scalar and vector penalty projection methods

7/30/2019 scalar and vector penalty projection methods

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Numerical Flow Models for Controlled Fusion - April 2007

Vector and scalar penalty-projection methodsfor incompressible and variable density flows

Philippe AngotUniversit de Provence, LATP - Marseille

with M. Jobelin [PhD, 2006], J.-C. LatchJ.-P. Caltagirone and P. Fabrie

Porquerolles, April 19 - 2007


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Motivations and objectives

Work focusing on the constraint of free divergence

How to deal efficiently with the free-divergence constraint withfractional-step methods (prediction-correction steps) ?

How to circumvent the major drawbacks of the usual projection

methods including a scalar correction step for the Lagrangemultiplier with solution of a Poisson-type equation ?

Example of fluid-type models with pressure as Lagrangemultiplier

solution of unsteady incompressible Navier-Stokes equations in theprimitive variables (velocity and pressure)

Introduction 2


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Outlines

1 Projection methods for incompressible flows

2 Scalar penalty-projection methods

3 Vector penalty-projection methods

4 Conclusion and perspectives

Introduction 3


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Outlines

1 Projection methods for incompressible flowsNon-homogeneous incompressible flowsSemi-implicit method (linearly implicit)Fractional-step and projection methods



4

Conclusion and perspectives

Projection methods for incompressible flows 4


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Navier-Stokes problem for incompressible flows

Incompressible and variable density flows of Newtonian fluidsNavier-Stokes equations with mixed boundary conditions :

Dirichlet on D and open (outflow) B.C. on N

t+ u = 0 in ]0, T[

ut + (u )u (u) + p = f in ]0, T[ u = 0 in ]0, T[u = uD on D]0, T[

pn + (u) n = fN on N]0, T[ = 0, and u = u0 in {0}

(u) = [(u + uT)], or u (for a constant viscosity)For an homogeneous fluid with constant density, we set = 1.

Projection methods for incompressible flows Non-homogeneous incompressible flows 5


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Semi-implicit method (linearly implicit)

Semi-discretization in time with low or high-order BDFschemes

for all n IN such that (n + 1)t T :

Dn+1

t+ u,n+1n+1 = 0 in

n+1Dun+1

t+ (u,n+1

)un+1

(un+1) + pn+1 = fn+1 in un+1 = 0 in

un+1 = un+1

Don

D

pn+1n + (un+1) n = fn+1N on N

Resolution of an elliptic problem in space (at each timestep) by methods of finite elements, finite volumes, DGM...

Projection methods for incompressible flows Semi-implicit method (linearly implicit) 6


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Semi-implicit method : coupled solver

Algebraic formulation of the Navier-Stokes system inf-sup stable discretization in space or with stabilization...

q

tMU

n+1 + A(Un, Un1)Un+1 + BTPn+1

= F(Un, Un1, fn+1, fn+1N , un+1D )

BUn+1 = G(un+1D )



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Semi-implicit method : coupled solver

Algebraic formulation of the Navier-Stokes system inf-sup stable discretization in space or with stabilization...

q

tMU

n+1 + A(Un, Un1)Un+1 + BTPn+1

= F(Un, Un1, fn+1, fn+1N , un+1D )

BUn+1 = G(un+1D )

Saddle-point problem at each time step : efficient solver ?

Fully-coupled solver with efficient multigrid preconditioner :Kortas, PhD 1997 - Cihl and Angot, 1999

Augmented Lagrangian method with iterative Uzawa algorithm :Fortin and Glowinski, 1983 - Khadra et al., IJNMF 2000

Projection methods using the Helmoltz-Hodge-Leraydecomposition of L2()d



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Fractional-step and projection-type methods

Projection scheme with scalar pressure-correction[Chorin, 1968 - Temam, 1969 - Goda, 1979 - Van Kan, 1986]See recent review : [Guermond, Minev, Shen, CMAME 2006]

Prediction step

n+1

Dun+1

t+ (u,n+1 )un+1

(un+1) + pn = fn+1

un+1 = un+1D on D

pnn + (un+1) n = fn+1N on NProjection step

qn+1u

n+1 un+1t

+ = 0

n = 0 on D (necessary since u n = u n) = 0 on N (sufficient by orthogonal projection onto H)

un+1 = 0

t

n+1

= q un+1

Pressure-correction step : pressure increment

pn+1

= pn

+ Projection methods for incompressible flows Fractional-step and projection methods 8

i l d j i h d


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Algebraic formulation for the Navier-Stokes system

q

tMU

n+1 + AUn+1 + BTPn = F

L =q

tBUn+1 G

Pn+1 = Pn +

MUn+1 = MU

n+1 +t

qBT

Projection methods for incompressible flows Fractional-step and projection methods 8

F ti l t d j ti t th d


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Major drawbacks of the incremental projection methods

Time order of the splitting error ?i.e. error between the numerical solutions of the implicit (orsemi-implicit) method and the fractional-step method

n = 0 on D

existence of an artificial pressure boundary layer in space

= 0 on N convergence in time and space spoiled for outflow boundaryconditions : splitting error varying like O(t1/2) (pressure) and nomore negligible (for both velocity and pressure) with respect to thetime and space discretization error

Pressure-correction step strongly dependent on density and viscosityfor non-homogeneous flows convergence very poor for large ratios of 1000.

Projection methods for incompressible flows Fractional-step and projection methods 8

N i l t t


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Numerical tests

Green-Taylor vortices : Navier-Stokes with Dirichlet B.C.Velocity error (discrete L2(0, T; L2()d) norm) versus time step t

1e-4 1e-3 1e-2 1e-1Pas de temps

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

normeL2erre

urvitesse

semi implicite

Time convergence in O(t2)Stagnation threshold = space discretization error in O(h

2

)Projection methods for incompressible flows Fractional-step and projection methods 9

N i l t t


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Numerical tests



1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

normeL2erre

urvitesse

semi implicite

incrementale


2

)Projection methods for incompressible flows Fractional-step and projection methods 9

O tli


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Outlines


2 Scalar penalty-projection methodsPenalty-projection methodsNumerical experimentsAnalysis of the scalar penalty-projection method for the Stokes problem


4

Conclusion and perspectives

Scalar p enalty-projection methods 10

Penalty projection methods


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Penalty-projection methods

[Jobelin et al., JCP 2006] : prediction step with augmentedLagrangian for r > 0 and consistent projection step by scalar

pressure-correction

[Shen, 1992] : r = 1/t2 with a different correction of pressure

[Caltagirone and Breil, 1999] : r > 0 with a singularprojection operator...

Scalar p enalty-projection methods Penalty-projection methods 11



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Penalty-prediction step : augmentation parameter r

0

n+1Dun+1

t+ (u,n+1 )un+1

(un+1)

r un+1 + pn = fn+1un+1 = un+1D on D

pnn + (un+1) n = fn+1N on NProjection step

qn+1u

n+1 un+1t

+ = 0

n = 0 on D (necessary since u n = u n) = 0 on N (sufficient by orthogonal projection onto H)

un+1 = 0

t

n+1

= q un+1

Pressure-correction step : consistent pressure increment

pn+1

= pn

r un+1

+ = pn+1

+ Scalar p enalty-projection methods Penalty-projection methods 11



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q

tMU

n+1 + rBTM1pl

BUn+1 G

+ BTPn

+AUn+1 = F

L =q

t BUn+1

G

Pn+1 = Pn + rM1pl

BUn+1 G

+

MUn+1 = MU

n+1 +t

qBT

Preconditioning the prediction step by one iteration ofaugmented Lagrangian and consistent scalar projection r = 0 : incremental projection method




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q

tMU

n+1 + BT (rM1pl

BUn+1 G

+ Pn)

Pn+1

+AUn+1 = F

L = qt

BUn+1 G

Pn+1 = Pn + rM1pl

BUn+1 G

Pn+1+

MUn+1 = MUn+1 + tq

BT

Preconditioning the prediction step by one iteration ofaugmented Lagrangian and consistent scalar projection

r = 0 : incremental projection method




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Direct implementation with the algebraic formulation or not...Why ?

Continuous term corresponding to the penalization Tp = r

u vBut Tp does not generally vanish for a velocity field with zero discrete

divergence !It depends on the spatial discretization...

Hence

Introduction of an additional error due to the space discretization

Error which increases with the augmentation parameter r > 0


Numerical experiments


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1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

normeL2erreurvitesse

semi implicite

incrementale


2

)Scalar p enalty-projection methods Numerical experiments 14



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1e-7

1e-6

1e-5

1e-4

1e-3

1e-2


semi implicite

incrementaler=1


2




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1e-7

1e-6

1e-5

1e-4

1e-3

1e-2


semi implicite

incrementaler=1r=10


2




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1e-7

1e-6

1e-5

1e-4

1e-3

1e-2


semi implicite

incrementaler=1r=10r=100


2




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Green-Taylor vortices : Navier-Stokes with Dirichlet B.C.Pressure error (discrete L2(0, T; L2()) norm) versus time step t


1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

normeL2erreu

rpression

semi implicite

incrementaler=1r=10r=100

Stagnation threshold = space discretization error in

O(h2)

Scalar p enalty-projection methods Numerical experiments 15



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p

Artificial pressure boundary layer : Stokes with Dirichlet B.C.on a disk

incremental projectionph pL() = 1.5 102

penalty-projection r=1ph pL() = 2.8 103




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p

Artificial pressure boundary layer : Stokes with Dirichlet B.C.on a disk

penalty-projection r=100ph pL() = 2.8 104

implicit schemeph pL() = 1.8 104




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p

Stokes with open boundary condition at a channel outflowVelocity error (discrete L2(0, T; L2()d) norm) versus time step t


1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

NormeL2erre

urvitesse

semi implicite

incrementaler=1r=10r=100r=1000r=10000


2




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Stokes with open boundary condition at a channel outflowPressure error (discrete L2(0, T; L2()) norm) versus time step t


1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

normeL2erreu

rpression

semi implicite

incrementale

r=1r=10r=100r=1000r=10000

Stagnation threshold = space discretization error in O(h2)Scalar p enalty-projection methods Numerical experiments 18



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Navier-Stokes open flow around a circular cylinder

Taylor-Hood P2-P1 finite elementsScheme 2nd order in timeReynolds number : Re = 100 ( = 0.01)Unstructured mesh with 4200 elements

Complete study of this configuration : see Khadra et al., IJNMF2000

with FVM (MAC mesh) and iterative augmented Lagrangian algorithm.

Splitting error varying as O( 1r

)




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Navier-Stokes open flow around a circular cylinder

1e+0 1e+1 1e+2 1e+3paramtre de pnalisation

1e-5

1e-4

1e-3

1e-2

1e-1

N

ormeL2erreur

defractionneme

nt

Pas de temps = 0.05 s


Pas de temps =0.01 s


Splitting error varying as O( 1r

)


Theoretical analysis for Stokes-Dirichlet problem


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Unsteady homogeneous model problem with = 1 connected and bounded open domain of IRd, Lipschitz boundary ,T > 0

u

t u + p = f dans ]0, T[

u = 0 dans

]0, T[

u = 0 sur ]0, T[u(x, 0) = u0(x) dans

u denotes the velocity vector, p the pressure field.

For f L2(0, T; L2()d) and u0 V given, there exists a uniquesolution u L(0, T; H) L2(0, T; V) and p L2(0, T; L20())See [Temam, 1979 - Girault and Raviart, 1986]

Scalar p enalty-projection methods Analysis of the scalar penalty-projection method 20



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[Shen, 1992-95 - Guermond, 1996] : standard projection(pressure-correction form)[Guermond and Shen, 2003] : standard projection (velocity-correction)

[Guermond and Shen, 2004] : rotational variant of [Timmermans et al.,1996]

[Angot, Jobelin, Latch, Preprint 2006] : penalty-projection

Analysis for small values of the penalty parameter r

Theorem (Splitting error - fully discrete case in time and space)Energy estimates of splitting errors compared to Euler implicit scheme

there exists c = c(, T , f , u0, h) > 0 such that : for 1 n N,

nk=0

t ||ek

||20

12

+ n

k=0t ||e

k

||20

12

c min(t2

,

t3/2

r1/2 )

n

k=0

t ||ek||20 12

+

n

k=0

t ||k||2012

c max(1, 1r1/2

)t3/2.




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Analysis for large values of the penalty parameter r

Theorem (Splitting error - fully discrete case in time and space)

Energy estimates of splitting errors compared to Euler implicit scheme

there exists c = c(, T , f , u0, h) > 0 such that : for 1 n N,

||en||0 + ||en||0 + n

k=0 t ||ek||20

12

c t12

r

n

k=0

t ||ek||2012

+

n

k=0

t ||ek||20 12

c tr

||n||h c1

r1/2n

k=0

t ||k||20 12

c 1r

.




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n

k=0t ||ek||20

12

: velocity splitting error en = un un

discrete L2(0, T; L2()d) norm versus time step t

1e-4 1e-3 1e-2 1e-1

Pas de temps

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

r = 0

r = 1

r = 5

r = 10

r = 50r = 100

r = 500

r = 1000




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n

k=0

t ||ek||20 12

: velocity splitting error en = un un

discrete L2

(0, T; L2

()d

) norm versus penalty parameter r > 0

1e+0 1e+1 1e+2 1e+3Paramtre de pnalisation

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

Pas de temps: 0.1

Pas de temps: 0.05

Pas de temps: 0.025

Pas de temps: 0.0125

Pas de temsp: 0.00625

Pas de temps: 3.125e-3







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n

k=0

t ||k||20 12

: pressure splitting error n = pn pn

discrete L2

(0, T; L2

()) norm versus penalty parameter r > 0

1e+0 1e+1 1e+2 1e+3Paramtre de pnalisation

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

Pas de temps: 0.1

Pas de temps: 0.05

Pas de temps: 0.025





Pas de temps: 7.813e-4Pas de temps: 3.906e-4


Outlines


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3

Vector penalty-projection methodsNew class of vector penalty-projection methodsAnalysis of the vector penalty-projection method for the Stokes problem


Vector penalty-projection methods 24

New class of vector penalty-projection methods


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Work in progress with Caltagirone and FabrieA two-parameter family with vector projection step

Penalty-prediction step : augmentation parameter r 0n+1

Dun+1

t+ (u,n+1 )un+1

(un+1)

r

un+1

+ pn = fn+1

un+1 = un+1D on D, and

pnn + (un+1)

n = fn+1N on N

pn+1 = pnr un+1Vector projection step : penalty parameter 0 < t

n+1

q

tun+1 + (u,n+1 )un+1

(un+1)

un+1 = un+1un+1 = 0 on D, and ( pn+1 pn)n + (un+1) n = 0 on NCorrection step :

un+1 = un+1 + un+1, and pn+1 = pn+1 1

un+1

Vector penalty-projection methods New class of vector p enalty-projection methods 25

New class of vector penalty-projection methods


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Why does it work well ?

Very nice conditioning properties of the vector penalty-correction

step as 0 since the right-hand side becomes more and moreadapted to the left-hand side operator :only a few iterations (2 or 3) of MILU0-BiCGStab solver whateverthe mesh step h !

Original boundary conditions not spoiled through a scalar projection

stepPenalty-correction step quasi-independent on the density or viscositywhen 0 for non-homogeneous incompressible flows

It is only an approximate projection scheme since

un+1

= 0

But both the velocity divergence and the splitting error can bemade negligible with respect to the time and spacediscretization errors when 0cheap method for small values of r < 1 and 1015 (zero machine)

Vector penalty-projection methods New class of vector p enalty-projection methods 26

Analysis for Stokes-Dirichlet problem with r = 0


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Error estimates of the one-parameter family : r = 0 and0 < tTheorem (Splitting error in the semi-discrete case)

Energy estimates of splitting errors w.r. to Euler implicit scheme

there exists C = C(, T , f , u0) > 0 such that : for 1 n N,

||en

||0 +

n

k=0

t||

ek

||2

0 12

C

t

n

k=0

t ||ek||2012

C t

||n||0 +

nk=0

t ||k||2012

C

|| un||0 = || en||0 C max1,

t t.Vector penalty-projection methods Analysis of the vector p enalty-projection method 27

Outlines


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Conclusion and perspectives 28

Conclusion


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Scalar penality-projection methods for incompressible flows

Reduce the splitting error up to make it negligible

Suppress pressure boundary layers for moderate values of r > 0

Optimal convergence for r > 0 with open boundary conditions

Generalization to dilatable and low Mach number flows :see [Jobelin et al., Preprint 2006]

Require efficient preconditioning for large values of r multi-level preconditioner for FVM with MAC mesh :see [Kortas, PhD 1997].


Perspectives


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Vector penality-projection methods for incompressible andnon-homogeneous flows

Small values of r < 1 are sufficient to get a good pressure field

Approximate projection with a vector correction step all the cheaperthan 0Same convergence properties as beforeVector correction step all the less dependent on density or viscositythan 0Other numerical experiments (in progress)

Error estimates : two-parameter class, Navier-Stokes, outflow B.C.,variable density flows...


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