2D Steady Heat Conduction Analysis With the Regular Hexagon Numerical Manifold Method
-
摘要: 发展了用于分析二维稳态热传导问题的多边形数值流形方法(numerical manifold method,NMM).根据热传导问题的控制方程、边界条件以及多边形NMM的温度近似函数,采用修正变分原理导出了多边形NMM求解稳态热传导问题的总体方程,给出了多边形单元上的域积分策略.考虑到NMM中数学覆盖系统可不与物理域边界一致以及规则单元的精度优势,采用Wachspress正六边形数学单元对两个典型热传导问题进行了仿真,计算结果与参考解能较好地吻合,表明多边形NMM可以很好地模拟平面稳态热传导问题.Abstract: The polygonal numerical manifold method (NMM) was developed to analyze two2dimensional (2D) steady heat conduction problems. Based on the governing equation, the boundary conditions and the NMM temperature approximation, the discrete NMM equations were deduced according to the modified variational principle. The domain integration schemes on the polygonal elements were presented. Due to the independence between the mathematical cover system and the physical domain and in virtue of the accuracy advantage of regular polygonal elements, the Wachspress regular hexagon mathematical elements were adopted in 2 typical examples, and the computed temperatures agreed well with the referential ones. The study shows that the regular hexagon NMM can well tackle 2D heat conduction problems.
-
Key words:
- numerical manifold method /
- polygonal element /
- steady heat conduction /
- temperature /
- 2D /
- regular hexagon
-
[1] Li B Y, Sun W W. Numerical analysis of heat and moisture transport with a finite difference method[J]. Numerical Methods for Partial Differential Equations,2013,29(1): 226-250. [2] Wang B L, Tian Z H. Application of finite element-finite difference method to the determination of transient temperature field in functionally graded materials[J]. Finite Elements in Analysis & Design,2005,41(4): 335-349. [3] Aguirre-Ramirez G, Oden J T. Finite element technique applied to heat conduction in solids with temperature dependent thermal conductivity[J]. International Journal for Numerical Methods in Engineering,1973,7(3): 345-355. [4] Adam Fic, Ryszard A B, Alain J K. Solving transient nonlinear heat conduction problems by proper orthogonal decomposition and the finite-element method[J]. Numerical Heat Transfer Fundamentals,2005,48(2): 103-124. [5] 欧贵宝, 费纪生. 解瞬态热传导问题的边界元法[J]. 核动力工程, 1991,12(4): 76-80.(OU Gui-bao, FEI Ji-sheng. Boundary element method for solving transient heat conduction[J].Nuclear Power Engineering,1991,12(4): 76-80.(in Chinese)) [6] Onyango T T M, Ingham D B, Lesnic D. Reconstruction of boundary condition laws in heat conduction using the boundary element method[J]. Computer & Mathematics With Applications,2009,57(1): 153-168. [7] Cheng C Z, Han Z L, Zhou H L, et al. Analysis of the temperature field in anisotropic coating-structures by the boundary element method[J]. Engineering Analysis With Boundary Elements,2015,60 (1): 115-122. [8] Liu Y, Zhang X, Lu M W. Meshless least-squares method for solving the steady-state heat conduction equation[J]. Journal of Tsinghua University Science and Technology,2005,10(1): 61-66. [9] Wang H, Qin Q H, Kang Y L. A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media[J].Archive of Applied Mechanics,2005,74(8): 563-579. [10] 王冰冰, 高欣, 段庆林. 稳态热传导的二阶一致无网格法[J]. 应用数学和力学, 2013,34(7): 750-755.(WANG Bing-bing, GAO Xin, DUAN Qing-lin. Quadratically consistent meshfree methods for heat conduction in steady state[J]. Applied Mathematics and Mechanics,2013,34(7): 750-755.(in Chinese)) [11] Merle R, Dolbow J. Solving thermal and phase change problems with the extended finite element method[J]. Computational Mechanics,2002,28(5): 339-350. [12] Liu G W, Hu Y, Li Q B, et al. XFEM for thermal crack of massive concrete[J]. Mathematical Problems in Engineering,2013,2013(12): 261-294. [13] Liu J T, Gu S T, Monteiro E, et al. A versatile interface model for thermal conduction phenomena and its numerical implementation by XFEM[J]. Computational Mechanics,2014,53(4): 825-843. [14] Stapor P. The XFEM for nonlinear thermal and phase change problems[J]. International Journal of Numerical Methods for Heat & Fluid Flow,2015,25(2): 400-421. [15] Shi G H. Manifold method of material analysis[C]// Transaction of 〖STBX〗9th Army Conference on Applied Mathematics and Computing . 1991: 57-76. [16] 魏高峰, 冯伟. 热传导问题的非协调数值流形方法[J]. 力学季刊, 2005,26(3): 451-454.(WEI Gao-feng, FENG Wei. Incompatible numerical manifold method based on heat exchange problem[J]. Chinese Quarterly of Mechanics,2005,26(3): 451-454.(in Chinese)) [17] 林绍忠, 明峥嵘, 祁勇峰. 用数值流形法分析温度场及温度应力[J]. 长江科学院院报, 2007,24(5): 72-75.(LIN Shao-zhong, MING Zheng-rong, QI Yong-feng. Thermal field and thermal stress analysis based on numerical manifold method[J]. Journal of Yangtze River Scientific Research Institute,2007,24(5): 72-75.(in Chinese)) [18] 明峥嵘. 数值流形法在大体积混凝土结构温度应力仿真计算中的应用研究[D]. 硕士学位论文. 武汉: 长江科学院, 2007.(MING Zheng-rong. Research on the application of numerical manifold method in thermal stress simulation of mass concrete structures[D]. Master Thesis. Wuhan: The Yangtze River Academy of Sciences, 2007.(in Chinese)) [19] 刘泉声, 刘学伟. 裂隙岩体温度场数值流形方法初步研究[J]. 岩土工程学报, 2013,35(4): 635-642.(LIU Quan-sheng, LIU Xue-wei. Preliminary research on numerical manifold method for temperature field of fractured rock mass[J]. Chinese Journal of Geotechnical Engineering,2013,35(4): 635-642.(in Chinese)) [20] 刘学伟, 刘泉声, 黄诗冰, 等. 裂隙岩体温度-渗流耦合数值流形方法[J]. 四川大学学报(工程科学版), 2013,45(2): 77-83.(LIU Xue-wei, LIU Quan-sheng, HUANG Shi-bing, et al. Study on numerical manifold method of coupled thermo-hydraulic of fractured rock masses[J]. Journal of Sichuan University(Engineering Science ), 2013,45(2): 77-83.(in Chinese)) [21] 刘学伟, 刘泉声, 卢超波, 等. 温度-应力耦合作用下岩体裂隙扩展的数值流形方法研究[J]. 岩石力学与工程学报, 2014 ,33(7): 1432-1441.(LIU Xue-wei, LIU Quan-sheng, LU Chao-bo, et al. A numerical manifold method for fracture propagation of rock mass considering thermo-mechanical coupling[J]. Chinese Journal of Rock Mechanics & Engineering,2014,33(7): 1432-1441.(in Chinese)) [22] Zhang H H, Ma G W, Ren F. Implementation of the numerical manifold method for thermo-mechanical fracture of planar solids[J]. Engineering Analysis With Boundary Elements,2014,〖STHZ〗 44(1): 45-54. [23] 李义. 基于独立覆盖数值流形法的大体积混凝土温度场仿真计算[D]. 硕士学位论文. 武汉: 长江科学院, 2015.(LI Yi. Simulation of mass concrete temperature field based on the numerical manifold method with independent cover[D]. Master Thesis. Wuhan: The Yangtze River Academy of Sciences, 2015.(in Chinese)) [24] 王兆清. 有理单元法研究[D]. 博士学位论文. 上海: 上海大学, 2004.(WANG Zhao-qing. Study on rational element method[D]. PhD Thesis. Shanghai: Shanghai University,2004.(in Chinese)) [25] Sukumar N, Tabarraei A. Conforming polygonal finite elements[J]. International Journal for Numerical Methods in Engineering,2004,61(12): 2045-2066. [26] Tabarraei A, Sukumar N. Extended finite element method on polygonal and quadtree meshes[J]. Computer Methods in Applied Mechanics & Engineering,2008,197(5): 425-438. [27] 张慧华, 严家祥. 基于蜂窝数值流形元的静弹性力学问题求解[J]. 南昌航空大学学报(自然科学版), 2011,25(4): 1-8.(ZHANG Hui-hua, YAN Jia-xiang. Investigation of elastostatic problems by the numerical manifold method on honeycomb elements[J]. Journal of Nanchang Hangkong University (Natural Science Edition),2011,25(4): 1-8.(in Chinese)) [28] Zhang H H, Zhang S Q. Extract of stress intensity factors on honeycomb elements by the numerical manifold method[J]. Finite Elements in Analysis and Design,2012,59(10): 55-65. [29] 张慧华, 祝晶晶. 复杂裂纹问题的多边形数值流形方法研究[J]. 固体力学学报, 2013,34(1): 38-46.(ZHANG Hui-hua, ZHU Jing-jing. Numerical manifold analysis of complex crack problems on polygonal elements[J]. Chinese Journal of Solid Mechanics,2013,34(1): 38-46.(in Chinese)) [30] Zhang H H, Chen Y L, Li L X, et al. Accuracy comparison of rectangular and triangular mathematical elements in the numerical manifold method[C]// Proceeding of the 〖STBX〗9th International Conference on Analysis of Discontinuous Deformation . 2010: 297-303. [31] 王勖成. 有限单元法[M]. 北京: 清华大学出版社, 2003.(WANG Xu-cheng. Finite Element Method [M]. Beijing: Tsinghua University Press, 2003.(in Chinese)) [32] Wachspress E L. A Rational Finite Element Basis [M]. New York: Academic Press, 1975. [33] 余天堂, 万林林. 非均质材料热传导问题的扩展有限元法[J]. 计算力学学报, 2011,28(6): 884-890.(YU Tian-tang, WAN Lin-lin. Extended finite element method for heat transfer problems in heterogeneous material[J]. Chinese Journal of Computational Mechanics,2011,28(6): 884-890.(in Chinese)) [34] Liu Y, Zhang X, Lu M W. A meshless method based on least-squares approach for steady and unsteady state heat conduction problems[J]. Numerical Heat Transfer Part B: Fundamentals,2007,47(3): 257-275. [35] Moaveni S. Finite Element Analysis Theory and Application With ANSYS [M]. 3rd ed. New Jersey: Pearson Prentice Hall, 2007.
点击查看大图
计量
- 文章访问数: 866
- HTML全文浏览量: 93
- PDF下载量: 818
- 被引次数: 0