A Numerical Manifold Method for Solving 2D Transient Nonlinear Heat Conduction Problems

ZHANG Limei; NIE Zhibao; ZHANG Nan; ZHENG Hong; ZHAO Shuaixing; YANG Long

doi:10.21656/1000-0887.460033

Volume 47 Issue 5

May 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(5): 589-604

ZHANG Limei, NIE Zhibao, ZHANG Nan, ZHENG Hong, ZHAO Shuaixing, YANG Long. A Numerical Manifold Method for Solving 2D Transient Nonlinear Heat Conduction Problems[J]. Applied Mathematics and Mechanics, 2026, 47(5): 589-604. doi: 10.21656/1000-0887.460033

Citation:

PDF( 9842 KB)

A Numerical Manifold Method for Solving 2D Transient Nonlinear Heat Conduction Problems

doi: 10.21656/1000-0887.460033

1. China Institute of Geo-Environment Monitoring, Beijing 100081, P.R. China;
2. School of Engineering and Technology, China University of Geosciences, Beijing 100083, P.R. China;
3. State Grid Electric Power Engineering Research Institute, Beijing 100163, P.R. China;
4. Faculty of Architecture, Civil and Transportation Engineering, Beijing University of Technology, Beijing 100124, P.R. China

Received Date: 2025-02-24
Rev Recd Date: 2025-06-05

Available Online: 2026-06-04

Publish Date: 2026-05-01

Abstract

Abstract

The numerical manifold method (NMM) effectively realizes the unified treatment of continuous and discontinuous problems through introduction of 2 cover systems: the mathematical cover for constructing the partition of unity functions and the physical cover for constructing the local approximation function. The application of the NMM to 2D transient nonlinear heat conduction problems was investigated. Firstly, based on the governing equations for the transient nonlinear heat conduction, along with the initial and boundary conditions, the weak form of the initial-boundary value problem was established. Subsequently, an NMM approximate expression for the temperature field was presented and the global discretization form was derived with the Galerkin method. For the time discretization, the backward Euler difference method was used and combined with the Newton-Raphson iterative method to solve the algebraic equations. From simulation of typical discontinuous plates with irregular boundaries and holes, the results show that, the NMM not only has high computational accuracy (the maximum error is not more than 0.6%) and good robustness, but also can handle complex geometries and discontinuous plates more effectively, which provides an innovative and efficient new method for numerical computation in this field.
- nonlinear heat conduction,
- numerical manifold method,
- Galerkin method,
- Newton-Raphson-method,
- temperature field

FullText(HTML)

References(27)

References

[2]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. (SHI Ce. Heat conduction equation finite difference method to achieve the MATLAB[J].Journal of Xianyang Normal University,2009,24(4): 27-29. (in Chinese))

张继红, 栾舒含, 梁波. 具非线性对流项热传导方程的有限差分法[J]. 大连交通大学学报, 2022,43(5): 115-117.

(ZHANG Jihong, LUAN Shuhan, LIANG Bo. Study on finite difference method of heat conduction equation with nonlinear convection term[J].Journal of Dalian Jiaotong University,2022,43(5): 115-117. (in Chinese))

[3]LU Q Y, RIZWAN-UDDIN. A finite element approach for nonlinear, transient heat conduction problems with convection, radiation or contact boundary conditions[J].Annals of Nuclear Energy,2023,193: 110009.

[4]张凯, 王克用, 齐东平. 热传导问题杂交基本解有限元法虚拟源点的探究[J]. 应用数学和力学, 2023,44(4): 431-440. (ZHANG Kai, WANG Keyong, QI Dongping. Research on the fictitious source points of the hybrid fundamental solution-based finite element method for heat conduction problems[J].Applied Mathematics and Mechanics,2023,44(4): 431-440. (in Chinese))

[5]YANG K, PENG H F, WANG J, et al. Radial integration BEM for solving transient nonlinear heat conduction with temperature-dependent conductivity[J].International Journal of Heat and Mass Transfer,2017,108: 1551-1559.

[6]ZHOU L, Lü J, CUI M, et al. A polygonal element differential method for solving two-dimensional transient nonlinear heat conduction problems[J].Engineering Analysis With Boundary Elements,2023,146: 448-459.

[7]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.(in Chinese))

[8]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.

[9]刘思敏, 张慧华, 韩尚宇, 等. 连续及不连续各向异性热传导问题的数值流形方法求解[J]. 应用数学和力学, 2020,41(6): 591-603. (LIU Simin, ZHANG Huihua, HAN Shangyu, et al. Solutions of continuous and discontinuous anisotropic heat conduction problems with the numerical manifold method[J].Applied Mathematics and Mechanics,2020,41(6): 591-603. (in Chinese))

[10]WU W, JIAO Y, ZHENG F, et al. NMM-based computational homogenization for nonlinear transient heat conduction in imperfectly bonded heterogeneous media[J].International Communications in Heat and Mass Transfer,2025,162: 108599.

[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]周保良, 李志远, 黄丹. 二维瞬态热传导的PDDO分析[J]. 应用数学和力学, 2022,43(6): 660-668. (ZHOU Baoliang, LI Zhiyuan, HUANG Dan. PDDO analysis of 2D transient heat conduction problems[J].Applied Mathematics and Mechanics,2022,43(6): 660-668. (in Chinese))

[13]BOBARU F, DUANGPANYA M. A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities[J].Journal of Computational Physics,2012,231(7): 2764-2785.

[14]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.

[15]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 Baojing, GAO Xiaowei, 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))

[16]MIERZWICZAK M, CHEN W, FU Z J. 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]吴泽艳, 郑保敬, 叶永, 等. 非线性热传导方程MLPG/RBF-FD无网格数值模拟[J]. 工程热物理学报, 2022,43(1): 164-172. (WU Zeyan, ZHENG Baojing, YE Yong, et al. Numerical simulation for the nonlinear heat conduction equations based on MLPG/RBF-FD meshless method[J].Journal of Engineering Thermophysics,2022,43(1): 164-172. (in Chinese))

[18]SHI G H. Manifold method of material analysis[C]//Transactions of the 9th Army Conference on Applied Mathematics and Computing,1991.

[19]陈远强, 郑宏, 屈新. 基于数值流形法的降雨入渗与坡面径流耦合算法研究[J]. 应用数学和力学, 2023,44(12): 1499-1511. (CHEN Yuanqiang, ZHENG Hong, QU Xin. A coupling analysis of rainfall infiltration and slope surface runoff based on the numerical manifold method[J].Applied Mathematics and Mechanics,2023,44(12): 1499-1511.(in Chinese))

[20]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 Guodong, ZHANG Huihua, TAN Yuxin. 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))

[21]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))

[22]TAN F, TONG D F, LIANG J W, et al. Two-dimensional numerical manifold method for heat conduction problems[J].Engineering Analysis With Boundary Elements,2022,137: 119-138.

[23]徐栋栋, 郑宏, 夏开文, 等. 高阶扩展数值流形法在裂纹扩展中的应用[J]. 岩石力学与工程学报, 2014,33(7): 1375-1387. (XU Dongdong, ZHENG Hong, XIA Kaiwen, et al. Application of higher-order enriched numerical manifold method to crack propagation[J].Chinese Journal of Rock Mechanics and Engineering,2014,33(7): 1375-1387. (in Chinese))

[24]ZHENG H, XU D. New strategies for some issues of numerical manifold method in simulation of crack propagation[J].International Journal for Numerical Methods in Engineering,2014,97(13): 986-1010.

[25]HU M, WANG Y, 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: 17-29.

[26]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.

[27]ZHOU L, SUN C B, XU B B, et al. A new general analytical PBEM for solving three-dimensional transient nonlinear heat conduction problems with spatially-varying heat generation[J].Engineering Analysis With Boundary Elements,2023,152: 334-346.

Relative Articles

Supplements(0)

Cited By

Proportional views