4 Evolution Problems

7/27/2019 4 Evolution Problems

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ROMs based on POD plus Galerkin projectionSome strategies for improvement

Acceleration of numerical integration

Reduced Order Modeling Applications

Applications to evolution problems

ECMI Summer School 2013

Leganes, July 18 Dr. Filippo Terragni

Dr. Filippo Terragni Reduced Order Modeling Applications 1 / 5 2
http://find/http://goback/


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Outline

1 ROMs based on POD plus Galerkin projectionSettingDifficulties

2 Some strategies for improvementLocal POD updatingTruncation instabilities control

3

Acceleration of numerical integrationThe complex Ginzburg-Landau equationThe lid-driven cavity flow

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SettingDifficulties

Outline



3

Acceleration of numerical integrationThe complex Ginzburg-Landau equationThe lid-driven cavity flow

http://find/


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SettingDifficulties

Goal

ROMs for time dependent problems intend to decrease the

computational effort required by standard numerical simulations


ROM b d POD l G l ki j ti


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SettingDifficulties

Goal

ROMs for time dependent problems intend to decrease the

computational effort required by standard numerical simulations

appealing idea: spectral method with optimally customized modes

existing ROMs for a variety of scientific/industrial applications

major challenge: developing effective ROMs for turbulent flows


ROMs based on POD plus Galerkin projection


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SettingDifficulties

Problem, CFD solver, and snapshots

Consider the general (parabolic) problem

tq = Lq + f(q, t) (1)

with suitable boundary and initial conditions, where q is defined on abounded domain , L (elliptic) is a linear operator, f is a nonlinearoperator (e.g., the Burgers equation u

t

= 2u

x2 u

u

x

with non-small ).




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SettingDifficulties

Problem, CFD solver, and snapshots

Consider the general (parabolic) problem

tq = Lq + f(q, t) (1)

with suitable boundary and initial conditions, where q is defined on abounded domain , L (elliptic) is a linear operator, f is a nonlinearoperator (e.g., the Burgers equation u

t

= 2u

x2 u

u

x

with non-small ).

Equation (1) can be regarded as finite dimensional upon discretizationby a numerical CFD solver, which is used to calculate N snapshots

q1 = q(t1), . . . , qN = q(tN) .

numerical spatial solutions of (1) at different time instants (vectors)

representative of the dynamics in the time interval where they are computed

used to perform POD by the method of snapshots (Sirovich, 1987)


ROMs based on POD plus Galerkin projectionS i
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SettingDifficulties

POD modes & singular values

Form the (symmetric, positive definite) covariance matrix Rwith

Rij = qi, qj , where for instance q1, q2 = 1M Mk=1 q1(xk) q2(xk) .Then, calculate its eigenvalues (i)

2 and orthonormal eigenvectors ifrom

Nj=1 Rkj

ji = (i)

2ki .

1 2 . . . N 0 are the POD singular valuesQi =

1i

Nk=1

ki qk are the orthonormal POD modes


ROMs based on POD plus Galerkin projectionS tti
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SettingDifficulties

POD modes & singular values

Form the (symmetric, positive definite) covariance matrix Rwith

Rij = qi, qj , where for instance q1, q2 = 1M Mk=1 q1(xk) q2(xk) .Then, calculate its eigenvalues (i)

2 and orthonormal eigenvectors ifrom

Nj=1 Rkj

ji = (i)

2ki .

1 2 . . . N 0 are the POD singular valuesQi =

1i

Nk=1

ki qk are the orthonormal POD modes

few POD modes yield optimal reconstructions of the snapshots

the number n of POD modes for accuracy is determined as thesmallest integer satisfying

RRMSENn =

Nj=n+1(j)

2

Nj=1(j)

2

<


ROMs based on POD plus Galerkin projection Setting
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p p jSome strategies for improvement


SettingDifficulties

Galerkin projection

Goal: a low dimensional model for tq = Lq + f(q, t)


ROMs based on POD plus Galerkin projection Setting


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p p jSome strategies for improvement


SettingDifficulties

Galerkin projection


Firstly, we assume that q is a linear combination of the n mostenergetic POD modes (with the largest singular values), that is

q(x, t) qnGS(x, t) =n

j=1

Aj(t)Qj(x) (2)

unknown coefficients Aj (mode amplitudes) depend on t

POD works as a separation of variables method


ROMs based on POD plus Galerkin projectionS i f i

Setting
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Some strategies for improvementAcceleration of numerical integration

SettingDifficulties

Galerkin projection


Firstly, we assume that q is a linear combination of the n mostenergetic POD modes (with the largest singular values), that is

q(x, t) qnGS(x, t) =n

j=1

Aj(t)Qj(x) (2)

unknown coefficients Aj (mode amplitudes) depend on t

POD works as a separation of variables method

Substitute (2) into the problem and multiply by Qi, to get

dAi

dt=

nj=1

LGSij Aj + fGSi (A1, . . . , An, t) , for i = 1, . . . , n ,

where LGSij =Qi,LQj

, fGSi =

Qi,f

nk=1 AkQk, t

.


ROMs based on POD plus Galerkin projectionS t t i f i t

Setting
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Sett gDifficulties

Galerkin system

Galerkin system (GS): dA/dt = LGS

A + fGS

(A, t)



Setting
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gDifficulties

Galerkin system


A + fGS

(A, t)

it is a system of n (= number of modes) ODEs



Setting
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Difficulties

Galerkin system


A + fGS

(A, t)


n is generally much smaller than the number of original

discretized equations (= number of mesh points)



SettingDiffi l i
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Difficulties

Galerkin system


A + fGS

(A, t)




it can be constructed using , based on few mesh points(to speed up fGS computation)



SettingDiffi lti
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S g pAcceleration of numerical integration

Difficulties

Galerkin system


A + fGS

(A, t)





it can be constructed starting from a discretization of the

problem (not the exact one)



SettingDifficulties
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g pAcceleration of numerical integration

Difficulties

Galerkin system


A + fGS

(A, t)





it can be constructed starting from a discretization of the

problem (not the exact one) it has to be time integrated to compute the amplitudes vector A

(GS integration will be much faster if n is sufficiently small)



SettingDifficulties
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Acceleration of numerical integrationDifficulties

Outline



3 Acceleration of numerical integrationThe complex Ginzburg-Landau equationThe lid-driven cavity flow



A l i f i l i i

SettingDifficulties
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Acceleration of numerical integrationDifficulties

It is not so easy ...

Nonhomogeneous boundary conditions have to be includedin the GS, which can be hard

Dr. Filippo Terragni Reduced Order Modeling Applications 10 / 52


A l ti f i l i t ti

SettingDifficulties
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Acceleration of numerical integrations



The POD modes are good to describe the solution in the timeinterval where snapshots are computed (by construction)

if we usen

j=1 Aj(t)Qj for future & different dynamics?




SettingDifficulties
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The POD modes are good to describe the solution in the timeinterval where snapshots are computed (by construction)

if we usen

j=1 Aj(t)Qj for future & different dynamics?

The GS solution can be spurious (somewhat unpredictably)




SettingDifficulties
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Imposing boundary conditions

Homogeneous boundary conditions are implicitly satisfied

For instance, for u(1, t) = 0, we have

u(1, t) =n

j=1

Aj(t)Uj(1) =n

j=1

Aj(t)1

j

Nk=1

kj uk(1) = 0




SettingDifficulties
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Homogeneous boundary conditions are implicitly satisfied

For instance, for u(1, t) = 0, we have

u(1, t) =n

j=1

Aj(t)Uj(1) =n

j=1

Aj(t)1

j

Nk=1

kj uk(1) = 0

Similarly, linear equations are automatically satisfiedif the POD modes are properly selected

For instance, for xu + yv = 0, choosing joint modes for the twovelocity components, u =

nj=1 AjUj and v =

nj=1 AjVj , we have

xu+yv =n

j=1

AjxUj+n

j=1

AjyVj =n

j=1

Aj

1

j

Nk=1

kj xuk +1

j

Nk=1

kj yvk

=n

j=1Aj

1

j

N

k=1kj (xuk + yvk) = 0




SettingDifficulties
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g


Nonhomogeneous boundary conditions are imposed in various ways

1 A change of variable can be used.

For instance, if u(0, t) = 1, then

the POD modes are computed from the snapshots uk u0(not from uk), where u0 is one fixed snapshot (satisfying the bc)

the solution is expanded as u(x, t) = u0(x) +n

j=1 Aj(t)Uj(x)

finally, we get

u(0, t) = u0(0) +n

j=1

Aj(t)Uj (0) = 1 +n

j=1

Aj(t)1

j

Nk=1

kj

uk(0) u0(0) 1 1 = 0

= 1




SettingDifficulties
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g



2 The discretized equations can be Galerkin-projected.

For instance, the equation ut

= 2ux2

with u(0) = 0 and u(1) = 5,

after discretization by finite differences, has the form

uk+1 uk

t= D

uk+1 + uk

+ b .

Expanding vector u in terms of some POD modes and projecting the

discretized equation, we directly account for all bcs (included in thenumerical scheme by vector b).


ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration

SettingDifficulties
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3 A new constraint equation can be added (penalty method).

For instance, boundary condition u = f(x, t), in the incompressible

2D Navier-Stokes equations, can be added asu

x+

v

y= 0

u

t= u

u

x v

u

y

p

x+

1

Reu

v

t = uv

x vv

y p

y +1

Rev

u

t= f(x, t) u (on the boundary)

where is a small parameter to be calibrated.



SettingDifficulties
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Errors estimation

How good is qnGS =n

j=1 Aj Qj for future dynamics?

The (instantaneous) spatial, relative error associated with qnGS is

Errorn =

q qnGSq

.



SettingDifficulties
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Errors estimation

How good is qnGS =n



Errorn =

q qnGSq

.

Now, if Errorn1 is sufficiently small for some n1 > n, then the quantity

En1n =

n1j=n+1(Aj)

2n1j=1(Aj)

2

is a good a priori estimate of Error

n

.



SettingDifficulties
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Errors estimation

How good is qnGS =n



Errorn =

q qnGSq

.

Now, if Errorn1 is sufficiently small for some n1 > n, then the quantity

En1n =

n1j=n+1(Aj)

2n1j=1(Aj)

2

is a good a priori estimate of Error

n

.

This is because, if Errorn1 is sufficiently small, then q n1

j=1 AjQj and thus

q

n

j=1

AjQj

2

n1

j=n+1

AjQj

2

=

n1

j=n+1

(Aj )2.



SettingDifficulties
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Truncation instability

Major drawback: the GS approximation may somewhat

unpredictably diverge from the actual solution



SettingDifficulties


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In qnGS =n

j=1 AjQj we are ignoring modes Qn+1,Qn+2, . . .(with higher order than n)



SettingDifficulties
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In qnGS =n


This introduces a truncation error in the GS



SettingDifficulties
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In qnGS =n



If, for some t, new features appear in the actual solution andthese are not recorded in the n POD modes, then qnGS will diverge



SettingDifficulties

T i i bili


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In qnGS =n




This is an intrinsic drawback of the POD plus Galerkin approach

for time dependent problems, which is still not well understood



SettingDifficulties

T ti i t bilit
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In qnGS =n




This is an intrinsic drawback of the POD plus Galerkin approach

for time dependent problems, which is still not well understood

It is called high-order modes truncation instability



SettingDifficulties

T ti i t bilit
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1D complex Ginzburg-Landau equation

tu = (1 + i)2xxu + u (1 + i)|u|

2u , with xu = 0 at x = 0, 1

(,,) = (30,1, 10), snapshots in (0, 0.1), 11 POD modes

plotting |u(0.5, t)| (the exact solution is numerically obtained)



Local POD updatingTruncation instabilities control

O tli


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Outline


2

Some strategies for improvementLocal POD updatingTruncation instabilities control

3 Acceleration of numerical integration

The complex Ginzburg-Landau equationThe lid-driven cavity flow




Setting
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Setting

The evolution problem tq = Lq + f(q, t) is considered

Dynamics in the time interval (0, T] are desiredA given CFD solver provides numerical solutions but is quite slow




Setting
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Setting



Time integration of the associated GS can be faster



Acceleration of numerical integrationLocal POD updatingTruncation instabilities control

Setting
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Setting



Time integration of the associated GS can be faster

Probably qnGS =n

j=1 Aj Qj will not be good in the whole timeinterval, but estimation En1n can predict the approximation error

we would like that En1n Errorn < (say, = 0.01)

We should also anticipate possible truncation instabilities




A basic algorithm


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A basic algorithm

1 Calculate some snapshots in the time interval ICFD withthe CFD solver

2 Apply POD and select the POD modes Q1, . . . ,Qn, . . . ,Qn1 ,where n and n1 > n are chosen according to the singular valuesspectrum (n and n1 should allow to extend the approximation validity)

3 Write q n1

j=1 AjQj and construct the GS

4 Integrate the GS over the time interval IGS, computing En1n for

each t. Continue until En1n occurs at some tstop

5 Go back to (1) if tstop < T or exit otherwise




A basic algorithm


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A basic algorithm

Anytime the approximation fails, the POD basis and the GS

are changed

In other words, a new set of snapshots has to be computed forthe new POD modes construction (in the ICFD intervals)




A basic algorithm
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A basic algorithm

Anytime the approximation fails, the POD basis and the GS

are changed

In other words, a new set of snapshots has to be computed forthe new POD modes construction (in the ICFD intervals)

Snapshots calculation is the most expensive part of the algorithm!

? Can we avoid to compute many snapshots?

We can exploit the information we already have, namely the old(previously used) POD modes




Updating the POD modes
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The POD basis is completely calculated in the first ICFD interval,

onlyupdated

in subsequent ICFD intervals (which can be veryshort

)




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The POD basis is completely calculated in the first ICFD interval,only updated in subsequent I

CFDintervals (which can be very short)

Updating is done by applying POD to

1Q1, . . . , nQn

old modes

, 1Q1, . . . , mQm

new modeswhere

j =jnk=1(k)

2, j =

jmk=1(k)

2

Thus, the new modes can be few & less snapshots are necessary.




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p g

Singular values are a natural weight, related to the importance

of the POD modes in the interval where snapshots are computed




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p g



Modes important in the past may not be important in the future





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p g




A way to eliminate the old modes that are no longer necessaryconsists in using the weights

j = min

jn

k=1(k)2

,|Aj|nk=1|Ak|

2

, |Aj | =

1

GS

IGS

|Aj | dt





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A way to eliminate the old modes that are no longer necessaryconsists in using the weights

j = min

jn

k=1(k)2

,|Aj|nk=1|Ak|

2

, |Aj | =

1

GS

IGS

|Aj | dt

We make the POD basis adaptive & dependent on the local dynamics





Outline
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Main idea
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There are a lot of works proposing different ways of dealing withthe high-order modes truncation instability

1 correcting the POD modes

2 correcting the GS





Main idea
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There are a lot of works proposing different ways of dealing withthe high-order modes truncation instability

1 correcting the POD modes

2 correcting the GS

The following approach adapts to the introduced algorithm

we monitor the behavior of the truncation error as long asthe GS is time integrated

if this error considerably grows, we update the POD basisas explained before





Using a second instrumental GS
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Consider two Galerkin systems

1 first GS based on n1 modes qn1GS = n1j=1 AjQj2 second GS based on n2 > n1 modes q

n2GS =

n2j=1 AjQj





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n2GS =

n2j=1 AjQj

the POD modes are the same in the two systems; note that inthe first GS we set An

1+1 = An

1+2 = . . . = An

2

= 0





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n2GS =

n2j=1 AjQj

the POD modes are the same in the two systems; note that inthe first GS we set An

1+1 = An

1+2 = . . . = An

2

= 0

truncation error should be smaller in the second GS





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n2GS =

n2j=1 AjQj

the POD modes are the same in the two systems; note that inthe first GS we set An1+1 = An1+2 = . . . = An2 = 0


if the contribution of the high-order modes Qn1+1, . . . ,Qn2 issmall, then the two systems will behave in the same way

truncation error in the first GS will not be growing





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n2GS =

n2j=1 AjQj

the POD modes are the same in the two systems; note that inthe first GS we set An1+1 = An1+2 = . . . = An2 = 0


if the contribution of the high-order modes Qn1+1, . . . ,Qn2 issmall, then the two systems will behave in the same way

truncation error in the first GS will not be growing

this can be controlled by the estimate

En2n1 =qn1GS q

n2GS

qn1GS





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1 first GS based on n1 modes qn1GS =n1

j=1 AjQj

2 second GS based on n2 > n1 modes qn2GS =

n2j=1 AjQj

Thus, point (4) of the algorithm can be replaced by

4 Integrate the two GSs over the time interval IGS, computingEn1n and E

n2n1

for each t. Continue until

En1n or En2n1

/100

occurs at some tstop





Local POD plus Galerkin projection method (Rapun & Vega 2010)


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Outline
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1

ROMs based on POD plus Galerkin projectionSettingDifficulties

2 Some strategies for improvement








The problem
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The 1D complex Ginzburg-Landau equation (CGLE) is

tu = (1 + i)2xxu + u (1 + i)|u|

2

u , with xu = 0 at x = 0, 1

where u is a complex variable and (,,) are real parameters.





The problem
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tu = (1 + i)2xxu + u (1 + i)|u|

2

u , with xu = 0 at x = 0, 1


This is a well-known equation, describing a variety of physical

phenomena (Aranson & Kramer, Rev. Mod. Phys. 74, 2002)





The problem
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tu = (1 + i)2xxu + u (1 + i)|u|

2

u , with xu = 0 at x = 0, 1



phenomena (Aranson & Kramer, Rev. Mod. Phys. 74, 2002) Symmetries are x 1 x , u u eic




The complex Ginzburg-Landau equation

The lid-driven cavity flow

The problem
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tu = (1 + i)2

xxu + u (1 + i)|u|2

u , with xu = 0 at x = 0, 1




Depending on the parameter values, different solutions appear






The problem
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tu = (1 + i)2

xxu + u (1 + i)|u|2

u , with xu = 0 at x = 0, 1




Depending on the parameter values, different solutions appear

For < 1 and larger than a critical value, the system may

exhibit complex behaviors (e.g., chaotic dynamics) at large time






Example of chaotic dynamics


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If (,,) = (90,2, 14) then the CGLE has

chaotic transient dynamics (sensitivity to perturbations)

1000 mesh points 2000 mesh points

Plots. Time evolution of |u| at x = 1/4 ( ), 3/4 ( ), and 1/2 ( )






The reduced order model


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Snapshots and reference solutions are computed by numerical

integration of the full CGLE (CFD solver)

spatial derivatives discretized by centered finite differences ona uniform mesh of 1001 points

time evolution described by Crank-Nicolson + Adams-Bashforthscheme with t = 10

4






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4

Required accuracy (ROM vs. CFD) is = 0.005






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4


The inner product is based on 100 uniformly selected mesh points








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4


The inner product is based on 100 uniformly selected mesh points

Galerkin systems are constructed as explained before (ROM)

projection of the exact equation

spatial & time discretization as above






Example 1: transient to a periodic solution

P t ( ) (18 20 10)


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Parameters are (,,) = (18,20, 10)

100 snapshots are computed in 0 < t 0.1 and POD is applied

n = 7 modes approximate the solutionn1 = 11 modes yield the GS & estimate the errorn2 = 13 modes control truncation instabilities

0 0.2 0.4 0.6 0.8 10

2

4

6

t

|u|

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

|Ui|

Left. Time evolution of |u| at x = 1/4 ( ), 3/4 ( ), and 1/2 ( )

Right. The 5 most energetic POD modes on 0 < x < 1






Example 1: transient to a periodic solution

E i 0 < t 0 1 th CFD l i


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Errors are zero in 0 < t 0.1 as the CFD solver is run

The GS is integrated in 0.1 < t 1, where the error is correctlypredicted by En1n = E

117 and remains smaller than = 0.005

The starting POD basis provides a good approximation with 7modes without any updating (there are no truncation instabilities)

0 0.2 0.4 0.6 0.8 110

7

106

105

104

103

t

E

Plot. Estimated ( ) and exact ( ) relative RMS error (vs. CFD) of the

GS solution with 7 POD modes; control of truncation instabilities ( )






Example 2: chaotic-like solution

Parameters are ( ) (145 10 2)
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Parameters are (,,) = (145,10, 2)

100 snapshots are computed in 0 < t 0.1 and POD is applied

n = 11 modes approximate the solutionn1 = 15 modes yield the GS & estimate the errorn2 = 19 modes control truncation instabilities

Dynamics are highly unstable

0 0.2 0.4 0.6 0.8 10

5

10

15

t

|u|

Plot. Time evolution of |u| at x = 1/4 ( ), 3/4 ( ), and 1/2 ( )






Example 2: chaotic-like solution

Errors are zero in 0 < t 0 1 as the CFD solver is run (also in
http://find/


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Errors are zero in 0 < t 0.1 as the CFD solver is run (also insmaller intervals where few snapshots must be computed)

A GS is integrated until t 0.3 ; then the approximation fails,the POD modes are updated, and a new GS is constructed

The starting POD basis is updated 7 times (truncation instabilitiesarise); modes number & structure change (finally, n = 9)

0 0.2 0.4 0.6 0.8 110

7

106

105

104

103

t

E


GS solution with n POD modes; control of truncation instabilities ( )






Outline
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1

ROMs based on POD plus Galerkin projectionSettingDifficulties

2 Some strategies for improvement









The problem

Th i ibl fl id ti i 2D it h t ll
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The incompressible fluid motion in a 2D square cavity whose top wallis moved by an external shear forcing (closed laminar flow)

x

y

10

1

Left. Steady state streamlines at Re = 400

Right. Steady state vorticity at Re = 800






Unsteady formulation

The flow is described by the incompressible Navier Stokes equations
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The flow is described by the incompressible Navier-Stokes equations

v = 0v

t+ (v )v = p +

1

Rev

in the spatial domain 0 < x < 1, 0 < y < 1, with boundary conditions

v = 0 at x = 0, 1 and y = 0 , v = (h(t)g(x), 0) at y = 1 .

Here, v = (vx, vy) and p are velocity field and pressure; the Reynoldsnumber is defined as Re = uL/; the function g(x) = 16 x2(1 x)2

smooths out the flow singularity near the upper corners.






Unsteady formulation

The flow is described by the incompressible Navier Stokes equations
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The flow is described by the incompressible Navier-Stokes equations

v = 0v

t+ (v )v = p +

1

Rev

in the spatial domain 0 < x < 1, 0 < y < 1, with boundary conditions

v = 0 at x = 0, 1 and y = 0 , v = (h(t)g(x), 0) at y = 1 .

Here, v = (vx, vy) and p are velocity field and pressure; the Reynoldsnumber is defined as Re = uL/; the function g(x) = 16 x2(1 x)2

smooths out the flow singularity near the upper corners.

Standard: h = 1, steady flow at large-time unless Re > Rec 104

Unsteady: h = h(t), unsteady dynamics even at moderate Re






The numerical CFD solver

Snapshots and reference solutions are computed by direct numerical
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S ps s s s p ysimulation of the full lid-driven cavity problem.

spatial derivatives discretized by finite differences on threestaggered grids

time evolution described by a fractional-step method

intermediate variables, averaged values, unphysical terms and bcsimplemented for numerical reasons (numerical artifacts)

low accuracy but fast performance






Goal
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Speed up simulations of the unsteady flow in the driven cavity fromthe quiescent state (t = 0) to the final asymptotic state (t = T Re)

How? Constructing an efficient ROM 1

Difficulties? Snapshots (the information we need) are not veryaccurate and the CFD solver (our competitor) is quite fast

the required accuracy (ROM vs. CFD) will be = 0.01

1Terragni et al., SIAM J. Sci. Comput. 33 (2011)







The considered inner product is
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v1, v2 = 1card(Ivx) (i,j)Ivx v1

x(xi, yj) v

2

x(xi, yj) +

1

card(Ivy ) (i,j)Ivy v1

y(xi, yj) v

2

y(xi, yj)

where Ivx and Ivy are two sets of indices corresponding to points on the

vx-mesh and vy-mesh, respectively.







The considered inner product is
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v1, v2 = 1card(Ivx) (i,j)Ivx v1

x

(xi, yj) v2

x

(xi, yj) +1

card(Ivy ) (i,j)Ivy v1

y

(xi, yj) v2

y

(xi, yj)

where Ivx and Ivy are two sets of indices corresponding to points on the

vx-mesh and vy-mesh, respectively.

Selected points

concentrated in regions of strong flow activity (lateral/upper sides)

not close to the upper wall where snapshots exhibit large errors

for instance, 400 among 66, 000 at Re = 800 (see vx-mesh below)

x

y1

0 1







The GS is constructed from the exact equations,
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GS a q ,ignoring all the numerical artifacts of the CFD solver

Exact Navier-Stokes equations

v = 0

v

t+ (v )v = p +

1

Rev

v = 0 at x = 0, 1 and y = 0 , v = (h(t)g(x), 0) at y = 1







The GS is constructed from the exact equations,
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q ,ignoring all the numerical artifacts of the CFD solver

Exact Navier-Stokes equations

v = 0

v

t+ (v )v = p +

1

Rev

v = 0 at x = 0, 1 and y = 0 , v = (h(t)g(x), 0) at y = 1

Time discretization by Crank-Nicolson + Adams-Bashforth ( f(v) (v )v )

vk+1 = 0

vk+1 vk

t= 3f(vk) f(vk

1)2

pk+1 + (2xx + 2yy)(vk+1 + vk)2 Re

vk+1 = 0 at x = 0, 1 and y = 0 , vk+1 =

hk+1g(x), 0

at y = 1







Spatial discretization by finite differences (compact notation including bcs


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Spatial discretization by finite differences (compact notation, including bcs,reordering terms) as in the CFD code

I

tL1

2Re2s

(vk+1 vk) = t

2L1vk + (hk + hk+1)G

2Re2s

tL3pk+1

s

t3F(vk) + 3hkL2vk F(vk1) hk1L2vk1

2s

the continuity equation is linear and does not have to be imposed(with a careful modes selection)

pay attention to staggered grids

L2 and G account for the nonhomogeneous boundary condition

Dr. Filippo Terragni Reduced Order Modeling Applications 44 / 52 ROMs based on POD plus Galerkin projection

Some strategies for improvement





Spatial discretization by finite differences (compact notation including bcs
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Spatial discretization by finite differences (compact notation, including bcs,reordering terms) as in the CFD code

I

tL1

2Re2s

(vk+1 vk) = t

2L1vk + (hk + hk+1)G

2Re2s

tL3pk+1

s

t3F(vk) + 3hkL2vk F(vk1) hk1L2vk1

2s

the continuity equation is linear and does not have to be imposed(with a careful modes selection)

pay attention to staggered grids

L2 and G account for the nonhomogeneous boundary condition

Snapshots for the velocity field only + introduced inner product

Joint POD modes such that vk =n

j=1 AkjVj and p

k =n

j=1 AkjPj

Only one set of amplitudes to be determined







Projection of the vector equation by means of V i, yields
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Projection of the vector equation by means of Vi, yields

I

tLGS1

2Re2s+

tLGS3

s

Ak+1 = Ak + t

LGS1 Ak + (hk + hk+1)GGS

2Re2s

t3FGS(Ak) + 3hkLGS2 A

k FGS(Ak1) hk1LGS2 Ak1

2s

where

LGS1,ij = Vi, L1Vj , LGS2,ij = Vi, L2Vj , L

GS3,ij = Vi, L3Pj ,

GGSi = Vi, G , FGSi (A

k) =

Vi,F

nj=1

AkjVj

this GS provides the vector Ak+1 of n mode amplitudes at time instant tk+1






Example 1: periodic flow at Re = 800

Forcing h(t) = sin(t)
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Snapshots computed on 256 256 staggered grids (t = 0.0025)

ROM based on 400 mesh points only

After a long transient, the flow becomes periodic around t 250

130 150 170 190 210 230 2500.15

0.1

0.05

0

0.05

0.1

0.15

t

vx

Plot. Time evolution of vx at two points near left ( ) and right ( )

upper corners, the center ( ), and one point near right lower corner ( )







Errors are zero in small intervals where the CFD solver is run tocompute some snapshots (for the POD basis updating)
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compute some snapshots (for the POD basis updating)

The GS is integrated over quite large IGS intervals; the error iscorrectly predicted by En1n and remains smaller than = 0.01

The numbers of POD modes oscillate, being (n, n1, n2) = (12, 18, 24)at t = 0 and (n, n1, n2) = (10, 19, 22) at t = 250

Acceleration 9 (transient + asymptotic), 32 (asymptotic only)



Dr Filippo Terragni Reduced Order Modeling Applications 47 / 52 ROMs based on POD plus Galerkin projection





The POD modes change with time & adapt to the actual dynamics.
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mode 1 mode 2 mode 3 mode 4

Plot. The 4 most energetic POD modes for vx in the basis used at the beginning

(top) and the end (bottom) of time integration





Example 2: quasi-periodic flow at Re = 800

Forcing h(t) = sin(t/4) cos(t/16)
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Snapshots computed on 256 256 staggered grids (t = 0.0025)

ROM based on 400 mesh points only

Two timescales are induced in the flow, which is quasi-periodicaround t 250 more POD modes & updating are expected

130 150 170 190 210 230 250

0.15

0.1

0.05

0

0.05

0.1

0.15

t

vx

Plot. Time evolution of vx at two points near left ( ) and right ( )

upper corners, the center ( ), and one point near right lower corner ( )





Example 2: quasi-periodic flow at Re = 800

Errors are zero in small intervals where the CFD solver is run tocompute some snapshots (for the POD basis updating)
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( )

The GS is integrated over quite small IGS intervals; the POD basis isfrequently updated and there are truncation instabilities

The numbers of POD modes oscillate, being (n, n1, n2) = (9, 14, 19)at t = 0 and (n, n1, n2) = (18, 35, 37) at t = 250

Acceleration 4 (transient + asymptotic)







Some references

1 D. RempferOn low-dimensional Galerkin models for fluid flow
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On low dimensional Galerkin models for fluid flowTheor. Comp. Fluid Dyn. 14 (2000), pp. 7588

2 S. Sirisup, G. E. Karniadakis, D. Xiu & I. G. KevrekidisEquation-free/Galerkin-free POD-assisted computation of incompressible

flowsJ. Comput. Phys. 207 (2005), pp. 568587

3 M. Bergmann, C.-H. Bruneau & A. IolloEnablers for robust POD models

J. Comput. Phys. 228 (2009), pp. 516538

4 F. Terragni, E. Valero & J. M. VegaLocal POD plus Galerkin projection in the unsteady lid-driven cavity problemSIAM J. Sci. Comput. 33 (2011), pp. 35383561

5 M. L. Rapun & J. M. VegaReduced order models based on local POD plus Galerkin projectionJ. Comput. Phys. 229 (2010), pp. 30463063

6 I. S. Aranson & L. KramerThe world of the complex Ginzburg-Landau equationRev. Mod. Phys. 74 (2002), pp. 99143





Some references

7 P. N. Shankar & M. D. DeshpandeFluid mechanics in the driven cavity


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yAnnu. Rev. Fluid Mech. 32 (2000), pp. 93136

8 D. Ahlman, F. Soderlund, J. Jackson, A. Kurdila & W. ShyyProper orthogonal decomposition for time-dependent lid-driven cavity flowsNumer. Heat Tr. B - Fund. 42 (2002), pp. 285306

9 W. Cazemier, R. W. C. P. Verstappen & A. E. P. VeldmanProper orthogonal decomposition and low-dimensional models for drivencavity flows

Phys. Fluids 10 (1998), pp. 16851699

Dr Filippo Terragni Reduced Order Modeling Applications 52 / 52
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4 Evolution Problems

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