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

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

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

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
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- 被引次数: 0
出版历程
- 收稿日期: 2025-02-24
- 修回日期: 2025-06-05
- 网络出版日期: 2026-06-04
- 刊出日期: 2026-05-01

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

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

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

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