1. Materials Process Design and Control Laboratory A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.eduhttp://mpdc.mae.cornell.edu Xiang Ma and Nicholas Zabaras

2. Materials Process Design and Control Laboratory Outline of the presentation 1.Motivation and problem definition 2.A stabilized second-order projection method 3.Representation of stochastic processes 4.Stochastic finite element method formulation 5.Numerical examples

3. Materials Process Design and Control Laboratory Transport in heterogeneous media - Thermal and fluid transport in heterogeneous media are ubiquitous, e.g. solidification - Range from large scale systems (geothermal systems) to the small scale - Complex phenomena - How to represent complex structures? - How to make them tractable? - Are simulations believable? - How does uncertainty propagate through them? To apply physical processes on these heterogeneous systems - worst case scenarios - variations on physical properties

4. Materials Process Design and Control Laboratory Problem of interest = 1 u = v =0 = 0 u = v =0 = 0, u = v =0 Natural convection in random porous media Deterministic governing equation: Pr -Prandtl number Ra- Raleigh number Da- Darcy number

5. Materials Process Design and Control Laboratory Problem of interest = 1 u = v =0 = 0 u = v =0 = 0, u = v =0 Natural convection in random porous media Modeling porosity as a Gaussian random field Stochastic governing equation:

6. Materials Process Design and Control Laboratory Problem of interest: Some difficulties solid Mushy zone q liquid ~ 10 -2 m This model is important in alloy solidification process. To understand the uncertainty propagation in this process can help understand error in the experiment results. A SUPG/PSPG formulation has been proposed to solve this complex problem. ( Deep & Zabaras, 2004). However, due to its coupling between velocity and pressure, it resulted in a very large linear system. very hard to extend to stochastic formulation due to the high degree of freedom. Especially, when dealing with real porous media, it results in a very high stochastic dimension.

7. Materials Process Design and Control Laboratory Problem of interest: Some difficulties and solutions  The projection method for the incompressible Navier-Stokes equations, also known as the fractional step method or operator splitting method, has attracted widespread popularity.  The reason for this lies on the decoupling of the velocity and pressure computation.  It was first introduced by Chorin, as the first-order projection method ( also called non-incremental pressure-correction method). Later, it was extended to the second-order scheme ( also called incremental pressure-correction scheme ) in which part of the pressure gradient is kept in the momentum equation.  Le Maitre et.al(2002) have developed a stochastic projection method to model natural convection in a closed cavity based on first order method.  It is really helpful when solving the high stochastic dimension problem.

8. Materials Process Design and Control Laboratory  Let us review first order method is Problem of interest: Some difficulties and solutions solve for intermediate velocity: projection step: eliminate : Once a finite element discretization is performed, the matrix form is eliminate :  The term may be understood as the difference between two discrete Laplacian operators computed in difference manner. This matrix turns out to be positive semi-definite, which increases the stability of the numerical method, thus allows equal-order interpolation.

9. Materials Process Design and Control Laboratory Problem of interest: Some difficulties and solutions  For the porous media problem considered here, due to the porosity dependence of the pressure gradient term in the momentum equation, we cannot omit the pressure term as is the case in the first-order projection method. We need to use second-order scheme. porosity dependence  However, if using FEM, second-order projection method is known that it exhibits spatial oscillation in the pressure field if not using a mixed finite element formulation for velocity and pressure. (J.L.Guermond et al. 1998) (J.L.Guermond et al. 1998)

10. Materials Process Design and Control Laboratory Problem of interest: Some difficulties and solutions  In order to utilize the advantage of the incremental projection method which retains the optimal space approximation property of the finite element and allows equal-order finite element interpolation, Codina and co-workers developed a pressure stabilized finite element second- order projection formulation for the incompressible Navier-Stokes equation. (Codina et al.2000) (Codina et al.2000)  The method consists in adding to the incompressibility equation the divergence of the difference between the pressure gradient and its projection onto the velocity space, both multiplied by algorithmic parameters defined element-wise. pressure field in a lid-driven cavity problem with stabilization (left) and without stabilization (right).

11. Materials Process Design and Control Laboratory Pressure stabilized formulation  We take these parameters as where is the viscosity, is the local size of the element, is the local velocity and and are algorithmic constants, that we take as and for linear elements.  Accordingly, the continuity equation is modified as follows: where is the stabilized parameter as discussed before and the new auxiliary variable is the projection of the pressure gradient onto the velocity space.

12. Materials Process Design and Control Laboratory Pressure stabilized formulation A matrix version of this form is weak form: Eliminating from equation, it is found that Let’s see how it works: which is similar to  So this stabilized method mimics the stable effect of the first-order method. Thus it allows the equal-order interpolation and stabilize the pressure.

13. Materials Process Design and Control Laboratory Stabilized second-order projection method Step 1: Solve for the intermediate velocity in the momentum equation: Fully decoupled, can be solved one by one. Step 2: Projection step Step 2: update step

14. Materials Process Design and Control Laboratory A numerical example Free fluid Porous Medium = 1 u = v =0 = 0 u = v =0 = 0, u = v =0  The computation region is a square domain.  The dimensionless length is taken as  The wall porosity is taken as 0.4. The porosity increases linearly from 0.4 at the wall to 1.0 (pure liquid) at the core.  The Rayleigh number is, Prandtl number is 1.0 and the Darcy number is  The problem was solved with a mesh discretization and the time step was chosen as  The simulation was run up to non-dimensional time 1.0.

15. Materials Process Design and Control Laboratory A numerical example Streamline Isothermal First-order solution Second-order solution The Streamline pattern is symmetric

16. Materials Process Design and Control Laboratory A numerical example First- order solution Second -order solution u velocity v velocity Pressure The velocities are mainly distributed in the free fluid region

17. Materials Process Design and Control Laboratory Representation of stochastic processes  For special kinds of stochastic processes that have finite variance-covariance function, we have mean-square convergent expansions Series expansions  Known covariance function  Unknown covariance function  Best approximation in mean-square sense  Useful typically for input uncertainty modeling  Can yield exponentially convergent expansions  Used typically for output uncertainty modeling Karhunen-Loeve Generalized polynomial chaos

18. Materials Process Design and Control Laboratory Karhunen-Loeve expansion Stochastic process Mean function ON random variables Deterministic functions  is the mean of the process and let denote its covariance function, which is needed to be known a priori.  is a set of i.i.d. random variables, whose distribution depends on the type of stochastic process.  and are the eigenvalues and eigenvectors of the covariance kernel. That is,they are the solution of the integral equation:  In practice, we truncate KLE to first M terms.

19. Materials Process Design and Control Laboratory Generalized polynomial chaos  Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input Stochastic output Askey polynomials in input Deterministic functions Stochastic input  Askey polynomials ~ type of input stochastic process  Usually, Hermite (Gaussian), Legendre (Uniform), Jacobi etc.

20. Materials Process Design and Control Laboratory Generalized polynomial chaos  The polynomial chaos forms a complete orthogonal basis in the of the Gaussian random variables, i.e. where denotes the expectation or ensemble average. It is the inner product in the Hilbert space of the Gaussian random variables where the weighting function has the form of the multi-dimensional independent Gaussian probability distribution with unit variance. 1 dimension: 2 dimensions: n dimensions:

21. Materials Process Design and Control Laboratory PC computation PC computation (B. Debusschere et al. 2004, Ma & Zabaras, 2007):B. Debusschere et al. Ma & Zabaras  Quadratic product:  Consider two random variables, and, with their respective GPCE:  We want to find the coefficients of :  The coefficient are obtained with a Galerkin projection, which minimizes the error of the resulting PC representation within the space spanned by the basis function up to order P.

22. Materials Process Design and Control Laboratory PC computation  Quadratic product:  Thus we can obtain: with  A similar procedure could also be used to determine the product of three stochastic variables with  Triplet product:

23. Materials Process Design and Control Laboratory PC computation Sampling approach for general nonpolynomial function evaluations:  We consider a general nonlinear function, where  is We need to express this function as Thus we obtain  Using the same method as before (Galerkin projection), we have High dimension integration, use Latin Hypercube sampling.

24. Materials Process Design and Control Laboratory Stochastic Finite Element Formulation  Since there are nonlinear functions of in the governing equation, we need to first express them in the polynomial basis using the method discussed before:  Since the input uncertainty is taken as a Gaussian random field, we use Hermite polynomials to represent the solution:

25. Materials Process Design and Control Laboratory Stochastic Finite Element Formulation  Substitute the above equation into Eqs (1)-(5)and then performing a Galerkin projection of each equation by

26. Materials Process Design and Control Laboratory Stochastic Finite Element Formulation  In the non-linear drag term we assume that the magnitude of the velocity is determined by the mean velocity  From the pressure update equation the stabilized parameter is determined by the kth coefficient of the spectral expansion. So we denote it as to emphasize that it is a function of the k-th coefficient

27. Materials Process Design and Control Laboratory Stochastic Finite Element Formulation  It is time consuming to evaluate the fourth order product term directly using the method discussed before. To simplify this calculation, we introduce an auxiliary random variable as follows: such that So our term of interest can be reduced to This form is now easier to calculate, since it only evolves third-order product terms, which is pre-computed.

28. Materials Process Design and Control Laboratory Some computational details 1.All calculations are performed using numerical algorithms provided in PETSc. (Portable, Extensible Toolkit for Scientific computation). The default Krylov solver was used for the linear solver). PETSc 2.For the numerical computation of KLE, the SLEPc is used. ( Scalable Library for Eigenvalue Problem Computation). It is based on the PETSc data structure and it is highly parallel.SLEPc 3.In order to further reduce the computational cost, we solve each expansion coefficient individually. That means we split the systems of equations into (P+1)(3d+2) deterministic scalar equations. 4.The computation is performed using 30 nodes on a local Linux cluster.

29. Materials Process Design and Control Laboratory Natural convection in heterogeneous media: large dimension = 1 u = v =0 = 0, u = v =0 = 0 u = v =0  The porosity is modeled as a Gaussian random filed. And the correlation function is extracted from a real porous media – sandstone 1. The mean is 0.6.  The computation region is a square domain.  The Rayleigh number is, Prandtl number is 1.0 and the Darcy number is  The problem was solved with a mesh discretization and the time step was chosen as  The simulation was run up to non-dimensional time 1.0. Alloy solidification, thermal insulation, petroleum prospecting Look at natural convection through a realistic sample of heterogeneous material 1. Reconstruction of random media using Monte Carlo methods, Manwat and Hilfer, Physical Review E. 59 (1999)

30. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Material: Sandstone Numerically computed Eigen spectrum Experimental correlation for the porosity of the sandstone. Eigen spectrum is peaked. Requires large dimensions to accurately represent the stochastic space KLE is truncated after first 6 terms. So the stochastic dimension is 6. Correlation function

31. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Eigenmodes for a 2-D domain Mode 1 Mode 2 Mode 4 Mode 6

32. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Two realizations of the porosity fields with 6 terms:

33. Materials Process Design and Control Laboratory Natural convection in heterogeneous media  We have already obtained the KLE of the Gaussian random fields.  Now We want to express the following non-polynomial functions in terms of polynomial chaos.  We can evaluate the coefficients of the expansions for the three non- linear functions using LHS. Then we can sample the calculated GPCE expansion to obtain the PDF.  The optimal order is determined such that the GPCE expansion can accurately represent the PDF of these non-linear functions.  We compare the results with the “Direct Sampling” approach, where the PDF of the functions is obtained by sampling form the standard normal distribution, then calculate the realization of each sample and plot the PDF.

34. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Probability A second-order GPCE is enough to capture all of the input uncertainties Thus, we choose a 6- dimension second order GPCE expansion to represent the solution process, which give a total of 28 modes.

35. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Mean

36. Materials Process Design and Control Laboratory Natural convection in heterogeneous media First order u velocity

37. Materials Process Design and Control Laboratory Natural convection in heterogeneous media First order v velocity

38. Materials Process Design and Control Laboratory Natural convection in heterogeneous media First order Temperature

39. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Second order Modes

40. Materials Process Design and Control Laboratory Natural convection in heterogeneous media velocity Probability velocity Probability T emperature PDF at one point

41. Materials Process Design and Control Laboratory Natural convection in heterogeneous media Standard deviation

42. Materials Process Design and Control Laboratory Natural convection in heterogeneous media GPCE Collocation Monte Carlo Compare Mean

43. Materials Process Design and Control Laboratory Natural convection in heterogeneous media GPCE Collocation Monte Carlo Compare standard deviation

44. Materials Process Design and Control Laboratory 3-D Natural convection in heterogeneous media One realization of the random porosity random field in 3-D contour porosity iso-surface porosity slice of the xz plane at y=0.5 The definition is the same as 2-D, except here we consider a covariance function KLE is truncated after two terms, so the stochastic dimension is 2.

45. Materials Process Design and Control Laboratory 3-D Natural convection in heterogeneous media Probability A second-order GPCE is not enough to capture all of the input uncertainties. At least, we need a third-order GPCE. Thus, we choose a 2- dimension third order GPCE expansion to represent the solution process, which give a total of 10 modes.

46. Materials Process Design and Control Laboratory Mean slice of the xz plane at y=0.5 Collocation GPCE Left: GPCE Right: Collocati on

47. Materials Process Design and Control Laboratory Standard deviation slice of the xz plane at y=0.5 Probability Temperature Probability velocity GPCE Collocation Left: GPCE Right: Colloca tion PDF at one point

48. Materials Process Design and Control Laboratory Conclusions  A second-order stabilized stochastic projection method was used to model uncertainty propagation in natural convection in random porous media.  For the problems examined, the computation cost of the SSFEM and sparse grid simulation were similar. The MC simulations were significantly more expensive. In the SSFEM, it was shown that it is rather easy to identify dominant stochastic models in the solution and investigate how the uncertainty propagates from porosity to the velocity and temperature random fields.  The key ingredient in the implementation of the algorithms presented here includes the development of a stochastic modeling library based on the GPCE formulation. This library includes computation tools for the implementation of the Askey polynomials, a parallel K-L expansion eigen solver, and a post-processing class for calculation of higher-order solution statistics such as standard deviation and probability density function.