Solutions of Continuous and Discontinuous Anisotropic Heat Conduction Problems With the Numerical Manifold Method

LIU Simin; ZHANG Huihua; HAN Shangyu; LIU Qiang

doi:10.21656/1000-0887.400289

Volume 41 Issue 6

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2020 > 41(6): 591-603

LIU Simin, ZHANG Huihua, HAN Shangyu, LIU Qiang. Solutions of Continuous and Discontinuous Anisotropic Heat Conduction Problems With the Numerical Manifold Method[J]. Applied Mathematics and Mechanics, 2020, 41(6): 591-603. doi: 10.21656/1000-0887.400289

Citation:

PDF( 2999 KB)

Solutions of Continuous and Discontinuous Anisotropic Heat Conduction Problems With the Numerical Manifold Method

doi: 10.21656/1000-0887.400289

School of Civil Engineering and Architecture, Nanchang Hangkong University, Nanchang 330063, P.R.China

Funds: The National Natural Science Foundation of China(11462014)

Received Date: 2019-09-25
Rev Recd Date: 2019-10-25
Publish Date: 2020-06-01

Abstract

Abstract

The heat conduction is a common problem in engineering practice. Compared with those of isotropic materials, the heat conduction problem of anisotropic materials is more complicated, so it is of great significance to accurately predict the internal temperature distribution. The numerical manifold method (NMM) was developed to solve typical continuous and discontinuous heat conduction problems in anisotropic materials. According to the governing differential equation, boundary conditions and variational principles, the NMM discrete equations for such problems were derived. Several representative examples involving continuous and discontinuous situations were analyzed with the uniform mathematical cover independent of all physical boundaries. The results prove the feasibility and accuracy of the method and indicate that the NMM can simulate the heat conduction problem of anisotropic materials well. Besides, the influences of the material properties and crack configurations on the temperatures were also investigated.
- anisotropic,
- steady heat conduction,
- numerical manifold method,
- temperature,
- continuous and discontinuous problem

FullText(HTML)

References(25)

References

[1]	PRESTINI D, FILIPPINI G, ZDANSKI P S B, et al. Fundamental approach to anisotropic heat conduction using the element-based finite volume method[J].Numerical Heat Transfer, Part B: Fundamentals,2017,71(4): 327-345.
[2]	闫相桥, 武海鹏. 正交各向异性材料三维热传导问题的有限元列式[J]. 哈尔滨工业大学学报, 2003,35(4): 405-409. (YAN Xiangqiao, WU Haipeng. Finite element formulation of 3-D heat transfer in orthotropic materials[J]. Journal of Harbin Institute of Technology,2003,35(4): 405-409.(in Chinese))
[3]	MERA N S, ELLIOTT L, INGHAM D B, et al. A comparison of boundary element method formulations for steady state anisotropic heat conduction problems[J]. Engineering Analysis With Boundary Elements,2001,25(2): 115-128.
[4]	刘俊俏, 苗福生, 李星. 二维各向异性功能梯度材料热传导的边界元分析[J]. 西安交通大学学报, 2013,47(5): 77-81.(LIU Junqiao, MIAO Fusheng, LI Xing. Boundary element analysis of the two-dimensional heat conduction equation for anisotropic fuctionally graded materials[J]. Journal of Xi’an Jiaotong University,2013,47(5): 77-81.(in Chinese))
[5]	陈闽慷, 杜涛, 苏雪, 等. 二维非线性正交各向异性材料的瞬态热传导反问题数值方法[J]. 国防科技大学学报, 2017,39(1): 194-198.(CHEN Minkang, DU Tao, SU Xue, et al. A numerical method for two-dimensional nonlinear transient inverse heat conduction problems for orthotropic material[J]. Journal of National University of Defense Technology,2017,39(1): 194-198.(in Chinese))
[6]	谢佳萱, 李冬明, 聂峰华, 等. 正交各向异性稳态热传导问题的ICVEFG方法[J]. 固体力学学报, 2019,40(1): 74-81.(XIE Jiaxuan, LI Dongming, NIE Fenghua, et al. The ICVEFG method for steady-state heat conduction problems in orthotropic media[J]. Chinese Journal of Solid Mechanics,2019,40(1): 74-81.(in Chinese))
[7]	SHI G H. Manifold method of material analysis[C］// Proceedings of the 〖STBX〗9th Army Conference on Applied Mathematics and Computing . Minneapolis, Minnesota, 1991.
[8]	MA G W, AN X M, ZHANG H H, et al. Modeling complex crack problems using the numerical manifold method[J]. International Journal of Fracture,2009,156(1): 21-35.
[9]	KOUREPINIS D, PEARCE C, BICANIC N. Higher-order discontinuous modeling of fracturing in concrete using the numerical manifold method[J]. International Journal of Computational Methods,2010,7(1): 83-106.
[10]	YANG Y T, ZHENG H. A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis[J]. Engineering Fracture Mechanics,2016,162: 51-75.
[11]	WU Z J, JIANG Y L, LIU Q S, et al. Investigation of the excavation damaged zone around deep TBM tunnel using a Voronoi-element based explicit numerical manifold method[J]. International Journal of Rock Mechanics and Mining Sciences,2018,112: 158-170.
[12]	WU Z J, XU X Y, LIU Q S, et al. A zero-thickness cohesive element-based numerical manifold method for rock mechanical behavior with micro-Voronoi grains[J]. Engineering Analysis With Boundary Elements,2018,96: 94-108.
[13]	YANG S K, CAO M S, REN X H, et al. 3D crack propagation by the numerical manifold method[J]. Computers & Structures,2018,194: 116-129.
[14]	YANG Y T, TANG X H, ZHENG H, et al. Hydraulic fracturing modeling using the enriched numerical manifold method[J]. Applied Mathematical Modelling,2018,53: 462-486.
[15]	GUO H W, ZHENG H, ZHUANG X Y. Numerical manifold method for vibration analysis of Kirchhoff’s plates of arbitrary geometry[J]. Applied Mathematical Modelling,2019,66: 695-727.
[16]	ZHENG H, YANG Y T, SHI G H. Reformulation of dynamic crack propagation using the numerical manifold method[J]. Engineering Analysis With Boundary Elements,2019,105: 279-295.
[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 Shaozhong, MING Zhengrong, QI Yongfeng. 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]	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 Quansheng, LIU Xuewei. 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))
[19]	谭育新, 张慧华, 胡国栋. 二维稳态热传导问题的正六边形流形元研究[J]. 应用数学和力学, 2017,38(5): 594-604.(TAN Yuxin, ZHANG Huihua, HU Guodong. 2D steady heat conduction analysis with the regular hexagon numerical manifold method[J]. Applied Mathematics and Mechanics,2017,38(5): 594-604.(in Chinese))
[20]	ZHANG H H, LIU S M, HAN S Y, et al. Modeling of 2D cracked FGMs under thermo-mechanical loadings with the numerical manifold method[J]. International Journal of Mechanical Sciences,2018,148: 103-117.
[21]	ZHANG H H, LIU S M, HAN S Y, et al. The numerical manifold method for crack modeling of two-dimensional functionally graded materials under thermal shocks[J]. Engineering Fracture Mechanics,2019,208: 90-106.
[22]	BABUSKA I, MELENK J M. The partition of unity method[J]. International Journal for Numerical Methods in Engineering,1997,40(4): 727-758.
[23]	BAYESTEH H, AFSHAR A, MOHAMMDI S. Thermo-mechanical fracture study of inhomogeneous cracked solids by the extended isogeometric analysis method[J]. European Journal of Mechanics A: Solids,2015,51: 123-139.
[24]	ZHANG H H, MA G W. Fracture modeling of isotropic functionally graded materials by the numerical manifold method[J]. Engineering Analysis With Boundary Elements,2014,38: 61-71.
[25]	ZHANG H H, HAN S Y, FAN L F. Modeling 2D transient heat conduction problems by the numerical manifold method on Wachspress polygonal elements[J].Applied Mathematical Modelling,2017,48: 607-620.