A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations

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1 An Extended Abstract submitted for the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada January 26 Preferred Session Topic: Uncertainty quantification and stochastic methods for CFD A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations S. Hosder, R. Perez, and R. W. Walters Department of Aerospace and Ocean Engineering Virginia Tech, Blacksburg, VA th June 25 1 Introduction In this extended abstract, we present a Non-Intrusive Polynomial Chaos (PC) method for the propagation of input uncertainty in Computational Fluid Dynamics (CFD) simulations. By the non-intrusive term, we specify a method which does not modify the original deterministic code used in the simulations. In our proposed paper, we focus on investigating such a method which uses deterministic solutions in a stochastic model to simulate the propagation of the input uncertainties for obtaining various statistics of output variables. In a previous study [1], an intrusive PC formulation was implemented to a deterministic Euler code. This intrusive method involves substituting the PC expansions into fluxes and Jacobians, and projecting them onto a random basis function in the form of Hermite Polynomials. Despite its power in uncertainty propagation modeling, this intrusive method can be relatively difficult, expensive, and time consuming to implement to complex problems such as the full Navier-Stokes simulation of 3-D, viscous, turbulent flows around realistic aerospace vehicles or multi-system level simulations which include the interaction of many different codes from different disciplines. In this abstract, we give a brief description of a non-intrusive PC method, which is relatively easy to implement. Our preliminary results with this method includes two stochastic flow problems: (a) steady, subsonic, 2-D, zero-pressure gradient laminar boundary layer over a flat plate, which has the free-stream dynamic viscosity as the uncertain parameter and (b) inviscid, steady, supersonic, 2-D flow over a wedge which has an uncertainty in the wedge Postdoctoral Research Associate, Member AIAA ( shosder@vt.edu). Graduate Research Assistant, Student Member AIAA ( rperez@vt.edu). Professor and Department Head, Associate Fellow AIAA ( rwalters@vt.edu). 1

2 angle. In the final version of our paper, we will present more test cases to further investigate the performance, advantages, and limitations of the non-intrusive PC approach for modeling the uncertainty propagation in flow simulations. 2 A Non-Intrusive Polynomial Chaos Method 2.1 Basics of Hermite Polynomial Chaos Several papers can be found in the literature that investigate the spectral representations of uncertainty (See [2], [3], [4], [5], [6],and [7]). An important aspect of spectral representation of uncertainty is that one may decompose a random function (or variable) into separable deterministic and stochastic components. For example, for any generic variable (i.e., α ) with random fluctuations, we can write, α (x, y, ξ) = P α i (x, y)ψ i (ξ) (1) i= where α i (x, y) is the deterministic component and Ψ i (ξ) is the random basis function corresponding to the i th mode. Effectively, α i (x, y) is the amplitude of the i th fluctuation. The discrete sum is taken over the number of output modes, P = (n+p)! 1, which is a function of the order of polynomial chaos, p, and the number of random dimensions, n. For n!p! the basis function, we use multi-dimensional Hermite Polynomial to span the n-dimensional random space, although many other choices are possible. A convenient form of the Hermite polynomials is given by δ n H n (ξ i1,..., ξ in ) = e 1 2 ξt ξ ( 1) n e δξ i1...δξ in 1 2 ξt ξ (2) where ξ = (ξ i1,..., ξ in ) is the n-dimensional random variable vector. The Hermite polynomials form a complete orthogonal set of basis functions in the random space. In terms of the inner product, f(ξ)g(ξ) = f(ξ)g(ξ)p N (ξ)dξ (3) with the weight function p N (ξ) taking the form of an n-dimensional Gaussian distribution with unit variance defined by 1 p N (ξ) = 1 (2π) n e 2 ξt ξ, (4) the inner product of the basis functions is zero with respect to each other, i.e. where δ ij is the Kronecker delta function. Ψ i Ψ j = Ψ 2 i δij (5) 2

3 2.2 Description of the Non-Intrusive Method The Non-Intrusive PC method utilizes the polynomial expansion in Equation 1. Let us assume a one-dimensional random space for the ease of description (Note that P = p for one-dimensional random space). For a given PC of order p, one can pick a set of random points, {ξ, ξ 1, ξ 2,, ξ p }, and use the elements of this set as the sampling points at which the deterministic code is evaluated (Here deterministic code may also be labelled as the blackbox). With the left hand side of Equation (1) known from the solutions of deterministic evaluations, one can form a system of linear equations as, Ψ (ξ ) Ψ 1 (ξ ) Ψ p (ξ ) α α Ψ (ξ 1 ) Ψ 1 (ξ 1 ) Ψ p (ξ 1 ) α 1 = α 1 (6) Ψ (ξ p ) Ψ 1 (ξ p ) Ψ p (ξ p ) Ψ α = r The spectral modes of the random variable, α, is found by finding the inverse of the linear system above, α = Ψ 1 r. (7) The method described above is non-intrusive in the sense that no modification to the deterministic code is required in any form. In contrast, for the intrusive method, the deterministic code is altered through the substitution of PC expansions and spectral projections to model the uncertainty propagation. The non-intrusive method is very simple to implement, requires a single LU decomposition on a relatively small matrix, minimally requires P + 1 deterministic evaluations to estimate statistics, and can be shown to converge to the projected PC expansion coefficients under certain conditions. However, in general, the solution obtained by this non-intrusive approach is not unique. Therefore, non-intrusive method can be interpreted as a model for estimating the projected coefficients. 3 Results of Initial Studies In this extended abstract, we present some of the non-intrusive polynomial chaos (PC) results obtained from the study of two stochastic fluid dynamics problems: The first problem is the steady, subsonic, 2-D, zero-pressure gradient laminar boundary layer over a flat plate, which has the free-stream dynamic viscosity as the uncertain parameter. The second test case is the inviscid, steady, supersonic, 2-D flow over a wedge which has an uncertainty in the wedge angle. In the final version of our paper, we plan to present more test cases to further investigate the performance, advantages, and limitations of the non-intrusive PC approach for modeling the uncertainty propagation in flow simulations. One such study will come from our current work on modeling the uncertainty propagation in 3-D transonic flow over a wing with a given uncertainty in an input parameter or in the geometry. We will use both the non-intrusive PC method and the Monte Carlo simulations, and analyze the statistics of the results obtained from both approaches. 3 α p α p

4 3.1 Laminar Boundary Layer Flow over a Flat Plate The stochastic problem for this flow is formulated by modeling the dynamic viscosity of the free-stream, µ, as a normally distributed stochastic parameter with a coefficient of variation of 2%. All the other free-stream flow properties including the density, velocities, and the pressure were kept constant. The free-stream Mach number was chosen as M =.3. The Reynolds number obtained with the mean value of the dynamic viscosity was Re mean = at x =.7375 (Figure 1). The histogram of the free-stream dynamic viscosity is shown in Figure 2, which was created by taking 1, samples from its normally distributed population. The uncertainty propagation in the laminar boundary layer problem was modeled with two different approaches: Monte Carlo simulations and the non-intrusive PC method. A total number of 1, Monte Carlo realizations were acquired to get the statistics of the uncertainty in different output quantities. Each realization was obtained by applying the incompressible Blasius solution to the problem (See [8]) for each dynamic viscosity sample taken from the normal distribution shown in Figure 2. A fourth-order polynomial chaos expansion was chosen to model the output uncertainty in the non-intrusive PC method. The required five deterministic solutions were evaluated at ξ i =, 1, 1, 2, 2 (i =, 1, 2, 3, 4) where the normalized random variable ξ i is defined as ξ i = (µ ) i µ. (8) σ Here µ is the mean value of the free-stream dynamic viscosity (µ ) and σ is the standard deviation. The deterministic solutions used in the non-intrusive PC method were obtained from a Thin-Layer Navier-Stokes (TLNS) Code. The deterministic TLNS simulations were run at different grid levels to examine the grid convergence and the discretization error. Table 1 gives the dimensions of the grid levels. Grid 1 is the finest and Grid 4 is the coarsest mesh used in the computations. In this table, i corresponds to the streamwise (x) and j to the normal (y) directions. Grid 2 and Grid 4 were obtained from Grid 1 by deleting every other grid line in both directions. A measure of mesh spacing is defined by h k = N 1 /N k where k is the grid level and N k = (n i n j ) k. In TLNS simulations, no-slip, adiabatic wall boundary condition was applied on the solid wall. For the in-flow boundary (Figure 1), total pressure (P ) and the total temperature (T ) were kept constant at their free-stream values. The static pressure was extrapolated from the first interior cell and the other flow variables were calculated using the isentropic flow relations. For the far-field (top boundary) and out-flow boundary, the static pressure was fixed at its free-stream value and the rest of the primitive variables (density and velocity components) were extrapolated from the interior cells. To obtain asymptotic convergence to a steady-state solution for each TLNS simulation, time integration steps were carried out until the L 2 norm of the residual was reduced 1 orders of magnitude. The statistics of the uncertainty distributions of different output variables obtained from the non-intrusive PC method and Monte Carlo simulations are compared at x = Figure 3 shows the standard deviation of u/u through the boundary layer at this station. Here, u is the velocity component in x-direction and U is the free-stream velocity. In this figure, non-intrusive PC results are given for all the grid levels used in the deterministic TLNS calculations. Starting from Grid 2, the PC results approach to the results of the Monte 4

5 Table 1: Grids used in deterministic TLNS runs Grid Level # of i cells (n i ) # of j cells (n j ) h Grid Grid Grid Grid Table 2: Statistics of the boundary layer quantities at x = The non-intrusive PC results are presented for the finest grid level. BL Thickness Disp. Thickness Mom. Thickness PC MC PC MC PC MC Mean 1 3 m STD 1 5 m CoV Carlo simulations. From Figure 4, a good agreement can be seen between the non-intrusive PC Grid 1 profile and MC results. The distributions of the boundary layer thickness (δ), displacement thickness (δ ), and the momentum thickness (θ) obtained at x =.7375 using the non-intrusive PC method are given in Figures 5, 6, and 7. Examining the shape and statistics (mean and standard deviation) of these distributions at different grid levels, one can see that the standard deviation remains approximately the same for all the grid levels whereas the mean is shifted slightly with the mesh refinement. All boundary layer quantities exhibit a normal distribution with a coefficient of variation of approximately 1%. Grid convergence of the mean boundary layer quantities can be seen from Figure 8. Note that this convergence becomes non-monotonic as the mesh is refined. In Figure 9, the distributions of the boundary layer quantities obtained with the non-intrusive PC method at the finest mesh level are compared to the distributions obtained from the Monte Carlo simulations. The numerical results are tabulated in Table 2. Slight difference between the mean values may be due to the physical modeling errors, such as the incompressible flow assumption by the Blasius solution. The non-intrusive PC approach used deterministic TLNS solutions obtained at M =.3, whereas Monte Carlo simulations utilized incompressible Blasius solution at the same Mach number. Another possible source of the difference may be due to the uncertainty originating from the boundary conditions applied at the far-field and the outflow boundaries. We will investigate this subject more and include our findings in the final version of our paper. Considering the accuracy level desired in most engineering fields, the results indicate that non-intrusive polynomial chaos method can accurately model the propagation of an input uncertainty and accurately predict the statistics of an output variable for this laminar boundary layer flow problem. 5

6 3.2 2-D Supersonic Flow over a Wedge For the formulation of a stochastic supersonic flow over a wedge, a geometric uncertainty was introduced through the wedge angle (θ) that was uncertain and described by a Gaussian distribution. The mean wedge angle was 1 and the coefficient of variation was 1%. The free-steam Mach number was M =.3 and the angle of attack was set to zero degrees. Uncertainty propagation in supersonic wedge flow has been modeled using two approaches: non-intrusive and intrusive PC methods. The deterministic solutions required in non-intrusive PC method were obtained from deterministic Euler simulations. Intrusive PC method was applied to this problem by using a stochastic Euler code (See [9]) which utilizes Hermite Polynomial Chaos for uncertainty propagation. In Euler computations, all flow variables were kept fixed at their free-stream values at the in-flow boundary. For the far-field (top) and out-flow surfaces, all flow variables were determined by first-order extrapolation from the interior cells. A tangency boundary condition was prescribed along the bottom surface. Preliminary results for this problem were obtained at a coarse grid (11 21) shown in Figure 1. Currently, both the deterministic and the stochastic Euler simulations are being performed on finer grid levels. The results of these studies will be available in the final version of the paper. Figure 11 illustrates a comparison of the mean and the standard deviation of static pressure between the non-intrusive and intrusive PC results. The order of the PC is five for both methods. The mean values are in close agreement with each other, however there is a slight difference in standard deviation for the region after the shock. We suspect that this difference is due to the use of a very coarse grid and expect that the fine grid results will give a more accurate comparison between two methods. 4 Conclusions This extended abstract presented a non-intrusive PC method for the propagation of input uncertainty in CFD simulations. A brief description of the method including its derivation and implementation was given. The non-intrusive scheme has been applied to two stochastic flow problems: (a) steady, subsonic, 2-D, zero-pressure gradient laminar boundary layer over a flat plate, which has the free-stream dynamic viscosity as the uncertain parameter and (b) inviscid, steady, supersonic, 2-D flow over a wedge which has an uncertainty in the wedge angle. Monte Carlo simulations were also performed for the laminar boundary layer problem and the statistics of various output quantities obtained from this approach were compared to those of the non-intrusive PC method. Results obtained for this problem show that non-intrusive PC method is capable of accurately modelling the propagation of an input uncertainty and predicting the statistics of an output variable with desired accuracy. Both intrusive and non-intrusive PC methods were applied to the supersonic wedge flow problem. Despite the coarse grid level used in the computations, mean pressure distributions were in good agreement. Fine grid runs are being performed for the same problem to achieve a more accurate comparison of the higher order statistics. 6

7 In the final version of our paper, we will present more test cases to further investigate the performance, advantages, and limitations of the non-intrusive PC approach for modeling the uncertainty propagation in flow simulations. One such study will come from our current work on modeling the uncertainty propagation in 3-D transonic flow over a wing geometry. We will use both the non-intrusive PC method and the Monte Carlo simulations, and analyze the statistics of the results obtained from both approaches. References [1] R.W. Walters. Towards stochastic fluid mechanics via polynomial choas-invited, AIAApaper In 41 st AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 23. CD-ROM. [2] Roger Ghanem. Stochastic finite elements with multiple random non-gaussian properties. Journal of Engineering Mechanics, pages 26 4, January [3] Roger Ghanem and P. D. Spanos. Polynomial chaos in stochastic finite elements. Journal of Applied Mechanics, 57:197 22, March 199. [4] Roger G. Ghanem. Ingredients for a general purpose stochastic finite element formulation. Computational Methods in Applied Mechanical Engineering, 168:19 34, [5] Roger G. Ghanem and P. D. Spanos. Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York, [6] D. Xiu and G. E. Karniadakis. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 21. in review. [7] D. Xiu and G. E. Karniadakis. The W iener Askey polynomial chaos for stochastic differential equations. Technical Report 1-1, Center for Fluid Mechanics, Division of Applied Mathematics, Brown University, Providence, RI, 21. [8] Frank M. White. Viscous Fluid Flow. McGraw-Hill, Inc., [9] R. Perez and R.W. Walters. An implicit compact polynomial chaos formulation for the euler equations, AIAA-paper In 43 rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 25. CD-ROM. 7

8 Laminar Boundary Layer (BL) Flow over a Flat Plate in-flow y x= M =.3 Problem Definition far-field Problem Definitio x y M(Re mean =2 1 5 =.3 ) x x=.7375 Figure 1: Description of the stochastic laminar boundary layer problem which has an input un- Parameter certainty in the dynamic with viscosity Uncertainty of the free-stream. x= = Dynamic The uncertainty Viscosity in u/u profiles of freestream and different boundary quantities are analyzed at x=.7375 m where the mean Reynolds number is Re mean = 2 Modeled 1 as a normally distributed stochastic parameter with 5. Parameter with Uncertainty = Dynamic Visc coefficient of variation Modeled (CoV) of as 2% a normally distributed stoch coefficient of variation (CoV) of 2% Uncertainty propagation in results obtained with Uncertainty propagation in resu 1. Monte Carlo (MC) Simulation 2. Black Box (No 1. 1, Monte Carlo samples (MC) from Simulation Polynomial Ch 3 µ population 1, samples 4 th degree from 25 Used incompressible µ 5 Determini population 2 Blasius Solution to get Used incompressib obtained from 15 results (output) for Blasius for Solution x=, 1, to -1 g 1 each sample results (output) where for x= 5 each sample µ (kg / m s) µ (kg / m s) Laminar Boundary Layer (BL) Flow Figure 2: Histogram showing the uncertainty in the free-stream dynamic viscosity which is modelled as a normally distributed stochastic parameter with a coefficient of variation (CoV ) of 2%. out-flow Analyze result x=.7375 m (Re mean =2 1 8

9 Standard Deviation of u/u at x=.7375 m (Solutions with all grid levels) Standard Deviation of u/u at x=.7375 m y/δ mean PC (grid 1, fine grid) PC (grid 2 ) PC (grid 3 ) PC (grid 4, coarse grid) MC (Incompressible Blasius Solution) Figure 3: Standard deviation of u/u at x =.7375 m. Non-Intrusive PC results are shown for all grid levels used in the Thin-Layer Navier Stokes (TLNS) computations. Monte Carlo (MC) simulations are performed for the incompressible Blasius solution. 3 9

10 Standard Deviation of u/u at x=.7375 m (comparison with fine grid solution).5 Standard Deviation of u/u at x=.7375 m PC (grid 1, fine grid) MC (Incompressible Blasius Solution) y/δ mean Figure 4: Standard deviation of u/u at x =.7375 m. Non-Intrusive PC results are shown only4 for the finest mesh level used in the TLNS computations. Monte Carlo (MC) simulations are obtained using the incompressible Blasius solution. 1

11 PC results at different grid levels - BL Thickness δ Mean x Mean x Mean x Mean x Grid 4 (Coarsest grid) STDx Grid 3 STDx Grid 2 STDx CoVx CoVx CoVx Grid 1 (Finest grid) STDx1 5 CoVx Figure 5: Histograms of the boundary layer thickness obtained at x =.7375 m using non-intrusive PC method at different grid levels. 11

12 PC results at different grid levels Disp. Thickness δ Mean x Mean x Mean x Mean x Grid 4 (Coarsest grid) STDx Grid 3 STDx Grid 2 STDx CoVx CoVx CoVx Grid 1 (Finest grid) STDx1 5 CoVx Figure 6: Histograms of the displacement thickness obtained at x =.7375 m using non-intrusive PC method at different grid levels. 6 12

13 PC results at different grid levels Mom. Thickness θ Mean x Mean x Mean x Mean x Grid 4 (Coarsest grid) STDx Grid 3 STDx Grid 2 STDx STDx CoVx CoVx CoVx Grid 1 (Finest grid) CoVx Figure 7: Histograms of the momentum thickness at obtained x =.7375 m using non-intrusive PC method at different grid levels. 7 13

14 Mean δ (a) h Mean δ (b) h Mean θ (c) Figure 8: Mean values of (a) boundary layer thickness (b) displacement thickness, and (c) momentum thickness at x =.7375 vs. mesh size, h. These quantities are calculated using the solutions of the Thin-Layer Navier Stokes (TLNS) simulations. h 14

15 Fine Grid PC results compared to Blasius MC BL Thickness Disp. Thickness Mom. Thickness MC 12 MC 3 MC δ Figure 9: Histograms of the boundary BL Thickness layer quantitiesdisp. at x = Thickness.7375 obtained Mom. with Thickness the non-intrusive PC method at the fine grid level and PC with the MC Monte PC Carlo simulations MC of PC incompressible MC Blasius solution. Mean x STD x 1 5 PC PC PC CoV x δ θ Y.4.3 Y X X Figure 1: The Grid (11 21) used in supersonic wedge flow computations. Figure 1: 15

16 Y 2D Euler PC (Blackbox PC) mean[pressure] Y 2D Euler PC (Blackbox PC) StdDev[pressure] X X Y 2D Euler PC (Intrusive PC) mean[pressure] Y 2D EulerPC (Intrusive PC) StdDev[pressure] X X Figure 11: Fifth-order PC statistics comparison (Mean and Standard Deviation) between the nonintrusive PC (Blackbox PC) and intrusive PC for the supersonic wedge flow problem. 16

A Non-Intrusive Polynomial Chaos Method for Uncertainty Propagation in CFD Simulations

Missouri University of Science and Technology Scholars' Mine Mechanical and Aerospace Engineering Faculty Research & Creative Works Mechanical and Aerospace Engineering --6 A Non-Intrusive Polynomial Chaos