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二维瞬态热传导的PDDO分析

周保良 李志远 黄丹

周保良,李志远,黄丹. 二维瞬态热传导的PDDO分析 [J]. 应用数学和力学,2022,43(6):660-668 doi: 10.21656/1000-0887.420150
引用本文: 周保良,李志远,黄丹. 二维瞬态热传导的PDDO分析 [J]. 应用数学和力学,2022,43(6):660-668 doi: 10.21656/1000-0887.420150
ZHOU Baoliang, LI Zhiyuan, HUANG Dan. PDDO Analysis of 2D Transient Heat Conduction Problems[J]. Applied Mathematics and Mechanics, 2022, 43(6): 660-668. doi: 10.21656/1000-0887.420150
Citation: ZHOU Baoliang, LI Zhiyuan, HUANG Dan. PDDO Analysis of 2D Transient Heat Conduction Problems[J]. Applied Mathematics and Mechanics, 2022, 43(6): 660-668. doi: 10.21656/1000-0887.420150

二维瞬态热传导的PDDO分析

doi: 10.21656/1000-0887.420150
基金项目: 国家自然科学基金(12072104;51679077;11932006);国家重点研发计划(2018YFC0406703)
详细信息
    作者简介:

    周保良(1998—),男,硕士(E-mail:zhoubaoliang2020@126.com

    黄丹(1979—),男,教授(通讯作者. E-mail:danhuang@hhu.edu.cn

  • 中图分类号: O302

PDDO Analysis of 2D Transient Heat Conduction Problems

  • 摘要:

    采用近场动力学微分算子(peridynamic differential operator, PDDO)理论求解了二维瞬态热传导问题。将热传导方程和边界条件由其局部微分形式重构为非局部积分形式,引入Lagrange乘数法,采用变分原理的概念,建立了二维瞬态热传导问题的非局部分析模型。通过误差与收敛性分析,与其他数值方法计算结果进行比较,验证了本模型的准确性。在此基础上,将本模型应用于计算不规则边界板和内部含微缺陷(裂纹和圆孔)板的二维瞬态热传导问题。结果表明该方法计算精度高、适用范围广、具有较好的收敛性,为计算二维瞬态热传导问题提供了新的思路。

  • 图  1  物质点在任意形状近场范围内的相互作用

    Figure  1.  Interaction of material points within an arbitrary-shape near field

    图  2  方形板及边界条件

    Figure  2.  A square plate and its boundary conditions

    图  3  瞬态温度场的PDDO误差测量值

    Figure  3.  The error measure of the PDDO solution for the transient temperature field

    图  4  A、B、C板的几何形状和边界条件

    Figure  4.  Geometric shapes and boundary conditions of plates A, B and C

    图  5  t=0.8 s时,A、B、C三种形状的板在不同方法下沿指定路径的温度分布

    Figure  5.  Temperature distributions of plates A, B and C along a specified path for different methods at t=0.8 s

    图  6  稳态时,A板在不同数值方法下的温度分布:(a) FEM[21];(b) RBF-DQ[21];(c) PDDO

    Figure  6.  Temperature distributions of plate A for different numerical methods in a steady state: (a) FEM[21]; (b) RBF-DQ[21]; (c) PDDO

    图  7  稳态时,B板在不同数值方法下的温度分布:(a) FEM[21];(b) RBF-DQ[21];(c) PDDO

    注 为了解释图中的颜色,读者可以参考本文的电子网页版本,后同。

    Figure  7.  Temperature distributions of plate B for different numerical methods in a steady state: (a) FEM[21]; (b) RBF-DQ[21]; (c) PDDO

    图  8  稳态时,C板在不同数值方法下的温度分布:(a) FEM[21];(b) RBF-DQ[21];(c) PDDO

    Figure  8.  Temperature distributions of plate C for different numerical methods in a steady state: (a) FEM[21]; (b) RBF-DQ[21]; (c) PDDO

    图  9  含裂纹板和含圆孔板模型

    Figure  9.  The models for plates containing a crack or a hole

    图  10  稳态时,含裂纹板在不同数值方法下的温度分布:(a) FEM;(b) PDDO

    Figure  10.  Temperature distributions of plates containing cracks for different numerical methods in steady states: (a) FEM; (b) PDDO

    图  11  稳态时,含圆孔板在不同数值方法下的温度分布:(a) FEM;(b) PDDO

    Figure  11.  Temperature distributions of plates containing holes for different numerical methods in steady states: (a) FEM; (b) PDDO

    图  12  含裂纹板和含圆孔板在不同数值方法下的温度变化曲线:(a) 含裂纹板测点处,温度随时间的变化曲线;(b) 稳态时,温度沿裂纹的下边界变化的曲线;(c) 含圆孔板测点处,温度随时间的变化曲线

    Figure  12.  The temperature variation curves of plates containing a crack or a hole for different numerical methods: (a) temperature change curves with time at the measuring point of the plate containing a crack; (b)temperature curves along the lower boundary of the crack in a steady state; (c) temperature change curves with time at the measuring point of the plate containing a hole

    表  1  t=1.2 h时不同方法的数值结果比较

    Table  1.   Comparison of numerical results between different methods at t =1.2 h

    point(x, y)/mPDDOBKMFEMBEMDRBEMTrefftz FEMexact solution
    a(2.4,1.5)1.0811.0811.1391.1141.0991.1031.065
    b(2.4,2.4)0.6370.6310.6700.6570.6450.6600.626
    c(1.8,1.5)1.7451.7791.8431.7981.7841.7971.723
    d(1.8,1.8)1.6601.6911.7531.7131.6951.7151.639
    e(1.5,1.5)1.8341.8711.9381.8871.8771.8941.812
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-06-02
  • 修回日期:  2021-10-07
  • 网络出版日期:  2022-05-30
  • 刊出日期:  2022-06-30

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