Stabilization for convection dominate problems...Stabilizationfor convectiondominate problemsd...
Transcript of Stabilization for convection dominate problems...Stabilizationfor convectiondominate problemsd...
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
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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)
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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
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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