

AU - Rabczuk, Timon

N1 - Funding Information: The authors would like to thank the support by the European Union through the FP7-grant ITN (Marie Curie Initial Training Networks) INSIST (Integrating Numerical Simulation and Geometric Design Technology) PITN-GA-2011-289361 . Y. Zhang was supported in part by the PECASE Award N00014-14-1-0234 and NSF CAREER Award OCI-1149591 . Publisher Copyright: © 2014 Elsevier B.V.

PY - 2015/2/1

Y1 - 2015/2/1

N2 - We develop an IGA collocation method modified by collocating at points other than the standard Greville abscissae. The method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory, therefore we refer to this method as "super-collocation" (IGA-SC). By carefully choosing the collocation points, it can be seen that the IGA-SC converges in the first derivative (energy) norms at rates similar to that of the Galerkin solution. This is different from the collocation at Greville abscissae (IGA-C), where the convergence in energy norm for odd polynomial degrees is typically suboptimal. The method is tested on 1D, 2D and 3D numerical examples, in which it is compared to IGA-C and Galerkin's method (IGA-G). The comparison includes a detailed cost vs. accuracy analysis, which shows an improved efficiency of the proposed method in particular for odd polynomial degrees.

AB - We develop an IGA collocation method modified by collocating at points other than the standard Greville abscissae. The method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory, therefore we refer to this method as "super-collocation" (IGA-SC). By carefully choosing the collocation points, it can be seen that the IGA-SC converges in the first derivative (energy) norms at rates similar to that of the Galerkin solution. This is different from the collocation at Greville abscissae (IGA-C), where the convergence in energy norm for odd polynomial degrees is typically suboptimal. The method is tested on 1D, 2D and 3D numerical examples, in which it is compared to IGA-C and Galerkin's method (IGA-G). The comparison includes a detailed cost vs. accuracy analysis, which shows an improved efficiency of the proposed method in particular for odd polynomial degrees.

KW - Galerkin method

KW - Greville abscissae

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KW - Least-squares

KW - Orthogonal collocation

KW - Superconvergence

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M3 - Article

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JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -

An isogeometric collocation method using superconvergent points

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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