Introduction There has been recently anintense interest in verification and validation of large-scale simulations and in modeling uncertainty [,, ]. I

Transcription

1 Modeling Uncertainty in Flow Simulations via Polynomial Chaos Dongbin Xiu and George Em Karniadakis Λ Division of Applied Mathematics Brown University Providence, RI 9 Submitted to Journal of Computational Physics February 8, Abstract We present a new algorithm based on Polynomial Chaos in order to model the input uncertainty and its propagation in incompressible flow simulations. Specifically, the stochastic input is represented spectrally by employing the Wiener-Hermite polynomials. Randomness is thus expressed as a new continuous variable, in addition to the space-time domain. A standard Galerkin projection is performed and the resulting set of coupled equations is then solved to obtain the solution for each random mode. We implement the method in the context of the spectral/hp element method, which maintains control of the numerical error while its spectral trial basis is from the same Askey family of polynomials as the Polynomial Chaos. The algorithm is applied to micro-channel flows with random wall boundary conditions, and to external flows with random freestream. Both Gaussian and non-gaussian inputs are considered, and the results are in good agreement with exact solutions and Monte Carlo simulations. The Polynomial Chaos based stochastic simulation, although more expensive than the corresponding deterministic simulation, it generates all the solution statistics in a single run. Compared with the Monte Carlo simulation, which requires at least thousands of runs to obtain accurate statistics, the Polynomial Chaos expansion promises a substantial speed-up (at least one thousand in the current simulations), and thus it is a very effective approach in modeling uncertainty propagation in flow simulations. Λ Corresponding author, gk@cfm.brown.edu

2 Introduction There has been recently anintense interest in verification and validation of large-scale simulations and in modeling uncertainty [,, ]. In simulations, just like in the experiments, we often question the accuracy of the results and we construct aposteriori error bounds, but the new objective is to model uncertainty from the beginning of the simulation and not simply as an afterthought. Numerical accuracy and error control have been employed in simulations for some time now, at least for the modern discretizations, e.g. [, ]. However, there is still an uncertainty component associated with the physical problem, and specifically with such diverse factors as constitutive laws, boundary and initial conditions, transport coefficients, source and interaction terms, geometric irregularities (e.g. roughness), etc.. Most of the research effort in CFD research sofarhas been in developing efficient algorithms for different applications, assuming an ideal input with precisely defined computational domains. With the field reaching now some degree of maturity, we naturally pose the more general question of how to model uncertainty and stochastic input, and how to formulate algorithms in order for the simulation output to reflect accurately the propagation of uncertainty. To this end, the Monte Carlo approach can be employed but it is computationally expensive and it is only used as the last resort. The sensitivity method is an alternative more economical approach, based on moments of samples, but it is less robust and it depends strongly on the modeling assumptions [6]. There are other more suitable methods for physical applications, and there has already been good progress in other fields, most notably in seismology and structural mechanics. Anumber of papers and books have been devoted to this subject, e.g. [7],[8], [9],[], [],[],[],[]. The most popular technique for modeling stochastic engineering systems is the perturbation method where all stochastic quantities are expanded around their mean via a Taylor series. This approach, however, is limited to small perturbations and does not readily provide information on high-order statistics of the response. Another approach is based on expanding the inverse of the stochastic operator in a Neumann series, but this too is limited to small fluctuations, and even combinations with the Monte Carlo method seem to result in computationally prohibitive algorithms for complex systems []. A more effective approach pioneered by Ghanem and Spanos [] in the context of finite elements for solid mechanics is based on a spectral representation of the uncertainty. This allows high-order representation, not just first-order as in most perturbation-based methods, at high computational efficiency. It is based on the original ideas of Wiener (98) on homogeneous chaos [6, 7] and the Hermite polynomials. The classical Grad's method of moments for solutions of Boltzmann equation is based on Hermite polynomial expansions

3 as well [8]. The effectiveness of Hermite expansions was also recognized by Chorin in his early work [9], who employed Wiener-Hermite series to substantially improve both accuracy and computational efficiency of Monte Carlo algorithms. However, the limitation of prematurely truncated Wiener-Hermite expansions, especially in applications of turbulence [], has also been recognized. The spectral representation of the uncertainty is based on a trial basis Ψ( ) where denotes the random event. For example, the vorticity (! in two-dimensions) has the following finite-dimensional representation!(x;t; )= PX! i (x;t)ψ i (ο( )): i= Here! i (x;t) represents the deteministic part and can be interpreted as a scale (i) of vorticity fluctuation. The random trial basis is expressed in terms of multi-dimensional Hermite polynomials in powers of the random variable ο( ), which defines a specific probability distribution; for example, ο( ) may be a Gaussian function. The polynomial trial basis constructed in this way has been termed Polynomial Chaos (PC) by Wiener. It is a functional, as it is a function of ο which is a function of the random parameter [; ]. The theory of orthogonal functionals plays a key role in the algorithms that we develop in this work. Note that the Monte Carlo algorithm is a subcase of the above representation corresponding to the collocation procedure where the test basis is Ψ i ( ) =ffi( i ); with ffi the Kronecker delta function, and i referring to an isolated random event. For convergence, the Monte Carlo method requires a great number of such events as it ignores interactions between the various scales unlike the Galerkin representation. In this work we propose a Polynomial Chaos (PC) expansion procedure to represent the uncertainty of the input in flow simulations. The algorithms we develop are general but in this paper we concentrate on modeling the uncertainty associated with boundary conditions. This situation is encountered, for example, in micro-channel flows [] but also in classical flows such, e.g. the freestream of flow past bluff bodies. We assume that the velocity at the boundaries is described by amean value and a random perturbation with certain propability distribution function (PDF). This distribution can be assumed to have a known analytical form, e.g. Gaussian or lognormal distribution, or it can be a measured distribution provided in tabulated form. The PC procedure can handle both Gaussian and non- Gaussian representations, although for certain distributions there may exist a better representation than the original Wiener-Hermite polynomial basis. To this end, we can replace the Wiener-Hermite expansions by the best spectral polynomials of the Askey family [] that match specific distribution functions. For example, the appropriate set for

4 Poisson distributions is the Charlier polynomials, for Gamma distributions the Laugerre polynomials, for binomial distributions the Krawtchouk polynomials, for the beta kernel the Jacobi polynomials, etc. This can be appreciated by the fact that the orthogonality weights of these polynomials match the probability measure of the corresponding polynomials, and the same relationship exists of course for the Hermite polynomials and the Gaussian distribution. The work we present in this paper is based on the pioneering work in solid mechanics of Ghanem & Spanos [] but applied here to incompressible Navier-Stokes equations. In addition, we employ the spectral/hp element method in order to obtain a better control of the numerical error []. Our approach leads to homogeneous representations as the mixed inner products required in the random and spatial disretizations are all spectral polynomials. In particular, for certain distributions, e.g. beta kernels the corresponding Askey polynomial for the Wiener expansion is the Jacobi polynomial, which is exactly the same as the trial basis employed in the spectral/hp element method []. In the next section we review the PC expansion, and in section we address its implementation details as applied to Navier-Stokes equations. In section we present computational results for Gaussian and lognormal distributions, and we compare with analytical stochastic solutions of Navier-Stokes we have obtained as well as with corresponding Monte Carlo simulations. We conclude the paper with a discussion on possible extensions and applications of the stochastic spectral/hp element method. We also include two appendices that explain how to generate partially correlated Gaussian fields and how to approximate the lognormal distribution that may be useful to CFD readers. Representation of Random Processes In this section we briefly review the PC expansion along with the Karhunen-Loeve (KL) expansion, another classical technique for representating random processes. The KL expansion will be used to represent the known stochastic fields, e.g. the input. In the following analysis, we will use the symbol ο to denote the standard Gaussian random variable, i.e., Gaussian random variable with zero mean and unit variance.. Polynomial Chaos Expansion The PC expansion was first proposed by Wiener [6]. According to the theorem by Cameron & Martin [], it can be used to approximate any functional in L (C) andconverges to the functional in the L (C) sense. Therefore, PC provides a means for expanding second-order random processes in terms of orthogonal polynomials. Second-order random processes are processes with finite variance, and this applies to all viscous flow processes. Thus, a secondorder random process X( ), viewed as a function of the independent random variable, can be represented in the

5 form X( ) = a H X i = X a i H (ο i ( )) i X i = i = X Xi Xi i = i = i = a ii H (ο i ( );ο i ( )) a ii i H (ο i ( );ο i ( );ο i ( )) + ; () where H n (ο i ;:::;ο in ) denotes the polynomial chaos of order n in the variables (ο i ;:::;ο in ). The above equation is the discrete version of the original Wiener-Hermite expansion, where the continuous integrals are replaced by summations. The general expression of the polynomials is given n H n (ο i ;:::;ο in )=e οt ο ( ) n e οt ο ; in where ο denotes the vector consisting of n random variables (ο i ;:::;ο in ). For notational convenience, equation () can be rewritten as X( ) = X j= ^a j Ψ j (ο); () where there is a one-to-one correspondence between the functions H n (ο i ;:::;ο in )andψ j (ο). The Polynomial Chaos forms a complete orthogonal basis in the L space of random variables, i.e., < Ψ i Ψ j >=< Ψ i >ffi ij; () where ffi ij is the Kronecker delta and < ; > denotes the ensemble average. This is the inner product in the Hilbert space of random variables <f(ο)g(ο) >= Z f(ο)g(ο)w (ο)dο: () and it is justified by the Cameron & Martin theorem []. The weighting function is W (ο) = p (ß) n e οt ο : (6) What distinguishes the Wiener-Hermite expansion from many other possible complete sets of expansions is that the polynomials here are orthononormal with respect to the weighting function W (ο) that has the form of a n- dimensional independent Gaussian probability distribution with unit variance. For example, the one-dimensional

6 (n = ) polynomials are: Ψ =; Ψ = ο; Ψ = ο ; Ψ = ο ο; ::: (7) The orthogonality condition takes the specific form < Ψ i Ψ j >=< Ψ i >ffi ij =(i!)ffi ij : (8). Karhunen-Loeve Expansion The Karhunen-Loeve (KL) expansion [] is another way of representing a random process. It is based on the spectral expansion of the covariance function of the process. Let us denote the process by h(x; ) and its covariance function by R hh (x; y), where x and y are the spatial or temporal coordinates. By definition, the covariance function is real, symmetric and positive definite. All eigenfunctions are mutually orthogonal and form a complete set spanning the function space to which h(x; ) belongs. The KL expansion then takes the following form: h(x; )= μ h(x)+ X p i ffi i (x)ο i ( ); (9) i= where μ h(x) denotes themeanoftherandom process, and ο i ( ) forms a set of orthogonal random variables. Also, ffi i (x) and i are the set of eigenfunctions and eigenvalues of the covariance function, respectively, i.e., Z R hh (x; y)ffi i (y)dy = i ffi i (x): () If the random process itself h(x; ) is a Gaussian process, then the random variables ο i form an orthonormal Gaussian vector. Among many possible decompositions of a random process, the KL expansion is optimal in the sense that the mean-square error resulting from a finite representation of the process is minimized. Its use, however, is limited as the covariance function of the solution process is often not known apriori. Nevertheless, the KL expansion still provides an effective means of representation of the input random processes when the covariance structure is known. In this paper we will employ the KL procedure to represent the stochastic boundary conditions in the case of Gaussian distributions. For non-gaussian distributions we will employ Polynomial Chaos representations directly. Polynomial Chaos Expansion for Navier-Stokes Equations In this section we discuss the solution procedure for solving the stochastic Navier-Stokes equations by PC expansion. The randomness in the solution can be introduced through boundary conditions, initial conditions, forcing, etc. 6

7 . Governing Equations We employ the incompressible Navier-Stokes equations r u = +(u r)u = rπ+re r u; () where Π is the pressure and Re the Reynolds number. All flow quantities, i.e., velocity and pressure are considered stochastic processes. A random dimension, denoted by the parameter, is introduced in addition to the spatialtemporal dimensions (x;t), thus u = u(x;t; ); Π=Π(x;t; ) () We then apply the Polynomial Chaos expansion () to these quantities to obtain u(x;t; ) = PX i= u i (x;t)ψ i ( ); Π(x;t; ) = PX i= Π i (x;t)ψ i ( ); () where we have replaced the infinite summation in () by a finite term summation. The total number of expansion terms, (P + ), depends on the number of random dimensions (n) and the highest order of the polynomials chaos (p) []: P =+ px s= Y s (n + r): () s! The most important aspect of the above expansion is that all random processes have been decomposed into a set of deterministic functions in the spatial-temporal variables multiplying random coefficients that are independent of these variables. Substituting () into Navier-Stokes equations (() and ()) and noting that the partial derivatives are taken in physical space and thus they commute with the operations in random space, we obtain the following equations PX i (x;t) Ψ i PX PX i= j= PX i= r= r u i (x;t)ψ i = ; (6) [(u i r)u j )]Ψ i Ψ j = PX i= rπ i (x;t)ψ i + ν PX i= r u i Ψ i : (7) We then project the above equations onto the random space spanned by the basis polynomials fψ i g by taking the inner product of above equation with each basis.by taking < ; Ψ k > and utilizing the orthogonality condition (), we obtain the following set of equations: 7

8 For each k + < Ψ k > PX PX i= j= r u k = ; (8) e ijk [(u i r)u j )] = rπ k + νr u k ; (9) where e ijk =< Ψ i Ψ j Ψ k >. Together with < Ψ i >, these coefficients can be determined analytically from the definition of Ψ i and equation (). This set of equations consists of (P + ) system of Navier-Stokes equations for each random mode coupled through the convection term. The solution associated with k = term represents the mean of the random solution. The solution corresponding to the first-order polynomial chaos, i.e., the k = ;:::;n terms where n is the dimension of random space, represents the Gaussian part of the solution. The rest of the solution terms represent the nonlinear interactions between the mean and each Gaussian part of the solution and are non-gaussian.. Numerical Formulation.. Temporal Discretization We employ the semi-implicit high-order fractional step method, which for the standard deterministic Navier-Stokes equations (() and ()) has the form [6]: ^u P J q= ff qu n q t ^u ^u t fl u n+ ^u t = JX q= fi q [(u r)u] n q ; () = rπ n+ ; () = Re r u n+ ; () where J is order of accuracy in time and ff; fi and fl are the coefficients of the integration weights. A pressure Poisson equation is obtained by enforcing the discrete divergence-free condition r u n+ = r Π n+ = t r ^u; () with the appropriate pressure boundary condition = n ^u + Re r! n+λ ; () where n is the outward unit normal vector and! = r u is the vorticity. The method is stiffly-stable and achieves third-order accuracy in time; the coefficients for the integration weights can be found in []. In order to discretize the stochastic Navier-Stokes equations, we apply the same approach to the coupled set of equations (8) and (9): 8

9 For each k =;:::;P, ^u k P J q= ff qu n q k t ^u k ^u k t fl u n+ k ^u k t = < Ψ k > JX q= fi q P X PX i= j= n q e ijk (u i r)u j ; () = rπ n+ k ; (6) = Re r u n+ k : (7) The discrete divergence-free condition for each moder u n+ k each pressure mode with appropriate pressure boundary condition derived similarly as in [6] = results in a set of consistent Poisson equations for r Π n+ k = t r ^u k; k =;:::;P; = n ^u Λ k + Re r! n+ k ; k =;:::;P; (9) where n is the outward unit normal vector along the boundary, and! k = r u k is the vorticity for each random mode... Spatial Discretization Spatial discretization can be carried out by any method, but here we employ the spectral/hp element method in order to have better control of the numerical error []. In addition, the all-spectral discretization in space and along the random direction leads to homogeneous inner products, which in turn result in more efficient ways of inverting the algebraic systems. In particular, the spatial discretization is based on Jacobi polynomials on triangles or quadrilaterals in two-dimensions, and tetrahedra, hexahedra or prisms in three-dimensions.. Post-Processing The coefficients in the expansion of the solution process () are obtained after solving equations () to (9). We then obtain the analytical form (in random space) of the solution process. It is possible to preform a number of analytical operations on the stochastic solution in order to carry out a sensitivity analysis. Specifically, the mean solution is contained in the expansion term with index of zero. The second-moment, i.e., the covariance function is given by R uu (x ;t ; x ;t ) = < u(x ;t ) u(x ;t ); u(x ;t ) u(x ;t ) > = PX ui (x ;t )u i (x ;t ) < Ψ >Λ i : () i= 9

10 Note that the summation starts from index (i = )instead of to exclude the mean, and that the orthogonality of the PC basis fψ i g has been used in deriving the above equation. The variance of the solution, also called the `mean-square' value, is obtained as Var(u(x;t)) =< X P u(x;t) u(x;t) >= u i (x;t) < Ψ >Λ i ; () i= and the root-mean-square (rms) is simply the square root of the variance. Similar expressions can be obtained for the pressure field. Stochastic Flow Simulations In this section we presentnumerical results of the Polynomial Chaos solution to the Navier-Stokes equations. We first consider a micro-channel flow where there is uncertainty associated with the wall boundary conditions. Subsequently, we simulate a laminar flow past a circular cylinder with uncertain freestream. We model the uncertainty in the boundary conditions both as Gaussian and as lognormal distributions in order to evaluate the convergence of the PC method. In both cases we employ a two-dimensional Polynomial Chaos expansion (n = ) with polynomial order up to p = ; these polynomial coefficients are listed in table. Index p, Order of k th Polynomial Chaos k Polynomial Chaos Ψ k p = p = ο ο p = ο ο ο 6 p = ο ο ο 7 ο ο ο 8 ο ο ο 9 ο ο p = ο 6ο + ο ο ο ο ο ο ο ο + ο ο ο ο ο 6ο + Table : Polynomial terms for two-dimensional Polynomial Chaos.. Micro-channel Flow: Gaussian Input We consider a micro-channel flow as shown in figure. In micro-flows the no-slip boundary condition is not always valid and appropriate slip boundary conditions should be used []. For gas flows Maxwell's boundary condition on

11 velocity slip or its high-order variants are appropriate at least in the slip flow regime. However, for liquids there is no existing models for the molecule layering observed in molecular dynamics simulations [7]. Also, the effect of roughness complicates further the validity or not of the no-slip boundary condition. Here we model such uncertainty at the boundary by assuming that u =+r where r is a random variable. y= u=u y x F=ν y=- u=u Figure : Schematic of the domain for micro-channel pressure-driven flow. The domain (see figure ) has dimensions such thaty [ ; ] and x [ ; ]. The pressure gradient, acting like a driving force, equals to twice the kinematic viscosity, and thus for a no-slip condition a parabolic profile is a solution with centerline velocity equals to unity. Assuming that the two walls are moving with constant velocities u and u, then the exact solution is so it is a superposition of the Poiseuille and two Couette type flows... Fully-Correlated Random Boundary Conditions u(x; y) =( y )+ y u + +y u () Let us now assume that the boundary conditions are random and spatially uniform, i.e., u = ffο and u = ff ο ; where ο and ο are two independent standard Gaussian random variables with zero mean and unit variance and ff, ff are the standard deviations of u and u, respectively. In this case, the exact solution of equation () still

12 applies and thus u(x; y) =( y )+ y ff ο + +y ff ο () It can be seen from table that the solution consists of the parabolic mean u and only the first two random modes u and u, linearly distributed across the channel. We will use this solution to validate the stochastic Navier-Stokes solver. u u u..7. u.. u, u y Figure : Solution profile across the channel with uniform Gaussian boundary conditions. Figure shows the solution profile across the channel. The solution is obtained by setting ff = : and ff =:. Atwo-dimensional PC expansion is used (n =)withpolynomial Chaos of order p =. The numerical results correctly obtain the distribution profiles of the mean, the first and the second random modes. If p>is employed, all higher-order random modes are identically zero... Partially-Correlated Random Boundary Conditions Next we consider the case of non-uniform random boundary conditions, i.e. the boundary points are only partiallycorrelated and thus the random contribution to the boundary condition varies along the walls. The boundary conditions are assumed to be Gaussian random processes with correlation function in the form C(x ;x )=e jx x j b ; ()

13 6 where b is the correlation length. This correlation function has been employed extensively to model processes in a variety of fields and its use can be justified on theorectical grounds [8]. Given the exponential form of the correlation function, the Karhunen-Loeve expansion of equation (9) can be used to decompose the input random process. The corresponding eigenvalue problem of equation () can be solved analytically. The eigenvalues are: and the eigenfunctions are: b n = ; n =; ;::: () +b! n f n (x) = 8 >< >: cos(! q n x) a+ sin(!na)!n sin(! n x) q a sin(!na)!n if n is odd, if n is even, where [ a; a] is the size of the domain, and! n is determined by ρ b! n tan(! n a)= if n is odd,! n + b tan(! na) = if n is even. The details of the derivations can be found in []. If the correlation function is not in the exponential form as in equation () and the eigenvalue problem cannot be solved analytically, the correlation function can be constructed numerically and a standard eigenvalue solver can be employed. (6) E- -.8E E E-.87E- -.87E E- -9.9E Figure : Deviation of mean solution from a parabolic profile in micro-channel flow with partially-correlated random boundary conditions at the lower wall; Upper: u-velocity, Lower:v-velocity. Here a =andre = ; also the correlation length in () is set to b =. The eigenvalues from equation () are =9:67; =:966; =:7; ::: Due to the fast decay of the eigenvalues, we usethefirsttwo terms in the Karhunen-Loeve expansion (9). A fourthorder Polynomial Chaos is used (p = ) and fifteen expansion terms (P = ) are needed. We also assume that

14 only the lower wall is considered to have the non-uniform random boundary condition. The upper wall is set to be stationary and deterministic. The variance of Gaussian process of the lower wall is ff =:. A mesh with elemnts is employed and nineth-order Jacobi polynomial is used as the basis polynomial in each element. A parabolic (no-slip) profile is used at the inlet and fully-developed conditions are asummed at the outlet Figure : Contours of rms of u velocity (upper) and v velocity (lower). Figure shows the velocity contour plot of the deviation of the mean solution at steady-state from a parabolic profile. The mean of u-veclocity remains close to the parabolic shape and the mean of v-velocity, although small in magnitude, is non-zero, in contrast to the case where boundary conditions are uniformly random or deterministic. Figure shows steady-state solutions of the rms (root-mean-square) of u and v-velocity. It is clear that theere is a developing region in the form of a boundary layer emanating at the inlet. Unlike the fully-correlatred case, here all the higher-order expansion terms are non-zero, which implies that although the random input is a Gaussian process, the solution output is not Gaussian. Since there is no analytic solution for this flow with non-uniform random boundary conditions, we employ Monte Carlo (MC) simulation to validate the Polynomial Chaos solution. The deterministic solver of the Navier-Stokes equations is used to compute the flow with each realization of the Gaussian process from the exponential correlation function () as the boundary condition at the lower wall. The MC solution is then obtained by gathering the statistics from the large ensemble of the solutions from such realizations. A brief discussion on the generation of spatially correlated Gaussian fields can be found in Appendix A. For the comparison, we plot the non-dimensionalized variance of velocities, u ms and v ms, along the centerline of the channel. It is defined as Var(u)=ff where Var(u) is defined in equation () and ff is the variance of the input Gaussian field. Figure shows the solution of u ms and v ms along the centerline from MC simulation with,, and

15 U ms. V ms.. PC MC MC MC MC. PC MC MC MC MC - - x - - x Figure : Variance of velocities along the centerline of the channel; Left: u-velocity, Right: v-velocity. realizations, together with the Polynomial Chaos (PC) solution. It is seen that the MC result converges as the number of realizations increases and the solution converges to the PC solution. The advantage of Polynomial Chaos expansion is evident here since the PC solution, with enough terms included for accurate approximation of the randomness, is equivalent to the MC solution with infinite number of realizations. While each single run of PC solution is more expensive than a single deterministic run, in this case about times more expensive asthere are terms in the expansion, it is substantially faster than the Monte Carlo solution which requires thousands of realizations in order to obtain solution statistics with comparable accuracy as the PC solutions. Note that here we employ the standard Monte Carlo simulation without any acceleration algorithms, such asvariance reduction [9].. Micro-channel Flow: Non-Gaussian Input In this section we revisit the pressure-driven micro-channel flow discussed above but with non-gaussian random boundary conditions. Non-Gaussian distributions are not uncommon in micro-fluidics where reactive ion etching techniques cause anisotropies and preferential directions. The Polynomial Chaos expansion procedure can also be applied to non-gaussian processes. The particular non-gaussian input we consider here corresponds to a lognormal distribution. This process has been considered by Ghanem in [7] and [8]. It is chosen because its projection onto the Polynomial Chaos basis can be obtained analytically. If this is not possible, as for most non-gaussian processes, a numerical projection procedure can instead be applied. The lognormal process l is defined as l(x) =e g(x) ; (7) where g is a Gaussian process. We first apply the Karhunen-Loeve expansion of equation (9) to decompose the

16 Gaussian process g(x) as follows g(x) =μg(x)+ nx i= g i (x)ο i ; (8) where g i (x) = p i ffi i (x) according to equation (9). We then expand the lognormal process l(x) by the Polynomial Chaos l(x) = PX i= l i (x)ψ i (ο): (9) The expansion coefficients can be obtained by projecting the lognormal process onto the PC basis l i (x) = <l(x)ψ i > < Ψ i > = <eg(x) Ψ i > < Ψ i > : () By using the Karhunen-Loeve expansion of g(x) (8), the above projection can be obtained analytically: where the terms < Φ i > are listed below, < Φ i >= 8 >< >: l i (x) = < Φ i > < Ψ i > μg(x)+ nx j= g i when Ψ i = ο i ; g i g j when Ψ i = ο i ο j ffi ij ; g i g j g k when Ψ i = ο i ο j ο k ο i ffi jk ο j ffi ki ο k ffi ij ; ::: :::: g j A ; () In case the process g is reduced to a random variable, the expansion of the lognormal variable can be simplified to l = exp ψ μg + ff g! PX j= ff j g j! Ψ j; () where ff g is the variance of the corresponding Gaussian variable g. A more detailed discussion about the lognormal distribution and its Polynomial Chaos approximations can be found in Appendix B... Fully-Correlated Random Boundary Conditions We first consider the case where the boundary conditions at the walls are random but spatially uniform, i.e. all grid points at the boundary are fully-correlated. The exact solution of equation () then applies. By substituting the expansion of lognormal random variable () into the formula, we find that the exact solution consists of the mean u and the random modes corresponding to the inputs from the two walls ο and ο, linearly distributed across the channel. There is no interaction between the two random walls so the cross terms, i.e., terms associated with ο i οj (i 6= j), in the expansion remain zero. Figure 6 shows the velocity modes across the channel. The standard deviations of the underlying Gaussian variables at lower and upper walls are ff = : and ff = :, respectively. Fourth-order Polynomial Chaos is employed (p = ) and the total number of expansion terms is (P = ). Only the nonzero terms, i.e., terms 6

17 ....8 u u u u u 6 u 9 u u..8.. u, u u, u u 6, u u 9, u.7.. 6E- - y - y Figure 6: Distribution of velocity random modes across the channel; Left: first- and second-order (non-zero) random modes, Right: third- and fourth-order (non-zero) random modes. associated with ο or ο only, are plotted (see table for the indices of these terms). We see the linear distribution of the different random modes across the channel. All the cross terms, i.e., terms with the form ο i οj (i 6= j), are zero as expected. The highest order terms, u and u, which are associated with the terms ο and ο, respectively, are of the order of 6. This implies that the fourth-order PC expansion is adequate in this case... Partially-Correlated Random Boundary Conditions Next we consider spatially non-uniform input. The wall boundary condition is considered as a lognormal random process as in equation (7). The underlying Gaussian process g(x) is exactly the same as the one in section.. and the same two-term Karhunen-Loeve expansion is applied to decompose g(x). Monte Carlo simulation is conducted since no exact formula is known. Again only the lower wall is considered to be random and the standard deviation of the underlying Gaussian process is ff = :. The non-dimensionalized variance of velocities, defined similarly as in section.., is plotted along the centerline of the channel in figure 7. Monte Carlo solutions corresponding to, and, realizations are plotted, together with the solution from the Polynomial Chaos expansion. Itcanbeseenthatasthenumber of realizations increases, the MC solution converges to PC solution. The convergence rate is relatively slow due to the large variance of the random input.. Flow Past a Circulat Cylinder In this section we simulate two-dimensional incompressible flow past a circular cylinder with random correlated fluctuations superimposed to the freestream. More specifically, the inflow takes the form u in =μu + g; 7

18 ... U ms. PC MC MC MC V ms. PC MC MC MC - - x - - x Figure 7: Variance of velocities along the centerline of the channel; Left: u-velocity, Right: v-velocity. where g is a Gaussian random variable or process. The size of the computational domain is [ ; ] [ 9; 9] and the cylinder is at the origin with diameter of D =. The Reynolds number Re = based on the mean value of the inflow velocity μu. The domain consists of triangular elements and the Jacobi polynomial in each element for spatial discretization is sixth-order... Fully-Correlated Gaussian Freestream In this case u in = μu + g and g = ffο is a Gaussian random variable, where ff = : is its standard deviation. This results in a one-dimensional PC expansion and we employ fourth-order PC in the simulation. Figure 8 shows the history of pressure signal at the rear stagnation point on the cylinder surface. The signal of the corresponding deterministic simulation is also plotted and it is denoted as P D in dotted line, for reference purposes. It is seen that the amplitude of the mean pressure signal is smaller compared to the deterministic signal due to diffusion induced by the randomness. The higher modes, which describe the stochastic component of the solution, all start from zero then develop gradually from P to P over time. A stationary periodic state is reached after t = in convective time units. Because only the mean and first random modes (Gaussian part) have non-zero boundary inputs, the development to non-zero value of the higher random modes is due to the interactions of the random modes through the nonlinear convective terms in equation (9). Figure 9 shows the instantaneous vorticity distribution from the deterministic simulation and the mean solution from the stochastic simulation at the same time t = 7;both simulations started with identical initial conditions corresponding to a converged periodic state at Re =. Both plots use exactly the same contour levels so that the randomness-induced diffusion in the stochastic solution can be identified in the wake of the cylinder. In figure a 8

19 snapshot of rms of vorticity for the stochastic simulation is shown at the same time t =7..8 P P P P. -. P D P Figure 8: Pressure signal of cylinder flow with uniform random inflow: ff = :. Upper: High modes. Lower: Zero mode (mean). It should be noted that this problem, although still relatively simple, is computationally much more complex than the channel problem, and the approach of Monte Carlo simulation is prohibitively expensive... Partially-Correlated Gaussian Freestream We consider now the more realistic case of partially-correlated freestream velocity. We describe this partial correlation by the exponential covariance kernel of equation () with variance ff =:. A relatively large correlation length is chosen (b = ) such that the first two eigenmodes are adequate to represent theprocess by Karhunen-Loeve expansion (9). Thus, we employ atwo-dimensional PC expansion and third-order polynomials (p = ) (see table for relevant terms). Figure shows the pressure signal, together with the deterministic signal for reference (denoted as P D in dotted line). We see that the stochastic mean pressure signal has a smaller amplitude and it is out of phase with respect to the deterministic signal. Although initially, the stochastic response follows the deterministic reponse, eventually there is a change in the Strouhal frequency as shown in figure. Specifically, the Strouhal frequency of the mean stochastic solution is lower than the deterministic one due to effective lowering of the Reynolds number induced by randomness. We now examine the response of the higher random modes. The first random mode ο is dominant and 9

20 6 y x 6 y x Figure 9: Instantaneous vorticity contours: Upper - Deterministic solution with uniform inflow; Lower - Mean solution with uniform Gaussian random inflow y x Figure : Contours of rms of vorticity field with uniform Gaussian random inflow.

21 P P P P 6 ο is subdominant. Correspondingly, in the higher modes, only the terms associated with ο, i.e., terms P ο ο and P 6 ο ο are active (see table ). These high modes are also plotted in figure (upper plot). In figure we presentvelocity profiles along the centerline for the deterministic and the mean stochastic solution at the same time instant. We see that significant quantitative differences emerge even with a relatively small % uncertainty in the freestream. In figure we plot instantaneous vorticity contours for the mean of the vorticity and compared it with the corresponding plot from the deterministic simulation. There is clearly a strong diffusive effect induced by the randomness. In figure we plot contours of the corresponding rms of vorticity. It shows that the uncertainty influences the most interesting region of the flow, i.e., the shear layers and the vortex street and not the far field..8. Pressure Signal -. Time -. P D P -. Pressure Signal Time Figure : Pressure signal of cylinder flow with non-uniform Gaussian random inflow. Upper: High modes. Lower: Zero mode (mean). Summary and Discussion We have developed a stochastic spectral method to model the uncertainty associated with boundary conditions in simulations of incompressible flows. This method can also be applied to model uncertainty in the boundary domain, e.g. a rough surface, and also to model the uncertainty associated with transport coefficients, e.g. the eddy viscosity in large eddy simulations or other transport models. It sets the foundation for a composite error bar in CFD [] that includes, in addition to the numerical error, contributions due to imprecise input to the simulation. Clearly, this uncertainty is propagated nonlinearly and the new method quantifies statistically the uncertainty in the solution. More specifically, we have employed Wiener-Hermite polynomials to represent the stochastic solution and in

22 . x Deterministic signal Mean random signal: σ=.. Spectrum Frequency Figure : Frequency spectrum for the deterministic (high peak) and stochastic simulation (low peak). U Deterministic U Mean V Deterministic V Mean x Figure : Instantaneous profiles of the two velocity components along the centerline for the deterministic and the mean stochastic solution.

23 y - x y - x Figure : Instantaneous vorticity field: Upper - Deterministic solution with uniform inflow; Lower - Mean solution with non-uniform Gaussian random inflow y - x Figure : Instantaneous contours of rms of vorticity field with non-uniform Gaussian random inflow.

24 particular to discretize the random dimension. We have verified in [] that exponential convergence is obtained for stochastic ordinary differential equations, for which exact solutions are available. On the other hand, it has been reported in the literature that Wiener-Hermite expansions may beconverging slower for certain distribution functions, e.g. for Poisson processes []. This, however, can be rectified by employing a more suitable spectral basis instead of the Hermite polynomials. For example in [] multivariate Charlier polynomials were found to best represent Poisson processes. This can be extended to many different known distributions by employing a basis from the general family of Askey polynomials [], which includes polynomials both for continuous as well as discrete processes. For example, ffl Laguerre polynomials are associated with the Gamma distribution, ffl Meixner polynomials with the Pascal distribution, ffl Krawtchouk polynomials with the Binomial distribution, ffl Jacobi polynomials with the Beta kernel, and ffl Hahn polynomials with the Hypergeometric distribution. For an arbitrary distribution any Wiener-Askey expansion would converge, in accord with the Cameron & Martin theorem [], however certain representations converge faster than others similarly to deterministic spatial expansions. As regards efficiency, a single Polynomial Chaos based simulation, albeit computationally more expensive than the deterministic Navier-Stokes solver, is able to generate the solution statistics equivalent tothemonte Carlo simulation. In the latter, thousands of realizations are required for converged statistics, which is prohibitively expensive for most CFD problems in practice. For example, to obtain converged statistics using the standard Monte Carlo approach for the two-dimensional flow past a cylinder that we presented in section. it would require currently morethana year of computation on a standard workstation compared to about six days for the Polynomial Chaos simulation. Of course a reduced variance version could be employed and also the Monte Carlo simulation is embarassingly parallel, and thus it can be accelerated greatly. However, the Polynomial Chaos based stochastic simuilation can also be done in parallel. One possible strategy is to distribute one random mode per processor, as the bulk of the work is parallel due to linearity, and transpose the data to perform the nonlinear products in a separate step. Even with a modest parallel efficiency, the aforementioned computation would be completed in less than a day on a fifteen-processor computer.

25 A Generation of Correlated Gaussian Random Fields While numerous routines are available for generating independent Gaussian random variables, it is often needed to generate a correlated Gaussian field (vector) with given correlation structure. In general, this can be performed by means of: () Spectral representation; () ARMA (Auto-Regressive Moving Average) modeling; and () Covariance matrix decomposition procedures. There are two approaches for covariance matrix decomposition methods: The Cholesky decomposition method and the modal decomposition method. In this section we briefly review the Cholesky decomposition method [] and the first-order ARMA method. A. The Cholesky Decomposition Method Without loss of generality, we consider the normal random vector x = (x ;x ; ;x n ) with zero mean, i.e., each x i (i =;:::;n) is a Gaussian random variable with zero mean. Let us denote its covariance matrix as C = We then call x the normal random vector of N(; C). 6 ff ff n..... ff n ff nn 7 : () Random field Independent Gaussian field Correlation length = Correlation length = 7 index Figure 6: Correlated Gaussian random field with exponential correlation structure (correlation length is nondimensionalized by the size of the domain) Let y be distributed N(; I n ) where I n is the unit matrix of size n, i.e., y is an independent (uncorrelated)

26 Gaussian random field. Let also x = Ay, then x is distributed N(; AA T )wherea T is the transpose of A. This result is a special case of a more general theorem by Anderson []. The problem of generating Gaussian random field with given correlation structure C is then transformed to the problem of finding matrix A such that AA T = C. Since matrix C is real and symmetric, this can be readily done by the Cholesky's decomposition, where A takes the form of a lower-triangular matrix: a i = ff i = p ff ;» i» n; a ii = q h P j ff ij ff ii P i k= a ik ; <i» n; i a ij = a k= ika jk =a jj ; <j<i» n; a ij = ; i<j» n: () Once the a ij 's have been determined according to the given ff ij 's, we need to generate an n-dimensional independent Gaussian vector y, and subsequently perform the transformation x = Ay. The resulting Gaussian random vector x will have the desired correlation structure C = fff ij g. Figure 6 shows the correlated Gaussian field of the exponential correlation function defined in () with unit variance. The correlation length is non-dimensionalized by the size of the domain. The two results with correlation length of and are both obtained by applying the above Cholesky decomposition method to the same independent Gaussian field also shown in the figure. We see that the original field is uncorrelated and by increasing the correlation length the correlation structure of the field becomes stronger. A. First-Order ARMA Process Method Another way of generating correlated Gaussian field is based on the idea of employing the first-order Auto-regressive (AR()) process, also called the Markov process. This method works specifically for the exponential covariance function () C(x ;x )=e jx x j b ; () where b is the correlation length. The stationary AR() process is defined as, x i = ffx i + ff y y i ; jffj < ; (6) where fy i g is the set of independent Gaussian variables with zero mean and unit variance. It can be shown that the stationary AR() process has the following properties []: E(x i ) = ; (7) Var(x i ) = ff y ff ; (8) C(x i ;x i±k ) = ff jkj ; k =; ±; ±;::: (9) 6

27 It is then straightforward to show that by choosing the parameters ff = e b ; ff y = p ff ; () the resulting random vector x = fx i g is a Gaussian random field with the exponential covariance kernel (equation ()). B Lognormal Distribution and its Approximations While the normal distribution arises from the sum of many small effects according to the Central Limit Theorem, the lognormal distribution arises as the result of a multiplicative mechanism. Let X denote a Gaussian random variable, the lognormal random variable is obtained by taking the exponential of X, Y = e X ; X =lny: () A random variable Y whose logarithms are normally distributed is said to have the lognormal distribution. The PDF (Probability Density Function) of the Gaussian variable X is: f X (x) = ff X p ß exp " x mx ff X # ;» x»; () where m X and ff X are the mean and variance of the Gaussian random variable X, respectively. Equation () defines a one-to-one monotonic transformation. It is well known that when such transformation in the form of Y = g(x) exists, the PDF of the new random variable Y is: f Y (y) = fi dg Therefore, upon substitution the PDF of the lognormal distribution is: f Y (y) = yff X p ß exp The moments of lognormal distribution can be calculated as follows [6]: fi fi fi fi fififi dy (y) f X (g (y)): () " ln y mx ff X # ; y : () <Y r >= e rmx e r ff X ; r =; ;::: () The Polynomial Chaos expansion to the lognormal random variable is given in equation (). Y =exp m X + ff X PX j= ff j X j! Ψ j; (6) 7

28 Upon substituting the one-dimensional Polynomial Chaos (equation (7)) we obtain the first few terms of the approximation:» Y = c +ff X ο + ff X! (ο ) + ff X! (ο ο)+ ; c = e mx + ff X : (7) The Taylor expansion-like formula approximates the lognormal distribution with a Gaussian distribution and higherorder non-gaussian correction terms. The PDF of the Polynomial Chaos approximation can be obtained via several common techniques such as the method of Cumulative Distribution Function (CDF) [7]:. First-order approximation: Y = c( + ff X ο); (8) f (y) = cff X f(ff); ff = y c : (9). Second-order approximation: where» Y = c +ff X ο + ff X! (ο ) ; (6) f (y) = cff X r [f(ff )+f(ff )] ; (6) ff ; = q ( ± r); r = +ffx ff +y=c: X. Third-order approximation: where r =» Y = c + ff X! (ο ) + ff X! (ο ο) ; (6)» f (y) = ffx + ff X ( ff X ) cff X b b + y=c r q 6y=c +9(y=c) ff X +ff X ff6 X ; b = ff X ( y=c + r) = ; f(ff); (6) and ff = + ff X ffx ff X b b : ffx In the above equations, the function f(x) = p ß e x = is the PDF for the standard Gaussian variable ο which has zero mean and unit variance. Figure 7 shows the PDFs of the first-, second- and third-order PC approximations to the lognormal distribution, together with the exact lognormal PDF. The results are obtained with m X = and ff X =:. It can be seen that the first-order approximation (8) is symmetric because it only contains the Gaussian term. As more high-order terms are added, the PDFs become asymmetric and the third-order expansion gives a very good approximation. 8

29 .. exact st order nd order rd order Figure 7: Probability density functions (PDF) of the lognormal distribution and its st-, nd- and rd-order Polynomial Chaos approximations. Acknowledgements We would like to thank Dr. M. Jardak, D. Lucor, Prof. C.-H. Su and Prof. R. Ghanem for useful discussions. This work was supported by ONR and computations were performed at Brown's TCASCV and NCSA's (University of Illinois) facilities. 9

30 References [] Workshop on Validation and Verification of Computational Mechanics Codes. Technical report, Caltech, December 998. [] Workshop on Predictability of Complex Phenomena, Los Alamos, 6-8 December 999. Technical report. [] Workshop on Decision Making Under Uncertainty, IMA, 6-7 September 999. Technical report. [] T.J. Oden, W. Wu, and M. Ainsworth. An a posteriori error estimate for finite element approximations of the Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng., :8, 99. [] L. Machiels, J. Peraire, and A.T. Patera. A posteriori finite element output bounds for the incompressible Navier-Stokes equations; application to a natural convection problem. J. Comp. Phys., to appear,. [6] R.G. Hills and T.G. Trucano. Statistical validation of engineering and scientific models: Background. Technical Report SAND99-6, Sandia National Laboratories, 999. [7] R.G. Ghanem. Stochastic finite elements for heterogeneous media with multiple random non-gaussian properties. ASCE J. Eng. Mech., ():6, 999. [8] R.G. Ghanem. Ingredients for a general purpose stochastic finite element formulation. Comp. Meth. Appl. Mech. Eng., 68:9, 999. [9] T.D. Fadale and A.F. Emery. Transient effects of uncertainties on the sensitivities of temperatures and heat fluxes using stochastic finite elements. J. Heat Trans., 6:88 8, 99. [] R.G. Ghanem and P. Spanos. Stochastic finite elements: a spectral approach. Springer-Verlag, 99. [] W.K. Liu, G. Besterfield, and A. Mani. Probabilistic finite elements in nonlinear structural dynamics. Comp. Meth. Appl. Mech. Eng., 6:6 8, 986. [] W.K. Liu, A. Mani, and T. Belytschko. Finite element methods in probabilistic mechanics. Probabilistic Eng. Mech., ():, 987. [] M. Shinozuka and E. Leone. A probabilistic model for spatial distribution of material properties. Eng. Fracture Mech., 8:7 7, 976. [] M. Shinozuka. Structural response variability. J. Eng. Mech., (6):8 8, 987.