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

2/ 34 G. Rozza Stabilization for Convection Dominated Problems

Advection-Diffusion problem

• Advection-diffusion equations depending on parameter:

−ε(µ)∆u + β(µ) · ∇u = f

3/ 34 G. Rozza Stabilization for Convection Dominated Problems

Advection-Diffusion problem

• Advection-diffusion equations depending on parameter:

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

3/ 34 G. Rozza Stabilization for Convection Dominated Problems

Advection-Diffusion problem

• Advection-diffusion equations depending on parameter:

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

⇓a(uh, vh;µ) = F (vh) ∀vh ∈ Vh

3/ 34 G. Rozza Stabilization for Convection Dominated Problems

Advection-Diffusion problem

• Advection-diffusion equations depending on parameter:

(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

3/ 34 G. Rozza Stabilization for Convection Dominated Problems

Advection-Diffusion problem

• Advection-diffusion equations depending on parameter:

(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

• Stabilization methods for advection dominate problem

3/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

4/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

5/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

5/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

5/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

5/ 34 G. Rozza Stabilization for Convection Dominated Problems

Numerical test

• β(µ) = (1, 1), ε(µ) = 1µ

⇒ Pe = µ

• µ ∈ [100, 1000]

6/ 34 G. Rozza Stabilization for Convection Dominated Problems

Numerical test

• β(µ) = (1, 1), ε(µ) = 1µ

⇒ Pe = µ

• µ ∈ [100, 1000]

6/ 34 G. Rozza Stabilization for Convection Dominated Problems

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)

6/ 34 G. Rozza Stabilization for Convection Dominated Problems

Numerical Results

Figure: RB solution for Pe = 600

7/ 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

8/ 34 G. Rozza Stabilization for Convection Dominated Problems

Steady Stokes equations

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

∇ · u = 0 in Ω

9/ 34 G. Rozza Stabilization for Convection Dominated Problems

Steady Stokes equations

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

9/ 34 G. Rozza Stabilization for Convection Dominated Problems

Steady Stokes equations

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

• Bilinear forms:

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

9/ 34 G. Rozza Stabilization for Convection Dominated Problems

Steady Stokes equations

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

• Bilinear forms:

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

• Discrete inf-sup condition:

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

supvh∈Vh

b(vh, qh;µ)‖vh‖1‖qh‖0

9/ 34 G. Rozza Stabilization for Convection Dominated Problems

Steady Stokes equations

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

• Bilinear forms:

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

• 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 .

9/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

10/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

10/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

10/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

10/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

10/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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 without supremizer

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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 without supremizer

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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 without supremizer Offline stabilization only with supremizer

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

Offline-online stabilization with supremizer

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

11/ 34 G. Rozza Stabilization for Convection Dominated Problems

Lid-driven Cavity

ΓD0

ΓD0

ΓDg

ΓD0

(0, 1) (µ2, 1)

(0, 0) (µ2, 0)

Figure: Domain Ω with the different boundaries identified.

12/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

13/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

13/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

13/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

13/ 34 G. Rozza Stabilization for Convection Dominated Problems

Numerical solutions

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

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

14/ 34 G. Rozza Stabilization for Convection Dominated Problems

Error evolution

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

15/ 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

16/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

Parametrized steady Navier-Stokes equations

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

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

Parametrized steady Navier-Stokes equations

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

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

Parametrized steady Navier-Stokes equations

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

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

Parametrized steady Navier-Stokes equations

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

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

Parametrized steady Navier-Stokes equations

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

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

Parametrized steady Navier-Stokes equations

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

• Stabilization terms for momentum and continuity equations:

sconv (vh;µ) = δ∑K∈Th

h2K

(− 1

Re ∆uh + uh · ∇uh +∇ph, uh · ∇vh)K

spres(qh;µ) = δ∑K∈Th

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

17/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

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

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

• Different stabilizations as in Stokes problem

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

18/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

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

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

• Different stabilizations as in Stokes problem

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

18/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

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

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

• Different stabilizations as in Stokes problem

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

18/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

19/ 34 G. Rozza Stabilization for Convection Dominated Problems

Numerical solutions

• Online solution for µ = 200

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

20/ 34 G. Rozza Stabilization for Convection Dominated Problems

Error evolution

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

21/ 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

22/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

23/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

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.

23/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

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.

23/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

as(uh; uh, vh;µ) =∑K∈Th

(νT (uh)∇uh,∇vh)K

• Nonlinear eddy viscosity: νT (uh) = (CS hK )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.

23/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

24/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

24/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

]−1

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

24/ 34 G. Rozza Stabilization for Convection Dominated Problems

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.

24/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

25/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

25/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

25/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

25/ 34 G. Rozza Stabilization for Convection Dominated Problems

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)

26/ 34 G. Rozza Stabilization for Convection Dominated Problems

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)

26/ 34 G. Rozza Stabilization for Convection Dominated Problems

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)

26/ 34 G. Rozza Stabilization for Convection Dominated Problems

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)

27/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

28/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

29/ 34 G. Rozza Stabilization for Convection Dominated Problems

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)

30/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

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FE and RB velocity solution

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

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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

33/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

34/ 34 G. Rozza Stabilization for Convection Dominated Problems

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

34/ 34 G. Rozza Stabilization for Convection Dominated Problems