Numerical Manifold Method for Nonlinear Steady-state Heat Conduction Problems

[Objective] This study addresses the numerical modeling of nonlinear steady-state heat conduction processes where the thermal conductivity varies with temperature. The governing equation for such problems is a second-order quasi-linear partial differential equation, whose nonlinear nature makes analytical solutions extremely challenging, thus necessitating efficient numerical approaches. This work employs the Numerical Manifold Method (NMM) based on quadrilateral mesh covers to analyze and solve two-dimensional nonlinear steady-state heat conduction problems. [Methods] Within the NMM over traditional methods framework, a discrete formulation suitable for nonlinear steady-state heat conduction was established by incorporating three typical boundary conditions: Dirichlet, Neumann, and Robin. The classical Newton-Raphson iterative algorithm was adopted to solve the resulting nonlinear system of equations. A complete numerical solution procedure was implemented on the MATLAB platform to ensure algorithm stability and computational efficiency. To systematically verify the accuracy and robustness of the proposed NMM in handling nonlinear heat conduction, a series of representative numerical examples were designed and conducted. These examples covered various scenarios, including continuous homogeneous materials, discontinuous media containing circular holes, and heterogeneous materials. The simulation results were compared against analytical solutions, existing literature data, or Finite Element Method (FEM) solutions. [Results] 1) Compared to the traditional Finite Element Method (FEM), NMM demonstrates significant theoretical and practical advantages when simulating problem domains with complex geometries or internal discontinuities. This advantage primarily stems from its distinctive numerical characteristics: in NMM, the interpolation subdomains are independent of the subdomains used for numerical integration, whereas in FEM they coincide entirely on the same mesh. Furthermore, FEM is prone to mesh distortion when handling complex boundaries, which can degrade accuracy and impair computational efficiency. Leveraging its physical cover system, NMM can accurately describe complex geometric boundaries. At material interfaces, the different heat conduction behaviors across materials are naturally captured through physical covers and local functions without introducing additional interface conditions, thereby simplifying the computational process and enhancing efficiency. 2) The proposed NMM not only achieves high accuracy in temperature field and heat flux distribution across all examples but also exhibits excellent stability and convergence when dealing with discontinuous interfaces and complex geometries, fully validating the method’s effectiveness and reliability for nonlinear steady-state heat conduction problems. [Conclusion] This study successfully applies NMM to solve two-dimensional nonlinear steady-state heat conduction problems. Through comprehensive comparative analysis and numerical validation, the unique advantages of this method in handling complex engineering thermal problems are highlighted. It provides a novel solution for the numerical simulation of nonlinear heat conduction problems and extends the application scope of NMM in the field of computational thermal physics.