## 留言板

 引用本文: 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. doi: 10.21656/1000-0887.410111
 Citation: 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.

## 二维瞬态热传导问题的无单元Galerkin法分析

##### doi: 10.21656/1000-0887.410111

###### 作者简介:王红（1994—），女，硕士（E-mail: 1072835817@qq.com）；李小林（1983—），男，教授（通讯作者. E-mail: lxlmath@163.com）.
• 中图分类号: O241.82

## Analysis of 2D Transient Heat Conduction Problems With the Element-Free Galerkin Method

Funds: The National Natural Science Foundation of China（11971085）
• 摘要: 采用无单元Galerkin(element-free Galerkin，EFG)法求解具有混合边界条件的二维瞬态热传导问题.首先采用二阶向后微分公式离散热传导方程的时间变量，将该问题转化为与时间无关的混合边值问题；然后采用罚函数法处理Dirichlet边界条件，建立了二维瞬态热传导问题的无单元Galerkin法；最后基于移动最小二乘近似的误差结果，详细推导了无单元Galerkin法求解二维瞬态热传导问题的误差估计公式.给出的数值算例表明计算结果与解析解或已有数值解吻合较好，该方法具有较高的计算精度和较好的收敛性.
•  [1] 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. [2] 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. [3] ANNAFIT A, GYEABOUR A A, AKAHO E H K, et al. Finite difference analysis of the transient temperature profile within GHARR-1 fuel element[J]. Annals of Nuclear Energy,2014,68: 204-208. [4] GU Y, WANG L, CHEN W, et al. Application of the meshless generalized finite difference method to inverse heat source problems[J]. International Journal of Heat and Mass Transfer,2017,108: 721-729. [5] CLARAC F, GOUSSARD F, TERESI L, et al. Do the ornamented osteoderms influence the heat conduction through the skin? A finite element analysis in Crocodylomorpha[J]. Journal of Thermal Biology,2017,69: 39-53. [6] KHAJEHPOUR S, HEMATIYAN M R, MARIN L. A domain decomposition method for the stable analysis of inverse nonlinear transient heat conduction problems[J]. International Journal of Heat and Mass Transfer,2013,58: 125-134. [7] CHEN J T, YUEH C Y, CHANG Y L, et al. Why dual boundary element method is necessary?[J]. Engineering Analysis With Boundary Elements,2017,76: 59-68. [8] TADEU A, PRATA J, SIMES N. Dynamic simulation of three-dimensional heat conduction through cylindrical inclusions using a BEM model formulated in the frequency domain[J]. Applied Mathematics and Computation,2015,261: 397-407. [9] 周焕林, 严俊, 余波. 识别含热源瞬态热传导问题的热扩散系数[J]. 应用数学和力学, 2018,39(2): 160-169. (ZHOU Huanlin, YAN Jun, YU Bo. Identification of thermal diffusion coefficients for transient heat conduction problems with heat sources[J]. Applied Mathematics and Mechanics,2018,39(2): 160-169. (in Chinese)) [10] CHEN L, MA H P, CHENG Y M. Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems[J]. Chinese Physics B,2013,22(5): 67-74. [11] 李煜冬, 王发杰, 陈文. 瞬态热传导的奇异边界法及其MATLAB实现[J]. 应用数学和力学, 2019,40(3): 259-268. (LI Yudong, WANG Fajie, CHEN Wen. MATLAB implementation of a singular boundary method for transient heat conduction[J]. Applied Mathematics and Mechanics,2019,40(3): 259-268. (in Chinese)) [12] ZHOU H M, ZHOU W H, QIN G, et al. Transient heat conduction analysis for distance-field-based irregular geometries using the meshless weighted least-square method[J]. Numerical Heat Transfer Fundamentals,2017,71(5): 1-11. [13] CHEN L, LIEW K M. A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems[J]. Computational Mechanics,2011,47(4): 455-467. [14] LI X L, ZHANG S G, WANG Y, et al. Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations[J]. Computers & Mathematics With Applications,2016,71(8): 1655-1678. [15] TANG Y Z, LI X L. Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems[J]. Chinese Physics B,2017,26(3): 215-225. [16] ZHAO N, REN H P. The interpolating element-free Galerkin method for 2D transient heat conduction problems[J]. Mathematical Problems in Engineering,2014,15(2): 181-198. [17] LI X L, LI S L. A meshless Galerkin method with moving least square approximations for infinite elastic solids[J]. Chinese Physics B,2013,22(8): 245-252. [18] YANG C X. Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems[J]. Computers and Mathematics With Applications,2015,70(3): 265-281. [19] LI X L. Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces[J]. Applied Numerical Mathematics,2016,99: 77-97. [20] LI X L, LI S L. On the stability of the moving least squares approximation and the element-free Galerkin method[J]. Computers & Mathematics With Applications,2016,72(6): 1515-1531. [21] ZHU T, ATLURI S N. A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method[J]. Computational Mechanics,1998,21(3): 211-222. [22] 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 Fundamentals,2018,73(1): 1-18. [23] GAO H F, WEI G F. Complex variable meshless manifold method for transient heat conduction problems[J]. International Journal of Applied Mechanics,2017,9(5): 1-20. [24] CHEN L, CHENG Y M. The complex variable reproducing kernel particle method for two-dimensional elastodynamics[J]. Chinese Physics B,2010,19(9): 1-12. [25] ZHANG H H, HAN S Y, FAN L F, et al. The numerical manifold method for 2D transient heat conduction problems in functionally graded materials[J]. Engineering Analysis With Boundary Elements,2018,88: 145-155.
##### 计量
• 文章访问数:  1069
• HTML全文浏览量:  234
• PDF下载量:  222
• 被引次数: 0
##### 出版历程
• 收稿日期:  2020-04-20
• 修回日期:  2020-07-07
• 刊出日期:  2021-05-01

/

• 分享
• 用微信扫码二维码

分享至好友和朋友圈