Numerical manifold method for steady-state nonlinear heat conduction using Kirchhoff transformation

The numerical manifold method (NMM) introduces the mathematical and physical cover to solve both continuum and discontinuum problems in a unified manner. In this study, the NMM for solving steady-state nonlinear heat conduction problems is presented, and heat conduction problems consider both convection and radiation boundary conditions. First, the nonlinear governing equation of thermal conductivity, which is dependent on temperature, is transformed into the Laplace equation by introducing the Kirchhoff transformation. The transformation reserves linearity of both the Dirichlet and the Neumann boundary conditions, but the Robin and radiation boundary conditions remain nonlinear. Second, the NMM is employed to solve the Laplace equation using a simple iteration procedure because the nonlinearity focuses on parts of the problem domain boundaries. Finally, the temperature field is retrieved through the inverse Kirchhoff transformation. Typical examples are analyzed, demonstrating the advantages of the Kirchhoff transformation over the direct solution of nonlinear equations using the Newton-Raphson method. This study provides a new method for calculating nonlinear heat conduction.