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A unified approach to construct generalized B-splines for isogeometric applications

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Abstract

Generalized B-splines have been employed as geometric modeling and numerical simulation tools for isogeometric analysis (IGA for short). However, the previous models used in IGA, such as trigonometric generalized B-splines or hyperbolic generalized B-splines, are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper, a unified approach to construct the generalized non-uniform B-splines over the space spanned by {α(t), β(t), ξ(t), η(t), 1, t, · · ·, t n–4} is proposed, and the corresponding isogeometric analysis framework for PDE solving is also studied. Compared with the NURBS-IGA method, the proposed frameworks have several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form. Furthermore, with the proposed spline models, isogeometric analysis can be performed on the computational domain bounded by transcendental curves/surfaces, such as the involute of circle, the helix/helicoid, the catenary/catenoid and the cycloid. Several numerical examples for isogeometric heat conduction problems are presented to show the effectiveness of the proposed methods.

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Authors and Affiliations

  1. Department of Computer Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Gang Xu, Ningning Sun & Jinlan Xu

  2. Key Laboratory of Complex Systems Modeling and Simulation, Ministry of Education, Hangzhou, 310018, China

    Gang Xu, Ningning Sun & Jinlan Xu

  3. Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China

    Kin-Chuen Hui

  4. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Guozhao Wang

Authors
  1. Gang Xu
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  2. Ningning Sun
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  3. Jinlan Xu
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  4. Kin-Chuen Hui
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  5. Guozhao Wang
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Corresponding author

Correspondence to Gang Xu.

Additional information

This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LR16F020003, the National Nature Science Foundation of China under Grant Nos. 61472111, 61602138, and the Open Project Program of the State Key Lab of CAD&CG (A1703), Zhejiang University.

This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.

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Xu, G., Sun, N., Xu, J. et al. A unified approach to construct generalized B-splines for isogeometric applications. J Syst Sci Complex 30, 983–998 (2017). https://doi.org/10.1007/s11424-017-6026-7

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  • DOI: https://doi.org/10.1007/s11424-017-6026-7

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