materials process design and control laboratory a stabilized stochastic finite element second-order...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


A stabilized stochastic finite element second-order

projection method for modeling natural convection in random

porous media

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu

Xiang Ma and Nicholas Zabaras

http://mpdc.mae.cornell.edu/

http://mpdc.mae.cornell.edu/




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




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




Problem of interest



= 1

u = v =0

= 0

u = v =0

= 0, u = v =0n

= 0, u = v =0n

Natural convection in random porous media

Deterministic governing equation:

Pr -Prandtl number

Ra- Raleigh number

Da- Darcy number




Problem of interest



= 1

u = v =0

= 0

u = v =0

= 0, u = v =0n

= 0, u = v =0n

( )w

Natural convection in random porous media

Modeling porosity as a Gaussian random field

( )w

Stochastic governing equation:




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.




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.




Let us review first order method is


1/ 2 1/ 21

( ) ( ) 0n n nu u N u udt

1 1/ 2 1

1

1( ) 0

0

n n n

n

u u pdt

u

2 1 1/ 21n np udt

1 1/ 2 1

1

1 1/ 2

1( ) 0

0

0

n n n

T n

n T n

M U U GPdt

G U

dtLP G U

1 1 1( ) 0T n T nG U dt L G M G P

solve for intermediate velocity:

projection step: eliminate : 1nu

Once a finite element discretization is performed, the matrix form is

eliminate : 1nu

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.

1TL G M G





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)

http://www.math.tamu.edu/~guermond/PUBLICATIONS/guermond_quartapelle_IJNMF_1998.pdf







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)

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).

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V29-3YF3VG9-J&_user=492137&_coverDate=02/18/2000&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000022719&_version=1&_urlVersion=0&_userid=492137&md5=a68c6b3d3cbb3a69870becdb9e0c92ca






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.

Kh K hu

1c 2c 1 4c 2 2c

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.

p




Pressure stabilized formulation

A matrix version of this form is

( , ) ( , ) ( , ) 0,

( , ) ( , ) 0 h h h h h h h

h h h h h h

p q q q v q Q

p v v v V

0 0

0

T TLP G G U

GP M

weak form:

Eliminating from equation, it is found that

Let’s see how it works:

10( ) 0T TL G M G P G U

which is similar to 1 1 1( ) 0T n T nG U dt L G M G P

So this stabilized method mimics the stable effect of the first-order method. Thus it allows the equal-order interpolation and stabilize the pressure.




Stabilized second-order projection method

Step 1: Solve for the intermediate velocity in the momentum equation: 1/ 2nv

(1)

Fully decoupled, can be solved one by one.

Step 2: Projection step

Step 2: update step

(2)

(3)

(4)

(5)




A numerical example

Free fluid

Porous Medium

= 1

u = v =0

= 0

u = v =0

= 0, u = v =0n

= 0, u = v =0n

l 1 w

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.

[0,1] [0,1]

l 0.3l

w

61 1076.665 10

50 50 55 10




A numerical example

-9

-8

-7

-6-5

-4

-4

-3

-3

-1

-1

1

2

2

4

-5.48

-1.73

-1.73

-0.0

9

-0.09

-0.0

9

-0.09

-0.0

1

-0.01

-0.01

Streamline Isothermal

First-order solution

Second-order solution

The Streamline pattern is symmetric




A numerical example

65000055000045000035000025000015000050000

806040200

-20-40-60

180160140120100806040200

-20

180000160000140000120000100000800006000040000200000

403020100

-10-20-30-40

50403020100

-10-20-30-40-50

First-order solution

Second-order

solution

u velocity v velocity Pressure

The velocities are mainly distributed in the free fluid region




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




Karhunen-Loeve expansion

1

( , , ) ( , ) ( ) ( )i i ii

X x t X x t f x

Stochastic process

Mean function

ON random variablesDeterministic 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.

( , )X tx 1 2( , )R x x

1{ }i i

i ( )if x

1 2 1 1 2( , ) ( ) ( )i i iDR x x f x dx f x

1( , , ) fn( , , , , )NX x t x t




Generalized polynomial chaos

Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input

0

( , , ) ( , ) (ξ( ))i ii

Z x t Z x t

Stochastic output

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NX x t x t

Askey polynomials ~ type of input stochastic process

Usually, Hermite (Gaussian), Legendre (Uniform), Jacobi etc.




Generalized polynomial chaos

The polynomial chaos forms a complete orthogonal basis in the of the Gaussian random variables, i.e.

2L

2

ii j ij

where denotes the expectation or ensemble average. It is the inner product in the Hilbert space of the Gaussian random variables

,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f g f g dP f g W d ξ ξ ξ ξ ξ ξ ξ ξ ξ

where the weighting function has the form of the multi-dimensional independent Gaussian probability distribution with unit variance.

( )W ξ

1 dimension:21

21

1( )

2W e

2 dimensions:2 21 2

1 1

2 22 1 2 1 1 1 2

1( ) ( ) ( )

2W W W e e

n dimensions:2 21

1( )

21 1 1 1

1( ) ( ) ( )

(2 )

n

n n n nW W W e




PC computation

PC computation (B. Debusschere et al. 2004, Ma & Zabaras, 2007):

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.

kc

kc

a b

c

http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=SJOCE3000026000002000698000001&idtype=cvips&prog=normal



http://mpdc.mae.cornell.edu/Publications/PDFiles/FlowInRandomPorousMedia.pdf

http://mpdc.mae.cornell.edu/Publications/PDFiles/FlowInRandomPorousMedia.pdf




PC computation

Quadratic product: Thus we can obtain:

with

A similar procedure could also be used to determine the product of

three stochastic variables d abc

with

Triplet product:




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

( , )f x

Using the same method as before (Galerkin projection), we have

High dimension integration, use

Latin Hypercube sampling.




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:

( , ) x

0

1ˆ ( )

P

i ii

ξ2

20

(1 )( )

P

i ii

ξ 2

0

1( )

P

i ii

ξ

Since the input uncertainty is taken as a Gaussian random field, we use Hermite polynomials to represent the solution:







Substitute the above equation into Eqs (1)-(5)and then performing a Galerkin projection of each equation by , k







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

nv

0nv

k

kv







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.




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).

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.

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.

http://acts.nersc.gov/petsc/

http://acts.nersc.gov/petsc/




Natural convection in heterogeneous media: large dimension

= 1

u = v =0

= 0, u = v =0n

= 0, u = v =0n

( )w = 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.

61 10

[0,1] [0,1]

76.665 10

30 30 31 10

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)




Natural convection in heterogeneous media

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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





0.0140.0130.0120.0110.010.0090.0080.0070.0060.005

0.010.0080.0060.0040.0020

-0.002-0.004-0.006-0.008-0.01

0.0080.0060.0040.0020

-0.002-0.004-0.006-0.008

Eigenmodes for a 2-D domain

Mode 1 Mode 2

Mode 4Mode 6

0.0060.0040.0020

-0.002-0.004-0.006

( )f x





0.6120.6080.6040.60.5960.5920.5880.5840.58

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.58

0.59

0.6

0.61

0.6120.6080.6040.60.5960.5920.5880.5840.58

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.58

0.59

0.6

0.61

Two realizations of the porosity fields with 6 terms:





We have already obtained the KLE of the Gaussian random fields.

1

( , ) ( , ) ( ) ( )M

i i ii

f

x x x

Now We want to express the following non-polynomial functions in terms of polynomial chaos.

0

1ˆ ( )

P

i ii

ξ2

20

(1 )( )

P

i ii

ξ 2

0

1( )

P

i ii

ξ

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.





1.5 1.6 1.7 1.80

2

4

6

8

10

order 1order 2order 3Direct Sampling

P

roba

bili

ty

1

( )

0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7 order 1order 2order 3Direct Sampling

P

roba

bili

ty

2

2

(1 ( ))

( )

0.8 1 1.2 1.40

1

2

3

4 order 1order 2order 3Direct Sampling

P

roba

bili

ty

2

1 ( )

( )

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.





Mean

7531

-1-3-5-7

86420

-2-4-6-8

0.950.850.750.650.550.450.350.250.150.05

650055004500350025001500500

u v

p





1u2u 3u

4u 5u 6u

0.350.250.150.05

-0.05-0.15-0.25-0.35

0.180.140.10.060.02

-0.02-0.06-0.1-0.14

0.160.120.080.040

-0.04-0.08-0.12-0.16

0.090.070.050.030.01

-0.01-0.03-0.05-0.07-0.09

0.070.050.030.01

-0.01-0.03-0.05-0.07

0.070.050.030.01

-0.01-0.03-0.05-0.07

First order u velocity





0.40.30.20.10

-0.1-0.2-0.3-0.4

1v

0.10.060.02

-0.02-0.06-0.1-0.14

0.140.10.060.02

-0.02-0.06-0.1-0.14-0.18

2v 3vFirst order v velocity

0.10.080.060.040.020

-0.02-0.04-0.06-0.08-0.1

0.080.060.040.020

-0.02-0.04-0.06-0.08

0.080.060.040.020

-0.02-0.04-0.06-0.08

4v 5v 6v





0.0080.0060.0040.0020

-0.002-0.004-0.006-0.008

0.00060.0002

-0.0002-0.0006-0.001-0.0014-0.0018

0.0010

-0.001-0.002-0.003-0.004-0.005

0.00140.0010.00060.0002

-0.0002-0.0006-0.001-0.0014

0.00160.00120.00080.00040

-0.0004-0.0008-0.0012-0.0016

0.00060.00040.00020

-0.0002-0.0004-0.0006

1 2 3

4 5 6

First order Temperature





0.0050.0040.0030.0020.0010

-0.001-0.002-0.003

0.0010.00080.00060.00040.00020

-0.0002-0.0004-0.0006-0.0008-0.001-0.0012

0.00060.00040.00020

-0.0002-0.0004-0.0006

0.0050.0040.0030.0020.0010

-0.001-0.002-0.003-0.004

0.00060.00040.00020

-0.0002-0.0004-0.0006-0.0008

0.0010.00080.00060.00040.00020

-0.0002-0.0004-0.0006

6E-052E-05

-2E-05-6E-05-0.0001-0.00014-0.00018

3.5E-052.5E-051.5E-055E-06

-5E-06-1.5E-05-2.5E-05-3.5E-05

8E-066E-064E-062E-060

-2E-06-4E-06-6E-06-8E-06

9u

9v

9

14u

14

14v

25u

25

25v

Second order Modes





5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

SSFEMMC

-0.5 0 0.50

0.5

1

1.5

2

2.5

3

SSFEMMC

velocityv

P

roba

bili

ty

velocityu

P

roba

bili

ty

P

roba

bili

ty

Temperature0.64 0.65 0.66 0.67

0

20

40

60

80

SSFEMMC

PDF at one point





0.380.340.30.260.220.180.140.10.060.02

0.460.420.380.340.30.260.220.180.140.10.060.02

0.00850.00750.00650.00550.00450.00350.00250.00150.0005

4642383430262218141062

Standard deviation

u v

p





7531

-1-3-5-7

0.950.850.750.650.550.450.350.250.150.05

0.950.850.750.650.550.450.350.250.150.05

7531

-1-3-5-7

0.950.850.750.650.550.450.350.250.150.05

7531

-1-3-5-7

GPCE

Collocation

Monte Carlo

Compare Mean





0.380.340.30.260.220.180.140.10.060.02

0.00850.00750.00650.00550.00450.00350.00250.00150.0005

0.380.340.30.260.220.180.140.10.060.02

0.380.340.30.260.220.180.140.10.060.02

0.00850.00750.00650.00550.00450.00350.00250.00150.0005

0.00850.00750.00650.00550.00450.00350.00250.00150.0005

GPCE

Collocation

Monte Carlo

Compare standard deviation




3-D Natural convection in heterogeneous media

0.5950.59450.5940.59350.5930.59250.5920.59150.5910.59050.59

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 2( ) 0.05 exp( )

10

rR r

KLE is truncated after two terms, so the stochastic dimension is 2.




3-D Natural convection in heterogeneous media

1.4 1.6 1.8 2 2.2 2.40

0.5

1

1.5

2

2.5

3order 1order 2order 3order 4Direct Sampling

P

roba

bili

ty

1

( )

P

roba

bili

ty

2

2

(1 ( ))

( )

0.4 0.8 1.20

0.5

1

1.5

2

2.5

order 1order 2order 3order 4Direct Sampling

1 2 30

0.2

0.4

0.6

0.8

1

1.2

1.4

order 1order 2order 3order 4Direct Sampling

P

roba

bili

ty

2

1 ( )

( )

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.




Mean slice of the xz plane at y=0.5

XY

Z

0.950.850.750.650.550.450.350.250.150.05

XY

Z

0.950.850.750.650.550.450.350.250.150.05

Collocation

GPCE

Left: GPCE

Right: Collocati

on




Standard deviation slice of the xz plane at y=0.5

0.43 0.44 0.45 0.46 0.47 0.48 0.490

10

20

30

40

P

roba

bili

ty

Temperature-0.3 -0.2 -0.1 0

0

2

4

6

8

10

P

roba

bili

ty

velocity

u

GPCE

Collocation

Left: GPCE

Right: Colloca

tion

PDF at one

point




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.

materials process design and control laboratory a stabilized stochastic finite element second-order...

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