A proper orthogonal decomposition analysis method for transient nonlinear heat conduction problems. Part 2: Advanced algorithm

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. The computational results as presented in Part 1 have revealed some signatures of the basic algorithms, in which poor efficiency is its main shortcoming. In this article, as Part 2, several advanced algorithms are derived to high efficiently deal with the POD-based ROM of transient nonlinear heat conduction problems with temperature-dependent thermal conductivity, including: (i) Element pre-conversion method (EPM), (ii) Multi-level linearization method (MLM), and (iii) EPM-MLM compound method. The accuracy and efficiency are verified by both two-dimensional (2-D) and three-dimensional (3-D) numerical examples with Neumann boundary condition. The results show that these methods can provide high accuracy and high solution quality in the transient nonlinear heat conduction problems. Most importantly, the computational efficiency of these methods has been improved to different degrees when compared to that of the basic algorithm. Among them, the EPM-MLM compound method is the most efficient one. The acceleration effect of POD-based ROM solved by using this compound method is around one order of magnitude higher than that of the basic algorithm, and more pronounced for larger problems.