stabilization for convection dominate problems...stabilizationfor convectiondominate problemsd...
TRANSCRIPT
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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
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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
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Table of contents
1 Advection-Diffusion problem
2 Steady Stokes equations
3 Steady Navier-Stokes equations
4 VMS-Smagorinsky turbulence model
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Advection-Diffusion problem
• Advection-diffusion equations depending on parameter:
−ε(µ)∆u + β(µ) · ∇u = f
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Advection-Diffusion problem
• Advection-diffusion equations depending on parameter:
(ε(µ)∇uh,∇vh)Ω + (β(µ) · ∇uh, vh)Ω = 〈f , vh〉
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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Numerical test
• β(µ) = (1, 1), ε(µ) = 1µ
⇒ Pe = µ
• µ ∈ [100, 1000]
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Numerical test
• β(µ) = (1, 1), ε(µ) = 1µ
⇒ Pe = µ
• µ ∈ [100, 1000]
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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)
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Numerical Results
Figure: RB solution for Pe = 600
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Table of contents
1 Advection-Diffusion problem
2 Steady Stokes equations
3 Steady Navier-Stokes equations
4 VMS-Smagorinsky turbulence model
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Steady Stokes equations
We define the steady Stokes equations, with ν the viscosity:−ν∆u +∇p = f in Ω
∇ · u = 0 in Ω
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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
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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
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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
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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 .
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Lid-driven Cavity
ΓD0
ΓD0
ΓDg
ΓD0
(0, 1) (µ2, 1)
(0, 0) (µ2, 0)
Figure: Domain Ω with the different boundaries identified.
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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
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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
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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
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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
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Numerical solutions
• Online solution for (µ1, µ2) = (0.6, 2)
Figure: FE solution (left) and RB solution (right), for velocity (top) and pressure (bottom)
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Error evolution
Figure: Error in Greedy algorithm for velocity (left) and pressure (right)
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Table of contents
1 Advection-Diffusion problem
2 Steady Stokes equations
3 Steady Navier-Stokes equations
4 VMS-Smagorinsky turbulence model
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Numerical solutions
• Online solution for µ = 200
Figure: FE solution (left) and RB solution (right), for velocity (top) and pressure (bottom)
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Error evolution
Figure: Error in Greedy algorithm for velocity (left) and pressure (right)
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Table of contents
1 Advection-Diffusion problem
2 Steady Stokes equations
3 Steady Navier-Stokes equations
4 VMS-Smagorinsky turbulence model
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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)
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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)
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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)
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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)
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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
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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
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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
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