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节点梯度光滑有限元配点法

樊礼恒 王东东 刘宇翔 杜洪辉

樊礼恒, 王东东, 刘宇翔, 杜洪辉. 节点梯度光滑有限元配点法[J]. 力学学报, 2021, 53(2): 467-481. doi: 10.6052/0459-1879-20-361
引用本文: 樊礼恒, 王东东, 刘宇翔, 杜洪辉. 节点梯度光滑有限元配点法[J]. 力学学报, 2021, 53(2): 467-481. doi: 10.6052/0459-1879-20-361
Fan Liheng, Wang Dongdong, Liu Yuxiang, Du Honghui. A FINITE ELEMENT COLLOCATION METHOD WITH SMOOTHED NODAL GRADIENTS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 467-481. doi: 10.6052/0459-1879-20-361
Citation: Fan Liheng, Wang Dongdong, Liu Yuxiang, Du Honghui. A FINITE ELEMENT COLLOCATION METHOD WITH SMOOTHED NODAL GRADIENTS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 467-481. doi: 10.6052/0459-1879-20-361

节点梯度光滑有限元配点法

doi: 10.6052/0459-1879-20-361
基金项目: 1) 国家自然科学基金资助项目(11772280);国家自然科学基金资助项目(12072302)
详细信息
    作者简介:

    2) 王东东, 教授, 主要研究方向: 计算力学与结构工程. E-mail: ddwang@xmu.edu.cn

    通讯作者:

    王东东

  • 中图分类号: O242.2

A FINITE ELEMENT COLLOCATION METHOD WITH SMOOTHED NODAL GRADIENTS

  • 摘要: 配点法构造简单、计算高效, 但需要用到数值离散形函数的高阶梯度,而传统有限元形函数的梯度在单元边界处通常仅具有C$^{0}$连续性,因此无法直接用于配点法分析. 本文通过引入有限元形函数的光滑梯度,提出了节点梯度光滑有限元配点法. 首先基于广义梯度光滑方法,定义了有限元形函数在节点处的一阶光滑梯度值,然后以有限元形函数为核函数构造了有限元形函数的一阶光滑梯度,进而对一阶光滑梯度直接求导并用一阶光滑梯度替换有限元形函数的标准梯度,即完成了有限元形函数二阶光滑梯度的构造.文中以线性有限元形函数为基础的理论分析表明,其光滑梯度不仅满足传统线性有限元形函数梯度对应的一阶一致性条件,而且在均布网格假定下满足更高一阶的二阶一致性条件.因此与传统线性有限元法相比,基于线性形函数的节点梯度光滑有限元法的$L_{2}$和$H_{1}$误差均具有二次精度,即其$H_{1}$误差收敛阶次比传统有限元法高一阶, 呈现超收敛特性.文中通过典型算例验证了节点梯度光滑有限元配点法的精度和收敛性,特别是其$H_{1}$或能量误差的精度和收敛率都明显高于传统有限元法.

     

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出版历程
  • 收稿日期:  2020-10-20
  • 刊出日期:  2021-02-10

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