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热力耦合问题数学均匀化方法的物理意义

朱晓鹏 黄俊 陈磊 邢誉峰

朱晓鹏, 黄俊, 陈磊, 等 . 热力耦合问题数学均匀化方法的物理意义[J]. 北京航空航天大学学报, 2019, 45(11): 2139-2151. doi: 10.13700/j.bh.1001-5965.2019.0088
引用本文: 朱晓鹏, 黄俊, 陈磊, 等 . 热力耦合问题数学均匀化方法的物理意义[J]. 北京航空航天大学学报, 2019, 45(11): 2139-2151. doi: 10.13700/j.bh.1001-5965.2019.0088
ZHU Xiaopeng, HUANG Jun, CHEN Lei, et al. Physical interpretation of mathematical homogenization method for thermomechanical problem[J]. Journal of Beijing University of Aeronautics and Astronautics, 2019, 45(11): 2139-2151. doi: 10.13700/j.bh.1001-5965.2019.0088(in Chinese)
Citation: ZHU Xiaopeng, HUANG Jun, CHEN Lei, et al. Physical interpretation of mathematical homogenization method for thermomechanical problem[J]. Journal of Beijing University of Aeronautics and Astronautics, 2019, 45(11): 2139-2151. doi: 10.13700/j.bh.1001-5965.2019.0088(in Chinese)

热力耦合问题数学均匀化方法的物理意义

doi: 10.13700/j.bh.1001-5965.2019.0088
详细信息
    作者简介:

    朱晓鹏  男, 本科, 教授级高级工程师。主要研究方向:结构工程、多尺度方法

    陈磊  男, 博士。主要研究方向:复合材料力学

    通讯作者:

    陈磊.E-mail:chenlei2019@buaa.edu.cn

  • 中图分类号: O302

Physical interpretation of mathematical homogenization method for thermomechanical problem

More Information
  • 摘要:

    针对复合材料周期结构热力耦合问题,通过构造各阶摄动项的全解耦格式,推导了高阶数学均匀化方法(MHM)的数学表达式,并使用加权残量方法将其转换为易于编程实现的矩阵列式。将弹性影响函数和热影响函数分别比拟为弹性虚拟位移和热虚拟位移,通过弹性虚拟载荷和热虚拟载荷的自平衡特性、量纲分析及几何直观等角度揭示了各阶影响函数和摄动位移的物理意义,并指出二阶摄动位移对于细观结构分析的必要性。数值计算结果验证了高阶MHM矩阵列式及物理意义分析的正确性。

     

  • 图 1  包含3块夹杂的二维周期复合材料单胞结构

    Figure 1.  2D periodical composites unit cell structures with 3 inclusions

    图 2  一阶虚拟载荷矢量图

    Figure 2.  First-order virtual load vector

    图 3  二阶虚拟载荷矢量图

    Figure 3.  Second-order virtual load vector

    图 4  三阶虚拟载荷矢量图

    Figure 4.  Third-order virtual load vector

    图 5  包含一块夹杂的周期复合材料杆单胞结构

    Figure 5.  Periodical composites rod unit cell structure with one inclusion

    图 6  二维周期复合材料单胞结构

    Figure 6.  Unit cell of 2D periodical composite structure

    图 7  沿纵线ABCD上节点位移曲线

    Figure 7.  Nodal displacement curves along longitudinal lines A, B, C and D

    图 8  二维周期复合材料多胞结构

    Figure 8.  2D periodical composite multi-cell structure

    图 9  沿纵线A′、B′、C′、D′上节点位移曲线

    Figure 9.  Nodal displacement curves along longitudinal lines A′, B′, C′ and D

    表  1  摄动位移及影响函数控制方程

    Table  1.   Perturbation displacements and governing equation of influence functions

    摄动位移 影响函数控制方程
    一阶弹性影响函数
    一阶热影响函数
    二阶弹性影响函数
    二阶热影响函数
    三阶弹性影响函数
    三阶热影响函数
    s阶弹性影响函数
    s阶热影响函数
    下载: 导出CSV

    表  2  二维周期复合材料单胞结构的势能泛函

    Table  2.   Potential energy function for 2D periodical composites unit cell

    摄动位移 Π/(10-6J)
    MHM FEM
    MHM1 -2.475 15.8
    MHM2 -2.916 -2.939 0.78
    MHM3 -2.937 0.068
    下载: 导出CSV

    表  3  二维周期复合材料多胞结构的势能泛函

    Table  3.   Potential energy function for 2D periodical composite multi-cell structure

    摄动位移 Π/(10-8J)
    MHM FEM
    MHM1 -3.547 7 20
    MHM2 -4.335 6 -4.436 1 2.27
    MHM3 -4.396 8 0.89
    下载: 导出CSV
  • [1] BERTHELOT J M. Composite materials:Mechanical behavior and structural analysis[M]. New York:Springer, 1999.
    [2] KALIDINDI S R, ABUSAFIEH A.Longitudinal and transverse moduli and strengths of low angle 3-D braided composites[J]. Journal of Composite Materials, 1996, 30(8):885-905. doi: 10.1177/002199839603000802
    [3] BABUSKA I.Solution of interface problems by homogenization, Parts I[J]. SIAM Journal on Mathematical Analysis, 1976, 7(5): 603-634. doi: 10.1137/0507048
    [4] BENSSOUSAN A, LIONS J L.Asymptotic analysis for periodic structures[M]. Amsterdam:North-Holland, 1978.
    [5] STROUBOULIS T, BABUSKA I, COPPS K.The generalized finite element method:An example of its implementation and illustration of its performance[J]. International Journal for Numerical Methods in Engineering, 2000, 47(8):1401-1417. doi: 10.1002/(SICI)1097-0207(20000320)47:8<1401::AID-NME835>3.0.CO;2-8
    [6] BABUSKA I, OSBOM J.Generalized finite element methods:Their performance and their relation to mixed methods[J]. SIAM Journal on Numerical Analysis, 1983, 20(3):510-536. doi: 10.1137/0720034
    [7] HOU T Y, WU X H.A multiscale finite element method for elliptic problems in composite materials and porous media[J]. Journal of Computational Physics, 1997, 134(1):169-189. http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_1211.3614
    [8] HOU T Y, WU X H, CAI Z Q.Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients[J]. Mathematics of Computation, 1999, 68(227):913-943. doi: 10.1090/S0025-5718-99-01077-7
    [9] E W, ENGQUIST B.The heterogeneous multiscale methods[J]. Communications in Mathematical Sciences, 2003, 1:87-132. doi: 10.4310/CMS.2003.v1.n1.a8
    [10] E W, ENGQUIST B, LI X T, et al.Heterogeneous multiscale methods:A review[J]. Communications in Computational Physics, 2007, 2(3):367-450. http://d.old.wanfangdata.com.cn/Periodical/gtlxxb-e201101002
    [11] XING Y F, YANG Y.An eigenelement method of periodical composite structures[J]. Composite Structures, 2011, 93(2):502-512. doi: 10.1016/j.compstruct.2010.08.029
    [12] XING Y F, YANG Y, WANG X M.A multiscale eigenelement method and its application to periodical composite structures[J]. Composite Structures, 2010, 92:2265-2275. doi: 10.1016/j.compstruct.2009.08.006
    [13] TERADA K, KURUMATANI M, USHIDAI N, et al.A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer[J]. Computational Mechanics, 2010, 46(2):269-285. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=4f89028c5dee36d135f023f62a36c8eb
    [14] MUHAMMAD R, ERDATA N, NAOYUKI W, et al.A novel asymptotic expansion homogenization analysis for 3-D composite with relieved periodicity in the thickness direction[J]. Composites Science and Technology, 2014, 97:63-73. doi: 10.1016/j.compscitech.2014.04.006
    [15] MUHAMMAD R, ERDATA N, NAOYUKI W, et al.Thermomechanical properties and stress analysis of 3-D textile composites by asymptotic expansion homogenization method[J]. Composites Part B:Engineering, 2014, 60:378-391. doi: 10.1016/j.compositesb.2013.12.038
    [16] BARROQUEIRO B, DIAS-DE-OLIVEIRA J, PINHO-DA-CRUZ J, et al.Practical implementation of asymptotic expansion homogenizationin thermoelasticity using a commercial simulation software[J]. Composite Structures, 2016, 141:117-131. doi: 10.1016/j.compstruct.2016.01.036
    [17] ZHAI J J, CHENG S, ZENG T, et al.Thermo-mechanical behavior analysis of 3D braided composites by multiscale finite element method[J]. Composite Structures, 2017, 176:664-672. doi: 10.1016/j.compstruct.2017.05.064
    [18] 李志青, 冯永平.一类小周期结构热力耦合问题的双尺度渐近分析[J].广州大学学报, 2016, 15(2):27-32. http://d.old.wanfangdata.com.cn/Periodical/gzdxxb-zkb201602006

    LI Z Q, FENG Y P.Two-scale asymptotic analysis on one class of thermoelastic coupling problem in small periodic structure[J]. Chinese Journal of Guangzhou University, 2016, 15(2):27-32(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/gzdxxb-zkb201602006
    [19] YANG Z Q, CUI J Z, ZHOU S.Thermo-mechanical analysis of periodic porous materials with microscale heat transfer by multiscale asymptotic expansion method[J]. International Journal of Heat and Mass Transfer, 2016, 92:904-919. doi: 10.1016/j.ijheatmasstransfer.2015.09.055
    [20] GUAN X F, LIU X, JIA X, et al.A stochastic multiscale model for predicting mechanical properties of fiber reinforced concrete[J]. International Journal of Solids and Structures, 2015, 56-57:280-289. doi: 10.1016/j.ijsolstr.2014.10.008
    [21] YANG Z Q, CUI J Z, MA Q.The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiationin periodic porous materials[J]. Discrete and Continous Dynamical System-Series B, 2014, 19(3):827-848. doi: 10.3934/dcdsb.2014.19.827
    [22] YANG Z Q, SUN Y, CUI J Z.A multiscale algorithm for heat conduction-radiation problems in porous materials with quasi-periodic structures[J]. Communications in Computational Physics, 2018, 24(1):204-233.
    [23] ALLAIRE G, HABIBI Z.Second order corrector in the homogenization of a conductive-radiative heat transfer problem[J]. Discrete and Continuous Dynamical System-Series B, 2013, 18(1): 1-36. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=2f321d65f48b55a82ee3504444f70394
    [24] WAN X, CAO L Q, WONG Y S.Multiscale computation and convergence for coupled thermoelastic system in composite materials[J]. Multiscale Model & Simulation, 2015, 13(2):661-690. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=fe1fbc708d43da7895d1e22626940bca
    [25] YANG Z Q, CUI J Z, SUN Y, et al.Multiscale analysis method for thermo-mechanical performance of periodic porous materials with interior surface radiation[J]. International Journal for Numerical Methods in Engineering, 2016, 105(5):323-350. doi: 10.1002/nme.4964
    [26] HAN F, CUI J Z, YU Y.The statistical second-order two-scale method for thermomechanical properties of statistically inhomogeneous materials[J]. Computational Materials Science, 2009, 46(3):654-659. doi: 10.1016/j.commatsci.2009.03.026
    [27] HAN F, CUI J Z, YU Y.The statistical second-order two-scale method for mechanical properties of statistically inhomogeneous materials[J]. International Journal for Numerical Methods in Engineering, 2010, 84(8):972-988. doi: 10.1002/nme.2928
    [28] XING Y F, CHEN L.Physical interpretation of multi-scale asymptotic expansion method[J]. Composite Structures, 2014, 116:694-702. doi: 10.1016/j.compstruct.2014.06.004
    [29] 郑健龙, 李友云, 钱国平.多尺度计算方法-均匀化和平均化[M].北京:科学出版社, 2010.

    ZHENG J L, LI Y Y, QIAN G P.Multi-scale calculation methods-homogenization and averaging[M]. Beijing:Science Press, 2010(in Chinese).
    [30] XING Y F, GAO Y H, CHEN L, et al.Solution methods for two key problems in multiscale asymptotic expansion method[J]. Composite Structures, 2017, 160:854-866. doi: 10.1016/j.compstruct.2016.10.104
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出版历程
  • 收稿日期:  2019-03-11
  • 录用日期:  2019-05-28
  • 网络出版日期:  2019-11-20

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