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

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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

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

doi: 10.21656/1000-0887.460033

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

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

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

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References(27)

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