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CN 34-1054/N

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Suitable for the construction of degenerate smooth interpolated surfaces patches in isogeometric analysis

  • 1.

    School of Mathematics, Hefei University of Technology, Hefei 230061, China

  • 2.

    School of Statistics and Mathematics, Nanjing Audit University, Nanjing 210000, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.03.011
  • Received Date: 05 December 2019
  • Accepted Date: 12 January 2020
  • Rev Recd Date: 12 January 2020
  • Publish Date: 31 March 2020
    Abstract

  • Abstract

    Different from constructing the degenerate smooth surface patches in the field of classical geometric modeling, the paper constructs an interpolation operator based on the singular H regularity parameterization in the isogeometric analysis. First, the definition of H regularity is introduced based on singular parameterization. Then, the definition of D-patch and interpolation operator is discussed. Finally, the Gsmoothness of the interpolation surface is verified and numerical examples are given.

    Abstract

    Different from constructing the degenerate smooth surface patches in the field of classical geometric modeling, the paper constructs an interpolation operator based on the singular H regularity parameterization in the isogeometric analysis. First, the definition of H regularity is introduced based on singular parameterization. Then, the definition of D-patch and interpolation operator is discussed. Finally, the Gsmoothness of the interpolation surface is verified and numerical examples are given.
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    • References(29)
  • [1]
    CRISFIELD MA. Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics[M]. London: John Wiley & Sons, Inc. 1997.
    [2]
    高曙明, 何发智. 异构CAD系统集成技术综述[J].计算机辅助设计与图形学学报, 2009, 21(5): 561-568.
    GAO S M, HE F Z. A survey of heterogeneous CAD system integration[J]. Journal of Computer Aided Design & Computer Graphics, 2009, 21(5): 561-568.
    [3]
    HUGHES T J R, COTTRELL J A, BAZILEVS Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39-41): 4135-4195.
    [4]
    张汉杰, 王东东, 轩军厂. 薄梁板结构NURBS几何精确有限元分析[J]. 力学季刊, 2010, 31(4): 469-477.
    ZHANG H J, WANG D D, XUAN J C. Non-uniform rational B spline-based isogeometric finite element analysis of thin beams and plates[J]. Chinese Quarterly of Mechanics, 2010, 31(4): 469-477.
    [5]
    许华强. 面向等几何分析的样条参数体生成方法研究[D].杭州: 杭州电子科技大学, 2012.
    [6]
    BAZILEVS Y，BEIRAO D V L, COTTRELL J A, et al. Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes[J]. Mathematical Models and Methods in Applied Science, 2006, 16(7): 1031-1090.
    [7]
    徐岗，李新，黄章进，等. 面向等几何分析的几何计算[J]. 计算机辅助设计与图形学学报,2015, 27(4): 570-581.
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    [8]
    吴紫俊，黄正东，左兵权，等. 等几何分析研究概述[J]. 机械工程学报,2015, 51(5): 114-129.
    WU Z J, HUANG Z D, ZUO B Q, et al. Perspectives on isogeometric analysis[J]. Journal of Mechanical Engineering, 2015, 51(5): 114-129.
    [9]
    COHEN E, MARTIN T, KIRBY R M , et al. Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5-8): 334-356.
    [10]
    XU G, MOURRAIN B, RGIS D, et al. Parameterization of computational domain in isogeometric analysis: Methods and comparison[J]. Computer Methods in Applied Mechanics & Engineering, 2011, 200(23-24): 2021-2031.
    [11]
    XU G, LI M, MOURRAIN B, et al. Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 175-200.
    [12]
    DED L, QUARTERONI A. Isogeometric analysis for second order partial differential equations on surfaces[J]. Computer Methods in Applied Mechanics & Engineering, 2015, 284: 807-834.
    [13]
    LU J. Circular element: Isogeometric elements of smooth boundary[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(30-32): 2391-2402.
    [14]
    MARTIN T, COHEN E, KIRBY R M. Volumetric parameterization and trivariate B-spline fitting using harmonic functions[J]. Computer Aided Geometric Design, 2009, 26: 648-664.
    [15]
    WANG D, XUAN J. An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40): 2425-2436.
    [16]
    WU M , WANG Y , MOURRAIN B , et al. Convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis[J]. Computer Aided Geometric Design, 2017, 52-53:170-189.
    [17]
    TAKACS T. Construction of smooth isogeometric function spaces on singularly parameterized domains[J]. Curves and Surfaces, 2014, 9213: 433-451.
    [18]
    NGUYEN T, PETERS J. Refinable C spline elements for irregular quad layout[J]. Computer Aided Geometric Design. 2016, 43: 123-130.
    [19]
    TOSHNIWAL D, SPELEERS H, Hiemstra R.R, et al. Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis[J]. Computer Methods in Applied Mechanics & Engineering, 2017, 316: 1005-1061.
    [20]
    TAKACS T. Approximation properties of isogeometric function spaces on singularly parameterized domains[J].Mathematics, 2015:arXiv:1507.08095v1.
    [21]
    WU M, MOURRAIN B, GALLIGO A, et al. Hermite type spline spaces over rectangular meshes with complex topological structures[J]. Communications in Computational Physics, 2017, 21(03): 835-866.
    [22]
    JEONG J W, OH H S, KANG S, et al. Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 334-352.
    [23]
    OH HS, KIM H, JEONG J W. Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners[J]. International Journal for Numerical Methods in Engineering, 2014, 97(3): 149-180.
    [24]
    WU M, MOURRAIN B, GALLIGO A, et al.H-parameterizations of complex planar physical domains in isogeometric analysis[J].Computer Methods in Applied Mechanics and Engineering, 2017, 318: 296-318.
    [25]
    WU M, WANG X H. A H-integrability condition of surfaces with singular parametrizations in isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 136-156.
    [26]
    HEBEY E. Sobolev Spaces on Riemannian Manifolds[M]. Berlin: Springer, 1996: 1635.
    [27]
    施法中. 计算机辅助几何设计与非均匀有理B样条[M].北京: 高等教育出版社, 2013: 217-258.
    [28]
    TAKACS T, JTTLER B. Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis[J]. Computer Methods in Applied Mechanics & Engineering, 2011, 200(49-52): 3568-3582.
    [29]
    REIF U. A refineable space of smooth spline surfaces of arbitrary topological genus[J]. Journal of Approximation Theory, 1997, 90(2): 174-199.)
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    [1]
    CRISFIELD MA. Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics[M]. London: John Wiley & Sons, Inc. 1997.
    [2]
    高曙明, 何发智. 异构CAD系统集成技术综述[J].计算机辅助设计与图形学学报, 2009, 21(5): 561-568.
    GAO S M, HE F Z. A survey of heterogeneous CAD system integration[J]. Journal of Computer Aided Design & Computer Graphics, 2009, 21(5): 561-568.
    [3]
    HUGHES T J R, COTTRELL J A, BAZILEVS Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39-41): 4135-4195.
    [4]
    张汉杰, 王东东, 轩军厂. 薄梁板结构NURBS几何精确有限元分析[J]. 力学季刊, 2010, 31(4): 469-477.
    ZHANG H J, WANG D D, XUAN J C. Non-uniform rational B spline-based isogeometric finite element analysis of thin beams and plates[J]. Chinese Quarterly of Mechanics, 2010, 31(4): 469-477.
    [5]
    许华强. 面向等几何分析的样条参数体生成方法研究[D].杭州: 杭州电子科技大学, 2012.
    [6]
    BAZILEVS Y，BEIRAO D V L, COTTRELL J A, et al. Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes[J]. Mathematical Models and Methods in Applied Science, 2006, 16(7): 1031-1090.
    [7]
    徐岗，李新，黄章进，等. 面向等几何分析的几何计算[J]. 计算机辅助设计与图形学学报,2015, 27(4): 570-581.
    XU G, LI X, HUANG Z J, et al. Geometric computing for isogeometric analysis[J]. Journal of Computer Aided Design & Computer Graphics, 2015, 27(4): 570-581.
    [8]
    吴紫俊，黄正东，左兵权，等. 等几何分析研究概述[J]. 机械工程学报,2015, 51(5): 114-129.
    WU Z J, HUANG Z D, ZUO B Q, et al. Perspectives on isogeometric analysis[J]. Journal of Mechanical Engineering, 2015, 51(5): 114-129.
    [9]
    COHEN E, MARTIN T, KIRBY R M , et al. Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5-8): 334-356.
    [10]
    XU G, MOURRAIN B, RGIS D, et al. Parameterization of computational domain in isogeometric analysis: Methods and comparison[J]. Computer Methods in Applied Mechanics & Engineering, 2011, 200(23-24): 2021-2031.
    [11]
    XU G, LI M, MOURRAIN B, et al. Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 175-200.
    [12]
    DED L, QUARTERONI A. Isogeometric analysis for second order partial differential equations on surfaces[J]. Computer Methods in Applied Mechanics & Engineering, 2015, 284: 807-834.
    [13]
    LU J. Circular element: Isogeometric elements of smooth boundary[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(30-32): 2391-2402.
    [14]
    MARTIN T, COHEN E, KIRBY R M. Volumetric parameterization and trivariate B-spline fitting using harmonic functions[J]. Computer Aided Geometric Design, 2009, 26: 648-664.
    [15]
    WANG D, XUAN J. An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40): 2425-2436.
    [16]
    WU M , WANG Y , MOURRAIN B , et al. Convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis[J]. Computer Aided Geometric Design, 2017, 52-53:170-189.
    [17]
    TAKACS T. Construction of smooth isogeometric function spaces on singularly parameterized domains[J]. Curves and Surfaces, 2014, 9213: 433-451.
    [18]
    NGUYEN T, PETERS J. Refinable C spline elements for irregular quad layout[J]. Computer Aided Geometric Design. 2016, 43: 123-130.
    [19]
    TOSHNIWAL D, SPELEERS H, Hiemstra R.R, et al. Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis[J]. Computer Methods in Applied Mechanics & Engineering, 2017, 316: 1005-1061.
    [20]
    TAKACS T. Approximation properties of isogeometric function spaces on singularly parameterized domains[J].Mathematics, 2015:arXiv:1507.08095v1.
    [21]
    WU M, MOURRAIN B, GALLIGO A, et al. Hermite type spline spaces over rectangular meshes with complex topological structures[J]. Communications in Computational Physics, 2017, 21(03): 835-866.
    [22]
    JEONG J W, OH H S, KANG S, et al. Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 334-352.
    [23]
    OH HS, KIM H, JEONG J W. Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners[J]. International Journal for Numerical Methods in Engineering, 2014, 97(3): 149-180.
    [24]
    WU M, MOURRAIN B, GALLIGO A, et al.H-parameterizations of complex planar physical domains in isogeometric analysis[J].Computer Methods in Applied Mechanics and Engineering, 2017, 318: 296-318.
    [25]
    WU M, WANG X H. A H-integrability condition of surfaces with singular parametrizations in isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 136-156.
    [26]
    HEBEY E. Sobolev Spaces on Riemannian Manifolds[M]. Berlin: Springer, 1996: 1635.
    [27]
    施法中. 计算机辅助几何设计与非均匀有理B样条[M].北京: 高等教育出版社, 2013: 217-258.
    [28]
    TAKACS T, JTTLER B. Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis[J]. Computer Methods in Applied Mechanics & Engineering, 2011, 200(49-52): 3568-3582.
    [29]
    REIF U. A refineable space of smooth spline surfaces of arbitrary topological genus[J]. Journal of Approximation Theory, 1997, 90(2): 174-199.)
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