On Computational Aspects of Least-Squares Projection-Based Model Reduction for Conductive–Radiative Systems

Abstract

Fast and accurate reduced order models (ROMs) of conductive–radiative systems are important for several industrial applications, such as spacecraft, radiant furnaces, solar collectors, etc. The nonlinear nature of radiative heat transfer limits the accuracy of the traditional proper orthogonal decomposition (POD)-based Galerkin projection approach, which works best in the linear realm. Optimal projection schemes based on least-squares minimization of time discrete residuals have shown great promise for solving nonlinear convection-diffusion problems. The accuracy and efficiency of the approach rely on the critical assumptions relating to the low-dimensional structure of the residuals, Jacobians, and a reduced sample mesh required for hyper-reduction. We argue that these assumptions may not hold true for the problems involving radiation. We investigate a coupled conduction and enclosure radiation problem to establish this claim. First, we demonstrate that least-squares Petrov–Galerkin reduced order model, (LSPG-ROM) indeed gives higher accuracy than Galerkin-ROM for the same reduced dimension. Further, we show that while hyper-reduction can be used to obtain significant computational gain for the residual approximation, this is not the case for the Jacobian approximation, as Jacobian snapshots exhibit very slow singular value decay. Moreover, we find that the sample mesh size is in fact close to the full order model (FOM) dimension, hence making the computational cost dependent on the FOM dimension. Finally, we reinforce the above observations by performing an exhaustive performance analysis to compare and characterize the computational cost of FOM, LSPG, and hyper-reduced LSPG-ROMs.