Reduced order model analysis method via proper orthogonal decomposition for nonlinear transient heat conduction problems

Abstract

A reduced order analysis method is proposed to solve the nonlinear transient heat conduction problems with temperature-dependent conductivity of material by using the proper orthogonal decomposition (POD) model reduction method and the finite element method. Firstly, the POD basis vectors are developed by calculating eigenvectors of an autocorrelation matrix composed of snapshots, which clustered by the given results of obtained from experiments or numerical methods for the linear transient heat transfer problem. Secondly, the finite element discrete scheme of the nonlinear transient heat conduction problem is reduced the order by using the POD basis vectors of the linear problem, and a lower order ordinary differential equation system of the reduced order model of the nonlinear transient heat conduction problem is obtained. Finally, the value of temperature field at each time is obtained by solving the reduced-order ordinary differential equations. Several numerical examples are given to verify the effectiveness of the proposed method. The results show that the proposed method has a good accuracy and the computational efficiency can be improved by 1–2 orders of magnitude.