A proper orthogonal decomposition analysis method for transient nonlinear heat conduction problems. Part 1: Basic algorithm

Abstract

The present work conducts a systematic and in-depth algorithm investigation for transient nonlinear heat conduction problems solved by using the proper orthogonal decomposition (POD) method. Part 1 of this two-part articles presents the process and characteristics of basic algorithms, including POD explicit and implicit time-marching methods. The accuracy and efficiency are verified by several numerical examples with various boundary conditions and element types. The results show that the POD-based reduced order model (ROM) can provide high quality temperature prediction of the transient nonlinear heat conduction problems when using implicit method. However, the computational time of implicit method is much longer than that of explicit one. The acceleration effect of POD-based ROM on the calculation of the transient nonlinear heat conduction problems is one order of magnitude lower than that of the corresponding transient linear heat conduction problems. The improvement of computational efficiency is not pronounced. Further studies of the more efficient advanced algorithms to deal with POD-based ROM for transient nonlinear heat conduction problems will be presented in Part 2. Additionally, an approximate POD-based ROM for transient nonlinear heat conduction problem is proposed, which can be constructed quickly by using POD modes obtained from the corresponding transient linear heat conduction system. It is confirmed to be feasible for allowing nonlinear behavior to be modeled at an acceptable level of accuracy. It has significant application potential of solving practical engineering problems.