We explore the use of generalized polynomial chaos (GPC) expansion with stochastic collocation (SC) for modeling the uncertainty in the noise radiated by a plate subject to turbulent boundary layer (TBL) forcing. The SC form of polynomial chaos permits re-use of existing computational models, while drastically reducing the number of evaluations of the deterministic code compared to Monte Carlo (MC) sampling, for instance. Further efficiency is attained through the application of new, efficient, quadrature rules to compute the GPC expansion coefficients. We demonstrate that our approach accurately reconstructs the statistics of the radiated sound power by propagating the input uncertainty through the computational physics model. The use of optimized quadrature rules permits these results to be obtained using far fewer quadrature nodes than with traditional methods, such as tensor product quadrature and Smolyak sparse grid methods. As each quadrature node corresponds to an expensive deterministic model evaluation, the computational cost of the analysis is seen to be greatly reduced.

References

References
1.
Lighthill
,
M. J.
,
1954
, “
On Sound Generated Aerodynamically—II: Turbulence as a Source of Sound
,”
Proc. R. Soc. London A
,
222
(
1148
), pp.
1
32
.
2.
Bull
,
M.
,
1996
, “
Wall Pressure Fluctuations Beneath Turbulent Boundary Layers: Some Reflections on Forty Years of Research
,”
J. Sound Vib.
,
190
(
3
), pp.
299
315
.
3.
Tennekes
,
H.
, and
Lumley
,
J.
,
1972
,
A First Course in Turbulence
,
Massachusetts Institute of Technology Press
, Cambridge, MA.
4.
Peltier
,
L.
, and
Hambric
,
S.
,
2007
, “
Estimating Turbulent-Boundary-Layer Wall-Pressure Spectra From CFD RANS Solutions
,”
J. Fluids Struct.
,
23
(
6
), pp.
920
937
.
5.
Bonness
,
W. K.
,
Fahnline
,
J. B.
,
Lysak
,
P. D.
, and
Shepherd
,
M. R.
,
2017
, “
Modal Forcing Functions for Structural Vibration From Turbulent Boundary Layer Flow
,”
J. Sound Vib.
,
395
, pp.
224
239
.
6.
Fishman
,
G.
,
2013
,
Monte Carlo: Concepts, Algorithms, and Applications
,
Springer Science & Business Media
, New York.
7.
Xiu
,
D.
,
2010
,
Numerical Methods for Stochastic Computations: A Spectral Method Approach
,
Princeton University Press
,
Princeton, NJ
.
8.
Wan
,
H.-P.
,
Mao
,
Z.
,
Todd
,
M. D.
, and
Ren
,
W.-X.
,
2014
, “
Analytical Uncertainty Quantification for Modal Frequencies With Structural Parameter Uncertainty Using a Gaussian Process Metamodel
,”
Eng. Struct.
,
75
, pp.
577
589
.
9.
Zhao
,
Y.-G.
, and
Ono
,
T.
,
2001
, “
Moment Methods for Structural Reliability
,”
Struct. Saf.
,
23
(
1
), pp.
47
75
.
10.
Xiu
,
D.
, and
Karniadakis
,
G. E.
,
2002
, “
The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,”
SIAM J. Sci. Comput.
,
24
(
2
), pp.
619
644
.
11.
Sepahvand
,
K.
,
Marburg
,
S.
, and
Hardtke
,
H.-J.
,
2007
, “
Numerical Solution of One-Dimensional Wave Equation With Stochastic Parameters Using Generalized Polynomial Chaos Expansion
,”
J. Comput. Acoust.
,
15
(
4
), pp.
579
593
.
12.
Ghanem
,
R.
, and
Spanos
,
P. D.
,
1993
, “
A Stochastic Galerkin Expansion for Nonlinear Random Vibration Analysis
,”
Probab. Eng. Mech.
,
8
(
3–4
), pp.
255
264
.
13.
Sepahvand
,
K.
,
2017
, “
Stochastic Finite Element Method for Random Harmonic Analysis of Composite Plates With Uncertain Modal Damping Parameters
,”
J. Sound Vib.
,
400
, pp.
1
12
.
14.
Xiu
,
D.
, and
Karniadakis
,
G. E.
,
2003
, “
Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos
,”
J. Comput. Phys.
,
187
(
1
), pp.
137
167
.
15.
Zhang
,
D.
, and
Lu
,
Z.
,
2004
, “
An Efficient, High-Order Perturbation Approach for Flow in Random Porous Media Via Karhunen–Loève and Polynomial Expansions
,”
J. Comput. Phys.
,
194
(
2
), pp.
773
794
.
16.
Rupert
,
C. P.
, and
Miller
,
C. T.
,
2007
, “
An Analysis of Polynomial Chaos Approximations for Modeling Single-Fluid-Phase Flow in Porous Medium Systems
,”
J. Comput. Phys.
,
226
(
2
), pp.
2175
2205
.
17.
DeGennaro
,
A. M.
,
Rowley
,
C. W.
, and
Martinelli
,
L.
,
2015
, “
Uncertainty Quantification for Airfoil Icing Using Polynomial Chaos Expansions
,”
J. Aircr.
,
52
(
5
), pp.
1404
1411
.
18.
Zhang
,
L.
,
Cui
,
T.
, and
Liu
,
H.
,
2009
, “
A Set of Symmetric Quadrature Rules on Triangles and Tetrahedra
,”
J. Comput. Math.
,
27
(1), pp.
89
96
.https://www.jstor.org/stable/43693493?seq=1#page_scan_tab_contents
19.
Witherden
,
F. D.
, and
Vincent
,
P. E.
,
2015
, “
On the Identification of Symmetric Quadrature Rules for Finite Element Methods
,”
Comput. Math. Appl.
,
69
(
10
), pp.
1232
1241
.
20.
Witherden, F. D.
, and
Vincent, P. E.
,
2015
, “
On the Development and Implementation of High-Order Flux Reconstruction Schemes for Computational Fluid Dynamics
,”
Ph.D. thesis
, Imperial College London, London.https://www.imperial.ac.uk/aeronautics/research/vincentlab/theses/witherden-thesis.pdf
21.
Papanicolopulos
,
S.-A.
,
2016
, “
Efficient Computation of Cubature Rules With Application to New Asymmetric Rules on the Triangle
,”
J. Comput. Appl. Math.
,
304
, pp.
73
83
.
22.
Papanicolopulos
,
S.-A.
,
2016
, “
New Fully Symmetric and Rotationally Symmetric Cubature Rules on the Triangle Using Minimal Orthonormal Bases
,”
J. Comput. Appl. Math.
,
294
, pp.
39
48
.
23.
Gratiet
,
L. L.
,
Marelli
,
S.
, and
Sudret
,
B.
,
2016
, “
Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes
,”
Handbook of Uncertainty Quantification
, Springer, Cham, Switzerland, pp.
1
37
.
24.
Garcia-Cabrejo
,
O.
, and
Valocchi
,
A.
,
2014
, “
Global Sensitivity Analysis for Multivariate Output Using Polynomial Chaos Expansion
,”
Reliab. Eng. Syst. Saf.
,
126
, pp.
25
36
.
25.
Hambric
,
S. A.
,
Hwang
,
Y. F.
, and
Bonness
,
W. K.
,
2002
, “
Vibrations of Plates With Clamped and Free Edges Excited by Highly Subsonic Turbulent Boundary Layer Flow
,”
ASME
Paper No. IMECE2002-32224.
26.
Hambric
,
S.
,
Hwang
,
Y.
, and
Bonness
,
W.
,
2004
, “
Vibrations of Plates With Clamped and Free Edges Excited by Low-Speed Turbulent Boundary Layer Flow
,”
J. Fluids Struct.
,
19
(
1
), pp.
93
110
.
27.
Chandiramani
,
K.
,
1977
, “
Vibration Response of Fluid-Loaded Structures to Low-Speed Flow Noise
,”
J. Acoust. Soc. Am.
,
61
(
6
), pp.
1460
1470
.
28.
Corcos
,
G. M.
,
1963
, “
Resolution of Pressure in Turbulence
,”
J. Acoust. Soc. Am.
,
35
(
2
), pp.
192
199
.
29.
Hwang
,
Y.
,
1998
, “
A Discrete Model of Turbulence Loading Function for Computation of Flow-Induced Vibration and Noise
,” American Society of Mechanical Engineers, New York.
30.
Hwang
,
Y.
, and
Maidanik
,
G.
,
1990
, “
A Wavenumber Analysis of the Coupling of a Structural Mode and Flow Turbulence
,”
J. Sound Vib.
,
142
(
1
), pp.
135
152
.
31.
Mellen
,
R. H.
,
1990
, “
On Modeling Convective Turbulence
,”
J. Acoust. Soc. Am.
,
88
(
6
), pp.
2891
2893
.
32.
Hambric
,
S. A.
,
Boger
,
D. A.
,
Fahnline
,
J. B.
, and
Campbell
,
R. L.
,
2010
, “
Structure- and Fluid-Borne Acoustic Power Sources Induced by Turbulent Flow in 90° Piping Elbows
,”
J. Fluids Struct.
,
26
(
1
), pp.
121
147
.
33.
Fahnline
,
J. B.
, and
Koopmann
,
G. H.
,
1996
, “
A Lumped Paramter Model for the Acoustic Power Output From a Vibrating Structure
,”
J. Acoust. Soc. Am.
,
100
(
6
), pp.
3539
3547
.
34.
Fahnline
,
J. B.
,
2016
, “
Boundary-Element Analysis
,”
Engineering Vibroacoustic Analysis: Methods and Applications
, Wiley, Chichester, UK, pp. 179–229.
35.
Chase
,
D.
,
1980
, “
Modeling the Wavevector–Frequency Spectrum of Turbulent Boundary Layer Wall Pressure
,”
J. Sound Vib.
,
70
(
1
), pp.
29
67
.
36.
Howe
,
M. S.
,
1998
,
Acoustics of Fluid-Structure Interactions
,
Cambridge University Press
, Cambridge, UK.
37.
Lysak
,
P. D.
,
2006
, “
Modeling the Wall Pressure Spectrum in Turbulent Pipe Flows
,”
ASME J. Fluids Eng.
,
128
(
2
), pp.
216
222
.
38.
Smol'yakov
,
A. V.
,
2006
, “
A New Model for the Cross Spectrum and Wavenumber-Frequency Spectrum of Turbulent Pressure Fluctuation in a Boundary Layer
,”
Acoust. Phys.
,
52
(
3
), pp.
331
337
.
39.
Hildebrand
,
F. B.
,
1987
,
Introduction to Numerical Analysis
, Dover, Mineola, NY.
40.
Cools
,
R.
,
1997
, “
Constructing Cubature Formulae: The Science Behind the Art
,”
Acta Numer.
,
6
, pp.
1
54
.
41.
Smolyak
,
S.
,
1963
, “
Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions
,”
Doklady Akademii Nauk
,
148
(5), pp. 1042–1045.http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=27586&option_lang=eng
42.
Zenger
,
C.
,
1990
,
Sparse Grids
,
Technische Universität
, Berlin.
43.
Novak
,
E.
, and
Ritter
,
K.
,
1996
, “
High Dimensional Integration of Smooth Functions Over Cubes
,”
Numer. Math.
,
75
(
1
), pp.
79
97
.
44.
Novak
,
E.
, and
Ritter
,
K.
,
1997
, “
The Curse of Dimension and a Universal Method for Numerical Integration
,”
Multivariate Approximation and Splines
,
Springer
, Basel, Switzerland, pp.
177
187
.
45.
Gerstner
,
T.
, and
Griebel
,
M.
,
1998
, “
Numerical Integration Using Sparse Grids
,”
Numer. Algorithms
,
18
(
3/4
), p.
209
.
46.
Bungartz
,
H.-J.
, and
Griebel
,
M.
,
2004
, “
Sparse Grids
,”
Acta Numer.
,
13
, pp.
147
269
.
47.
Garcke
,
J.
,
2012
, “
Sparse Grids in a Nutshell
,”
Sparse Grids and Applications
,
Springer
, Berlin, pp.
57
80
.
48.
Zhang
,
Z.
, and
Karniadakis
,
G.
,
2017
,
Numerical Methods for Stochastic Partial Differential Equations With White Noise
,
Springer
, Cham, Switzerland.
49.
Eldred
,
M.
,
2009
, “
Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design
,”
AIAA
Paper No. 2009-2274
.
50.
Nobile
,
F.
,
Tempone
,
R.
, and
Webster
,
C. G.
,
2008
, “
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations With Random Input Data
,”
SIAM J. Numer. Anal.
,
46
(
5
), pp.
2309
2345
.
51.
Cools
,
R.
, and
Kim
,
K. J.
,
2001
, “
Rotation Invariant Cubature Formulas Over the n-Dimensional Unit Cube
,”
J. Comput. Appl. Math.
,
132
(
1
), pp.
15
32
.
52.
Mantel
,
F.
, and
Rabinowitz
,
P.
,
1977
, “
The Application of Integer Programming to the Computation of Fully Symmetric Integration Formulas in Two and Three Dimensions
,”
SIAM J. Numer. Anal.
,
14
(
3
), pp.
391
425
.
53.
Espelid
,
T. O.
,
1987
, “
On the Construction of Good Fully Symmetric Integration Rules
,”
SIAM J. Numer. Anal.
,
24
(
4
), pp.
855
881
.
54.
Möller
,
H. M.
,
1979
, “
Lower Bounds for the Number of Nodes in Cubature Formulae
,”
Numerische Integration
,
Springer
, Basel, Switzerland, pp.
221
230
.
55.
Burkardt
,
J.
, and
Webster
,
C.
,
2007
, “
A Low Level Introduction to High Dimensional Sparse Grids
,” Sandia National Laboratories, Albuquerque, NM, accessed May 3, 2018, http://people.sc.fsu.edu/~jburkardt/presentations/sandia_2007.pdf
56.
Maestrello
,
L.
,
1965
, “
Measurement of Noise Radiated by Boundary Layer Excited Panels
,”
J. Sound Vib.
,
2
(
2
), pp.
100
115
.
57.
Wilby
,
J. F.
,
1967
, “
The Response of Simple Panels to Turbulent Boundary Layer Excitation
,” Air Force Flight Dynamics Laboratory, Technical Report No.
AFFDL-TR-67-70
.https://apps.dtic.mil/docs/citations/AD0824482
58.
Sobol
,
I. M.
,
1993
, “
Sensitivity Estimates for Nonlinear Mathematical Models
,”
Math. Modell. Comput. Exp.
,
1
(
4
), pp.
407
414
.http://max2.ese.u-psud.fr/epc/conservation/MODE/Sobol%20Original%20Paper.pdf
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