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求解固体力学问题的强-弱耦合形式单元微分法

胡凯 高效伟 徐兵兵

胡凯, 高效伟, 徐兵兵. 求解固体力学问题的强-弱耦合形式单元微分法. 力学学报, 待出版 doi: 10.6052/0459-1879-22-087
引用本文: 胡凯, 高效伟, 徐兵兵. 求解固体力学问题的强-弱耦合形式单元微分法. 力学学报, 待出版 doi: 10.6052/0459-1879-22-087
Hu kai, Gao Xiaowei, Xu Bingbing. Strong weak coupling form element differential method for solving solid mechanics problems. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-22-087
Citation: Hu kai, Gao Xiaowei, Xu Bingbing. Strong weak coupling form element differential method for solving solid mechanics problems. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-22-087

求解固体力学问题的强-弱耦合形式单元微分法

doi: 10.6052/0459-1879-22-087
基金项目: 国家自然科学基金资助项目(12,072,064, 11772083)
详细信息
    作者简介:

    高效伟, 教授, 主要研究方向: 计算力学. E-mail: xwgao@dlut.edu.cn

  • 中图分类号: O341

STRONG WEAK COUPLING FORM ELEMENT DIFFERENTIAL METHOD FOR SOLVING SOLID MECHANICS PROBLEMS

  • 摘要: 单元微分法是一种新型强形式有限单元法. 与弱形式算法相比, 该算法直接对控制方程进行离散, 不需要用到数值积分. 因此该算法有较简单的形式, 并且其在计算系数矩阵时具有极高的效率. 但作为一种强形式算法, 单元微分法往往需要较多网格或者更高阶单元才能达到满意的计算精度. 与此同时, 对于一些包含奇异点的模型, 如在多材料界面、间断边界条件、裂纹尖端等处, 传统单元微分法往往得不到较精确的计算结果. 为了克服这些缺点, 本文提出了将伽辽金有限元法与单元微分法相结合的强-弱耦合算法, 即整体模型采用单元微分法的同时, 在奇异点附近或某些关键部件采用有限元法. 该策略在保留单元微分法高效率与简洁形式等优点的同时, 确保了求解奇异问题的精度. 在处理大规模问题时, 针对关键部件采用有限元法, 其他部件采用单元微分法, 可以在得到较精确结果的同时, 极大提高整体计算效率. 在本文中, 给出了两个典型算例, 一个是具有切口的二维问题, 一个是复杂的三维发动机问题. 针对这两个问题, 分析了该耦合算法在求二维奇异问题和三维大规模问题时的精度与效率.

     

  • 图  1  单元微分法和有限元法耦合形式的模型离散方案

    Figure  1.  Model discretization scheme in coupling form of element differential method and finite element method

    图  2  单元内部点及单元边界点

    Figure  2.  Element internal nodes and element interface nodes

    图  3  顶部受拉的V型缺口平板模型

    Figure  3.  V-notch plate model with tension at the top

    图  4  顶部节点y方向的位移

    Figure  4.  Displacement of top node in y direction

    图  5  y = L/2线上y方向的位移

    Figure  5.  Displacement in y direction on y = L/2 line

    图  6  y = L/2线上von-Mises应力

    Figure  6.  von-Mises stress on y = L/2 line

    图  7  两种方法的von-Mises stress应力云图

    Figure  7.  von-Mises stress cloud maps of the two methods

    图  8  燃烧室模型及其重要尺寸

    Figure  8.  Combustion chamber model and its important parameters

    图  9  燃烧室模型的边界条件

    Figure  9.  Boundary conditions of combustion chamber model

    图  10  燃烧室模型的网格

    Figure  10.  Grid of combustion chamber model

    图  11  三种方法在AB线上的总位移

    Figure  11.  Total displacement of three methods on AB line

    图  12  三种方法在AB线上的von-Mises应力

    Figure  12.  von-Mises stress of three methods on AB line

    表  1  两种方法的具体刚度矩阵K及载荷向量b表达式

    Table  1.   The specific stiffness matrix K and load vector b expressions of the two methods

    Stiffness matrix KLoad vector b
    EDM$ \left\{\begin{array}{l}D_{i j k l}(\xi) \dfrac{\partial^{2} N_{\alpha}(\xi)}{\partial x_{j} \partial x_{l}} \quad \xi \in \Omega^{e} \\\displaystyle\sum_{e=1}^{N^{I}} \displaystyle\sum_{s=1}^{S^{e l}} D_{i j k l}\left(\xi^{e}\right) n_{j}^{s}\left(\xi^{e}\right) \dfrac{\partial N_{\alpha}\left(\xi^{e}\right)}{\partial x_{l}} \quad \xi^{e} \in \partial \Omega\end{array}\right.$$\left\{\begin{array}{l}-f_{i} \quad \xi \in \Omega^{e} \\0 \quad \text { or } \displaystyle\sum_{e=1}^{N^{I}} \displaystyle\sum_{s=1}^{S^{e I}} t_{i}^{s}\left(\boldsymbol{\xi}^{e}\right), \quad \boldsymbol{\xi}^{e} \in \Gamma_{t}\end{array}\right. $
    FEM$ \displaystyle\int_{\Omega^{\varepsilon}} \dfrac{\partial N_{*}}{\partial x_{j}} D_{i j k l} \dfrac{\partial N_{\alpha}}{\partial x_{l}} \mathrm{~d} \Omega$$\displaystyle\int_{\Gamma^{\varepsilon}} N_{*} t_{i} \mathrm{~d} \Gamma+\displaystyle\int_{\Omega^{\varepsilon}} N_{*} f_{i} \mathrm{~d} \Omega $
    下载: 导出CSV

    表  2  三种方法所得y方向最大位移结果对比

    Table  2.   Comparison of maximum displacement in Y direction obtained by three methods (mm)

    MethodFEMEDMEDM_FEM
    Max uy0.0616710.0637480.061699
    下载: 导出CSV

    表  3  三种方法组集方程及求解所用时间对比

    Table  3.   Time comparison of three methods (s)

    MethodTime of forming the matrixTime of solving equationTotal time
    EDM78380458
    FEM10963821478
    Coupled280387667
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-03-01
  • 录用日期:  2022-05-11
  • 网络出版日期:  2022-05-09

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