A general finite element method: Extension of variational analysis for nonlinear heat conduction with temperature-dependent properties and boundary conditions, and its implementation as local refinement

Abstract

In simulation of heat conduction with temperature-independent physical properties and boundary conditions (BCs), Galerkin residual analysis and variational analysis yield equivalent finite element method (FEM), the conventional FEM. However, if the properties and BCs are temperature-dependent, it is discovered that their derivatives further induce nonlinearity of FEM which consequently generates divergence between the two analyses. A general FEM, extension of variational analysis, is derived as general form of conventional FEM modeling nonlinear heat conduction. Numerical examples demonstrate that the general FEM produces results with considerably higher accuracy and stability and also possesses higher performances on conforming with both two analyses. Since general FEM degenerates to conventional FEM if derivatives are of small-amplitude or zero and its direct implementation to the entire domain is costive, general FEM is alternatively utilized as local refinement of governing equation only to points with significant derivatives. The strategy of local refinement is optimized to enhance efficiency.