Reduced Order Models for conduction and radiation inside semi-transparent media via the Modal Identification Method

Abstract

In the frame of characterization of thermophysical properties of semi-transparent gray media for which radiative transfers can be modelled by P1 approximation, the present study deals with development, construction and validation of conducto-radiative linear reduced order models (ROMs) explicitly parametrized by thermal conductivity and effective absorption coefficient. As variations of temperature are assumed to be limited in magnitude (about a few tenths of K) in the considered applications, the radiative contribution is linearized in the ROM formulation. ROMs are built through the Modal Identification Method (MIM): once their general form has been derived, they are identified using Particle Swarm Optimization and Ordinary Least Squares, from simulations coming from a nonlinear reference model. In the presented application, the latter is a two-dimensional axisymmetric unsteady model designed to model a "flash-type" experiment. The chosen range of values for thermal conductivity and effective absorption coefficient includes the values usually found for polymethyl methacrylate (PMMA). In comparison with the reference model, the computing time is considerably reduced with limited loss of accuracy.

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

“…( 81) satisfies the condition (83) when test basis functions are defined by Eq. (84). We use a weight matrix W (Eq.…”

“…ROMs for thermal analysis of spacecraft have been created with POD-Galerkin and trajectory piecewise-linear methods [81]. ROMs for grey nonlinear radiation diffusion problems have been formulated with the PGD [82], POD [83] and modal identification method [84]. A grey ROM for multigroup TRT problems was developed using POD-based representations of frequency averaged opacities and other coefficients [85].…”

“…A new approach based on data-driven reduced-order models (ROMs) has been gaining popularity in recent years which make use of data-based methodologies to dimensionality reduction. Data-driven models have been developed for (i) linear particle transport problems [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48] (ii) nonlinear RT problems [49,50,51,52,53,54,55,56,57,58,59,60], and (iii) various problems in nuclear reactor-physics [61,62,63,64,65,66,67,68,69,70,71]. The fundamental idea behind these ROMs is to leverage databases of solutions to their problems of interest (known a-priori) to develop some reduction in the dimensionality for their involved equations.…”

Optimal hybrid parameter selection for stable sequential solution of inverse heat conduction problem

“…Numerical methods include three major discretization methods, namely, finite element method (FEM) [14,20,21,22,26], finite difference method (FDM) [18,19,27], and boundary element method (BEM) [54,55,44]. In addition, model reduction techniques [30,31,40,42,43,60] and parallel algorithm [35] have been developed in thermal sciences to handle large and complex discretized thermal systems. In mathematics, the IHCP is called an ill-posed problem in the sense of Hadamard [1].…”

Thermo-rheological reduced order models for non-Newtonian fluid flows with power-law viscosity via the modal identification method

“…As in POD-Galerkin, MIM requires the knowledge of the local governing equations. In the MIM approach, the ROM equations may also be written down using a Galerkin projection [14] [18] [19] but only the form of these equations is important. Whereas in POD-Galerkin, the components of the ROM constitutive elements are computed using their literal expressions depending on the space functions obtained from POD, in the MIM these components are identified through a parameter estimation problem corresponding to the minimization of a quadratic functional based on the difference between some reference output data characterizing the system behavior, on the one hand, and the ROM outputs corresponding to the same inputs, on the other hand [15].…”