An enhanced tetrahedral quasi-smooth manifold element method for complex three-dimensional nonlinear heat conduction problems

In this paper, an enhanced tetrahedral quasi-smooth manifold element (QSME) is developed for three-dimensional nonlinear heat conduction analysis in complex geometries. By introducing nodal temperature gradients as additional degrees of freedom (DOFs), the proposed element achieves higher-order continuity and improved accuracy in complex temperature fields. To enhance solution robustness, a line–search Newton–Raphson (LSNR) scheme is employed, particularly for problems involving strongly nonlinear boundary conditions. Based on the proposed element and solution strategy, a Galerkin framework with consistent linearization of radiation boundary conditions is established to ensure numerical consistency and stability. Benchmark tests verify the performance of the proposed element and the effectiveness of the LSNR scheme. Numerical simulations show that, for an equivalent number of DOFs, the formulation reduces numerical errors by approximately 20–45% compared with the conventional finite element (FE) method, while achieving nearly the same level of accuracy with about 50% fewer DOFs. Natural-convection cooling experiments further validate its reliability and applicability in complex geometries, with errors below 15%. The proposed formulation provides an efficient and accurate computational framework for heat conduction problems with strongly nonlinear boundary conditions.