A Reduced-Order FEM Based on POD for Solving Non-Fourier Heat Conduction Problems under Laser Heating

Abstract

The study presents a novel approach called FEM-POD, which aims to enhance the computational efficiency of the Finite Element Method (FEM) in solving problems related to non-Fourier heat conduction. The present method employs the Proper Orthogonal Decomposition (POD) technique. Firstly, spatial discretization of the second-order hyperbolic differential equation system is achieved through the Finite Element Method (FEM), followed by the application of the Newmark method to address the resultant ordinary differential equation system over time, with the resultant numerical solutions collected in snapshot form. Next, the Singular Value Decomposition (SVD) is employed to acquire the optimal proper orthogonal decomposition basis, which is subsequently combined with the FEM utilizing the Newmark scheme to construct a reduced-order model for non-Fourier heat conduction problems. To demonstrate the effectiveness of the suggested method, a range of numerical instances, including different laser heat sources and relaxation durations, are executed. The numerical results validate its enhanced computational accuracy and highlight significant time savings over addressing non-Fourier heat conduction problems using the full order FEM with the Newmark approach. Meanwhile, the numerical results show that when the number of elements or nodes is relatively large, the CPU running time of the FEM-POD method is even hundreds of times faster than that of classical FEM with the Newmark scheme.