二维瞬态非线性热传导问题的数值流形法求解

张丽美; 聂治豹; 张楠; 郑宏; 赵帅星; 杨龙

doi:10.21656/1000-0887.460033

二维瞬态非线性热传导问题的数值流形法求解

doi: 10.21656/1000-0887.460033

张丽美^{1, 2, ,},
聂治豹³,
张楠¹,
郑宏⁴,
赵帅星¹,
杨龙^{1, 2}

1.
中国地质环境监测院，北京 100081
2.
中国地质大学工程技术学院，北京 100083
3.
国网电力工程研究院有限公司，北京 100163
4.
北京工业大学城市建设学部，北京 100124

(我刊编委郑宏来稿)

基金项目:

云南省重点研发计划 202403AA080001

国家电网有限公司总部科技项目 5200-202355156A-1-1-ZN

详细信息

通讯作者:
张丽美(1992—)，女，工程师，博士(通信作者. E-mail: 710907894@qq.com)

中图分类号: O302
计量
- 文章访问数: 49
- HTML全文浏览量: 15
- PDF下载量: 18
- 被引次数: 0
出版历程
- 收稿日期: 2025-02-24
- 修回日期: 2025-06-05
- 刊出日期: 2026-05-01

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

ZHANG Limei^{1, 2
, ,},
NIE Zhibao³,
ZHANG Nan¹,
ZHENG Hong⁴,
ZHAO Shuaixing¹,
YANG Long^{1, 2}

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

(Contributed by ZHENG Hong, Member of the Editorial Board of AMM)

摘要

摘要: 数值流形法(numerical manifold method，NMM)通过引入两套覆盖系统：数学覆盖用于构造单位分解函数；物理覆盖用于构造局部逼近函数，有效实现了连续与不连续问题的统一处理. 该文深入研究了NMM在二维瞬态非线性热传导问题的应用. 首先，根据瞬态非线性热传导的控制方程、初始条件以及边界条件，建立了初边值问题的弱形式. 随后，提出了温度场的NMM近似表达式，采用Galerkin法推导出全局离散格式. 在时间离散方面，采用Euler向后差分法，并结合Newton-Raphson迭代法求解了最终的代数方程组. 通过对具有不规则边界和含孔洞的不连续板等典型算例进行模拟，结果表明NMM不仅计算精度高(最大的误差不超过0.6%)、鲁棒性好，更能有效处理复杂几何形状和不连续性板，为该领域的数值计算提供了一种高效的新方法.
- 非线性热传导 /
- 数值流形法 /
- Galerkin法 /
- Newton-Raphson法 /
- 温度场
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
edited-by

edited-by
注释:

1) (我刊编委郑宏来稿)

HTML全文

图 1 瞬态非线性热传导的问题域

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. The transient problem domain of nonlinear heat conduction

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图 2 在计算区域内的数学片、物理片和流形单元

Figure 2. Mathematical patches, physical patches and manifold elements in the computational domain

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图 3 方板几何模型及边界条件

Figure 3. The geometric model of the square plate and the boundary condition

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图 4 含121个数学片的数学覆盖

Figure 4. The mathematical cover with 121 mathematical patches

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图 5 6 282个三角形单元的有限元网格

Figure 5. The finite element mesh with 6 282 triangular elements

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图 6 不同罚因子下沿着x₂=0.5 m在t=5.0 s时的温度值

Figure 6. Temperatures along x₂=0.5 m at t=5.0 s with different penalty factors

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图 7 不同时间步长Δt下沿着x₂=0.5 m的温度变化

Figure 7. Temperature variations along x₂=0.5 m at different time steps Δt

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图 8 板上沿着x₁方向的温度

Figure 8. Temperatures on the plate in the x₁ direction

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图 9 随时间分布的温度值

Figure 9. The temperature distribution vs. time

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图 10 不同时间的温度等值线图

Figure 10. Contour plots of the temperature at different moments

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图 11 板的几何尺寸和边界条件

Figure 11. Geometric dimensions and boundary conditions of the plate

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图 12 838个数学片覆盖板

Figure 12. The mathematical cover with 838 mathematical patches

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图 13 3 324个三角形单元的有限元网格

Figure 13. The finite element mesh with 3 324 triangular elements

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图 14 沿板上边界的计算温度

Figure 14. Computed temperatures along the upper boundary of the plate

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图 15 沿着圆环的温度

Figure 15. Computed temperatures along the circle

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图 16 M点的温度随时间的分布

Figure 16. The temperature distribution vs. time of point M

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图 17 不同时间的温度等值线图

Figure 17. Contour plots of the temperature at different moments

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图 18 算例3的几何模型

Figure 18. The geometric model for example 3

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图 19 含821个数学片的数学覆盖

Figure 19. The mathematical cover with 821 mathematical patches

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图 20 4 472个三角形单元的有限元网格

Figure 20. The finite element mesh with 4 472 triangular elements

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图 21 NMM和FEM计算t=2 s, 4 s, 6 s, 8 s的温度等值线图

Figure 21. Contour plots of the temperature fields by NMM and FEM at t=2 s, 4 s, 6 s, 8 s

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图 22 算例4的几何示意图

Figure 22. The geometric model for example 4

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图 23 含1 024个数学片的数学覆盖

Figure 23. The mathematical cover with 1 024 mathematical patches

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图 24 3 348个三角形单元的有限元网格

Figure 24. The finite element mesh with 3 348 triangular elements

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图 25 NMM和FEM计算t=5 s, 10 s的温度等值线图

Figure 25. Contour plots of the temperature fields by NMM and FEM at t=5 s, 10 s

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表 1 NMM和FEM计算不同时刻的A₁和A₂两点的温度值

Table 1. The temperatures of point A₁ and A₂ at different moments calculated by NMM and FEM

t/s	point A₁			point A₂
t/s	NMM	FEM	error/%	NMM	FEM	error/%
1	100.02	100.01	0.002 3	100.73	100.56	0.168 4
2	100.73	100.62	0.110 3	108.86	108.80	0.056 6
3	103.99	103.74	0.242 1	120.51	120.88	0.308 3
4	109.46	109.27	0.173 7	130.27	130.92	0.493 6
5	115.55	115.45	0.084 3	137.75	138.50	0.539 5
6	121.22	121.29	0.056 1	143.49	144.23	0.509 8
7	126.09	126.33	0.189 2	147.97	148.64	0.451 0
8	130.09	130.46	0.281 0	151.49	152.08	0.386 4
9	133.30	133.80	0.370 1	154.28	154.87	0.384 0
10	135.85	136.42	0.421 5	156.47	157.08	0.387 0

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表 2 B₁和B₂两点不同时刻的温度值

Table 2. The temperatures of point B₁ and B₂ at different moments

t/s	point B₁			point B₂
t/s	NMM	FEM	error/%	NMM	FEM	error/%
2	799.26	800.21	0.118 7	794.04	795.49	0.182 3
4	793.90	794.93	0.129 6	780.81	781.37	0.071 7
6	784.12	784.9	0.099 4	766.16	766.53	0.048 3
8	772.01	772.68	0.086 7	752.38	752.54	0.021 3
10	759.33	759.66	0.043 4	739.99	739.83	0.022 0

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表 3 B₃, B₄, B₅和B₆四个点不同时刻的温度值

Table 3. The temperatures of point B₃, B₄, B₅ and B₆ at different moments

point	FEM	PEDM	error/%	EDM	error/%	NMM	error/%
B₃	734.307	728.484	0.793	733.681	0.085	735.244	0.128
B₄	774.639	770.477	0.537	773.307	0.172	771.633	0.388
B₅	775.070	770.261	0.620	773.389	0.217	771.74	0.429
B₆	735.843	732.233	0.490	733.897	0.264	735.179	0.090

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表 4 C₁和C₂两点不同时刻的温度值

Table 4. The temperatures of point C₁ and C₂ at different moments

t/s	point C₁			point C₂
t/s	NMM	FEM	error/%	NMM	FEM	error/%
1	499.41	499.46	0.009 4	498.04	498.07	0.006 9
2	498.51	498.50	0.001 6	495.95	496.01	0.012 6
3	497.53	497.50	0.006 6	494.1	494.15	0.010 2
4	496.57	496.54	0.006 6	492.45	492.49	0.007 1
5	495.65	495.62	0.005 5	490.95	490.98	0.006 0
6	494.76	494.73	0.006 0	489.57	489.60	0.006 4
7	493.90	493.86	0.008 5	488.27	488.33	0.011 5
8	493.07	493.03	0.007 6	487.06	487.13	0.015 3
9	492.25	492.21	0.008 3	485.93	486.01	0.017 1
10	491.45	491.40	0.009 6	484.85	484.95	0.020 5

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参考文献(27)

[1]	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)
[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)
[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. doi: 10.1016/j.anucene.2023.110009
[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. doi: 10.21656/1000-0887.430077 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) doi: 10.21656/1000-0887.430077
[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. doi: 10.1016/j.ijheatmasstransfer.2017.01.030
[6]	ZHOU L, LV 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. doi: 10.1016/j.enganabound.2022.10.015
[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. doi: 10.21656/1000-0887.410111 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) doi: 10.21656/1000-0887.410111
[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. doi: 10.1016/j.apm.2011.02.039
[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. doi: 10.21656/1000-0887.400289 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) doi: 10.21656/1000-0887.400289
[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. doi: 10.1016/j.icheatmasstransfer.2025.108599
[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. doi: 10.1016/j.icheatmasstransfer.2022.106428
[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. doi: 10.21656/1000-0887.420150 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) doi: 10.21656/1000-0887.420150
[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. doi: 10.1016/j.jcp.2011.12.017
[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. doi: 10.1016/j.icheatmasstransfer.2020.104985
[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. doi: 10.1016/j.ijheatmasstransfer.2015.07.051
[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. doi: 10.21656/1000-0887.440115 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) doi: 10.21656/1000-0887.440115
[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. doi: 10.21656/1000-0887.370306 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) doi: 10.21656/1000-0887.370306
[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. doi: 10.1016/j.enganabound.2022.02.004
[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. doi: 10.1002/nme.4620
[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. doi: 10.1016/j.advwatres.2015.03.004
[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. doi: 10.1002/nme.5370
[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. doi: 10.1016/j.enganabound.2023.04.025