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Research Papers: Radiative Heat Transfer

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

[+] Author and Article Information
John Tencer

Sandia National Laboratories,
Albuquerque, NM 87123-0840
e-mail: jtencer@sandia.gov

Kevin Carlberg

Sandia National Laboratories,
Livermore, CA 94550-9159
e-mail: ktcarlb@sandia.gov

Marvin Larsen

Sandia National Laboratories,
Albuquerque, NM 87123-0825
e-mail: melarse@sandia.gov

Roy Hogan

Sandia National Laboratories,
Albuquerque, NM 87123-0836
e-mail: rehogan@sandia.gov

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 31, 2017; final manuscript received June 12, 2017; published online August 1, 2017. Assoc. Editor: Ravi Prasher.

J. Heat Transfer 139(12), 122701 (Aug 01, 2017) (9 pages) Paper No: HT-17-1057; doi: 10.1115/1.4037098 History: Received January 31, 2017; Revised June 12, 2017

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.

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References

Figures

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Fig. 8

Accuracy as a function of cumulative solution timeforaradially quadratic temperature distribution T(x,y,z)=300+400(x2 +y2 + z2)

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Fig. 5

Decay of singular values demonstrates that many additional modes are required to capture the behavior of the fully 3D problem

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Fig. 4

Convergence of angle-integrated intensity distributions for linear temperature profile (22) with increasing LOQ order in terms of (a) number of FOM evaluations (i.e., LOQ size) and (b) solution time. We plot these for the LOM (x), ROM 1 (•), and ROM 2 (°).

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Fig. 3

Convergence of angle-integrated intensity distributions for discontinuous temperature profile (21) with increasing LOQ order in terms of (a) number of FOM evaluations (i.e., LOQ size) and (b) solution time. We plot these for the LOM (x), ROM 1 (•), and ROM 2 (°).

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Fig. 2

Convergence of angle-integrated intensity distributions for quadratic temperature profile (20) with increasing LOQ order in terms of (a) number of FOM evaluations (i.e., LOQ size) and (b) solution time. We plot these for the LOM (x), ROM 1 (•), and ROM 2 (°).

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Fig. 1

Decay of singular values demonstrates the presence of a reduced-order basis for the intensity solution for 1D geometry where 20 modes are sufficient to capture all solution behavior

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Fig. 6

Accuracy as a function of cumulative solution time for a discontinuous temperature distribution T(x,y,z)=100 if y = 0, 0 else

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Fig. 7

Accuracy as a function of cumulative solution time for a linear temperature distribution, T(x,y,z)=300+700(1−y)

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Fig. 10

Convergence of adaptive ROM and associated error estimate of an angularly integrated quantity for a linear temperature distribution, T(x, y, z) = 300 + 700 (1 − y)

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Fig. 9

Example of construction of ROM error estimate from training data acquired during the greedy sampling algorithm for a linear temperature distribution, T(x, y, z) = 300 + 700 (1 − y)

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