Abstract

This paper presents a novel methodology to solve an inverse uncertainty quantification problem where only the variation of the system response is provided by a small set of experimental data. Furthermore, the method is extended for cases where the uncertainty of the response quantities is given by an incomplete set of statistical moments. For both cases, the uncertainty on the output space is represented by a minimum volume enclosing ellipsoid (MVEE). The actual inverse uncertainty quantification is conducted by identifying also a hyper-ellipsoid for the input parameters, which has an image on the output space that matches the MVEE as close as possible. Hence, the newly introduced approach is a contribution to the field of nonprobabilistic uncertainty quantification methods. Compared to literature, the new approach has often superior accuracy and especially an improved efficiency for high-dimensional problems. The method is validated first by an analytical test case and subsequently applied to a jet engine performance model, where this type of inverse uncertainty quantification has to be solved to allow for a consistent and integrated solution procedure. In both cases, the results are compared with an inverse method where the variability on the input side is quantified by a multidimensional interval. It can be shown that the hyper-ellipsoid approach is superior with respect to the computation time in high-dimensional problems encountered not only in jet engine design.

References

1.
Der Kiureghian
,
A.
, and
Ditlevsen
,
O.
,
2009
, “
Aleatory or Epistemic? Does It Matter?
,”
Struct. Saf.
,
31
(
2
)03, pp.
105
112
.10.1016/j.strusafe.2008.06.020
2.
Beck
,
J. L.
, and
Katafygiotis
,
L. S.
,
1998
, “
Updating Models and Their Uncertainties. i: Bayesian Statistical Framework
,”
J. Eng. Mech.
,
124
(
4
), pp.
455
461
.10.1061/(ASCE)0733-9399(1998)124:4(455)
3.
Moens
,
D.
, and
Hanss
,
M.
,
2011
, “
Non-Probabilistic Finite Element Analysis for Parametric Uncertainty Treatment in Applied Mechanics: Recent Advances
,”
Finite Elem. Anal. Des.
,
47
(
1
), pp.
4
16
.10.1016/j.finel.2010.07.010
4.
Fedele
,
F.
,
Muhanna
,
R. L.
,
Xiao
,
N.
, and
Mullen
,
R. L.
,
2015
, “
Interval-Based Approach for Uncertainty Propagation in Inverse Problems
,”
J. Eng. Mech.
,
141
(
1
), p.
06014013
.10.1061/(ASCE)EM.1943-7889.0000815
5.
Haag
,
T.
, and
Hanss
,
M.
,
2010
, “
Model Assessment Using Inverse Fuzzy Arithmetic
,”
International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems
,
Springer
, Dortmund, Germany, June 28–July 2, pp.
461
470
.
6.
Faes
,
M.
, and
Moens
,
D.
,
2020
, “
Recent Trends in the Modeling and Quantification of Non-Probabilistic Uncertainty
,”
Arch. Comput. Methods Eng.
,
27
(
3
), pp.
633
639
.10.1007/s11831-019-09327-x
7.
Faes
,
M.
,
Cerneels
,
J.
,
Vandepitte
,
D.
, and
Moens
,
D.
,
2017
, “
Identification and Quantification of Multivariate Interval Uncertainty in Finite Element Models
,”
Comput. Methods Appl. Mech. Eng.
,
315
, pp.
896
920
.10.1016/j.cma.2016.11.023
8.
Faes
,
M.
,
Broggi
,
M.
,
Patelli
,
E.
,
Govers
,
Y.
,
Mottershead
,
J.
,
Beer
,
M.
, and
Moens
,
D.
,
2019
, “
A Multivariate Interval Approach for Inverse Uncertainty Quantification With Limited Experimental Data
,”
Mech. Syst. Signal Process.
,
118
, pp.
534
548
.10.1016/j.ymssp.2018.08.050
9.
Faes
,
M.
,
Broggi
,
M.
,
Patelli
,
E.
,
Govers
,
Y.
,
Mottershead
,
J.
,
Beer
,
M.
, and
Moens
,
D.
,
2019
, “
Inverse Quantification of Epistemic Uncertainty Under Scarce Data: Bayesian or Interval Approach?
,” 13th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP13),
Seoul, South Korea
, May 26–30, Paper No.
71433
.https://www.researchgate.net/publication/333039280_Inverse_quantification_of_epistemic_uncertainty_under_scarce_data_Bayesian_or_Interval_approach
10.
Daub
,
M.
, and
Duddeck
,
F.
,
2019
, “
Maximizing Flexibility for Complex Systems Design to Compensate Lack-of-Knowledge Uncertainty
,”
ASCE-ASME J. Risk Uncertainty Eng. Syst., Part B Mech. Eng.
,
5
(
4
), p.
041008
.10.1115/1.4044045
11.
Daub
,
M.
, and
Duddeck
,
F.
,
2020
, “
A Decoupled Design Approach for Complex Systems Under Lack-of-Knowledge Uncertainty
,”
Int. J. Approx. Reason.
,
119
, pp.
408
420
.10.1016/j.ijar.2020.01.006
12.
Elishakoff
,
I.
, and
Sarlin
,
N.
,
2016
, “
Uncertainty Quantification Based on Pillars of Experiment, Theory, and Computation. Part II: Theory and Computation
,”
Mech. Syst. Signal Process.
,
74
, pp.
54
72
.10.1016/j.ymssp.2015.04.036
13.
Boyd
,
S.
, and
Vandenberghe
,
L.
,
2004
,
Convex Optimization
,
Cambridge University Press
,
New York
.
14.
Chen
,
X.
,
2007
, “
A New Generalization of Chebyshev Inequality for Random Vectors
,” arXiv:0707.0805.
15.
ARP,
2011
, “
Aircraft Propulsion System Performance Station Designation and Nomenclature
,”
SAE
Paper No.
775A
.10.4271/775A
16.
Cohen
,
H.
, and
Saravanamuttoo
,
H. I. H.
,
1987
,
Gas Turbine Theory
,
Longman Sc & Tech
,
New York
.
17.
Barber
,
C. B.
,
Dobkin
,
D. P.
, and
Huhdanpaa
,
H.
,
1996
, “
The Quickhull Algorithm for Convex Hulls
,”
ACM Trans. Math. Softw.
,
22
(
4
), pp.
469
483
.10.1145/235815.235821
18.
Khachiyan
,
L. G.
,
1996
, “
Rounding of Polytopes in the Real Number Model of Computation
,”
Math. Oper. Res.
,
21
(
2
), pp.
307
320
.10.1287/moor.21.2.307
You do not currently have access to this content.