留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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
  • [1] 王红, 李小林. 二维瞬态热传导问题的无单元Galerkin法分析[J]. 应用数学和力学, 2021, 42(5): 460-469. (WANG Hong, LI Xiaolin. Analysis of 2D transient heat conduction problems with the element-free Galerkin method[J]. Applied Mathematics and Mechanics, 2021, 42(5): 460-469.(in Chinese)

    WANG Hong, LI Xiaolin. Analysis of 2D transient heat conduction problems with the element-free Galerkin method[J]. Applied Mathematics and Mechanics, 2021, 42(5): 460-469. (in Chinese))
    [2] ZIENIUK E, SAWICKI D. Modification of the classical boundary integral equation for two dimensional transient heat conduction with internal heat source, with the use of NURBS for boundary modeling[J]. Journal of Heat Transfer, 2017, 139(8): 81-95.
    [3] BURLAYENKO V N, ALTENBACH H, SADOWSKI T, et al. Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements[J]. Applied Mathematical Modelling, 2017, 45(5): 422-438.
    [4] WU X H, TAO W Q. Meshless method based on the local weak-forms for steady state heat conduction problems[J]. International Journal of Heat and Mass Transfer, 2008, 51(11/12): 3103-3112.
    [5] BREBBIA C A, TELLES J C F, WROBEL L C. Boundary Element Techniques: Theory and Applications in Engineering[M]. Berlin: Springer Verlag, 1984.
    [6] 师晋红, 傅卓佳, 陈文. 边界节点法计算二维瞬态热传导问题[J]. 应用数学和力学, 2014, 35(2): 111-120. (SHI Jinhong, FU Zhuojia, CHEN Wen. Boundary knot method for 2D transient heat conduction problems[J]. Applied Mathematics and Mechanics, 2014, 35(2): 111-120.(in Chinese) doi: 10.3879/j.issn.1000-0887.2014.02.001

    SHI Jinhong, FU Zhuojia, CHEN Wen. Boundary knot method for 2D transient heat conduction problems[J]. Applied Mathematics and Mechanics, 2014, 35(2): 111-120. (in Chinese)) doi: 10.3879/j.issn.1000-0887.2014.02.001
    [7] PARTRIDGE P W, BREBBIA C A, WROBEL L C. The Dual Reciprocity Boundary Element Method[M]. Southampton: Computational Mechanics Publications, 1992.
    [8] JIROUSEK J, QIN Q H. Application of hybrid-Trefftz element approach to transient heat conduction analysis[J]. Computers & Structures, 1996, 58(1): 195-201.
    [9] SILLING S A, EPTON M, WECKNER O, et al. Peridynamic states and constitutive modeling[J]. Journal of Elasticity, 2007, 88(2): 151-184. doi: 10.1007/s10659-007-9125-1
    [10] 黄丹, 章青, 乔丕忠, 等. 近场动力学方法及其应用[J]. 力学进展, 2010, 40(4): 448-459. (HUANG Dan, ZHANG Qing, QIAO Pizhong, et al. A review on peridynamics(PD)method and its applications[J]. Advances in Mechanics, 2010, 40(4): 448-459.(in Chinese) doi: 10.6052/1000-0992-2010-4-J2010-002

    HUANG Dan, ZHANG Qing, QIAO Pizhong, et al. A review on peridynamics(pd)method and its applications[J]. Advances in Mechanics, 2010, 40(4): 448-459. (in Chinese)) doi: 10.6052/1000-0992-2010-4-J2010-002
    [11] MADENCI E, BARYT A, DORDUNCU M. Peridynamic Differential Operator for Numerical Analysis[M]. Switzerland: Springer Nature Switzerland AG, 2019.
    [12] MADENCI E, BARUT A, FUTCH M. Peridynamic differential operator and its applications[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 408-451. doi: 10.1016/j.cma.2016.02.028
    [13] BAZAZZADEH S, SHOJAEI A, ZACCARIOTTO M, et al. Application of the peridynamic differential operator to the solution of sloshing problems in tanks[J]. Engineering Computations, 2018, 36(1): 45-83. doi: 10.1108/EC-12-2017-0520
    [14] DORDUNCU M. Stress analysis of laminated composite beams using refined zigzag theory and peridynamic differential operator[J]. Composite Structures, 2019, 218: 193-203. doi: 10.1016/j.compstruct.2019.03.035
    [15] DORDUNCU M. Stress analysis of sandwich plates with functionally graded cores using peridynamic differential operator and refined zigzag theory[J]. Thin-Walled Structures, 2020, 146: 106468. doi: 10.1016/j.tws.2019.106468
    [16] GAO Y, OTERKUS S. Nonlocal modeling for fluid flow coupled with heat transfer by using peridynamic differential operator[J]. Engineering Analysis With Boundary Elements, 2019, 105: 104-121. doi: 10.1016/j.enganabound.2019.04.007
    [17] GAO Y, OTERKUS S. Nonlocal numerical simulation of low Reynolds number laminar fluid motion by using peridynamic differential operator[J]. Ocean Engineering, 2019, 179: 135-158. doi: 10.1016/j.oceaneng.2019.03.035
    [18] LI Z Y, HUANG D, XU Y P, et al. Nonlocal steady-state thermoelastic analysis of functionally graded materials by using peridynamic differential operator[J]. Applied Mathematical Modelling, 2021, 93: 294-313. doi: 10.1016/j.apm.2020.12.004
    [19] CUI M, XU B B, FENG W Z, et al. A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity[J]. Numerical Heat Transfer, 2018, 73: 1-18.
    [20] SOLEIMANI S, JALAAL M, BARARNIA H, et al. Local RBF-DQ method for two dimensional transient heat conduction problems[J]. International Communications in Heat and Mass Transfer, 2010, 37(9): 1411-1418. doi: 10.1016/j.icheatmasstransfer.2010.06.033
    [21] MUKHERJEE Y X, MUKHERJEE S. On boundary conditions in the element-free Galerkin method[J]. Computational Mechanics, 1997, 19(4): 264-270. doi: 10.1007/s004660050175
  • 加载中
图(12) / 表(1)
计量
  • 文章访问数:  641
  • HTML全文浏览量:  334
  • PDF下载量:  65
  • 被引次数: 0
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

目录

    /

    返回文章
    返回