Stabilization for convection dominate problems...Stabilizationfor convectiondominate problemsd GianluigiRozza mathLab,Mathematics Area, SISSA International Schoolfor AdvancedStudies,

Stabilization for convection dominated problems

Gianluigi Rozza

mathLab, Mathematics Area, SISSA International School for Advanced Studies, Trieste, Italy

Advanced Topicsin Comp.Mech.CISM Udine,

December 7 - 10, 2020

Outline

• FE and RB Stabilization for advection-diffusion problems

• Stabilization for fluids: Stokes and Navier-Stokes equations

• Increasing the Reynolds number: VMS-Smagorinsky RB model

1/ 34 G. Rozza Stabilization for Convection Dominated Problems

Table of contents

1 Advection-Diffusion problem

2 Steady Stokes equations

3 Steady Navier-Stokes equations

4 VMS-Smagorinsky turbulence model


Advection-Diffusion problem

• Advection-diffusion equations depending on parameter:

−ε(µ)∆u + β(µ) · ∇u = f




(ε(µ)∇uh,∇vh)Ω + (β(µ) · ∇uh, vh)Ω = 〈f , vh〉





⇓a(uh, vh;µ) = F (vh) ∀vh ∈ Vh






• High Péclet number: advection dominated probem

Pe = |β(µ)|2ε(µ) hK > 1






• High Péclet number: advection dominated probem

Pe = |β(µ)|2ε(µ) hK > 1

• Stabilization methods for advection dominate problem


Stabilization method for advection dominated problems

a(uh, vh;µ) + astab(uh, vh;µ) = F (vh) ∀vh ∈ Vh

with

astab(uh, vh;µ) =∑k∈Th

δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K

• Different stabilization method depending on the choice of ρ:

ρ = 0: Streamline Upwind Petrov-Galerkin (SUPG) method ρ = 1: Galerkin Least-Square (GLS) method ρ = −1: Douglas-Wang (DW) method

A. Quarteroni, A. Valli Numerical Approximation of Partial Differential Equations.Springer Science & Business Media. Volume 23. 2008.




with


δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K







with


δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K







with


δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K


ρ = 0: Streamline Upwind Petrov-Galerkin (SUPG) method

ρ = 1: Galerkin Least-Square (GLS) method ρ = −1: Douglas-Wang (DW) method





with


δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K


ρ = 0: Streamline Upwind Petrov-Galerkin (SUPG) method ρ = 1: Galerkin Least-Square (GLS) method

ρ = −1: Douglas-Wang (DW) method





with


δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K







with


δ

(ε(µ)∆uh + β(µ) · ∇uh,

hk

|β(µ)| (ρε(µ)∆vh + β(µ) · ∇vh))

K





Reduced Basis Stabilization

First issue: How stabilize the Reduced Basis problem?

• Offline stabilization only: VN = spanuh(µ1), . . . , uh(µN) and solveFind uN ∈ VN such thata(uN , vN ;µ) = F (vN) ∀vN ∈ VN

• Offline-Online stabilization:Find uN ∈ VN such thata(uN , vN ;µ) + astab(uN , vN ;µ) = F (vN) ∀vN ∈ VN

P. Pacciarini, G. Rozza Stabilized reduced basis method for parametrizedadvection-diffusion PDEs. Computer Methods in Applied Mechanics andEngineering, 274 (2014), pp. 1-18.




















Numerical test

• β(µ) = (1, 1), ε(µ) = 1µ

⇒ Pe = µ

• µ ∈ [100, 1000]


Numerical test

• β(µ) = (1, 1), ε(µ) = 1µ

⇒ Pe = µ

• µ ∈ [100, 1000]


Numerical test

• β(µ) = (1, 1), ε(µ) = 1µ

⇒ Pe = µ

• µ ∈ [100, 1000]

u = 0

u = 0

u = 1

u = 1

(0, 1) (1, 1)

(0, 0) (1, 0)


Numerical Results

Figure: RB solution for Pe = 600


Table of contents






Steady Stokes equations

We define the steady Stokes equations, with ν the viscosity:−ν∆u +∇p = f in Ω

∇ · u = 0 in Ω



We define the steady Stokes equations, with ν the viscosity:ν(∇uh,∇vh)Ω − (ph,∇ · vh)Ω = 〈f, vh〉 ∀vh ∈ Vh

(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh




(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh

• Bilinear forms:

a(u, v;µ) = ν(∇u,∇v)Ω, b(v, q;µ) = −(∇ · v, q)Ω




(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh

• Bilinear forms:


• Discrete inf-sup condition:

∃β0 such that 0 < β0 < βh(µ) = infqh∈Mh

supvh∈Vh

b(vh, qh;µ)‖vh‖1‖qh‖0




(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh

• Bilinear forms:


• Discrete inf-sup condition:

∃β0 such that 0 < β0 < βh(µ) = infqh∈Mh

supvh∈Vh

b(vh, qh;µ)‖vh‖1‖qh‖0

• Standard ROM stabilization: inner pressure supremizer .


Discrete Stokes equations with stabilization

a(uh, vh;µ) + b(vh, ph;µ) = 〈f, vh〉 ∀vh ∈ Vh

b(uh, qh;µ) + spres(qh;µ) = 0 ∀qh ∈ Mh

Different stabilization terms:

• Brezzi-Pitkaranta stabilization (1984)

spres(qh;µ) =∑K∈Th

h2K (∇ph,∇qh)K

• Hughes, Franca and Balestra stabilization (1986)

spres(qh;µ) = δ∑K∈Th

h2K (a0uh − ν∆uh +∇ph − f,∇qh)K

S. Ali Stabilized reduced basis methods for the approximation of parametrizedviscous flows. PhD Thesis, SISSA, 2018.








h2K (∇ph,∇qh)K












h2K (∇ph,∇qh)K












h2K (∇ph,∇qh)K












h2K (∇ph,∇qh)K






Reduced Basis Model for stabilized Stokes equations

• Reduced space by the Greedy algorithm selecting snapshots

u(µ1), . . . , u(µN), p(µ1), . . . , p(µN)

• Stabilization of the Reduced Basis problem Inner pressure supremizer ⇒ Enrich the velocity space

(Tµp p(µk), vh)V = b(vh, p(µk);µ) ∀vh ∈ Vh

Online stabilization term

spres(qN ;µ) =∑K∈Th

h2K (∇pN ,∇qN)K

• Inf-sup condition for the reduced stabilized problem:

∃β0 such that 0 < β0‖qh‖0 ≤ supvh∈Vh

b(vN , qN ;µ)‖vN‖1

+∑K∈Th

h2K (∇qN ,∇qN)K

• Different options for stabilization:

Offline stabilization only with supremizer Offline-online stablization without supremizer Offline-online stabilization with supremizer




u(µ1), . . . , u(µN), p(µ1), . . . , p(µN)• Stabilization of the Reduced Basis problem

Inner pressure supremizer ⇒ Enrich the velocity space(Tµp p(µk), vh)V = b(vh, p(µk);µ) ∀vh ∈ Vh



h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K






u(µ1), . . . , u(µN), p(µ1), . . . , p(µN)• Stabilization of the Reduced Basis problem Inner pressure supremizer ⇒ Enrich the velocity space




h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K










h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K










h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K










h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K

• Different options for stabilization: Offline stabilization only without supremizer









h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K

• Different options for stabilization: Offline stabilization only without supremizer









h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K

• Different options for stabilization: Offline stabilization only without supremizer Offline stabilization only with supremizer

Offline-online stablization without supremizer Offline-online stabilization with supremizer








h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K

• Different options for stabilization: Offline stabilization only without supremizer Offline stabilization only with supremizer Offline-online stablization without supremizer

Offline-online stabilization with supremizer








h2K (∇pN ,∇qN)K




+∑K∈Th

h2K (∇qN ,∇qN)K

• Different options for stabilization: Offline stabilization only without supremizer Offline stabilization only with supremizer Offline-online stablization without supremizer Offline-online stabilization with supremizer


Lid-driven Cavity

ΓD0

ΓD0

ΓDg

ΓD0

(0, 1) (µ2, 1)

(0, 0) (µ2, 0)

Figure: Domain Ω with the different boundaries identified.


Problem details

• 2 parameters: viscosity (µ1) and domain length (µ2)

• Non stable pair of FE: (P1− P1)

• Range of parameter domain: (µ1, µ2) ∈ [0.25, 0.75]× [1, 3]

Figure: RB pressure solution with offline only stabilization


Problem details






Problem details






Problem details






Numerical solutions

• Online solution for (µ1, µ2) = (0.6, 2)

Figure: FE solution (left) and RB solution (right), for velocity (top) and pressure (bottom)


Error evolution

Figure: Error in Greedy algorithm for velocity (left) and pressure (right)


Table of contents






Parametrized steady Navier-Stokes equations

We define the steady Navier-Stokes equations, with Re the Reynolds number: − 1Re ∆u + u · ∇u +∇p = f in Ω

∇ · u = 0 in Ω

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K


hK

(− 1

Re ∆uh + uh · ∇uh +∇ph,∇qh

)K

sdiv (vh;µ) = γ∑K∈Th

(∇ · uh,∇ · vh

)K

A. Brooks and T. Hughes Streamline Upwind/Petrov-Galerkin formulations forconvection dominated flows with particular emphasis on incompressibleNavier-Stokes equations. Computer Methods in Applied Mechanics and Engineering,32(13):199-259, 1982



We define the steady Navier-Stokes equations, with Re the Reynolds number:Find (uh, ph) ∈ Vh ×Mh such that1

Re (∇uh,∇vh)Ω + (uh · ∇uh, vh)Ω − (ph,∇ · vh)Ω = 〈f, vh〉 ∀vh ∈ Vh

(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh



h2K

(− 1



hK

(− 1


)K


(∇ · uh,∇ · vh

)K






(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh



h2K

(− 1



hK

(− 1


)K


(∇ · uh,∇ · vh

)K






(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh



h2K

(− 1



hK

(− 1


)K


(∇ · uh,∇ · vh

)K






(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh



h2K

(− 1



hK

(− 1


)K


(∇ · uh,∇ · vh

)K






(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh



h2K

(− 1



hK

(− 1


)K


(∇ · uh,∇ · vh

)K






(∇ · uh, qh)Ω = 0 ∀qh ∈ Mh



h2K

(− 1



hK

(− 1


)K


(∇ · uh,∇ · vh

)K



Stabilized Reduced Basis Navier-Stokes equations

Find (uh, ph) ∈ Vh ×Mh such that

a(uh, vh;µ) + c(uh, uh, vh;µ) + b(vh, ph;µ)+sconv (vh;µ) + sdiv (vh;µ) = 〈f, vh〉 ∀vh ∈ Vh


• Greedy algorithm for the snapshots selection ⇒ A posteriori error bound

• Different stabilizations as in Stokes problem



















Problem details

• Physical parameter: Reynolds number, (µ)• Non stable pair of FE: (P1− P1)• Range of parameter domain: µ ∈ [100, 500]

ΓD0

ΓD0

ΓDg

ΓD0

(0, 1) (1, 1)

(0, 0) (1, 0)

Saddam Hijazi, Shafqat Ali, Giovanni Stabile, Francesco Ballarin and GianluigiRozza. The Effort of Increasing Reynolds Number in Projection-Based ReducedOrder Methods: from Laminar to Turbulent Flows. Arxiv preprint. arXiv:1807.11370


Numerical solutions

• Online solution for µ = 200

Figure: FE solution (left) and RB solution (right), for velocity (top) and pressure (bottom)


Error evolution

Figure: Error in Greedy algorithm for velocity (left) and pressure (right)


Table of contents






Smagorinsky model

We define the steady Smagorinsky model, with Re the Reynolds number− 1

Re ∆u−∇ · (νT (u)∇(u)) + u · ∇u +∇p = f in Ω

∇ · u = 0 in Ω

as(uh; uh, vh;µ) =∑K∈Th

(νT (uh)∇uh,∇vh)K

• Non linear eddy viscosity: νT (uh) = (CShK )2|∇uh|

• VMS modelling for the eddy viscosity term

Chacón Rebollo, T., Gómez Mármol, M., Rubino S. Numerical analysis of a finiteelement projection-based VMS turbulence model with wall laws. Comput. MethodsAppl. Mech. Engrg. 285 (2015), 379–405.


Smagorinsky model

We define the steady Smagorinsky model, with Re the Reynolds numberFind (uh, ph) ∈ Vh ×Mh such that

a(uh, vh;µ) + c(uh, uh, vh;µ) + b(vh, ph;µ)+as(uh; uh, vh;µ) = 〈f, vh〉 ∀vh ∈ Vh

b(uh, qh;µ) = 0 ∀qh ∈ Mh







Smagorinsky model










Smagorinsky model






• Nonlinear eddy viscosity: νT (uh) = (CS hK )2|∇uh|




VMS-Smagorinsky model

We decompose the velocity and pressure spaces as

Yh = Yh ⊕ Y ′h Mh = Mh ⊕M′h

thus, uh = uh + u′h, u′h = (Id − σh)uh = σ∗huh, ph = ph + p′h

LES closure model: VMS-Smagorinsky

a′s(uh; uh, vh) =∫

Ω(CShK )2|∇(σ∗h (uh))|∇(σ∗h (uh)) : ∇(σ∗h (vh)) dΩ

Pressure stabilization:

spres(p, q) =∫

ΩτK ,p(µ) σ∗h (∇ph)σ∗h (∇qh) dΩ,

τK ,p(µ) =[

c11/Re + νT

h2K

+ c2UK

hK

]−1E. Delgado Ávila. Development of reduced numeric models to aero-thermic flows inbuildings. PhD Thesis, University of Seville, 2018.










spres(p, q) =∫


τK ,p(µ) =[

c11/Re + νT

h2K

+ c2UK

hK











spres(p, q) =∫


τK ,p(µ) =[

c11/Re + νT

h2K

+ c2UK

hK

]−1

E. Delgado Ávila. Development of reduced numeric models to aero-thermic flows inbuildings. PhD Thesis, University of Seville, 2018.










spres(p, q) =∫


τK ,p(µ) =[

c11/Re + νT

h2K

+ c2UK

hK



Stabilized VMS-Smagorinsky RB model

• Offline-Online stabilizationFind (uN , pN) ∈ VN ×MN such that

a(uN , vN ;µ) + c(uN , uN , vN ;µ) + b(vN , pN ;µ)+a′s(uN ; uN , vN ;µ) = 〈f, vN〉 ∀vN ∈ VN

b(uN , qN ;µ) + spres(pN , qN ;µ) = 0 ∀qN ∈ MN

• Greedy algorithm for the snapshots selection ⇒ A posteriori error estimator• EIM approximation for non linear terms:

∑K∈Th

(νT (σ∗huh)∇uh,∇vh)K ≈∑K∈Th

Mv∑j=1

(qvj ∇uh,∇vh)K

∑K∈Th

(τK ,p(µ) σ∗h (∇ph), σ∗h (∇qh))K ≈∑K∈Th

Mp∑j=1

(qpj σ∗h (∇ph), σ∗h (∇qh))K






• Greedy algorithm for the snapshots selection ⇒ A posteriori error estimator

• EIM approximation for non linear terms:

∑K∈Th


Mv∑j=1

(qvj ∇uh,∇vh)K

∑K∈Th


Mp∑j=1








∑K∈Th


Mv∑j=1

(qvj ∇uh,∇vh)K

∑K∈Th


Mp∑j=1








∑K∈Th


Mv∑j=1

(qvj ∇uh,∇vh)K

∑K∈Th


Mp∑j=1



A posteriori error estimator

Residual: R((vh, qh);µ) = F ((vh, qh);µ)− A((uN , pN), (vn, qN);µ)

εN(µ) = ‖R(·;µ)‖X ′ , τN(µ) = 4εN(µ)ρTβ2N

∆N(µ) = βN2ρT

[1−

√1− τN(µ)

]TheoremIf βN > 0 and τN(µ) ≤ 1, then there exists a unique solution (uh(µ), ph(µ)) to(FE) such that

‖(uh(µ), ph(µ))− (uN(µ), pN(µ))‖X ≤ ∆N(µ)





∆N(µ) = βN2ρT

[1−

√1− τN(µ)

]

TheoremIf βN > 0 and τN(µ) ≤ 1, then there exists a unique solution (uh(µ), ph(µ)) to(FE) such that






∆N(µ) = βN2ρT

[1−

√1− τN(µ)

]TheoremIf βN > 0 and τN(µ) ≤ 1, then there exists a unique solution (uh(µ), ph(µ)) to(FE) such that



Finite element details

• Reynolds range: µ ∈ [1000, 5100]

• Non stable pair of Finite Element. (P2− P2)

• Regular mesh (2601 nodes and 5000 triangles):

ΓD0

ΓD0

ΓDg

ΓD0

(0, 1) (1, 1)

(0, 0) (1, 0)


M5 10 15 20 25

10-4

10-3

10-2

10-1

100

‖νT (µ)− I[νT (µ)]‖∞‖τK,p(µ)− I[τK,p(µ)]‖∞

Figure: Infinity norm error for EIM greedy algorithm


N

1 2 3 4 5 6 7 810

0

101

102

103

104

105

maxµ∈D

τN (µ) without supremizer

maxµ∈D

τN (µ) with supremizer

Figure: Comparison with and without supremizer


N

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

10-4

10-3

10-2

10-1

100

101

102

103

104

105

maxµ∈D

τN (µ)

maxµ∈D

∆N (µ)

Figure: Maximum a posteriori error bound (without supremizer)


Re

1000 1500 2000 2500 3000 3500 4000 4500 500010

-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

∆N (µ)‖Uh(µ)− UN (µ)‖X

Figure: A posteriori error bound at N=16


FE and RB velocity solution

Figure: FE (left) and RB (right) velocity solution for µ = 4521


Results

FE dof: 30603

EIM dof: 25 (νT ) + 20 (τK ,p), RB dof: 32

Data µ = 1610 µ = 2751 µ = 3886 µ = 4521TFE 4083.19s 6918.53s 9278.51s 10201.7sTonline 0.71s 0.69s 0.69s 0.7sspeedup 5750 10026 13280 14459‖uh − uN‖T 2.4 · 10−5 4.129 · 10−6 3.14 · 10−5 3.23 · 10−5‖ph − pN‖0 2.17 · 10−7 1.99 · 10−8 5.38 · 10−8 6.36 · 10−8

Table: Data summary


Conclusions

• FE stabilization terms for convection dominated problems

• RB Offline-online stabilization the only consistent one

• No considering the inner pressure supremizer reduces the RB velocity spacedimension

• Good accuracy in the computation of the RB-Smagorinsky solution

THANK YOU FOR YOURATTENTION


Conclusions

• FE stabilization terms for convection dominated problems

• RB Offline-online stabilization the only consistent one

• No considering the inner pressure supremizer reduces the RB velocity spacedimension

• Good accuracy in the computation of the RB-Smagorinsky solution

THANK YOU FOR YOURATTENTION


Documents

Stabilization for convection dominate problems...Stabilizationfor convectiondominate problemsd GianluigiRozza mathLab,Mathematics Area, SISSA International Schoolfor AdvancedStudies,