The Complex Variable Dimension Coupling Method for 3D Inhomogeneous Transient Heat Conduction Problems

Based on the dimensional splitting method (DSM) and an improved complex variable element-free Galerkin (ICVEFG) method, the complex variable dimension coupling method (CVDCM) is proposed to analyze 3D inhomogeneous transient heat conduction problems. The original 3D governing equation is split into a collection of 2D forms by the dimensional splitting method (DSM), and the discrete equation of the 2D problem is derived via the ICVEFG method. The splitting direction is then treated by using the finite element method (FEM), while the time-dependent term in the governing equation is handled by the finite difference method (FDM). Finally, the numerical solution formula is obtained. To verify the accuracy of the CVDCM, the ratio of the L₂ norm to the true value is used as the relative error. The convergence of the proposed method is demonstrated by increasing the number of nodes and meshes. Five numerical examples of transient inhomogeneous heat conduction problems with spatially varying material properties (density, specific heat capacity, and thermal conductivity) are solved using the CVDCM; the results show that the proposed method achieves good convergence and achieves higher accuracy compared to the dimension coupling method (DCM) and the improved element-free Galerkin (IEFG) method in five examples.