Abstract

A projection-based reduced order model (pROM) methodology has been developed for transient heat transfer problems involving coupled conduction and enclosure radiation. The approach was demonstrated on two test problems of varying complexity. The reduced order models demonstrated substantial speedups (up to 185×) relative to the full order model with good accuracy (less than 3% L error). An attractive feature of pROMs is that there is a natural error indicator for the ROM solution: the final residual norm at each time-step of the converged ROM solution. Using example test cases, we discuss how to interpret this error indicator to assess the accuracy of the ROM solution. The approach shows promise for many-query applications, such as uncertainty quantification and optimization. The reduced computational cost of the ROM relative to the full-order model (FOM) can enable the analysis of larger and more complex systems as well as the exploration of larger parameter spaces.

References

1.
Modest
,
M. F.
,
2013
,
Radiative Heat Transfer
,
Academic Press
, London.
2.
Howell
,
J. R.
,
Mengüç
,
M. P.
,
Daun
,
K.
, and
Siegel
,
R.
,
2020
,
Thermal Radiation Heat Transfer
,
CRC Press
, Oxfordshire, UK.
3.
Han
,
D.
,
Yu
,
B.
,
Yu
,
G.
,
Zhao
,
Y.
, and
Zhang
,
W.
,
2014
, “
Study on a BFC-Based POD-Galerkin ROM for the Steady-State Heat Transfer Problem
,”
Int. J. Heat Mass Transfer
,
69
, pp.
1
5
.10.1016/j.ijheatmasstransfer.2013.10.004
4.
Han
,
D.
,
Yu
,
B.
, and
Zhang
,
X.
,
2014
, “
Study on a BFC-Based POD-Galerkin Reduced-Order Model for the Unsteady-State Variable-Property Heat Transfer Problem
,”
Numer. Heat Transfer, Part B Fundam.
,
65
(
3
), pp.
256
281
.10.1080/10407790.2013.849989
5.
Georgaka
,
S.
,
Stabile
,
G.
,
Rozza
,
G.
, and
Bluck
,
M. J.
,
2020
, “
Parametric POD-Galerkin Model Order Reduction for Unsteady-State Heat Transfer Problems
,”
Commun. Comput. Phys.
, 27(1), pp.
1
32
.10.4208/cicp.OA-2018-0207
6.
Gelsomino
,
F.
, and
Rozza
,
G.
,
2011
, “
Comparison and Combination of Reduced-Order Modelling Techniques in 3D Parametrized Heat Transfer Problems
,”
Math. Comput. Modell. Dyn. Syst.
,
17
(
4
), pp.
371
394
.10.1080/13873954.2011.547672
7.
Gartling
,
D. K.
, and
Hogan
,
R. E.
, “
Reduced Order Models for Thermal Analysis
,” LDRD, Report No. 137807.
8.
Zhu
,
Q.-H.
,
Liang
,
Y.
, and
Gao
,
X.-W.
,
2020
, “
A Proper Orthogonal Decomposition Analysis Method for Transient Nonlinear Heat Conduction Problems. Part 1: Basic Algorithm
,”
Numer. Heat Transfer, Part B Fundam.
,
77
(
2
), pp.
87
115
.10.1080/10407790.2019.1690378
9.
Wang
,
Y.
,
Yu
,
B.
,
Cao
,
Z.
,
Zou
,
W.
, and
Yu
,
G.
,
2012
, “
A Comparative Study of Pod Interpolation and Pod Projection Methods for Fast and Accurate Prediction of Heat Transfer Problems
,”
Int. J. Heat Mass Transfer
,
55
(
17–18
), pp.
4827
4836
.10.1016/j.ijheatmasstransfer.2012.04.053
10.
Gaonkar
,
A.
, and
Kulkarni
,
S.
,
2015
, “
Application of Multilevel Scheme and Two Level Discretization for Pod Based Model Order Reduction of Nonlinear Transient Heat Transfer Problems
,”
Comput. Mech.
,
55
(
1
), pp.
179
191
.10.1007/s00466-014-1089-y
11.
Zucatti
,
V.
,
Lui
,
H. F.
,
Pitz
,
D. B.
, and
Wolf
,
W. R.
,
2020
, “
Assessment of Reduced-Order Modeling Strategies for Convective Heat Transfer
,”
Numer. Heat Transfer, Part A Appl.
,
77
(
7
), pp.
702
729
.10.1080/10407782.2020.1714330
12.
Brenner
,
T. A.
,
Fontenot
,
R. L.
,
Cizmas
,
P. G.
,
O'Brien
,
T. J.
, and
Breault
,
R. W.
,
2012
, “
A Reduced-Order Model for Heat Transfer in Multiphase Flow and Practical Aspects of the Proper Orthogonal Decomposition
,”
Comput. Chem. Eng.
,
43
, pp.
68
80
.10.1016/j.compchemeng.2012.04.003
13.
Wang
,
Y.
,
Ma
,
H.
,
Cai
,
W.
,
Zhang
,
H.
,
Cheng
,
J.
, and
Zheng
,
X.
,
2020
, “
A POD-Galerkin Reduced-Order Model for Two-Dimensional Rayleigh-Bénard Convection With Viscoelastic Fluid
,”
Int. Commun. Heat Mass Transfer
,
117
, p.
104747
.10.1016/j.icheatmasstransfer.2020.104747
14.
Lorenzi
,
S.
,
Cammi
,
A.
,
Luzzi
,
L.
, and
Rozza
,
G.
,
2016
, “
POD-Galerkin Method for Finite Volume Approximation of Navier–Stokes and RANS Equations
,”
Comput. Methods Appl. Mech. Eng.
,
311
, pp.
151
179
.10.1016/j.cma.2016.08.006
15.
Park
,
H.
, and
Cho
,
D.
,
1996
, “
Low Dimensional Modeling of Flow Reactors
,”
Int. J. Heat Mass Transfer
,
39
(
16
), pp.
3311
3323
.10.1016/0017-9310(96)00038-5
16.
Ding
,
P.
,
Wu
,
X.-H.
,
He
,
Y.-L.
, and
Tao
,
W.-Q.
,
2008
, “
A Fast and Efficient Method for Predicting Fluid Flow and Heat Transfer Problems
,”
ASME J. Heat Transfer-Trans. ASME
,
130
(
3
), p. 032502.10.1115/1.2804935
17.
Park
,
H.
, and
Lee
,
M.
,
1998
, “
An Efficient Method of Solving the Navier–Stokes Equations for Flow Control
,”
Int. J. Numer. Methods Eng.
,
41
(
6
), pp.
1133
1151
.10.1002/(SICI)1097-0207(19980330)41:6<1133::AID-NME329>3.0.CO;2-Y
18.
Gunes
,
H.
,
2002
, “
Low-Dimensional Modeling of Non-Isothermal Twin-Jet Flow
,”
Int. Commun. Heat Mass Transfer
,
29
(
1
), pp.
77
86
.10.1016/S0735-1933(01)00326-8
19.
Buchan
,
A. G.
,
Calloo
,
A.
,
Goffin
,
M. G.
,
Dargaville
,
S.
,
Fang
,
F.
,
Pain
,
C. C.
, and
Navon
,
I. M.
,
2015
, “
A POD Reduced Order Model for Resolving Angular Direction in Neutron/Photon Transport Problems
,”
J. Comput. Phys.
,
296
, pp.
138
157
.10.1016/j.jcp.2015.04.043
20.
Tencer
,
J.
,
Carlberg
,
K.
,
Larsen
,
M.
, and
Hogan
,
R.
,
2017
, “
Accelerated Solution of Discrete Ordinates Approximation to the Boltzmann Transport Equation for a Gray Absorbing–Emitting Medium Via Model Reduction
,”
ASME J. Heat Transfer-Trans. ASME
, (
12
), p.
139
.10.1115/1.4037098
21.
Soucasse
,
L.
,
Buchan
,
A. G.
,
Dargaville
,
S.
, and
Pain
,
C. C.
,
2019
, “
An Angular Reduced Order Model for Radiative Transfer in Non Grey Media
,”
J. Quant. Spectrosc. Radiat. Transfer
,
229
, pp.
23
32
.10.1016/j.jqsrt.2019.03.005
22.
Yan
,
X.
,
Han
,
J.
,
Yun
,
H.
, and
Chen
,
X.
,
2020
, “
Reduced-Order Models for Radiative Heat Transfer of Hypersonic Vehicles
,”
Proc. Inst. Mech. Eng., Part G J. Aerosp. Eng.
,
234
(
11
), pp.
1836
1848
.10.1177/0954410020926730
23.
Rowley
,
C. W.
,
2005
, “
Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition
,”
Int. J. Bifurcation Chaos
,
15
(
3
), pp.
997
1013
.10.1142/S0218127405012429
24.
Sirovich
,
L.
,
1987
, “
Turbulence and the Dynamics of Coherent Structures. I. Coherent Structures
,”
Q. Appl. Math.
,
45
(
3
), pp.
561
571
.10.1090/qam/910462
25.
Park
,
H.
, and
Cho
,
D.
,
1996
, “
The Use of the Karhunen-Loève Decomposition for the Modeling of Distributed Parameter Systems
,”
Chem. Eng. Sci.
,
51
(
1
), pp.
81
98
.10.1016/0009-2509(95)00230-8
26.
Ryckelynck
,
D.
,
2009
, “
Hyper-Reduction of Mechanical Models Involving Internal Variables
,”
Int. J. Numer. Methods Eng.
,
77
(
1
), pp.
75
89
.10.1002/nme.2406
27.
Carlberg
,
K.
,
Farhat
,
C.
,
Cortial
,
J.
, and
Amsallem
,
D.
,
2013
, “
The GNAT Method for Nonlinear Model Reduction: Effective Implementation and Application to Computational Fluid Dynamics and Turbulent Flows
,”
J. Comput. Phys.
,
242
, pp.
623
647
.10.1016/j.jcp.2013.02.028
28.
Farhat
,
C.
,
Avery
,
P.
,
Chapman
,
T.
, and
Cortial
,
J.
,
2014
, “
Dimensional Reduction of Nonlinear Finite Element Dynamic Models With Finite Rotations and Energy-Based Mesh Sampling and Weighting for Computational Efficiency
,”
Int. J. Numer. Methods Eng.
,
98
(
9
), pp.
625
662
.10.1002/nme.4668
29.
Farhat
,
C.
,
Chapman
,
T.
, and
Avery
,
P.
,
2015
, “
Structure-Preserving, Stability, and Accuracy Properties of the Energy-Conserving Sampling and Weighting Method for the Hyper Reduction of Nonlinear Finite Element Dynamic Models
,”
Int. J. Numer. Methods Eng.
,
102
(
5
), pp.
1077
1110
.10.1002/nme.4820
30.
Ravichandran
,
T. K.
,
2016
, “
Investigation of Accuracy, Speed and Stability of Hyper-Reduction Techniques for Nonlinear FE
,” Masters thesis, Department of Precision and Microsystems Engineering, TU Delft, Delft, The Netherlands.
31.
Hernandez
,
J. A.
,
Caicedo
,
M. A.
, and
Ferrer
,
A.
,
2017
, “
Dimensional Hyper-Reduction of Nonlinear Finite Element Models Via Empirical Cubature
,”
Comput. Methods Appl. Mech. Eng.
,
313
, pp.
687
722
.10.1016/j.cma.2016.10.022
32.
Blonigan
,
P. J.
,
Carlberg
,
K.
,
Rizzi
,
F.
,
Howard
,
M.
, and
Fike
,
J. A.
,
2020
, “
Model Reduction for Hypersonic Aerodynamics Via Conservative LSPG Projection and Hyper-Reduction
,”
AIAA
Paper No. 2020-0104.10.2514/6.2020-0104
33.
Rother
,
S.
, and
Beitelschmidt
,
M.
,
2018
, “
Load Snapshot Decomposition to Consider Heat Radiation in Thermal Model Order Reduction
,”
IFAC-PapersOnLine
,
51
(
2
), pp.
667
672
.10.1016/j.ifacol.2018.03.113
34.
Rizzi
,
F.
,
Blonigan
,
P. J.
, and
Carlberg
,
K. T.
,
2020
, “
Pressio: Enabling Projection-Based Model Reduction for Large-Scale Nonlinear Dynamical Systems
,” preprint arxiv https://arxiv.org/abs/2003.07798.
35.
Sierra Thermal Fluid Development Team
, and
Lamb
,
J. M.
,
2019
, “
SIERRA Multimechanics Module: Aria User Manual—Version 4.52
,” Sandia National Laboratory, Albuquerque, NM, Technical Report No. SAND2019-3786.
36.
Subia
,
S. R.
,
Overfelt
,
J. R.
, and
Baur
,
D. G.
,
2017
,
SIERRA Multimechanics Module: Aria Thermal Theory Manual (Version 4.46)
, Sandia National Laboratory, Albuquerque, NM, Technical Report No. SAND-2017-10396.
37.
Swischuk
,
R.
,
Kramer
,
B.
,
Huang
,
C.
, and
Willcox
,
K.
,
2020
, “
Learning Physics-Based Reduced-Order Models for a Single-Injector Combustion Process
,”
AIAA J.
,
58
(
6
), pp.
2658
2672
.10.2514/1.J058943
38.
Carlberg
,
K.
,
Barone
,
M.
, and
Antil
,
H.
,
2017
, “
Galerkin v. least-Squares Petrov–Galerkin Projection in Nonlinear Model Reduction
,”
J. Comput. Phys.
,
330
, pp.
693
734
.10.1016/j.jcp.2016.10.033
39.
Astrid
,
P.
,
Weiland
,
S.
,
Willcox
,
K.
, and
Backx
,
T.
,
2008
, “
Missing Point Estimation in Models Described by Proper Orthogonal Decomposition
,”
IEEE Trans. Autom. Control
,
53
(
10
), pp.
2237
2251
.10.1109/TAC.2008.2006102
40.
Hinze
,
M.
, and
Volkwein
,
S.
,
2005
, “
Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control
,”
Dimension Reduction of Large-Scale Systems
,
Springer
, Berlin, Heidelberg, pp.
261
306
.
41.
Rathinam
,
M.
, and
Petzold
,
L. R.
,
2003
, “
A New Look at Proper Orthogonal Decomposition
,”
SIAM J. Numer. Anal.
,
41
(
5
), pp.
1893
1925
.10.1137/S0036142901389049
42.
Drohmann
,
M.
,
Haasdonk
,
B.
, and
Ohlberger
,
M.
,
2012
, “
Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations Based on Empirical Operator Interpolation
,”
SIAM J. Sci. Comput.
,
34
(
2
), pp.
A937
A969
.10.1137/10081157X
43.
Drohmann
,
M.
, and
Carlberg
,
K.
,
2015
, “
The ROMES Method for Statistical Modeling of Reduced-Order-Model Error
,”
SIAM/ASA J. Uncertainty Quantif.
,
3
(
1
), pp.
116
145
.10.1137/140969841
44.
Pagani
,
S.
,
Manzoni
,
A.
, and
Carlberg
,
K. T.
,
2022
, “
Statistical Closure Modeling for Reduced-Order Models of Stationary Systems by the ROMES Method
,” Int. J. Uncertainty Quantification, 12(1).
45.
Freno
,
B. A.
, and
Carlberg
,
K. T.
,
2019
, “
Machine-Learning Error Models for Approximate Solutions to Parameterized Systems of Nonlinear Equations
,”
Comput. Methods Appl. Mech. Eng.
,
348
, pp.
250
296
.10.1016/j.cma.2019.01.024
46.
Parish
,
E. J.
, and
Carlberg
,
K. T.
,
2020
, “
Time-Series Machine-Learning Error Models for Approximate Solutions to Parameterized Dynamical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
365
, p.
112990
.10.1016/j.cma.2020.112990
47.
Berry
,
M. W.
,
Pulatova
,
S. A.
, and
Stewart
,
G.
,
2005
, “
Algorithm 844: Computing Sparse Reduced-Rank Approximations to Sparse Matrices
,”
ACM Trans. Math. Software (TOMS)
,
31
(
2
), pp.
252
269
.10.1145/1067967.1067972
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