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弹性力学有限元分析中的平衡与协调理论

傅向荣 陈璞 孙树立 袁明武

傅向荣, 陈璞, 孙树立, 袁明武. 弹性力学有限元分析中的平衡与协调理论[J]. 工程力学, 2023, 40(2): 8-16. doi: 10.6052/j.issn.1000-4750.2021.08.0647
引用本文: 傅向荣, 陈璞, 孙树立, 袁明武. 弹性力学有限元分析中的平衡与协调理论[J]. 工程力学, 2023, 40(2): 8-16. doi: 10.6052/j.issn.1000-4750.2021.08.0647
FU Xiang-rong, CHEN Pu, SUN Shu-li, YUAN Ming-wu. RESEARCH ON EQUILIBRIUM AND CONFORMING THEORY OF THE FINITE ELEMENT METHOD IN ELASTICITY[J]. Engineering Mechanics, 2023, 40(2): 8-16. doi: 10.6052/j.issn.1000-4750.2021.08.0647
Citation: FU Xiang-rong, CHEN Pu, SUN Shu-li, YUAN Ming-wu. RESEARCH ON EQUILIBRIUM AND CONFORMING THEORY OF THE FINITE ELEMENT METHOD IN ELASTICITY[J]. Engineering Mechanics, 2023, 40(2): 8-16. doi: 10.6052/j.issn.1000-4750.2021.08.0647

弹性力学有限元分析中的平衡与协调理论

doi: 10.6052/j.issn.1000-4750.2021.08.0647
基金项目: 国家自然科学基金项目(11672362,11472014)
详细信息
    作者简介:

    陈 璞(1962−),男,重庆人,教授,博士,主要从事计算力学、结构动力学方面的研究(E-mail: chenpu@pku.edu.cn)

    孙树立(1968−),男,河北人,教授,博士,主要从事计算力学方面的研究(E-mail: SUNSL@mech.pku.edu.cn)

    袁明武(1939−),男,江苏人,教授,博士,主要从事计算力学方面的研究(E-mail: yuanmw@pku.edu.cn)

    通讯作者:

    傅向荣(1972−),男,湖南人,教授,博士,主要从事计算力学研究(E-mail: fuxr@cau.edu.cn)

  • 中图分类号: O343

RESEARCH ON EQUILIBRIUM AND CONFORMING THEORY OF THE FINITE ELEMENT METHOD IN ELASTICITY

  • 摘要: 有限元分析中的单元可以遵循不同的方法构造,该文提出以单元模型的平衡性与协调性进行分类,并对弹性力学平面问题中的几种经典单元进行了分析比较,总结了协调元、非协调元和超协调元的协调性方法,以及基于解析试函数法的平衡型方法。单元的协调性理论思路包含单纯形格式协调元和非单纯形格式协调元,以及相应的非协调和超协调元格式,关注的重点是单元边界的协调。单元的平衡性理论思路包含解析试函数和权函数的高阶完备性,关注的重点研究是单元内部及边界平衡性。研究表明:针对弹性力学中平衡性和协调性要求,两类理论给出的不同单元格式各具特点,而既能保证单元内部平衡性,又能考虑单元界面协调性的单元类型给出了更精确、合理的计算结果。
  • 图  1  纯弯梁问题

    Figure  1.  Pure bending beam

    图  2  三角形元网格

    Figure  2.  Triangular Meshes

    图  3  网格畸变敏感性测试

    Figure  3.  Distorted mesh test

    图  4  网格畸变敏感性测试

    Figure  4.  Distorted mesh test

    表  1  三角形网格下各类单元的挠度精度分析(精确解为1.0000)

    Table  1.   Precision of triangular elements

    网格数 2×22×42×82×16
    CST0.01280.04890.16540.4058
    GT90.10450.31560.63710.8510
    GT9M0.10490.31910.65040.8692
    TR30.99060.99030.99110.9961
    下载: 导出CSV

    表  2  网格畸变敏感性测试精度分析

    Table  2.   Precision in distorted mesh

    d5β7βP-SATF-GCQ4ATF-Q4
    0.01.00000.31511.00001.00001.0000
    1.00.26700.19000.62900.55121.0000
    2.00.18760.15740.55000.43861.0000
    3.00.19110.16810.54700.44311.0000
    4.00.21180.17010.53100.42971.0000
    4.90.34960.14130.49800.42671.0000
    下载: 导出CSV

    表  3  网格畸变敏感性测试计算结果

    Table  3.   Precision in distorted mesh

    网格结果Q8HSF-Q8
    1$ {\sigma _x} $(A)56.447120.000
    $ {\sigma _x} $(B)−74.863−120.000
    $w ({\rm C})$−2.328−12.000
    2$ {\sigma _x} $(A)13.665120.000
    $ {\sigma _x} $(A)5.262120.000
    $ {\sigma _x} $(B)−5.665−120.000
    $ {\sigma _x} $(B)−14.299−120.000
    $w ({\rm C})$−0.477−12.000
    下载: 导出CSV
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  • 收稿日期:  2021-08-20
  • 修回日期:  2022-03-24
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  • 刊出日期:  2023-02-01

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