0
Research Papers

Generalized Polynomial Chaos With Optimized Quadrature Applied to a Turbulent Boundary Layer Forced Plate

[+] Author and Article Information
Andrew S. Wixom

Applied Research Laboratory,
Pennsylvania State University,
State College, PA 16804
e-mail: axw274@psu.edu

Gage S. Walters, Sheri L. Martinelli

Applied Research Laboratory,
Pennsylvania State University,
State College, PA 16804

David M. Williams

Department of Mechanical Engineering,
Pennsylvania State University,
State College, PA 16804

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 31, 2018; final manuscript received October 10, 2018; published online January 7, 2019. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 14(2), 021010 (Jan 07, 2019) (9 pages) Paper No: CND-18-1243; doi: 10.1115/1.4041772 History: Received May 31, 2018; Revised October 10, 2018

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

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. [CrossRef]
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. [CrossRef]
Tennekes, H. , and Lumley, J. , 1972, A First Course in Turbulence, Massachusetts Institute of Technology Press, Cambridge, MA.
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. [CrossRef]
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. [CrossRef]
Fishman, G. , 2013, Monte Carlo: Concepts, Algorithms, and Applications, Springer Science & Business Media, New York.
Xiu, D. , 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ.
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. [CrossRef]
Zhao, Y.-G. , and Ono, T. , 2001, “ Moment Methods for Structural Reliability,” Struct. Saf., 23(1), pp. 47–75. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Xiu, D. , and Karniadakis, G. E. , 2003, “ Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys., 187(1), pp. 137–167. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
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
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. [CrossRef]
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
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
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.
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. [CrossRef]
Chandiramani, K. , 1977, “ Vibration Response of Fluid-Loaded Structures to Low-Speed Flow Noise,” J. Acoust. Soc. Am., 61(6), pp. 1460–1470. [CrossRef]
Corcos, G. M. , 1963, “ Resolution of Pressure in Turbulence,” J. Acoust. Soc. Am., 35(2), pp. 192–199. [CrossRef]
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.
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. [CrossRef]
Mellen, R. H. , 1990, “ On Modeling Convective Turbulence,” J. Acoust. Soc. Am., 88(6), pp. 2891–2893. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Fahnline, J. B. , 2016, “ Boundary-Element Analysis,” Engineering Vibroacoustic Analysis: Methods and Applications, Wiley, Chichester, UK, pp. 179–229.
Chase, D. , 1980, “ Modeling the Wavevector–Frequency Spectrum of Turbulent Boundary Layer Wall Pressure,” J. Sound Vib., 70(1), pp. 29–67. [CrossRef]
Howe, M. S. , 1998, Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, UK.
Lysak, P. D. , 2006, “ Modeling the Wall Pressure Spectrum in Turbulent Pipe Flows,” ASME J. Fluids Eng., 128(2), pp. 216–222. [CrossRef]
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. [CrossRef]
Hildebrand, F. B. , 1987, Introduction to Numerical Analysis, Dover, Mineola, NY.
Cools, R. , 1997, “ Constructing Cubature Formulae: The Science Behind the Art,” Acta Numer., 6, pp. 1–54. [CrossRef]
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
Zenger, C. , 1990, Sparse Grids, Technische Universität, Berlin.
Novak, E. , and Ritter, K. , 1996, “ High Dimensional Integration of Smooth Functions Over Cubes,” Numer. Math., 75(1), pp. 79–97. [CrossRef]
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.
Gerstner, T. , and Griebel, M. , 1998, “ Numerical Integration Using Sparse Grids,” Numer. Algorithms, 18(3/4), p. 209. [CrossRef]
Bungartz, H.-J. , and Griebel, M. , 2004, “ Sparse Grids,” Acta Numer., 13, pp. 147–269. [CrossRef]
Garcke, J. , 2012, “ Sparse Grids in a Nutshell,” Sparse Grids and Applications, Springer, Berlin, pp. 57–80.
Zhang, Z. , and Karniadakis, G. , 2017, Numerical Methods for Stochastic Partial Differential Equations With White Noise, Springer, Cham, Switzerland.
Eldred, M. , 2009, “ Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design,” AIAA Paper No. 2009-2274.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Espelid, T. O. , 1987, “ On the Construction of Good Fully Symmetric Integration Rules,” SIAM J. Numer. Anal., 24(4), pp. 855–881. [CrossRef]
Möller, H. M. , 1979, “ Lower Bounds for the Number of Nodes in Cubature Formulae,” Numerische Integration, Springer, Basel, Switzerland, pp. 221–230.
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
Maestrello, L. , 1965, “ Measurement of Noise Radiated by Boundary Layer Excited Panels,” J. Sound Vib., 2(2), pp. 100–115. [CrossRef]
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
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

Figures

Grahic Jump Location
Fig. 1

Projected images of the abscissas for candidate quadrature rules in dimension D = 4 and with degree ϕ = 7: (a) FSG—57 points, (b) GLS—201 points, (c) CCS—137 points, and (d) tensor product—256 points

Grahic Jump Location
Fig. 2

Convergence of radiated sound power variance as the number of 1D basis functions is increased from two to ten. The colors are from light to dark corresponding to increasing number of basis functions. The highlighed lines in both plots show when the variance is converged to O(10−4). The vertical dashed lines represent the natural frequencies of the plate. In the second plot, the vertical dashed lines are associated with a mean value of E. The relative errors are computed between each number of basis function with coefficients obtained using tensor product quadrature in increasing order (e.g., difference between 2 and 3 basis functions, 3 and 4 basis functions, etc.). (a) Flow parameters only (αβδ*) and (b) all four uncertain variables (αβδ*, E).

Grahic Jump Location
Fig. 3

Convergence of radiated sound power statistics for both flow parameters (α, β, δ*) and flow plus structural parameters (α, β, δ*, E). Both mean and variance convergence are plotted.

Grahic Jump Location
Fig. 4

Radiated sound power output using seven 1D basis functions and coefficient computed by three different quadrature rules: (a) tensor product with 2401 points, (b) GLS with 5257 points, and (c) FSG with 720 points

Grahic Jump Location
Fig. 5

Radiated sound power in one-third octave bands using seven 1D basis functions: (a) comparison of results generated in this study to those experimentally measured by Maestrello [56] and (b) possibility region plotted with the mean set to zero

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In