Accelerated Solution of Discrete Ordinates Approximation to the Boltzmann Transport Equation for a Gray Absorbing–Emitting Medium Via Model Reduction

Abstract

This work applies a projection-based model-reduction approach to make high-order quadrature (HOQ) computationally feasible for the discrete ordinates approximation of the radiative transfer equation (RTE) for purely absorbing applications. In contrast to traditional discrete ordinates variants, the proposed method provides easily evaluated error estimates associated with the angular discretization as well as an efficient approach for reducing this error to an arbitrary level. In particular, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model (ROM) constructed from this reduced basis. This strategy results in a much more accurate solution than might have been achieved using only the ordinate directions used to construct the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.

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

“…Substituting Eq. ( 53) in the corner balance equation (44) and dividing the resulting equation by ϕ i,c we obtain the discretized NBTE in the i th cell given by…”

“…In recent years, many data-driven ROMs have been developed using these techniques for linear problems involving the BTE. Dimensionality reduction in the angular variable has been formulated using both the POD and reduced-basis methods [43,44,45,46,47]. POD-Petrov-Galerkin projections have been formulated for the steady-state BTE in 1D geometry [48,49].…”

A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation

“…POD has also been applied to resolve the angular dependence of the neutron transport equation, 29 which was followed by similar work to resolve the angular dimension in radiative heat transfer problems. 30 More recent work has applied POD to nonlinear feedback effects within lead-cooled reactors, 31 fuel burnup within benchmark systems, 32 molten salt reactor analysis, 33 and natural thermohydraulic circulation. 34 This article presents a new discretization method using a POD-based ROM to resolve the angular dimension of the Boltzmann transport equation.…”

“…A comparison of POD to modal methods in transient analysis is given in Reference 27, and the use of ROMs to resolve subchannels in lead cooled fast reactors is developed in Reference 28. POD has also been applied to resolve the angular dependence of the neutron transport equation, 29 which was followed by similar work to resolve the angular dimension in radiative heat transfer problems 30 . More recent work has applied POD to nonlinear feedback effects within lead‐cooled reactors, 31 fuel burnup within benchmark systems, 32 molten salt reactor analysis, 33 and natural thermohydraulic circulation 34 …”

An angular reduced order model for radiative transfer in non grey media

“…Angular POD basis functions were derived from high order spherical harmonics expansion solutions and applied to solve neutron transport problems. Tencer et al [18,19] followed a similar approach to build optimal angular basis functions from high order discrete ordinates expansions. They developed a greedy algorithm for improving the snapshot sampling, that selects angular regions with the highest angular residual.…”