工程力学 ›› 2018, Vol. 35 ›› Issue (11): 17-25.doi: 10.6052/j.issn.1000-4750.2017.08.0602

• 基本方法 • 上一篇    下一篇

复杂三维曲梁结构的无闭锁等几何分析算法研究

夏阳, 廖科   

  1. 大连理工大学运载工程与力学学部汽车工程学院, 工业装备结构分析国家重点实验室, 辽宁, 大连 116024
  • 收稿日期:2017-08-15 修回日期:2018-01-16 出版日期:2018-11-07 发布日期:2018-11-07
  • 通讯作者: 夏阳(1987-),男,河南人,讲师,博士,主要从事等几何分析和增材制造工艺力学研究(E-mail:yangxia@dlut.edu.cn). E-mail:yangxia@dlut.edu.cn
  • 作者简介:廖科(1993-),男,四川人,硕士生,主要从事等几何分析研究(E-mail:liaokk11@dlut.edu.cn).
  • 基金资助:
    中国自然科学基金项目(11702056,61572021);中央高校基本科研业务费专项资金项目(DUT17JC32)

LOCKING-FREE ISOGEOMETRIC ANALYSIS OF COMPLEX THREE-DIMENSIONAL BEAM STRUCTURES

XIA Yang, LIAO Ke   

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, School of Automotive Engineering, Dalian University of Technology, Dalian 116024, China
  • Received:2017-08-15 Revised:2018-01-16 Online:2018-11-07 Published:2018-11-07

摘要: 梁结构在工程中应用广泛,梁结构的仿真分析是计算力学的一个重要研究内容。该文研究了复杂三维曲梁结构的等几何分析方法,首次应用拟协调有限元中的多套函数技术,使用降阶基函数逼近梁内应变项,解决复杂三维曲梁结构仿真中的闭锁问题。利用全局坐标系列式方法,避免了单元刚度阵组装时的复杂坐标变换过程,提高计算效率。使用多片NURBS (非均匀有理B样条)数据表示复杂三维梁结构,可精确描述曲梁结构的几何形状,与有限元方法等仿真技术相比避免了网格生成过程,减少了几何误差。数值结果表明该文算法可有效解决闭锁问题,适于复杂三维曲梁结构的仿真分析。

关键词: 等几何分析, 三维曲梁, 闭锁问题, 多套函数, 拟协调有限元

Abstract: The beam structure is widely used in engineering. The numerical simulation of beam structures is an important topic in computational mechanics. In this paper, the locking-free isogeometric analysis of complex three-dimensional beam structures is investigated. The technique of multiple sets of approximation functions originated from quasi-conforming finite element method is first applied to the isogeometric analysis of three-dimensional beam structures to solve the locking problem. Order-reduced approximation functions are applied to simulate the strains of beams. Global formulation of beam strains is applied, and the stiffness matrices of beam elements and patches can be combined without transformation between local and global coordinate systems. The beam structure is described by multi-patch non-uniform rational B-spline functions. The geometry is exactly described, and the geometrical error introduced by finite element mesh can be avoided. The numerical experiments prove that the proposed algorithm can effectively avoid the locking problem in Timoshenko beam formulation, and is suitable for the analysis of complex three-dimensional beam structures.

Key words: isogeometric analysis, three-dimensional beam, locking free, multiple sets of approximation functions, quasi-conforming finite element method

中图分类号: 

  • O241.8
[1] 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(39/40/41):4135-4195.
[2] Bazilevs Y, Da Veiga LB, Cottrell JA, et al. Isogeometric analysis:approximation, stability and error estimates for h-refined meshes[J]. Mathematical Models & Methods in Applied Sciences, 2006, 16(7):1031-1090.
[3] 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(5):114-129. Wu Zijun, Huang Zhengdong, Zuo Bingquan, et al. Perspectives on isogeometric analysis[J]. Journal of Mechanical Engineering, 2015(5):114-129. (in Chinese)
[4] Zhang G, Alberdi R, Khandelwal K. Analysis of three-dimensional curved beams using isogeometric approach[J]. Engineering Structures, 2016, 117:560-574.
[5] Luu A T, Kim N I, Lee J. Isogeometric vibration analysis of free-form Timoshenko curved beams[J]. Meccanica, 2015, 50(1):169-187.
[6] Cottrell J A, Reali A, Bazilevs Y, Hughes T J R. Isogeometric analysis of structural vibrations[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195(41/42/43):5257-5296.
[7] Cazzani A, Malagu M, Turco E. Isogeometric analysis:A powerful numerical tool for the elastic analysis of historical masonry arches[J]. Continuum Mechanics and Thermodynamics, 2016, 28(1/2):139-156.
[8] 王勖成. 有限单元法[M]. 北京:清华大学出版社, 2003:354-364. Wang Xucheng. Finite element method[M]. Beijing:Tsinghua University Press, 2003:354-364. (in Chinese)
[9] Bouclier R, Elguedj T, Combescure A. Locking free isogeometric formulations of curved thick beams[J]. Computer Methods in Applied Mechanics and Engineering, 2012, 245:144-162.
[10] Beirao da Veiga L, Lovadina C, Reali A. Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods[J]. Computer Methods In Applied Mechanics And Engineering, 2012, 241/242/243/244:38-51.
[11] Marino E. Locking-free isogeometric collocation formulation for three-dimensional geometrically exact shear-deformable beams with arbitrary initial curvature[J]. Computer Methods in Applied Mechanics And Engineering, 2017, 324:546-572.
[12] Greco L, Cuomo M, Contrafatto L, et al. An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 324:476-511.
[13] Adam C, Bouabdallah S, Zarroug M, et al. Improved numerical integration for locking treatment in isogeometric structural elements, Part I:Beams[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 279:1-28.
[14] 唐立民, 陈万吉, 刘迎曦. 有限元分析中的拟协调元[J]. 大连工学院学报, 1980, 19(2):19-35. Tang Limin, Chen Wanji, Liu Yingxi. Quasi-conforming elements for finite element analysis[J]. Journal of Dalian Institute of Technology, 1980, 19(2):19-35. (in Chinese)
[15] 张鸿庆, 王鸣. 多套函数有限元逼近与拟协调板元[J]. 应用数学和力学, 1985, 6(1):41-52. Zhang Hongqing, Wang Ming. Finite element approximations with multiple sets of functions and quasi-conforming elements for plate bending problems[J]. Applied Mathematics and Mechanics, 1985, 6(1):41-52. (in Chinese)
[16] 胡清元, 夏阳, 胡平, 等. 假设位移拟协调平面单元应变离散算法研究[J]. 工程力学, 2016, 33(9):30-39. Hu Qingyuan, Xia Yang, Hu Ping, et al. Research on the strain discretization algorithmin assumed displacement quasi-conforming plane finite element method[J]. Engineering Mechanics, 2016, 33(9):30-39. (in Chinese)
[17] Piegl L A, Tiller W. The NURBS book[M]. 2nd ed., Germany:Springer Verlag, 1997.
[18] 过斌, 葛建立, 杨国来, 等. 三维实体结构NURBS等几何分析[J]. 工程力学, 2015, 32(9):42-48. Guo Bin, Ge Jianli, Yang Guolai, et al. Nurbs-based isogeometric analysis of three-dimensional solid structures[J]. Engineering Mechanics, 2015, 32(9):42-48. (in Chinese)
[19] Hu P, Hu Q, Xia Y. Order reduction method for locking free isogeometric analysis of Timoshenko beams[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 308:1-22.
[20] 曾森, 陈少峰, 曲婷, 等. 大位移小转角空间曲梁的弹性力学方程[J]. 工程力学, 2010, 27(12):14-20. Zeng Sen, Chen ShaoFeng, Qu Ting, et al. Elasticity equations for spatial curved beams with large displacement and small rotation[J]. Engineering Mechanics, 2010, 27(12):14-20. (in Chinese)
[21] Hu Q, Xia Y, Zou R, Hu P. A global formulation for complex rod structures in isogeometric analysis[J]. International Journal of Mechanical Sciences, 2016, 115/116:736-745.
[22] 张勇, 林皋, 胡志强. 等几何分析方法中重控制点问题的研究与应用[J]. 工程力学, 2013, 30(2):1-7. Zhang Yong, Lin Gao, Hu Zhiqiang. Repeated control points issue in isogeometric analysis and its application[J]. Engineering Mechanics, 2013, 30(2):1-7. (in Chinese)
[1] 薛冰寒, 林皋, 胡志强, 张勇. 求解摩擦接触问题的IGA-B可微方程组方法[J]. 工程力学, 2016, 33(10): 35-43.
[2] 过斌, 葛建立, 杨国来, 吕加. 三维实体结构NURBS等几何分析[J]. 工程力学, 2015, 32(9): 42-48.
[3] 关玉璞;唐立民. 拟协调九结点四边形退化壳单元[J]. 工程力学, 1995, 12(3): 22-30.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
[1] 肖映雄;周志阳;舒 适. 几类典型网格下三维弹性问题的代数多层网格法[J]. 工程力学, 2011, 28(6): 11 -018 .
[2] 鞠 伟;岑 松;傅向荣;龙驭球. 基于哈密顿解法的厚板边界效应典型算例分析[J]. 工程力学, 2008, 25(2): 0 -008 .
[3] 刘春梅, 肖映雄, 舒适, 钟柳强. 弹性力学问题自适应有限元及其局部多重网格法[J]. 工程力学, 2012, 29(9): 60 -67,91 .
[4] 程永锋, 朱照清, 卢智成, 张富有. 运动简谐振子作用下地基梁体系振动特性的半解析研究[J]. 工程力学, 2018, 35(7): 18 -23 .
[5] 姜忻良 张崇祥 姜南 罗兰芳. 设备-结构-土体系振动台实时子结构试验方法探讨[J]. 工程力学, 0, (): 0 .
[6] 袁全, 袁驷, 李易, 闫维明, 邢沁妍. 线性元时程积分按最大模自适应步长公式的证明[J]. 工程力学, 2018, 35(8): 9 -13 .
[7] 王贞, 李强, 吴斌. 实时混合试验的自适应时滞补偿方法[J]. 工程力学, 2018, 35(9): 37 -43 .
[8] 丁杰, 邹昀, 蔡鑫, 李天祺, 郑黎君, 赵桃干. 损伤可控型钢框架边节点的试验研究[J]. 工程力学, 2018, 35(S1): 107 -112 .
[9] 潘天林, 吴斌. 基于桁架单元的能量一致积分方法[J]. 工程力学, 2018, 35(10): 1 -9,36 .
[10] 郑欣, 刘宇斌, 陈璞, 沈峰, 张圣君, 傅向荣. 基于弯扭耦合理论的颤振频率计算方法[J]. 工程力学, 2018, 35(S1): 1 -5,12 .
X

近日,本刊多次接到来电,称有不法网站冒充《工程力学》杂志官网,并向投稿人收取高额费用。在此,我们郑重申明:

1.《工程力学》官方网站是本刊唯一的投稿渠道(原网站已停用),《工程力学》所有刊载论文必须经本刊官方网站的在线投稿审稿系统完成评审。我们不接受邮件投稿,也不通过任何中介或编辑收费组稿。

2.《工程力学》在稿件符合投稿条件并接收后会发出接收通知,请作者在接到版面费或审稿费通知时,仔细检查收款人是否为“《工程力学》杂志社”,千万不要汇款给任何的个人账号。请广大读者、作者相互转告,广为宣传!如有疑问,请来电咨询:010-62788648。

感谢大家多年来对《工程力学》的支持与厚爱,欢迎继续关注我们!

《工程力学》杂志社

2018年11月15日