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.

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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



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