4 evolution problems
TRANSCRIPT
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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/ -
7/27/2019 4 Evolution Problems
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
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
Dr. Filippo Terragni Reduced Order Modeling Applications 2 / 5 2
http://find/ -
7/27/2019 4 Evolution Problems
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
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
Dr. Filippo Terragni Reduced Order Modeling Applications 3 / 5 2
http://find/ -
7/27/2019 4 Evolution Problems
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
Goal
ROMs for time dependent problems intend to decrease the
computational effort required by standard numerical simulations
Dr. Filippo Terragni Reduced Order Modeling Applications 4 / 5 2
ROM b d POD l G l ki j ti
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
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
Dr. Filippo Terragni Reduced Order Modeling Applications 4 / 5 2
ROMs based on POD plus Galerkin projection
http://find/http://goback/ -
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
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 ).
Dr. Filippo Terragni Reduced Order Modeling Applications 5 / 5 2
ROMs based on POD plus Galerkin projection
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7/27/2019 4 Evolution Problems
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
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)
Dr. Filippo Terragni Reduced Order Modeling Applications 5 / 5 2
ROMs based on POD plus Galerkin projectionS i
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
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
Dr. Filippo Terragni Reduced Order Modeling Applications 6 / 5 2
ROMs based on POD plus Galerkin projectionS tti
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7/27/2019 4 Evolution Problems
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
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
<
Dr. Filippo Terragni Reduced Order Modeling Applications 6 / 5 2
ROMs based on POD plus Galerkin projection Setting
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7/27/2019 4 Evolution Problems
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p p jSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
Galerkin projection
Goal: a low dimensional model for tq = Lq + f(q, t)
Dr. Filippo Terragni Reduced Order Modeling Applications 7 / 5 2
ROMs based on POD plus Galerkin projection Setting
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p p jSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
Galerkin projection
Goal: a low dimensional model for tq = Lq + f(q, t)
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
Dr. Filippo Terragni Reduced Order Modeling Applications 7 / 5 2
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
Goal: a low dimensional model for tq = Lq + f(q, t)
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
.
Dr. Filippo Terragni Reduced Order Modeling Applications 7 / 5 2
ROMs based on POD plus Galerkin projectionS t t i f i t
Setting
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Some strategies for improvementAcceleration of numerical integration
Sett gDifficulties
Galerkin system
Galerkin system (GS): dA/dt = LGS
A + fGS
(A, t)
Dr. Filippo Terragni Reduced Order Modeling Applications 8 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Setting
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Some strategies for improvementAcceleration of numerical integration
gDifficulties
Galerkin system
Galerkin system (GS): dA/dt = LGS
A + fGS
(A, t)
it is a system of n (= number of modes) ODEs
Dr. Filippo Terragni Reduced Order Modeling Applications 8 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Setting
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7/27/2019 4 Evolution Problems
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Some strategies for improvementAcceleration of numerical integration
Difficulties
Galerkin system
Galerkin system (GS): dA/dt = LGS
A + fGS
(A, t)
it is a system of n (= number of modes) ODEs
n is generally much smaller than the number of original
discretized equations (= number of mesh points)
Dr. Filippo Terragni Reduced Order Modeling Applications 8 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
SettingDiffi l i
http://find/ -
7/27/2019 4 Evolution Problems
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Some strategies for improvementAcceleration of numerical integration
Difficulties
Galerkin system
Galerkin system (GS): dA/dt = LGS
A + fGS
(A, t)
it is a system of n (= number of modes) ODEs
n is generally much smaller than the number of original
discretized equations (= number of mesh points)
it can be constructed using , based on few mesh points(to speed up fGS computation)
Dr. Filippo Terragni Reduced Order Modeling Applications 8 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
SettingDiffi lti
http://find/ -
7/27/2019 4 Evolution Problems
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S g pAcceleration of numerical integration
Difficulties
Galerkin system
Galerkin system (GS): dA/dt = LGS
A + fGS
(A, t)
it is a system of n (= number of modes) ODEs
n is generally much smaller than the number of original
discretized equations (= number of mesh points)
it can be constructed using , based on few mesh points(to speed up fGS computation)
it can be constructed starting from a discretization of the
problem (not the exact one)
Dr. Filippo Terragni Reduced Order Modeling Applications 8 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
18/95
g pAcceleration of numerical integration
Difficulties
Galerkin system
Galerkin system (GS): dA/dt = LGS
A + fGS
(A, t)
it is a system of n (= number of modes) ODEs
n is generally much smaller than the number of original
discretized equations (= number of mesh points)
it can be constructed using , based on few mesh points(to speed up fGS computation)
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)
Dr. Filippo Terragni Reduced Order Modeling Applications 8 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
19/95
Acceleration of numerical integrationDifficulties
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
Dr. Filippo Terragni Reduced Order Modeling Applications 9 / 5 2
ROMs based on POD plus Galerkin projectionSome strategies for improvement
A l i f i l i i
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
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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
ROMs based on POD plus Galerkin projectionSome strategies for improvement
A l ti f i l i t ti
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
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Acceleration of numerical integrations
It is not so easy ...
Nonhomogeneous boundary conditions have to be includedin the GS, which can be hard
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?
Dr. Filippo Terragni Reduced Order Modeling Applications 10 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
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Acceleration of numerical integration
It is not so easy ...
Nonhomogeneous boundary conditions have to be includedin the GS, which can be hard
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)
Dr. Filippo Terragni Reduced Order Modeling Applications 10 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
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Acceleration of numerical integration
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
Dr. Filippo Terragni Reduced Order Modeling Applications 11 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
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Acceleration of numerical integration
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
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
Dr. Filippo Terragni Reduced Order Modeling Applications 11 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
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7/27/2019 4 Evolution Problems
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g
Imposing boundary conditions
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
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ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
SettingDifficulties
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g
Imposing boundary conditions
Nonhomogeneous boundary conditions are imposed in various ways
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).
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ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
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Imposing boundary conditions
Nonhomogeneous boundary conditions are imposed in various ways
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 14 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
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
.
Dr. Filippo Terragni Reduced Order Modeling Applications 15 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
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
.
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
.
Dr. Filippo Terragni Reduced Order Modeling Applications 15 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
30/95
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
.
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 15 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
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Truncation instability
Major drawback: the GS approximation may somewhat
unpredictably diverge from the actual solution
Dr. Filippo Terragni Reduced Order Modeling Applications 16 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
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7/27/2019 4 Evolution Problems
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Truncation instability
Major drawback: the GS approximation may somewhat
unpredictably diverge from the actual solution
In qnGS =n
j=1 AjQj we are ignoring modes Qn+1,Qn+2, . . .(with higher order than n)
Dr. Filippo Terragni Reduced Order Modeling Applications 16 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
33/95
Truncation instability
Major drawback: the GS approximation may somewhat
unpredictably diverge from the actual solution
In qnGS =n
j=1 AjQj we are ignoring modes Qn+1,Qn+2, . . .(with higher order than n)
This introduces a truncation error in the GS
Dr. Filippo Terragni Reduced Order Modeling Applications 16 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
http://find/ -
7/27/2019 4 Evolution Problems
34/95
Truncation instability
Major drawback: the GS approximation may somewhat
unpredictably diverge from the actual solution
In qnGS =n
j=1 AjQj we are ignoring modes Qn+1,Qn+2, . . .(with higher order than n)
This introduces a truncation error in the GS
If, for some t, new features appear in the actual solution andthese are not recorded in the n POD modes, then qnGS will diverge
Dr. Filippo Terragni Reduced Order Modeling Applications 16 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
T i i bili
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7/27/2019 4 Evolution Problems
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Truncation instability
Major drawback: the GS approximation may somewhat
unpredictably diverge from the actual solution
In qnGS =n
j=1 AjQj we are ignoring modes Qn+1,Qn+2, . . .(with higher order than n)
This introduces a truncation error in the GS
If, for some t, new features appear in the actual solution andthese are not recorded in the n POD modes, then qnGS will diverge
This is an intrinsic drawback of the POD plus Galerkin approach
for time dependent problems, which is still not well understood
Dr. Filippo Terragni Reduced Order Modeling Applications 16 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
T ti i t bilit
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7/27/2019 4 Evolution Problems
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Truncation instability
Major drawback: the GS approximation may somewhat
unpredictably diverge from the actual solution
In qnGS =n
j=1 AjQj we are ignoring modes Qn+1,Qn+2, . . .(with higher order than n)
This introduces a truncation error in the GS
If, for some t, new features appear in the actual solution andthese are not recorded in the n POD modes, then qnGS will diverge
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
Dr. Filippo Terragni Reduced Order Modeling Applications 16 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
SettingDifficulties
T ti i t bilit
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Truncation instability
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)
Dr. Filippo Terragni Reduced Order Modeling Applications 17 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
O tli
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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 integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
Dr. Filippo Terragni Reduced Order Modeling Applications 18 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
Dr. Filippo Terragni Reduced Order Modeling Applications 19 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
Setting
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7/27/2019 4 Evolution Problems
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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
Time integration of the associated GS can be faster
Dr. Filippo Terragni Reduced Order Modeling Applications 19 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
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
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
Dr. Filippo Terragni Reduced Order Modeling Applications 19 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
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
Dr. Filippo Terragni Reduced Order Modeling Applications 20 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
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)
Dr. Filippo Terragni Reduced Order Modeling Applications 21 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
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
Dr. Filippo Terragni Reduced Order Modeling Applications 21 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
Updating the POD modes
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Updating the POD modes
The POD basis is completely calculated in the first ICFD interval,
onlyupdated
in subsequent ICFD intervals (which can be veryshort
)
Dr. Filippo Terragni Reduced Order Modeling Applications 22 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
Updating the POD modes
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Updating the POD modes
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 22 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
Updating the POD modes
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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
Dr. Filippo Terragni Reduced Order Modeling Applications 23 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integrationLocal POD updatingTruncation instabilities control
Updating the POD modes
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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
Modes important in the past may not be important in the future
Dr. Filippo Terragni Reduced Order Modeling Applications 23 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
Updating the POD modes
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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
Modes important in the past may not be important in the future
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
Dr. Filippo Terragni Reduced Order Modeling Applications 23 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
Updating the POD modes
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Singular values are a natural weight, related to the importance
of the POD modes in the interval where snapshots are computed
Modes important in the past may not be important in the future
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
Dr. Filippo Terragni Reduced Order Modeling Applications 23 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
Outline
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1 ROMs based on POD plus Galerkin projectionSettingDifficulties
2 Some strategies for improvementLocal POD updatingTruncation instabilities control
3 Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
Dr. Filippo Terragni Reduced Order Modeling Applications 24 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
Dr. Filippo Terragni Reduced Order Modeling Applications 25 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
Dr. Filippo Terragni Reduced Order Modeling Applications 25 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
Dr. Filippo Terragni Reduced Order Modeling Applications 26 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
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
Dr. Filippo Terragni Reduced Order Modeling Applications 26 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
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
Dr. Filippo Terragni Reduced Order Modeling Applications 26 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
the POD modes are the same in the two systems; note that inthe first GS we set An1+1 = An1+2 = . . . = An2 = 0
truncation error should be smaller in the second GS
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
Dr. Filippo Terragni Reduced Order Modeling Applications 26 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
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
the POD modes are the same in the two systems; note that inthe first GS we set An1+1 = An1+2 = . . . = An2 = 0
truncation error should be smaller in the second GS
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
Dr. Filippo Terragni Reduced Order Modeling Applications 26 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
Using a second instrumental GS
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Consider two Galerkin systems
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
Dr. Filippo Terragni Reduced Order Modeling Applications 27 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
Local POD updatingTruncation instabilities control
Local POD plus Galerkin projection method (Rapun & Vega 2010)
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Dr. Filippo Terragni Reduced Order Modeling Applications 28 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
Outline
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1
ROMs based on POD plus Galerkin projectionSettingDifficulties
2 Some strategies for improvement
Local POD updatingTruncation instabilities control
3 Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
Dr. Filippo Terragni Reduced Order Modeling Applications 29 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 30 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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.
This is a well-known equation, describing a variety of physical
phenomena (Aranson & Kramer, Rev. Mod. Phys. 74, 2002)
Dr. Filippo Terragni Reduced Order Modeling Applications 30 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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.
This is a well-known equation, describing a variety of physical
phenomena (Aranson & Kramer, Rev. Mod. Phys. 74, 2002) Symmetries are x 1 x , u u eic
Dr. Filippo Terragni Reduced Order Modeling Applications 30 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The problem
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The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2
u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
This is a well-known equation, describing a variety of physical
phenomena (Aranson & Kramer, Rev. Mod. Phys. 74, 2002) Symmetries are x 1 x , u u eic
Depending on the parameter values, different solutions appear
Dr. Filippo Terragni Reduced Order Modeling Applications 30 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The problem
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The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2
u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
This is a well-known equation, describing a variety of physical
phenomena (Aranson & Kramer, Rev. Mod. Phys. 74, 2002) Symmetries are x 1 x , u u eic
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
Dr. Filippo Terragni Reduced Order Modeling Applications 30 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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 ( )
Dr. Filippo Terragni Reduced Order Modeling Applications 31 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Dr. Filippo Terragni Reduced Order Modeling Applications 32 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Required accuracy (ROM vs. CFD) is = 0.005
Dr. Filippo Terragni Reduced Order Modeling Applications 32 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Required accuracy (ROM vs. CFD) is = 0.005
The inner product is based on 100 uniformly selected mesh points
Dr. Filippo Terragni Reduced Order Modeling Applications 32 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Required accuracy (ROM vs. CFD) is = 0.005
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
Dr. Filippo Terragni Reduced Order Modeling Applications 32 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Dr. Filippo Terragni Reduced Order Modeling Applications 33 / 52
ROMs based on POD plus Galerkin projection
Some strategies for improvementAcceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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 ( )
Dr. Filippo Terragni Reduced Order Modeling Applications 34 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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 ( )
Dr. Filippo Terragni Reduced Order Modeling Applications 35 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
Example 2: chaotic-like solution
Errors are zero in 0 < t 0 1 as the CFD solver is run (also in
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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
Plot. Estimated ( ) and exact ( ) relative RMS error (vs. CFD) of the
GS solution with n POD modes; control of truncation instabilities ( )
Dr. Filippo Terragni Reduced Order Modeling Applications 36 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
Outline
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1
ROMs based on POD plus Galerkin projectionSettingDifficulties
2 Some strategies for improvement
Local POD updatingTruncation instabilities control
3 Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
Dr. Filippo Terragni Reduced Order Modeling Applications 37 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Dr. Filippo Terragni Reduced Order Modeling Applications 38 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 39 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Dr. Filippo Terragni Reduced Order Modeling Applications 39 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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
Dr. Filippo Terragni Reduced Order Modeling Applications 40 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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)
Dr. Filippo Terragni Reduced Order Modeling Applications 41 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 42 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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
Dr. Filippo Terragni Reduced Order Modeling Applications 42 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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
Dr. Filippo Terragni Reduced Order Modeling Applications 43 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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
Dr. Filippo Terragni Reduced Order Modeling Applications 43 / 52
ROMs based on POD plus Galerkin projectionSome strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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
Dr. Filippo Terragni Reduced Order Modeling Applications 44 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
The reduced order model
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
Dr. Filippo Terragni Reduced Order Modeling Applications 45 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
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 ( )
Dr. Filippo Terragni Reduced Order Modeling Applications 46 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equation
The lid-driven cavity flow
Example 1: 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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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)
Plot. Estimated ( ) and exact ( ) relative RMS error (vs. CFD) of the
GS solution with n POD modes; control of truncation instabilities ( )
Dr Filippo Terragni Reduced Order Modeling Applications 47 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
Example 1: periodic flow at Re = 800
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
Dr Filippo Terragni Reduced Order Modeling Applications 48 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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 ( )
Dr Filippo Terragni Reduced Order Modeling Applications 49 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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)
Plot. Estimated ( ) and exact ( ) relative RMS error (vs. CFD) of the
GS solution with n POD modes; control of truncation instabilities ( )
Dr Filippo Terragni Reduced Order Modeling Applications 50 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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
Dr Filippo Terragni Reduced Order Modeling Applications 51 / 52 ROMs based on POD plus Galerkin projection
Some strategies for improvement
Acceleration of numerical integration
The complex Ginzburg-Landau equationThe lid-driven cavity flow
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
http://find/