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A Survey on Isogeometric Collocation Methods with Applications

Author

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  • Jingwen Ren

    (School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China
    State Key Lab. of CAD&CG, Zhejiang University, Hangzhou 310058, China)

  • Hongwei Lin

    (School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China
    State Key Lab. of CAD&CG, Zhejiang University, Hangzhou 310058, China)

Abstract

Isogeometric analysis (IGA) is an effective numerical method for connecting computer-aided design and engineering, which has been widely applied in various aspects of computational mechanics. IGA involves Galerkin and collocation formulations. Exploiting the same high-order non-uniform rational B-spline (NURBS) bases that span the physical domain and the solution space leads to increased accuracy and fast computation. Although IGA Galerkin provides optimal convergence, IGA collocation performs better in terms of the ratio of accuracy to computational time. Without numerical integration, by working directly with the strong form of the partial differential equation over the physical domain defined by NURBS geometry, the derivatives of the NURBS-expressed numerical solution at some chosen collocation points can be calculated. In this study, we survey the methodological framework and the research prospects of IGA. The collocation schemes in the IGA collocation method that affect the convergence performance are addressed in this paper. Recent studies and application developments are reviewed as well.

Suggested Citation

  • Jingwen Ren & Hongwei Lin, 2023. "A Survey on Isogeometric Collocation Methods with Applications," Mathematics, MDPI, vol. 11(2), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:469-:d:1037007
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    References listed on IDEAS

    as
    1. Cho, Durkbin, 2020. "Optimal multilevel preconditioners for isogeometric collocation methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 168(C), pages 76-89.
    2. Nguyen, Vinh Phu & Anitescu, Cosmin & Bordas, Stéphane P.A. & Rabczuk, Timon, 2015. "Isogeometric analysis: An overview and computer implementation aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 117(C), pages 89-116.
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