非线性稳态热传导问题的数值流形法研究

张丽美; 殷跃平; 郑宏; 朱赛楠; 魏云杰; 张楠; 杨龙

doi:10.11988/ckyyb.20241284

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长江科学院院报 ›› 0 DOI: 10.11988/ckyyb.20241284

非线性稳态热传导问题的数值流形法研究

张丽美^1,2, 殷跃平¹, 郑宏³, 朱赛楠¹, 魏云杰¹, 张楠¹, 杨龙^1,2

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Numerical Manifold Method for Nonlinear Steady-State Heat Conduction Problems

ZHANG Li-mei^1,2, YING Yue-ping¹, ZHENG Hong³, ZHU Sai-nan¹, WEI Yun-jie¹, ZHANG Nan¹, YANG Long^1,2

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摘要

数值流形法（Numerical Manifold Method, NMM）通过引入两套覆盖系统：数学覆盖和物理覆盖,可以统一的解决连续和非连续问题。本研究首次构建了基于NMM的非线性稳态热传导离散格式,并采用Newton-Raphson迭代算法对非线性方程组进行求解,从而实现了问题域内温度场和热流量场的精确计算。与有限元法相比,NMM在处理复杂边界条件和材料界面问题时展现出更强的适应性,其前处理过程也更为灵活便捷。为验证NMM的有效性,针对二维复杂边界和双层材料等问题域的热传导算例进行模拟,结果表明该方法具有较高的计算精度和良好的鲁棒性。

Abstract

Numerical manifold method (NMM) can solve continuous and discontinuous problems in a unified framework by introducing two covering systems: mathematical cover and physical cover. In this study, a nonlinear steady-state heat conduction discrete scheme based on NMM is constructed for the first time, and the Newton-Raphson iterative algorithm is used to solve the nonlinear equations, so that the temperature field and heat flux field in the problem domain can be accurately calculated. Compared with the Finite Element Method, NMM shows stronger adaptability in dealing with complex boundary conditions and material interface problems, and its pre-processing process is more flexible and convenient. In order to verify the effectiveness of NMM, heat transfer simulations are performed for problem domains such as two-dimensional complex boundaries and double-layer materials. The results show that the proposed method has high computational accuracy and good robustness.

导出引用

张丽美, 殷跃平, 郑宏, 朱赛楠, 魏云杰, 张楠, 杨龙. 非线性稳态热传导问题的数值流形法研究[J]. 长江科学院院报. 0 https://doi.org/10.11988/ckyyb.20241284

ZHANG Li-mei, YING Yue-ping, ZHENG Hong, ZHU Sai-nan, WEI Yun-jie, ZHANG Nan, YANG Long. Numerical Manifold Method for Nonlinear Steady-State Heat Conduction Problems[J]. Journal of Changjiang River Scientific Research Institute. 0 https://doi.org/10.11988/ckyyb.20241284

中图分类号： TK124

参考文献

[1] 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)).
[2] 任继勋, 佳琳, 阳建新, 等. 渗流条件下地下水含盐量对基坑坑底冻结温度场影响的数值模拟[J]. 长江科学院院报, 2024, 41(1): 151-158, 166.
(REN Ji-xun, JIA Lin, YANG Jian-xin, et al.Numerical Simulation On the Effect of Groundwater Salinity On Freezing Temperature Field at the Bottom of Foundation Pit Under Seepage Conditions[J]. Journal of Yangtze River Scientific Research Institute, 2024, 41(1): 151-158, 166. (in Chinese))
[3] 郭平业, 卜墨华, 张鹏, 等. 高地温隧道灾变机制与灾害防控研究进展[J]. 岩石力学与工程学报, 2023, 42(7): 1561-1581.
(GUO Ping-ye, BU Mo-hua, ZHANG Peng, et al.Review On Catastrophe Mechanism and Disaster Countermeasure of High Geotemperature Tunnels[J]. Chinese Journal of Rock Mechanics and Engineering, 2023, 42(7): 1561-1581. (in Chinese))
[4] 江文豪, 冯晨, 李江山. 饱和黏土一维非线性固结与热传导耦合模型[J]. 岩石力学与工程学报, 2023, 42(10): 2588-2600.
(JIANG Wen-hao, Feng Chen, LI Jiang-shan.Coupled Model for One-Dimensional Nonlinear Consolidation and Heat Conduction in Saturated Clay[J]. Chinese Journal of Rock Mechanics and Engineering, 2023, 42(10): 2588-2600. (in Chinese))
[5] 史策. 热传导方程有限差分法的Matlab实现[J]. 咸阳师范学院学报, 2009, 24(4): 27-29, 36.
(SHI Ce.Heat Conduction Equation Finite Difference Method to Achieve the Matlab[J]. Journal of Xianyang Normal University, 2009, 24(4): 27-29, 36. (in Chinese))
[6] Annasabi Z, Erchiqui F.Robust Kirchhoff Transformation Using B-Spline for Finite Element Analysis of the Non-Linear Heat Conduction[J]. International Communications in Heat and Mass Transfer, 2021, 120: 104985.
[7] Feng W, Gao X.An Interface Integral Equation Method for Solving Transient Heat Conduction in Multi-Medium Materials with Variable Thermal Properties[J]. International Journal of Heat and Mass Transfer, 2016, 98: 227-239.
[8] 王峰, 林皋, 郑保敬, 等. 非线性热传导问题的基于滑动Kriging插值的MLPG法[J]. 大连理工大学学报, 2014, 54(3): 339-344.
(WANG Feng, LIN Gao, ZHENG Baojing, et al.MLPG method based on moving Kriging interpolation for solving nonlinear heat conduction problems[J]. Journal of Dalian University of Technology. 2014(3): 339-344. (in Chinese))
[9] 吴泽艳, 郑保敬, 叶永, 等. 非线性热传导方程无网格MLPG/RBF-FD数值模拟[J]. 工程热物理学报, 2022, 43(1): 164-172.
(Wu Ze-yan, Zheng Bao-jing, Ye Yong, et al.Numerical simulation for the nonlinear heat conduction equations based on MLPG/RBF-FD meshless method[J]. 2022, 43(1): 164-172. (in Chinese))
[10] 李庆华, 冯子超, 陈莘莘, 等. 稳态非线性热传导问题的比例边界有限元法[J]. 华东交通大学学报, 2023, 40(6): 110-114.
(Li Qing-hua, Feng Zi-chao, Chen Shen-shen, et al.Scaled Boundary Finite Element Method for steady-state nonlinear heat conduction problem[J]. Journal of East China Jiaotong University. 2023, 40(6): 110-114. (in Chinese))
[11] Zhang L, Guo F, Zheng H.The MLS-Based Numerical Manifold Method for Nonlinear Transient Heat Conduction Problems in Functionally Graded Materials[J]. International Communications in Heat and Mass Transfer, 2022, 139: 106428.
[12] 李腊梅, 冯春. 一种非连续介质中热传导过程的数值模拟方法[J]. 工程力学, 2016, 33(1): 25-31, 46.
(LI La-mei, FENG Chun.A numerical simulation method for heat conduction in discontinuous media[J]. Engineering Mechanics. 2016, 33(1): 25-31, 46. (in Chinese))
[13] 刘承论, 秦忠诚. 三维非稳态热传导问题的边界元法[J]. 岩石力学与工程学报, 2004, 23(18): 3168-3173.
(Liu Cheng-lun, Qin Zhong-cheng.Boundary element method for 3D Non-steady heat conduction[J]. Chinese Journal of Rock Mechanics and Engineering. 2004, 23(18): 3168-3173. (in Chinese))
[14] 梁钰, 郑保敬, 高效伟, 等. 基于Pod模型降阶法的非线性瞬态热传导分析[J]. 中国科学(物理学力学天文学), 2018, 48(12): 32-41.
(LIANG Yu, ZHENG Bao-Jing, GAO Xiao-wei, et al.Reduced order model analysis method via proper orthogonal decomposition for nonlinear transient heat conduction problems[J]. SCIENTIA SINICA Physica, Mechanica & Astronomica. 2018, 48(12): 32-41. (in Chinese))
[15] Suvin V S, Ooi E T, Song C, et al.Temperature-Dependent Nonlinear Transient Heat Conduction Using the Scaled Boundary Finite Element Method[J]. International Journal of Heat and Mass Transfer, 2025, 243: 126780.
[16] Mierzwiczak M, Chen W, Fu Z.The Singular Boundary Method for Steady-State Nonlinear Heat Conduction Problem with Temperature-Dependent Thermal Conductivity[J]. International Journal of Heat and Mass Transfer, 2015, 91: 205-217.
[17] Yang K, Feng W, Wang J, et al.Ribem for 2D and 3D Nonlinear Heat Conduction with Temperature Dependent Conductivity[J]. Engineering Analysis with Boundary Elements, 2018, 87: 1-8.
[18] Khosravifard A, Hematiyan M R, Marin L.Nonlinear Transient Heat Conduction Analysis of Functionally Graded Materials in the Presence of Heat Sources Using an Improved Meshless Radial Point Interpolation Method[J]. Applied Mathematical Modelling, 2011, 35(9): 4157-4174.
[19] Shi G H.Manifold Method of Material Analysis[C]//. Transactions of the 9 Army Conference On Applied Mathematics and Computing, 1991.
[20] Zhang L, Kong H, Zheng H.Numerical Manifold Method for Steady-State Nonlinear Heat Conduction Using Kirchhoff Transformation[J]. Science China Technological Sciences, 2023, 67(4): 992-1006.
[21] 贾真, 杨冬梅, 郑宏. 数值流形法渗流分析中处理弱不连续界面的新方法[J]. 长江科学院院报, 2023: 1-7.
(JIA Zhen, YANG Dong-mei, Zheng Hong.A New Method to Deal with Weak Discontinuous Interface in Seepage Analysis Based On Numerical Manifold Method[J]. Journal of Yangtze River Scientific Research Institute, 2023: 1-7. (in Chinese))
[22] 李伟, 郑宏, 王海龙, 等. 求解断裂问题的新型无网格数值流形法[J]. 岩石力学与工程学报, 2020, 39(S1): 2655-2664.
(Li Wei, Zheng Hong, Wang Hai-long, et al.A new meshfree-numerical manifold method for solving the fracture problem[J]. 2020, 39(S1): 2655-2664. (in Chinese))
[23] 王方义, 郑宏. 无界域问题的数值流形法[J]. 长江科学院院报, 2023, 40(7): 110-117.
(WANG Fang-yi, Zheng Hong.Numerical Manifold Method for Unbounded Domain Problems[J]. Journal of Yangtze River Scientific Research Institute, 2023, 40(7): 110-117. (in Chinese))
[24] 陈远强, 郑宏, 陈涛. 基于数值流形法的重力坝抗滑稳定性分析[J]. 长江科学院院报, 2016, 33(9): 133-137.
(CHEN Yuan-qiang, Zheng Hong, Chen Tao.Analysis of Anti-Sliding Stability of Gravity Dam Using Numerical Manifold Method[J]. Journal of Yangtze River Scientific Research Institute, 2016, 33(9): 133-137. (in Chinese))
[25] 胡国栋, 张慧华, 谭育新. 功能梯度材料稳态热传导问题的数值流形方法研究[J]. 应用力学学报, 2017, 34(2): 311-317.
(Hu Guo-dong, Zhang Hui-hua, Tan Yu-xin.Numerical manifold study of steady heat conduction problems in functionally graded materials[J]. Chinese Journal of Applied Mechanics. 2017, 34(2): 311-317. (in Chinese))
[26] Tan F, Tong D, Liang J, et al.Two-Dimensional Numerical Manifold Method for Heat Conduction Problems[J]. Engineering Analysis with Boundary Elements, 2022, 137: 119-138.
[27] Wang K, Tang C A, Qian X, et al.Numerical Manifold Method with Local Mesh Refinement for Thermo-Mechanical Coupling Analysis in Rocks[J]. Computers and Geotechnics, 2025, 179: 107009.
[28] Li C L, Guo D L, Zhang H H.The Numerical Manifold Method for Piezoelectric Materials with Hole Flaws Under Electro-Mechanical Loadings[J]. Engineering Analysis with Boundary Elements, 2025, 173: 106149.
[29] Zhang Y, Zheng H, Lin S.Random Structure Modeling of Soil and Rock Mixture and Evaluation of its Permeability Using Three-Dimensional Numerical Manifold Method[J]. Computers and Geotechnics, 2025, 180: 107089.
[30] 苏海东, 颉志强, 龚亚琦, 等. 基于独立覆盖的流形法的收敛性及覆盖网格特性[J]. 长江科学院院报, 2016, 33(2): 131-136.
(SU Hai-dong, XIE Zhi-qiang, GONG Ya-qi, et al.Characteristics of Convergence and Cover Mesh in Numerical Manifold Method Based On Independent Covers[J]. Journal of Yangtze River Scientific Research Institute, 2016, 33(2): 131-136. (in Chinese))
[31] Wu W, Wan T, Yang Y, et al.Three-Dimensional Numerical Manifold Formulation with Continuous Nodal Gradients for Dynamics of Elasto-Plastic Porous Media[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 388: 114203.
[32] Zheng H, Liu F, Li C.The Mls-Based Numerical Manifold Method with Applications to Crack Analysis[J]. International Journal of Fracture, 2014, 190(1-2): 147-166.
[33] Mengsu H, Yuan W, Rutqvist J.On Continuous and Discontinuous Approaches for Modeling Groundwater Flow in Heterogeneous Media Using the Numerical Manifold Method; Model Development and Comparison[J]. Advances in Water Resources, 2015, 80(C): 17-29.
[34] Zheng H, Li W, Du X.Exact Imposition of Essential Boundary Condition and Material Interface Continuity in Galerkin‐Based Meshless Methods[J]. International Journal for Numerical Methods in Engineering, 2017, 110(7): 637-660.
[35] Yang K, Wang J, Du J, et al.Radial Integration Boundary Element Method for Nonlinear Heat Conduction Problems with Temperature-Dependent Conductivity[J]. International Journal of Heat and Mass Transfer, 2017, 104: 1145-1151.
[36] Zhang L, Yin Y, Zheng H, et al.Singularity Treatments in Transient Confined Seepage Using Numerical Manifold Method[J]. Engineering Analysis with Boundary Elements, 2025, 171: 106100.
[37] Yang K, Li H, Peng H, et al.New Interface Integration Bem for Solving Multi-Medium Nonlinear Heat Transfer Problems[J]. Engineering Analysis with Boundary Elements, 2020, 117: 66-75.