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基于赫林格−赖斯纳变分原理的一致高效无网格本质边界条件施加方法

吴俊超 吴新瑜 赵珧冰 王东东

吴俊超, 吴新瑜, 赵珧冰, 王东东. 基于赫林格−赖斯纳变分原理的一致高效无网格本质边界条件施加方法. 力学学报, 2022, 54(12): 3283-3296 doi: 10.6052/0459-1879-22-151
引用本文: 吴俊超, 吴新瑜, 赵珧冰, 王东东. 基于赫林格−赖斯纳变分原理的一致高效无网格本质边界条件施加方法. 力学学报, 2022, 54(12): 3283-3296 doi: 10.6052/0459-1879-22-151
Wu Junchao, Wu Xinyu, Zhao Yaobing, Wang Dongdong. A consistent and efficient method for imposing meshfree essential boundary conditions via hellinger-reissner variational principle. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3283-3296 doi: 10.6052/0459-1879-22-151
Citation: Wu Junchao, Wu Xinyu, Zhao Yaobing, Wang Dongdong. A consistent and efficient method for imposing meshfree essential boundary conditions via hellinger-reissner variational principle. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3283-3296 doi: 10.6052/0459-1879-22-151

基于赫林格−赖斯纳变分原理的一致高效无网格本质边界条件施加方法

doi: 10.6052/0459-1879-22-151
基金项目: 国家自然科学基金(12102138, 12072302)和福建省自然科学基金(2021 J02003)资助项目
详细信息
    作者简介:

    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

  • 中图分类号: O242.2

A CONSISTENT AND EFFICIENT METHOD FOR IMPOSING MESHFREE ESSENTIAL BOUNDARY CONDITIONS VIA HELLINGER-REISSNER VARIATIONAL PRINCIPLE

  • 摘要: 无网格法具有高阶连续光滑的形函数, 在结构分析中呈现出显著的精度优势. 但无网格形函数在节点处一般没有插值性, 导致伽辽金无网格法难以直接施加本质边界条件. 采用变分一致尼兹法施加边界条件的数值解具有良好的收敛性和稳定性, 因而得到了非常广泛的应用, 然而该方法仍然需要引入人工参数来保证算法的稳定性. 本文以赫林格−赖斯纳变分原理为基础, 建立了一种变分一致的本质边界条件施加方法. 该方法采用混合离散近似赫林格−赖斯纳变分原理弱形式中的位移和应力, 其中位移采用传统无网格形函数进行离散, 而应力则在背景积分单元中近似为相应阶次的多项式. 此时的无网格离散方程可视为一种新型的尼兹法施加本质边界条件, 其中修正变分项采用再生光滑梯度和无网格形函数进行混合离散, 稳定项则内嵌于赫林格−赖斯纳变分原理弱形式中, 无需额外增加稳定项, 消除了对人工参数的依赖性. 该方法无需计算复杂耗时的形函数导数, 并满足积分约束条件, 保证了数值求解的精度. 数值结果表明, 所提方法能够保证伽辽金无网格法的计算精度最优误差收敛率, 与传统的尼兹法相比明显提高了计算效率.

     

  • 图  1  无网格形函数

    Figure  1.  Meshfree shape functions

    图  2  背景积分单元示意图和优化的数值积分方案

    Figure  2.  Illustration of background integration cells and optimized quadrature rules

    图  3  分片试验无网格离散模型

    Figure  3.  Meshfree discretization for the patch test

    图  4  悬臂梁问题模型

    Figure  4.  Description of the cantilever beam problem

    图  5  悬臂梁问题节点离散

    Figure  5.  Meshfree discretizations of the cantilever beam problem

    图  6  悬臂梁问题误差对比

    Figure  6.  Error comparison for the cantilever beam problem

    图  7  悬臂梁问题效率对比

    Figure  7.  Efficiency comparison for the cantilever beam problem

    图  8  带孔无限大平板问题模型

    Figure  8.  Description of the plate with a hole problem

    图  9  带孔无限大平板问题无网格离散

    Figure  9.  Meshfree discretizations of the plate with a hole problem

    图  10  带孔无限大平板问题误差对比

    Figure  10.  Error comparison for the plate with a hole problem

    图  11  带孔无限大平板问题效率对比

    Figure  11.  Efficiency comparison for the plate with a hole problem

    图  12  带孔无限大平板问题$ {\sigma _{xx}} $应力云图

    Figure  12.  Contour plot of stress $ {\sigma _{xx}} $ for the plate with a hole problem

    表  1  二次基函数无网格法分片试验结果

    Table  1.   The results of patch test with quadratic basis functions

    Linear patch test Quadratic patch test
    L2-ErrHe-ErrL2-ErrHe-Err
    GI-penalty$ 7.7 \times {10^{ - 6}} $$ 2.7 \times {10^{ - 4}} $ $ 1.2 \times {10^{ - 5}} $$ 2.6 \times {10^{ - 4}} $
    GI-LM$ 1.0 \times {10^{ - 4}} $$ 4.7 \times {10^{ - 3}} $$ 1.5 \times {10^{ - 4}} $$ 4.3 \times {10^{ - 3}} $
    GI-Nitsche$ 8.1 \times {10^{ - 6}} $$ 2.8 \times {10^{ - 4}} $$ 1.3 \times {10^{ - 5}} $$ 2.8 \times {10^{ - 4}} $
    RKGSI-penalty$ 7.9 \times {10^{ - 8}} $$ 2.0 \times {10^{ - 6}} $$ 1.4 \times {10^{ - 7}} $$ 2.1 \times {10^{ - 6}} $
    RKGSI-LM$ 8.6 \times {10^{ - 5}} $$ 4.0 \times {10^{ - 3}} $$ 1.4 \times {10^{ - 4}} $$ 3.7 \times {10^{ - 3}} $
    RKGSI-Nitsche$ 2.1 \times {10^{ - 15}} $$ 4.0 \times {10^{ - 14}} $$ 2.2 \times {10^{ - 15}} $$ 2.7 \times {10^{ - 14}} $
    RKGSI-HR$ 2.0 \times {10^{ - 15}} $$ 3.2 \times {10^{ - 14}} $$ 2.2 \times {10^{ - 15}} $$ 2.1 \times {10^{ - 14}} $
    下载: 导出CSV

    表  2  三次基函数无网格法分片试验结果

    Table  2.   The results of patch test with cubic basis functions

    Quadratic patch test Cubic patch test
    L2-ErrHe-ErrL2-ErrHe-Err
    GI-penalty$ 9.1 \times {10^{ - 6}} $$ 2.1 \times {10^{ - 4}} $$ 1.2 \times {10^{ - 5}} $$ 2.0 \times {10^{ - 4}} $
    GI-LM$ 2.9 \times {10^{ - 4}} $$ 9.3 \times {10^{ - 3}} $$ 4.0 \times {10^{ - 4}} $$ 9.3 \times {10^{ - 3}} $
    GI-Nitsche$ 1.1 \times {10^{ - 5}} $$ 2.8 \times {10^{ - 4}} $$ 1.4 \times {10^{ - 5}} $$ 2.7 \times {10^{ - 4}} $
    RKGSI-penalty$ 1.4 \times {10^{ - 7}} $$ 2.1 \times {10^{ - 6}} $$ 2.0 \times {10^{ - 7}} $$ 2.7 \times {10^{ - 6}} $
    RKGSI-LM$ 3.0 \times {10^{ - 4}} $$ 9.8 \times {10^{ - 3}} $$ 4.2 \times {10^{ - 4}} $$ 9.8 \times {10^{ - 3}} $
    RKGSI-Nitsche$ 3.6 \times {10^{ - 15}} $$ 1.0 \times {10^{ - 13}} $$ 4.6 \times {10^{ - 15}} $$ 9.5 \times {10^{ - 14}} $
    RKGSI-HR$ 3.1 \times {10^{ - 15}} $$ 1.0 \times {10^{ - 13}} $$ 3.5 \times {10^{ - 15}} $$ 7.4 \times {10^{ - 14}} $
    下载: 导出CSV

    B1  二次基函数无网格法优化的数值积分方案

    B1.   The optimized quadrature rules for quadratic basis function

    $ \xi $$ \eta $$ \gamma $$ w $$ {w_B} $
    $\dfrac{ {\text{2} } }{ {\text{3} } }$$ \dfrac{{\text{1}}}{{\text{6}}} $$ \dfrac{{\text{1}}}{{\text{6}}} $$ \dfrac{{\text{1}}}{{\text{3}}} $
    $ \dfrac{{\text{1}}}{{\text{2}}} $$ \dfrac{{\text{1}}}{{\text{2}}} $0$ \dfrac{{\text{1}}}{{\text{6}}} $
    100$ \dfrac{{\text{2}}}{{\text{3}}} $$ \dfrac{{\text{1}}}{{\text{3}}} $
    下载: 导出CSV

    B2  三次基函数无网格法优化的数值积分方案

    B2.   The optimized quadrature rules for cubic basis function

    $ \xi $$ \eta $$ \gamma $$ w $$ {w_B} $
    $\eta_a $$ \dfrac{1-\xi_{a}}{2} $$ \dfrac{1-\xi_{a}}{2} $${w}_{a}$
    $ \eta_b $$ \eta_b $$ \dfrac{1-\xi_{b}}{2} $${w}_{b}$
    $\begin{aligned}{\xi_{a}=0.108\;103\;018\;168\;070}, \quad {w_{a}=0.223\;381\;589\;678\;011} \\{\xi_{b}=0.816\;847\;572\;980\;459}, \quad w_{b}=0.109\;951\;743\;655\;322\end{aligned}$
    $\dfrac{{\text{1}}}{{\text{3}}} $$ \dfrac{{\text{1}}}{{\text{3}}} $$ \dfrac{{\text{1}}}{{\text{3}}} $$ \dfrac{{\text{9}}}{{\text{20}}} $
    100$ -\dfrac{{\text{1}}}{{\text{30}}} $$ \dfrac{{\text{1}}}{{\text{20}}} $
    $ \dfrac{{\text{1}}}{{\text{2}}} $$ \dfrac{{\text{1}}}{{\text{2}}} $0$ \dfrac{{\text{4}}}{{\text{135}}} $$ \dfrac{{\text{16}}}{{\text{46}}} $
    $ \dfrac{7+\sqrt{21}}{14} $$ \dfrac{7+\sqrt{21}}{14} $0$ \dfrac{{\text{49}}}{{\text{540}}} $$ \dfrac{{\text{49}}}{{\text{180}}} $
    下载: 导出CSV
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  • 收稿日期:  2022-04-10
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  • 刊出日期:  2022-12-15

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