Abstract

Gaussian random field has been widely applied to quantify high-dimensional uncertainties in the spatial or temporal domain. A common practice in Gaussian random field modeling is to use the exponential function to represent the covariance matrix. However, the exponential function-based covariance formulation does not allow negative values, thus it cannot capture the negative correlation between two locations in the input domain. To resolve this issue, this work reports new formulations of the covariance matrix based on oscillating functions, and a process of reconstructing Gaussian random field models from observation data. The proposed covariance functions are compared with the traditional exponential covariance functions on two test cases, where the datasets show negative correlations. The results of comparative studies demonstrate that the proposed formulations improve the accuracy of Gaussian random field models effectively.

References

References
1.
Greene
,
M. S.
,
Xu
,
H.
,
Tang
,
S.
,
Chen
,
W.
, and
Liu
,
W. K.
,
2013
, “
A Generalized Uncertainty Propagation Criterion From Benchmark Studies of Microstructured Material Systems
,”
Comput. Methods Appl. Mech. Eng.
,
254
, p.
291
. 10.1016/j.cma.2012.10.023
2.
Bostanabad
,
R.
,
Zhang
,
Y.
,
Li
,
X.
,
Kearney
,
T.
,
Brinson
,
L. C.
,
Apley
,
D. W.
,
Liu
,
W. K.
, and
Chen
,
W.
,
2018
, “
Computational Microstructure Characterization and Reconstruction: Review of the State-of-the-art Techniques
,”
Prog. Mater. Sci.
,
95
, pp.
1
41
. 10.1016/j.pmatsci.2018.01.005
3.
Crevillen-Garcia
,
D.
,
Wilkinson
,
R.
,
Shah
,
A.
, and
Power
,
H.
,
2017
, “
Gaussian Process Modelling for Uncertainty Quantification in Convectively-Enhanced Dissolution Processes in Porous Media
,”
Adv. Water Resour.
,
99
, pp.
1
14
. 10.1016/j.advwatres.2016.11.006
4.
Wang
,
Y.
,
Lava
,
P.
,
Reu
,
P.
, and
Debruyne
,
D.
,
2016
, “
Theoretical Analysis on the Measurement Errors of Local 2D DIC: Part I Temporal and Spatial Uncertainty Quantification of Displacement Measurements
,”
Strain
,
52
(
2
), pp.
110
128
. 10.1111/str.12173
5.
Ostoja-Starzewski
,
M.
,
1998
, “
Random Field Models of Heterogeneous Materials
,”
Int. J. Solids Struct.
,
35
(
19
), pp.
2429
2455
. 10.1016/S0020-7683(97)00144-3
6.
Greene
,
M. S.
,
Liu
,
Y.
,
Chen
,
W.
, and
Liu
,
W. K.
,
2011
, “
Computational Uncertainty Analysis in Multiresolution Materials Via Stochastic Constitutive Theory
,”
Comput. Methods Appl. Mech. Eng.
,
200
(
1–4
), pp.
309
325
. 10.1016/j.cma.2010.08.013
7.
Xu
,
H.
,
Jiang
,
Z.
,
Apley
,
D. W.
, and
Chen
,
W.
,
2016
, “
New Metrics for Validation of Data-Driven Random Process Models in Uncertainty Quantification
,”
J. Verification, Validation Uncertainty Quantif.
,
1
(
2
), p.
021002
. 10.1115/1.4031813
8.
Xi
,
Z.
,
Youn
,
B. D.
,
Jung
,
B. C.
, and
Yoon
,
J. T.
,
2015
, “
Random Field Modeling With Insufficient Field Data for Probability Analysis and Design
,”
Struct. Multidiscip. Optim.
,
51
(
3
), pp.
599
611
. 10.1007/s00158-014-1165-0
9.
Bostanabad
,
R.
,
Kearney
,
T.
,
Tao
,
S.
,
Apley
,
D. W.
, and
Chen
,
W.
,
2018
, “
Leveraging the Nugget Parameter for Efficient Gaussian Process Modeling
,”
Int. J. Numerical Methods Eng.
,
114
(
5
), pp.
501
516
. 10.1002/nme.5751
10.
Zhang
,
Y.
,
Tao
,
S.
,
Chen
,
W.
, and
Apley
,
D. W.
,
2019
, “
A Latent Variable Approach to Gaussian Process Modeling With Qualitative and Quantitative Factors
,”
Technometrics
, pp.
1
12
.
11.
Franklin
,
J. N.
,
1965
, “
Numerical Simulation of Stationary and Non-Stationary Gaussian Random Processes
,”
SIAM Rev.
,
7
(
1
), pp.
68
80
. 10.1137/1007007
12.
Kramer
,
P. R.
,
Kurbanmuradov
,
O.
, and
Sabelfeld
,
K.
,
2007
, “
Comparative Analysis of Multiscale Gaussian Random Field Simulation Algorithms
,”
J. Comput. Phys.
,
226
(
1
), pp.
897
924
. 10.1016/j.jcp.2007.05.002
13.
Xu
,
H.
,
Li
,
Y.
,
Brinson
,
C.
, and
Chen
,
W.
,
2014
, “
A Descriptor-Based Design Methodology for Developing Heterogeneous Microstructural Materials System
,”
ASME J. Mech. Des.
,
136
(
5
), p.
051007
. 10.1115/1.4026649
14.
Mathelin
,
L.
,
Hussaini
,
M. Y.
, and
Zang
,
T. A.
,
2005
, “
Stochastic Approaches to Uncertainty Quantification in CFD Simulations
,”
Numer. Algorithms
,
38
(
1–3
), pp.
209
236
. 10.1007/s11075-004-2866-z
15.
Lu
,
J.
,
Zhan
,
Z.
,
Apley
,
D. W.
, and
Chen
,
W.
,
2019
, “
Uncertainty Propagation of Frequency Response Functions Using a Multi-Output Gaussian Process Model
,”
Comput. Structures
,
217
, pp.
1
17
. 10.1016/j.compstruc.2019.03.009
16.
Oliver
,
T. A.
, and
Moser
,
R. D.
,
2011
, “
Bayesian Uncertainty Quantification Applied to RANS Turbulence Models
,”
J. Phys.: Conf. Ser.
,
318
(
4
), p.
042032
.
17.
Zhu
,
H.
, and
Zhang
,
L.
,
2013
, “
Characterizing Geotechnical Anisotropic Spatial Variations Using Random Field Theory
,”
Can. Geotech. J.
,
50
(
7
), pp.
723
734
. 10.1139/cgj-2012-0345
18.
Azzimonti
,
D.
,
Bect
,
J.
,
Chevalier
,
C.
, and
Ginsbourger
,
D.
,
2016
, “
Quantifying Uncertainties on Excursion Sets Under a Gaussian Random Field Prior
,”
SIAM/ASA J. Uncertainty Quantif.
,
4
(
1
), pp.
850
874
. 10.1137/141000749
19.
Williams
,
C. K.
, and
Rasmussen
,
C. E.
,
2006
,
Gaussian Processes for Machine Learning
,
MIT Press
,
Cambridge, MA
.
20.
Chan
,
G.
,
1997
, “
Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields
,”
J. Royal Stat. Soc. Series C (Appl. Statist.)
,
46
(
1
), pp.
171
181
. 10.1111/1467-9876.00057
21.
Djian
,
D.
,
Alloin
,
F.
,
Martinet
,
S.
,
Lignier
,
H.
, and
Sanchez
,
J.-Y.
,
2007
, “
Lithium-Ion Batteries With High Charge Rate Capacity: Influence of the Porous Separator
,”
J. Power Sources
,
172
(
1
), pp.
416
421
. 10.1016/j.jpowsour.2007.07.018
22.
Xu
,
H.
,
Zhu
,
M.
,
Marcicki
,
J.
, and
Yang
,
X. G.
,
2017
, “
Mechanical Modeling of Battery Separator Based on Microstructure Image Analysis and Stochastic Characterization
,”
J. Power Sources
,
345
, pp.
137
145
. 10.1016/j.jpowsour.2017.02.002
23.
Xu
,
H.
, and
Bae
,
C.
,
2019
, “
Stochastic 3D Microstructure Reconstruction and Mechanical Modeling of Anisotropic Battery Separators
,”
J. Power Sources
,
430
, pp.
67
73
. 10.1016/j.jpowsour.2019.05.021
24.
Zhu
,
J.
,
Zhang
,
X.
,
Luo
,
H.
, and
Sahraei
,
E.
,
2018
, “
Investigation of the Deformation Mechanisms of Lithium-Ion Battery Components Using in-Situ Micro Tests
,”
Appl. Energy
,
224
, pp.
251
266
. 10.1016/j.apenergy.2018.05.007
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