Materials Process Design and Control Laboratory

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


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

169 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

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

NICHOLAS ZABARAS

and

B. VELAMUR ASOKAN

A STOCHASTIC VARIATIONAL MULTISCALE METHOD WITH EXPLICIT

SUBGRID MODELING FOR ADVECTION-DIFFUSION SYSTEMS




OUTLINE

Stochastic variational multiscale method

Uncertainty modeling – Overview and techniques

Stochastic advection-diffusion systems – ASGS modeling

Stochastic fluid-flow – ASGS modeling

Stochastic natural-convection – capturing unstable equilibrium

Stochastic multiscale elliptic equation

Stochastic multiscale advection-diffusion

Extensions and future research




MULTISCALE TRANSPORT SYSTEMS

Solidification Fluid-flow Diffusion in composites

Presence of various spatial and temporal length scales

Varied application areas – Engineering, Material science and other

Uncertainty manifests from impreciseness in boundary condition, material property specification and other modeling assumptions




IMPORTANCE OF UNCERTAINTY

Component

Meso

micro

Only statistical description of

material properties possible

Imprecise knowledge of

governing model

Imprecise boundary

conditions, initial

perturbations

Propagation and interaction of uncertainties have to be resolved




MODELING ASPECTS OF UNCERTAINTY

Probabilistic interpretation – Imprecise knowledge about boundary conditions, governing models, material properties described using stochastic processes

Uncertainty due to codes, algorithms, machine precision are not considered here

Physically, the uncertainty at progressively finer scales is higher [fluctuations]

Ideally, our computation paradigm should reflect above consideration

Computational approach should be orders of magnitude faster than other uncertainty analysis approaches that are sampling based




Uncertainty representation techniques

Introduction to spectral stochastic theory [Ghanem, Stochastic finite elements: A spectral approach]

Generalized polynomial chaos expansion [Karniadakis, J. Fluids Engrg., 125, 2001]

Support-space [stochastic Galerkin expansion] [Zabaras, JCP, 208, 2005], [Babuska SIAM J. Num. Anal., 42, 2005]




STOCHASTIC PROCESSES AS FUNCTIONS

A probability space is a triple comprising of collection of the sample space , the -algebra of subsets (events) of and the probability measure on .

FP

A real-valued random variable is a function that maps the probability space to a real line [regions in go to intervals in the real line]

F

X : Random variableX

: ( , , )X F P

A space-time stochastic process is can be represented as

: ( , , )W x t + other regularity conditions

F




SERIES REPRESENTATION 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




SERIES REPRESENTATION [CONTD] Karhunen-Loeve

1

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

W x t W x t W x t

Stochastic process

Mean function

ON random variablesDeterministic functions

The deterministic functions are based on the eigen-values and eigenvectors of the covariance function of the stochastic process.

The orthonormal random variables depend on the kind of probability distribution attributed to the stochastic process.

Any function of the stochastic process (typically the solution of PDE system with as input) is of the form( , , )W x t

1( , , ) fn( , , , , )NW x t x t




SERIES REPRESENTATION [CONTD] Generalized polynomial chaos expansion is used to represent quantities like

0

( , , ) ( , ) (ξ( ))i ii

W x t W x t

Stochastic process

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NW x t x t

The Askey polynomials depend on the kind of joint PDF of the orthonormal random variables

Typically: Gaussian – Hermite, Uniform – Legendre, Beta -- Jacobi polynomials

1, , N




NEED FOR SUPPORT-SPACE APPROACH

GPCE and Karhunen-Loeve are Fourier like expansions

Gibb’s effect in describing highly nonlinear, discontinuous uncertainty propagation

Onset of natural convection

[Zabaras JCP 208(1)] – Using support-space method

[Ghanem JCP 197(1)] – Using Wiener-Haar wavelets Finite element representation of stochastic processes [stochastic Galerkin method: Babuska et al]

Incorporation of importance based meshing concept for improving accuracy [support space method]




SUPPORT-SPACE [STOCHASTIC GALERKIN]

Let stochastic inputs be represented by ON random variables with a joint PDF

Support space is the region in the span of stochastic input that has a positive PDF

1( , , )N 1( , , )Nf

Example of a 2D input and associated support-space

PDF Grid

1( , , ) : ( ) 0NA f ξ ξ

2

2

1

ˆ

ˆ( ( ) ( )) ( )d

L

A

q

X X

X X f

Ch

ξ ξ ξ ξ

Piecewise polynomials defined on support-space grid




VMS – Basic idea

Algebraic subgrid modeling approaches

Illustration of the approach with derivation of subgrid problems for stochastic natural convection equation

Numerical examples

Stochastic advection-diffusion equation

Stochastic fluid-flow

Stochastic natural convection [GPCE approach and capturing unstable equilibrium using support-space method]




VMS – ILLUSTRATION [NATURAL CONVECTION]

g

2

0

( ) Pr( ) e

2Pr( ) ( )

1( ) [ ( ) ]

2T

v

vv v Ra

t

vt

pI v

v v v

Continuity

Momentum

Energy

Constitutive laws

hm D Dgm

gtht

.n h

gv v

g 0.n q




DEFINITION OF FUNCTION SPACES

1 2 2 22

22 1

( ) : ( ( ) )d ; ( ) : d

( ) : d ; ( ) : d

D D

T T

H D v v v x L D v v x

L T w w t L T w w t

Deterministic function spaces

Stochastic function spaces – The space of all second order random variables is critical to spectral uncertainty modeling

22 ( ) : dPL




DERIVED FUNCTION SPACES

Velocity function space

Test

Trial

Pressure function space

Test

Trial

Energy function space

Test

Trial

d12 2: ( ; ( ; ( ))) , g gmV v v L L T H D v v on

d10 2: ( ; ( )) , 0 gmV w w L H D w on

2 1 2: ( ; ( ; ( )))Q p p L L T L D

0 2 2: ( ; ( ))Q q q L L D

12 2: ( ; ( ; ( ))), g gtE L L T H D on

10 2: ( ; ( )), 0 gtE w w L H D w on




VARIATIONAL FORMULATION

v 0( , ) ( . , ) ( , ) ( , )htt w v w w q w

v g( , ) ( . , ) ( , ( )) ( , ) ( ( ) Pr( ) e , )

( , ) 0hmtv w v v w v h w Ra w

v q

Energy equation – Find such that, for all , the following holds

E 0w E

Momentum and continuity equation – Find such that, for all , the following holds

[ , ] [ , ]v p V Q0 0[ , ] [ , ]w q V Q

( )Ra ( )Pr Wherein, and are the random Rayleigh number and Prandtl number, respectively. These will be defined separately for each example considered.




VMS HYPTOTHESIS

VMS hypothesis: Exact solution can be written as a sum of coarse scale resolved components [bar quantities] and subgrid scale unresolved components [prime quantities]

Induced function space decomposition [Hughes 1995]: This induces a function space decomposition as follows

The coarse scale function spaces are to be approximated using finite element basis functions, the small scales are to be solved using Green’s functions, element Fourier transform and other

', ', 'v v v p p p

0 0 0

0 0 0 0 0 0

', ', '

', ', '

V V V V V V Q Q Q

Q Q Q E E E E E E




Stochastic VMS applied to the energy equation – The algebraic subgrid modeling approach

Scale decomposed variational formulation

Element Fourier transform [Codina, CMAME 191, 2002]

Algebraic subgrid scale model [Stochastic]

Modified coarse scale equation




ENERGY EQUATION – SCALE DECOMPOSITION

v 0( ', ) ( . . ', ) ( ', ) ( , )htt t w v v w w q w

v 0( ', ') ( . . ', ') ( ', ') ( , ')htt t w v v w w q w

Energy equation – Find and such that, for all and , the following holds

Coarse scale variational formulation

Subgrid scale variational formulation

These equations can be re-written in the strong form with assumption on regularity as follows

E E

0w E0w E

2 2' . ' ' ( . )t tv v R 2( ) : .L v




ASGS APPROACHES In VMS, the central idea is to solve the subgrid scale variational formulation in an approximate manner

Different techniques to generate an algebraic subgrid scale model [approximation] are

Green’s functions and Residual-free bubbles

Element Fourier transform

Two-level finite element methods

Spatial domain discretized

( )eD Spatial domain discretized into Nel disjoint finite element sub-domains




ELEMENT FOURIER TRANSFORM

( )eD( )

ˆ ( , ) exp( ) ( , )deD

k xg k i g x x

h

For a random field defined over a coarse element sub-domain, the element Fourier transform is defined as

The spatial derivative can now be represented as

( )

ˆ ˆexp( ) ( , )d ( , ) ( , )e

j jj

D

k kg k xn i g x i g k i g k

x h h h

Term is negligible for large wave numbers

Note – Subgrid solution denotes fluctuations and hence is captured with large wave number terms




ASGS MODEL FOR ENERGY EQUATION

2 2' . ' ' ( . )t tv v R

' '( )t n n nL R

2

' '2

1 1ˆ ˆˆn n n

kv ki R

t h h t

122 2

' '1 22

1 1 1,n t n n t

vR c c

t h t h

1 1

1( ), (1 )t n n n n n nf f f f f f

t Time integration

rule

Time-discretized equation

After application of Parseval’s and Mean value theorem




MODIFIED COARSE FORMULATION

2v v( ', ) ( ', ), ( ', ) ( ', )v w v w w w

We assume the following strong regularity conditions

Applying the ASGS model, we obtain the following modified coarse scale equation

Time integration choice plays a role in deciding the coarse scale formulation

v 0

2 2

1

' 2

1

( , ) ( , ) ( , ) ( , )

( , ( )

( , /( ) ) ) 0

htt n n n

Nel

t n n n te

Nel

n t te

w v w w q w

v w v w w

w t v w w w




VARIATIONAL MULTISCALE METHOD The nonlinear advection term is linearized using Picard assumption [accurate for moderate Reynolds numbers]

Similar to the energy equation, we can derive the subgrid scale strong form of equations as follows

'v v v v v v

g' ' ( ', ') ( ) Pr( ) e ( , )

'

t tv v v v p Ra v v v v p

v v

g( ) Pr( ) e ( , )mom t n n n nR Ra v v v v p

con nR v




ALGEBRAIC SUBGRID SCALE MODEL

2

' ' '2

'

1 1ˆˆ ˆ ˆPr( )

ˆ ˆ

n n mom n

ncon

k v k ki v i p R v

t h h h t

k vi R

h

After time discretization, application of element Fourier transform, we have

Application of mean value theorem and Parseval’s theorem1

22 222'

1 1

' 2'

1

, Pr( )

,

n c con c

nn m mom m

c

c v hhp R

c t c

v hv R

t c




MODIFIED COARSE FORMULATION

v v( ', ) ( ', ), (Pr( ) ( '), ( )) ( ', Pr( ) ( ))v v w v v w v w v w Assuming strong regularity conditions, we have

Modified coarse scale momentum equation

g1

'

1

'

( , ) ( , ) (2 Pr( ) ( ), ( )) ( , )

( ) Pr( ) e , Pr( ) ( )

( , ) , Pr( ) ( )

( , ) ( , ) 0

hmt n n n n

Nel

t n me

Neln

n n n me

n n c

v v v w p w v w h w

Ra v w v w w

vv v v p w v w w

t

v w v w




MODIFIED COARSE FORMULATION Modified coarse scale continuity equation

Wherein, the test function

g1

'

1

( , ) ( ) Pr( ) e ,

( , ) , 0

Nel

n t n me

Neln

n n n me

v q Ra v q

vv v v p q

t

/( )w w t




IMPLEMENTATION ISSUE - GPCE Assume that the input can be represented as a function of orthonormal random variables 1, , N

( )eDnbf

1

( , ) ( ) ( )f x f N x

Spatial random field

Random variablesGalerkin shape function

0

( ) (ξ( ))P

i ii

f f

The random variables are represented in a GPCE

GPCE coefficients

Askey chaos polynomials




IMPLEMENTATION ISSUES – SUPPORT-SPACE We use a two level grid approach

Again, we assume that the input can be represented as a function of orthonormal polynomials 1, , N

( )eDEach Gauss point has an underlying support-space grid

Finite element interpolation at spatial and support-space grid

Object oriented structure: every coarse element has nbf basis functions, support-space grid has nbf’ basis functions

nbf nbf ' nbf'

1 1 1

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

f x f N x f N N x




CURSE OF DIMENSIONALITY Both GPCE and support-space method are fraught with the curse of dimensionality

As the number of random input orthonormal variables increase, computation time increases exponentially

Support-space grid is usually in a higher-dimensional manifold (if the number of inputs is > 3), we need special tensor product techniques for generation of the support-space

Parallel implementations are currently performed using PETSc (Parallel scientific extensible toolkit )




Numerical examples

Stochastic fluid flow – GPCE implementation

Stochastic natural convection – GPCE implementation

Stochastic natural convection – capturing unstable equilibrium using support-space methodology




TRANSIENT ROTATING CONE PROBLEM

0

0

0 0

Advection velocity – pure rotation about origin

Fourth order Legendre chaos used for simulation

[ , ], 1

unif[0.9,1.1]

a y x

1200 bilinear elements used for solution, preconditioned GMRES

Time of simulation [0,9], time step [0.003]

Transient response is observed




MEAN SOLUTION

Mean shows extensive decrease with time. This is nature of solution and not excess diffusion

Contribution of mean to off-mean terms increases with increase in time




STANDARD DEVIATION OF SOLUTION

As time increases, uncertainty is progressively amplified along the characteristics

The final position of the cone vertex is uncertain




FLOW PAST A CIRCULAR CYLINDER

X

Y

0 5 10 15 200

2

4

6

8

Uin

let =

uni

f[-0

.9,1

.1]

Tra

ctio

n fr

ee

No-slip

No-slip

2000 bilinear elements, 3rd order Legendre chaos expansion for velocity and pressure, preconditioned GMRES

Time [0,180], time step [0.03], Kinematic viscosity [0.01]

Onset of vortex shedding, shedding, wake characteristic




VORTEX SHEDDING

X

Y

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

-0.50 -0.42 -0.34 -0.26 -0.18 -0.10 -0.02 0.06 0.14 0.22 0.30 0.39 0.47 0.55 0.63

X

Y

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

-0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 0.10 0.12

Mean pressure at time 79.2

Vortex shedding initiated [not in periodic shedding]

First Legendre chaos coefficient

Vortex shedding is periodic




FULLY DEVELOPED VORTEX SHEDDING Mean pressure

Second LCE coefficient

First LCE coefficient

Wake region in the mean pressure is diffusive in nature

Also, the vortices do not occur at regular intervals [Karniadakis J. Fluids. Engrg]




OTHER PLOTS – VELOCITIES AND FFT

Frequency

Am

plit

ud

e

0.1 0.2 0.3 0.4 0.50

0.03

0.06

0.09

0.12

0.15

X

V

5 8 11 14 17 20-0.6

-0.4

-0.2

0

0.2

0.4

0.6

DeterministicMean

FFT yields a Strouhal number of 0.162

Spectrum is diffuse in comparison to deterministic simulations

Mean X-velocity has superimposed frequencies and is lower in magnitude in comparison to deterministic simulations




2048 bilinear elements, 3rd order Legendre chaos expansion for velocity and pressure, preconditioned GMRES

NATURAL CONVECTION – GPCE

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Time [0,1.5]

Time step [0.002]

Rayleigh number [104]

Prandtl number [0.7]

Transient behavior response is observed

Insu

late

d

Insu

late

d

Hot wall = unif[0.9,1.1]

Cold wall = 0




TRANSIENT RESPONSE

Mean temperature

Steady heat conduction like state not reached

Second coefficient in LCE of temperature

First coefficient in LCE of temperature




CAPTURING UNSTABLE EQUILIBRIUM 1600 bilinear elements, 5th order Legendre chaos expansion for velocity and pressure, preconditioned GMRES

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1Cold wall = 0

Cold wall = 1

Insu

late

d

Insu

late

d

Time [0,1.5]

Time step [0.002]

Rayleigh number unif[1530, 1870]

Prandtl number [6.95]

Support-space mesh [10 elements]

Simulation about critical Rayleigh number [Ghanem JCP]

Failure of GPCE approach and use of support-space




FAILURE OF GPCE

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

19.2E-07 5.7E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05 3.0E-05 3.4E-05

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-6.4E-08 4.3E-07 9.3E-07 1.4E-06 1.9E-06 2.4E-06 2.9E-06 3.4E-06

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-5.0E-03 -3.6E-03 -2.1E-03 -7.1E-04 7.1E-04 2.1E-03 3.6E-03 5.0E-03

XY

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-3.2E-03 -2.0E-03 -8.0E-04 3.9E-04 1.6E-03 2.8E-03 4.0E-03 5.2E-03

X-v

eloc

ityX

-vel

ocity

Y-v

eloc

ityY

-vel

ocity

Mean X and Y velocity obtained from GPCE yield unphysical low values

Comparative X and Y velocity obtained using a deterministic simulation at Ra = 1870 (the upper limit)




PREDICTION BY SUPPORT-SPACE METHOD

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-5.0E-03 -3.6E-03 -2.1E-03 -7.1E-04 7.1E-04 2.1E-03 3.6E-03 5.0E-03

XY

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-3.2E-03 -2.0E-03 -8.0E-04 3.9E-04 1.6E-03 2.8E-03 4.0E-03 5.2E-03

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-5.0E-03 -3.6E-03 -2.1E-03 -7.1E-04 7.1E-04 2.1E-03 3.6E-03 5.0E-03

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-3.2E-03 -2.0E-03 -7.4E-04 4.9E-04 1.7E-03 2.9E-03 4.2E-03 5.4E-03

X-v

eloc

ityX

-vel

ocity

Y-v

eloc

ityY

-vel

ocity

X and Y velocity obtained using a support-space method at Ra = 1870

Comparative X and Y velocity obtained using a deterministic simulation at Ra = 1870 (the upper limit)




Explicit subgrid modeling

Stochastic elliptic equation – VMS implementation

Numerical examples

EXPLICIT SUBGRID MODELING




MODEL MULTISCALE ELLIPTIC EQUATION

( )K u f

Permeability of Upper Ness formation

DDomain

Boundary

gu uin

on D

Multiple scale variations in K

K is inherently random [property predictions are at best statistical]

Crystal microstructures

Composites

Diffusion processes




VMS [VARIATIONAL FORMULATION]

u U[V] Find such that, for all v V

( , ) ( , ); ( , ) : ( , )a u v f v a u v K u v [V] denotes the full variational formulation

U and V denote appropriate function spaces for the multiscale solution u and test function v respectively

VMS hypothesis:

Induced function space decomposition [Hughes 1995]

C Fu u u

C FU U U

Exact = coarse + fine

C FV V V




VMS [COARSE AND SUBGRID SCALES]

C Fu u u ,C F C FU U U V V V

Using and the induced function space

decomposition

( , ) ( , ) ( , )C C F C Ca u v a u v f v

C Cu UFind such that, for all C Cv VF Fu Uand

and F Fv V

( , ) ( , ) ( , )C F F F Fa u v a u v f v

Coarse [V]

Subgrid [V]

Solve subgrid [V] using Greens' functions, PU and other

Substitute the subgrid solution in coarse [V]




DEFINITIONS AND DISCRETIZATION Assume a finite element discretization of the spatial region D into NelC coarse elements

Each coarse element is further discretized by a subgrid mesh with NelF elements

In each coarse element, the coarse solution uC can be approximated as

span ( ) ( ) : 1, , nbf ; 0, ,C r Cu N x r P

nbf = number of spatial finite element basis functions in each coarse element

PC = number of terms in the GPCE of coarse solution




DEFINITIONS AND DISCRETIZATION [CONTD]

( , )

( , )

x y

x y

Coarse meshSubgrid mesh for element EC

All basis function

problems solved here




SOLUTION OF HOMOGENEOUS [V] CONTD

max

0

( , ); ( , ) ( ) ( )

max nbf ( 1) 1, ( 1)( 1)

N

C CN N N rN

C C

u u x x N x

N P N P r

Considering the following finite element – GPCE representation for coarse solution uC

The subgrid solution can be represented as followsmax

0

ˆ ( , )N

F CN NN

u u x

Since represents subgrid variations, a nonlinear coarse scale mapping, a higher order GPCE is used [implies more terms in GPCE of ]

ˆFu

ˆFu

ˆFu




SOLUTION OF HOMOGENEOUS [V] CONTD Now, we have

Thus, we end up with (Nmax+1) homogeneous subgrid problems in each coarse element D(e)

Following representation is used for approximation

ˆ( , ) ( , ) 0

( , ) ( , ) 0C C

C C

E C F E F F

E N F E N F

a u v a u v

a v a v

span ( ) ( ) : 1, , nbf ; 0, ,N r FN x r P

nbf = number of spatial finite element basis functions in each element defined on the subgrid mesh PF = number of terms in the GPCE. Also, PF >PC




SOLUTION OF AFFINE [V]

Affine [V]

This affine correction is unique to the VMS formulation [Arbogast et al] and is not obtained in MsFEM type of formulations

Crucial in case of localized sources and sinks

Again, similar to the homogeneous [V], we have

0( , ) ( , )C CE F F F Ea u v f v

0 span ( ) ( ) : 1, , nbf ; 0, ,F r Fu N x r P This affine correction solution has no dependence on coarse scale behavior

We solve this equation on each coarse element with zero boundary conditions




SOLVING REDUCED PROBLEM

Subgrid mesh

Mapping element

edge

sn

Each coarse element edge is mapped to a line grid

Line grid yields coordinates (s – along the line grid, n – normal to line grid)

The reduced problem specified below is solved on the line grid

( ) ( ) 0, ( 1)( 1)

[ ](x ) ( ),

[ ](x ) 0, 1

s s N s s N C

N N r C

N N C F

K K N P r

r P

P r P




SOLVING REDUCED PROBLEM

( ) ( ) 0, ( 1)( 1)

[ ](x ) ( ),

[ ](x ) 0, 1

s s N s s N C

N N r C

N N C F

K K N P r

r P

P r P

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

x x

GPCE coefficients of coarse + subgrid BC




DESCRIPTION OF NUMERICAL PROBLEMS

1x

2x

3x 4x

Coarse element D(e)

Subgrid mesh

Based on boundary conditions used and subgrid problems solved, we have three studies

MsFEM-L MsFEM-Os VMS-Os Linear boundary conditions are used

No affine correction

Reduced solution as BC

No affine correction

Reduced solution as BC

Affine correction explicitly solved




FEM FOR HOMOGENEOUS [V]

( , ) ( , ) 0N F N Fa v a v Thus in each coarse element EC, we can solve for the subgrid basis functions as follows

[ ]{ } 0

[ ] ( ) ( ) ( ) ( )dxdC

r s

E

K

K K N x N x P

Note that we solve for the sum of coarse + subgrid basis functions

The boundary conditions for this equation are obtained as the solution of the reduced problem on coarse element edges

DOF for the problem = (Nno-subgrid)(PF+1)




DISTINCTIONS BETWEEN FUNCTION SPACES

{ ( ), 1, , nbf , 0, , }r FN r P

The number of functions spanning the subgrid approximation space is given by the tensor product

The subgrid basis functions however are a subset of these functions and hence do not contain all subgrid information

Is there a contradiction?

h

Subgrid scale

solution

Coarse scale

solution

Actual solution

Idea of VMS is only to solve the subgrid approximately

We just look in an intelligent manner, a subset of the fine function space wherein the solution is best represented




STOCHASTIC [CASE I] PSEUDO-PERIODIC MEDIA

0 1

0

11

1

1

, unif[ 1,1]

(2 sin(2 / )) (2 sin(2 / ))2

(2 cos(2 / )) (2 sin(2 / ))

(2 sin(2 / ))

(2 cos(2 / ))

1, 1.9, 0.08, 0.04, 0.1

K K K

P x yK

P y P x

P xK A

P y

f P A

uniformly distributed diffusion coefficient

K0




STOCHASTIC [CASE I] RESULTSFEM MsFEM-Os VMS-Os

U0

U1

U2




STOCHASTIC [CASE I] ERROR MEASURES

(coarse/fine) grid ratio

L-2

and

L-i

nfe

rro

rs

1 2 3 40

5E-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

MsFEM-OsVMS-OsMsFEM-OsVMS-Os

L-inf error was calculated on the mean value

Again, VMS is consistently better than MsFEM-Os.

2

1

22( vms ex) dxd

E[ vms]-E[ ex]

LD

L

E U U P

E U U




STOCHASTIC [CASE I] EFFECT OF PC TERMS We have assumed that the fine scale solution has more PC terms in its expansion

While reconstructing the fully resolved solution from the fine scale solution, we can only reconstruct up to the PCC terms. Beyond those terms, the fine scale solution is no longer a one-to-one map, hence, we see abnormalities (still equal in L-2)

L2 e

rror

Linf

err

or

PF-PC

L-2

erro

r

0 1 22.86E-04

2.86E-04

2.87E-04

2.87E-04

2.88E-04

PF-PC

L-2

erro

r

0 1 2

1.80E-05

1.84E-05

1.88E-05




Transient multiscale heat conduction

VMS implementation

Formulation 1: no subgrid time evolution

Formulation 2: quasi-static subgrid




MODEL MULTISCALE DIFFUSION EQUATION

( )u

K ut

Chip cooling and diffusion heat transfer

DDomain

Boundary

gu uin

on D

Multiple scale variations in K

K is inherently random [property predictions are at best statistical]

Diffusion, heat conduction in

microstructural lengths scale, composites

First step towards a multiscale industrial process simulator




VMS [VARIATIONAL FORMULATION]

u U[V] Find such that, for all v V

( , ) ( , ) 0; ( , ) : ( , )tu v a u v a u v K u v [V] denotes the full variational formulation

U and V denote appropriate function spaces for the multiscale solution u and test function v respectively

VMS hypothesis:

Induced function space decomposition [Hughes 1995]

C Fu u u

C FU U U

Exact = coarse + fine

C FV V V

t Ct Ftu u u




VMS [COARSE AND SUBGRID SCALES]

C Fu u u ,C F C FU U U V V V

Using and the induced function space

decomposition

( , ) ( , ) ( , ) ( , ) 0Ct C Ft C C C F Cu v u v a u v a u v

C Cu UFind such that, for all C Cv VF Fu Uand

and F Fv V

Coarse [V]

Subgrid [V]

Solve subgrid [V] using Greens' functions, PU and other

Substitute the subgrid solution in coarse [V]

( , ) ( , ) ( , ) ( , ) 0Ct F Ft F C F F Fu v u v a u v a u v




ASSUMPTIONS Formulation 1

The projection of time derivative ut on the subgrid function space is zero

This assumption assumes that the phenomena on the subgrid level attains equilibrium at every instant

Formulation 2

Quasistatic subgrid solution

This renders the term (uFt,vF) zero

This assumption is accurate provided the time step size used in the integration of governing equations is small




DEFINITIONS

0

( )Nbf

C Cu u x

Considering the following finite element representation for coarse solution uC

The subgrid solution uF can be represented as follows

Note that in case of quasistatic subgrid scale, the subgrid solution is a map of both coarse scale solution and coarse scale time derivative of solution

0

( )Nbf

UF Cu u x

0 0

( ) ( )Nbf Nbf

U UtF C Ctu u x u x

Formulation 1

Formulation 2




WHY THE CHOICE?

The alternative is to use backward Euler time discretization for the fine scale (after some algebra)

0 0

( ) ( )Nbf Nbf

U UtF C Ctu u x u x

C

prev( , ) ( , ) ( , ) ( , )C F C F F F Fu v ta u v ta u v u v The subgrid homogeneous equation now yields

ˆ( , ) ( , ) ( , ) 0

ˆ( , ) ( , ) ( , ) 0C F C F F F

C F C F F F

u v ta u v ta u v

u v t K u v t K u v

This is a tougher reaction-diffusion equation (notice that the diffusion coefficient is multiplied with a small time step size). In practical experiments, this equation yields spurious solutions




SOLUTION OF SUBGRID EQUATION For the assumptions employed for the subgrid model, we have

( , ) ( , ) 0UF Fa v a v

( , ) ( , ) 0

( , ) ( , ) 0

UF F

UtF F

a v a v

v a v

Formulation 1

Formulation 2

Wherein, the index a ranges from 1 to Nbf, the number of coarse basis functions in each coarse spatial sub-domain D(e)

Since the governing model is linear, the basis functions need to be estimated only in the first time step. Hence, we just solve a coarse problem for all remaining time steps!!




DETERMINISTIC [CASE I] PERIODIC MEDIA

2 2

(2 sin(2 / )) (2 sin(2 / ))

(2 cos(2 / )) (2 sin(2 / ))

1.9, 0.08

cos(4 ), 0.25(x,0)

0, 0.25

( 0.5) ( 0.5)

P x yK

P y P x

P

r ru

r

r x y

K

Fully resolved solution was obtained using a 128x128 mesh with bilinear quad elements

Time step size = 0.0005, duration of simulation = [0,0.05]




RESULTS[ANIMATIONS]

VMS solution FEM solution

L-inf error




RESULTS – FORMULATION 1 For the trivial assumption that subgrid attains equilibrium at every instant, we have the following solution error behavior

Error values are small in comparison to the solution maximum error values are L-inf errors that are 8% of solution

Nor

mal

ized

Un-

norm

aliz

ed




RESULTS – FORMULATION 2 For the trivial assumption that subgrid attains equilibrium at every instant, we have the following solution error behavior

Error values are consistently lesser than the formulation 1, wherein, the subgrid projection of coarse time derivative is neglected

Nor

mal

ized

Un-

norm

aliz

ed




Open issues in VMS

Difficulties in adapting to convection-diffusion

Nonlinearities

Adaptive VMS formulations

Non-quasistatic subgrid phenomena

Open issues in stochastic modeling

Compatibility of statistics across length scales

Transferring information from further lower scales




CONVECTION-DIFFUSION [DIFFICULTIES] Need to make transition from diffusion problems to convection-diffusion problems

Advection skew to a mesh shown

Information propagates along the characteristics

Most practical problems have extremely low diffusion. A further stabilized subgrid formulation may be required

Boundary conditions applied to subgrid problems have to be enhanced




BOUNDARY CONDITIONS

A typical coarse element

Edge mapped to a line grid

In diffusion type problems, a reduced problem is solved in the line grid

The reduced problem is the set of governing differential equations with derivatives normal to the element edge equated to zero

In convection-diffusion problems, the multidimensional layers that form in the solution should be captured by the basis function [not possible with current oscillatory BC approach]

Internal layers proportional to diffusion

coefficient




NONLINEARITY AND ADAPTIVITY Adaptive VMS [Larson, Chalmers preprint]

Nonlinear MsFEM [Hou]

Multiscale regions

Transition regions

Coarse regions

Calculation of basis functions

Very fine subgrid meshCoarser subgrid mesh

Inexact Newton methods for solving subgrid problems calculate coarse

scale solution by iterative methods

calculate error in solution, dual of solution to find regions of varying physics




STATISTICS ACROSS LENGTH SCALES

Subgrid

Coarse

Lower scales

VM

S li

nkag

eLo

wer

ord

er

desc

ripto

rs

Simulations are usually discrete

Need to describe statistics and upscale statistical information

Governing equations not necessarily PDEs

Statistical information received

via MAXENT approach – that PDF that maximizes entropy

VMS then uses series

expansions to characterize uncertainty propagation across the coarse and subgrid scales

Materials Process Design and Control Laboratory

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