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On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
On reliability of higher order finite elementmethod in fluid-structure interaction problems
Svacek, P.
Department of Math. Sciences The University of Texas at El Paso
CTU, Fac. of Mech. Eng., Dep. of Technical Math.
REC’06, February 22-24
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Thanks to
M. Horacek, CAS, IT, Czech Rep.
M. Feistauer, Charles University, Czech Rep.
M. Cecrdle, J. Malecek, Aeronautical Test and ResearchInstitute.
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Outline
1 Introduction
2 Mathematical ModelGeneral ModelSimplificationsInterface Conditions
3 Numerical ApproximationTime DiscretizationSpace discretization of Fluid Model
4 Numerical ResultsFluid Flow ApproximationFluid Approximation over Moving StructureAeroelastic Simulations
5 Conclusions
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Fluid-structure interaction problem
Air Flows Interacts with Elastic WingWind Tunnel in Aeronautical Test and Research Institute,
Prague
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Computational Aeroelasticity
numerical simulation of both fluid and structure motion
fluid-structure mutual interaction
structure motion → fluid characterizationaerodynamical forces → structural motion
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Computational Aeroelasticity
numerical simulation of both fluid and structure motion
fluid-structure mutual interaction
structure motion → fluid characterizationaerodynamical forces → structural motion
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Computational Aeroelasticity
numerical simulation of both fluid and structure motion
fluid-structure mutual interaction
structure motion → fluid characterizationaerodynamical forces → structural motion
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Mathematical Model
Computational Aeroelasticity
fluid flow model
elastic structure
interface conditions
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Mathematical Model
Computational Aeroelasticity
fluid flow model
elastic structure
interface conditions
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Mathematical Model
Computational Aeroelasticity
fluid flow model
elastic structure
interface conditions
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Fluid Models
incompressible viscous flow (NS eq.)
RANS equations - turbulence models
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Navier-Stokes System of Equations
Navier-Stokes system
∂v
∂t+ (v · ∇) v +∇p − ν4v = 0
∇ · v = 0 in Ωt
v - fluid velocity
p - pressure
Navier-Stokes system of equations
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Reynolds Averaged Navier-Stokes equations
Navier-Stokes system
∂v
∂t+ v · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
Reynolds Averaging
v = V + v′, p = P + p′, such that v = V
RANS equations
∂Vi
∂t− ν4Vi + (V · ∇) Vi +
∂P
∂xi=
∑j
∂
∂xj
(−v ′i v
′j
),
Reynolds-Stresses σRij = −v ′i v
′j
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Reynolds Averaged Navier-Stokes equations
Navier-Stokes system
∂v
∂t+ v · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
Reynolds Averaging
v = V + v′, p = P + p′, such that v = V
RANS equations
∂Vi
∂t− ν4Vi + (V · ∇) Vi +
∂P
∂xi=
∑j
∂
∂xj
(−v ′i v
′j
),
Reynolds-Stresses σRij = −v ′i v
′j
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Reynolds Averaged Navier-Stokes equations
Navier-Stokes system
∂v
∂t+ v · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
Reynolds Averaging
v = V + v′, p = P + p′, such that v = V
RANS equations
∂Vi
∂t− ν4Vi + (V · ∇) Vi +
∂P
∂xi=
∑j
∂
∂xj
(−v ′i v
′j
),
Reynolds-Stresses σRij = −v ′i v
′j
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Reynolds Averaged Navier-Stokes equations
Navier-Stokes system
∂v
∂t+ v · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
Reynolds Averaging
v = V + v′, p = P + p′, such that v = V
RANS equations
∂Vi
∂t− ν4Vi + (V · ∇) Vi +
∂P
∂xi=
∑j
∂
∂xj
(−v ′i v
′j
),
Reynolds-Stresses σRij = −v ′i v
′j
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Turbulence model - SA model
Reynolds-Stresses approximation σRij = −v ′i v
′j
Reynolds stresses are approximated
σRij =
2
3kδij + νT
(∂Vi
∂xj+∂Vj
∂xi
), νT ≈ ν,
Turbulence Modelling - Spallart-Almaras Model
∂ν
∂t+ (V · ∇)ν =
1
σ
∂
∂xi
((ν + ν)
∂ν
∂xi
)+
cb2
σ(∇ν)2 + G − Y ,
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUT
time/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes system
RANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes system
RANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes system
RANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.
Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUT
the turbulent stresses requires further modelling(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling
(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Comparison of NS and RANS:
Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!
Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Structure Model
elasticity equation
ρ∂2ui
∂t2−
∑j
∂σij(u)
∂xj= fi
u - structure deflection
special cases: linear stationary elasticity
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Structure Model
elasticity equation
ρ∂2ui
∂t2−
∑j
∂σij(u)
∂xj= fi
u - structure deflection
special cases: linear stationary elasticity
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Structure Model
elasticity equation
ρ∂2ui
∂t2−
∑j
∂σij(u)
∂xj= fi
u - structure deflection
special cases: linear stationary elasticity
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Flexibly Supported Airfoil
T
EO
khh
a
h
x'
kaaL t( )
M t( )
U0
x
Airfoil motion equations
system ODEs
mh + Sαα + Khhh = −L
Sαh + Iαα+ Kααα = M3
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Flexibly Supported Airfoil
T
EO
khh
a
h
x'
kaaL t( )
M t( )
U0
x
Airfoil motion equations
system ODEs
mh + Sαα cosα− Sαα2 sinα+ Khhh = −L
Sαh cosα+ Iαα+ Kααα = M3
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Interface Conditions
elastic structure model
interface velocity wI
v = wI , u = wI
equality of fluid/elastic forces
flexibly supported airfoil model
airfoil surface condition
v = wI
aerodynamical fluid forces L - lift and M - torsionalmoment
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Interface Conditions
elastic structure model
interface velocity wI
v = wI , u = wI
equality of fluid/elastic forces
flexibly supported airfoil model
airfoil surface condition
v = wI
aerodynamical fluid forces L - lift and M - torsionalmoment
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Goals - revisited
Model
fluid flow
flexibly supported airfoil
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
how can we verify our results ?
compare numerical results to experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Goals - revisited
Model
fluid flow
flexibly supported airfoil
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
how can we verify our results ?
compare numerical results to experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Goals - revisited
Model
fluid flow
flexibly supported airfoil
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
how can we verify our results ?
compare numerical results to experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
General Model
Simplifications
InterfaceConditions
NumericalApproximation
NumericalResults
Conclusions
Goals - revisited
Model
fluid flow
flexibly supported airfoil
Goals
determine the safe region (critical velocity)
simulate post-critical regimes (nonlinear aeroelasticity)
how can we verify our results ?
compare numerical results to experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Numerical Approximation
time discretization
space discretization
interface conditions
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Computations on Moving Meshes
How to approximat the time derivative ? AVI format
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
How to approximate the time derivative?
Navier-Stokes system in ALE form
DA
Dtv + (v −wg ) · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
How to approximate the time derivative?
Arbitrary Lagrangian-Eulerian method
Navier-Stokes system in ALE form
DA
Dtv + (v −wg ) · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
How to approximate the time derivative?
Arbitrary Lagrangian-Eulerian method
Define ALE mapping At
At : Ωref 7→ Ωt
Domain velocity (grid velocity)
wg (t,Y ) =∂At(Y )
∂t
ALE derivative - time derivative on ALE trajectory
DA
Dtf =
∂f
∂t+ (wg · ∇)f
Navier-Stokes system in ALE form
DA
Dtv + (v −wg ) · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
How to approximate the time derivative?
Arbitrary Lagrangian-Eulerian method
Define ALE mapping At
At : Ωref 7→ Ωt
Domain velocity (grid velocity)
wg (t,Y ) =∂At(Y )
∂t
ALE derivative - time derivative on ALE trajectory
DA
Dtf =
∂f
∂t+ (wg · ∇)f
Navier-Stokes system in ALE form
DA
Dtv + (v −wg ) · ∇v +∇p − ν4v = 0
∇ · v = 0 in Ωt
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Weak Formulation
The ALE derivative is approximated
DAv
Dt≈ 3vn+1 − 4vn + vn−1
2∆t
Weak formulation: find vn+1 = v, p
Weak formulation
(3v
2∆t, ϕ
)+
([(v −wg ) · ∇] v, ϕ
)+ ν(∇v,∇ϕ)
−(p,∇T · ϕ) + (∇ · v, q) =
(4vn − vn−1
2∆t, ϕ
)
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Galerkin formulation
FEM gives unstable results?
Galerkin method is unstable → several sources ofinstabilities
Babuska-Brezzi (inf-sup) condition needs to be satisfied
supvh∈Xh
(qh,∇ · vh)
‖vh‖1,2,Ω≥ c‖qh‖0,2,Ω
very high Reynolds numbers → convection dominatedflows
Re locK =
h‖v‖Kν
> 1
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Galerkin formulation
FEM gives unstable results?
Galerkin method is unstable → several sources ofinstabilities
Babuska-Brezzi (inf-sup) condition needs to be satisfied
supvh∈Xh
(qh,∇ · vh)
‖vh‖1,2,Ω≥ c‖qh‖0,2,Ω
very high Reynolds numbers → convection dominatedflows
Re locK =
h‖v‖Kν
> 1
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Galerkin formulation
FEM gives unstable results?
Galerkin method is unstable → several sources ofinstabilities
Babuska-Brezzi (inf-sup) condition needs to be satisfied
supvh∈Xh
(qh,∇ · vh)
‖vh‖1,2,Ω≥ c‖qh‖0,2,Ω
very high Reynolds numbers → convection dominatedflows
Re locK =
h‖v‖Kν
> 1
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Galerkin formulation
FEM gives unstable results?
Galerkin method is unstable → several sources ofinstabilities
Babuska-Brezzi (inf-sup) condition needs to be satisfied
supvh∈Xh
(qh,∇ · vh)
‖vh‖1,2,Ω≥ c‖qh‖0,2,Ω
very high Reynolds numbers → convection dominatedflows
Re locK =
h‖v‖Kν
> 1
⇒ use Galerkin/Least-Squares stabilization
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Spatial discretization
Stabilization ψ = (w · ∇)ϕ+∇q
L(U,V ) =∑K
δK
(3
2∆tv − ν4v + (w · ∇) v +∇p, ψ
)K
,
F(V ) =∑K
δK
(4vn − vn−1
2∆t, ψ
)K
,
Stabilized problem
Galerkin terms, GALS stabilization, grad-div stabilization
a(U,V )+L(U,V )+∑K∈τh
τK
(∇ · v,∇ · ϕ
)= f (V )+F(V ).
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
TimeDiscretization
Spacediscretization ofFluid Model
NumericalResults
Conclusions
Other topics
solution of nonlinear problem (NS) - linearization
solution of linear problem (UMFPACK)
approximation of ODEs
interface conditions - coupling of fluid-structure models
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Numerical Results
Fluid Flow Approximation (fixed structure)
Fluid Flow over Moving structure (validation)
Aeroelastic Simulations
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Numerical Results
Fluid Flow Approximation (fixed structure)
Fluid Flow over Moving structure (validation)
Aeroelastic Simulations compare to NASTRAN
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Boundary layer approximation
Re = 2 · 105, comparison with the analytical Blasius solution
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Approximation of Boundary Layer - Taylor Hood
laminar flow (Re = 2 · 105)
FE Dimension: 16683 x 2 + 4242 = 37608
Nodes: 4242
Elements: 8200
1 1
224
5
3
1
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Blasius solution - Taylor-Hood
laminar flow (Re = 2 · 105)
FE Dimension: 16683 x 2 + 4242 = 37608
Nodes: 4242 Elements: 8200
Can we improve that?
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Blasius solution - Taylor-Hood
laminar flow (Re = 2 · 105)
FE Dimension: 16683 x 2 + 4242 = 37608
Nodes: 4242 Elements: 8200
Can we improve that?
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Blasius solution - Taylor-Hood
laminar flow (Re = 2 · 105)
FE Dimension: 16683 x 2 + 4242 = 37608
Nodes: 4242 Elements: 8200
Can we improve that? anisotropic mesh adaptation, use ofhigher order FEs
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Blasius solution - P3/P2 elements
laminar flow (Re = 2 · 105)
uniform p-distribution
FE Dimension: 4012 x 2 + 1809 = 9833
Nodes: 472
Elements: 866
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Blasius solution - P4/P3 elements
laminar flow (Re = 2 · 105)
FE Dimension: 2x13235 (Velocity) + 7483 (Pressure) =33953
Nodes: 866, Elements: 1629
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Flow over NACA 632 − 415
Fluid velocity isolines, Re = 5 · 105, AVI format
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Flow over NACA 632 − 415
Fluid velocity isolines, Re = 5 · 105 AVI format
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Flow over NACA 632 − 415
aerodynamical lift coefficient (time averaged values)
comparison with experimental data for NACA 632 − 415
−8 −6 −4 −2 0 2 4 6 8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
CL
α [°]
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Flow over NACA 632 − 415
aerodynamical moment coefficient (time averaged values)
comparison with experimental data for NACA 632 − 415
−8 −6 −4 −2 0 2 4 6 8−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
CM
α [°]
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Flow over NACA 632 − 415
aerodynamical lift coefficient (time averaged values)
comparison with experimental data for NACA 632 − 415
−0.5 0 0.5 1−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
CD
α [°]
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Prescribed vibrations of α, 30 Hz, amplitude 3,2,1degrees.
0 0.02 0.04 0.06 0.08 0.1 0.12−3
−2
−1
0
1
2
3
1
2
3
4
5
6
7
8
9
10
11
12
13
time [sec]
alph
a[de
gree
s]
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Prescribed vibrations of α, 30 Hz, amplitude 1degree.
t[s]
c p
0.2 0.25
-0.5
-0.25
0
0.25
t[s]
c p
0.2 0.25
-0.5
-0.25
0
0.25
Pressure coeficient (up/down) at x/c = 0.15 - dependence ontime.
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Pressure Coefficient
What is pressure coefficient?
cp =p − p012ρU
2∞
for prescribed vibrations
α = α0 · sin(2πft)
the pressure at airfoil surface is expected to behave like
cp = cmeanp + c ′p sin(2πft) + c ′′p cos(2πft)
comparison with experimental dataBenetka, J. et al, Tech. report 3418/02, ARTI, 2002 ,Triebstein, H., 1986., J. Aircraft 23.
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Comparison of cmeanp with experimental data
x/c
c p
0 0.25 0.5 0.75
-0.8
-0.6
-0.4
-0.2
0
0.2
Comparison of cmeanp with experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Comparison of c ′p with experimental data
x/c
c p’
0 0.25 0.5 0.75
-24
-20
-16
-12
-8
-4
0
Comparison of c ′p with experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Comparison of c ′′p with experimental data
x/c
c p’’
0 0.25 0.5 0.75
0
4
Comparison of c ′′p with experimental data
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Stall Flutter Approximation
α = (10 + 10 · sin(2πft)),Re = 5000
Naudasher, E., Rockwell, D., Flow-Induced Vibrations, 1994
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Stall Flutter Approximation
α = (10 + 10 · sin(2πft)),Re = 5000
α = 10.859 α = 18.3407 α = 19.9988
α = 19.4553 α = 12.045 α = 8.06Naudasher, E., Rockwell, D., Flow-Induced Vibrations, 1994
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Stall Flutter
Fluid velocity isolines, Re = 5 · 105 AVI(1) format
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-60
-40
-20
0
20
40
60
U∞ = 4 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-60
-40
-20
0
20
40
60
U∞ = 8 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-60
-40
-20
0
20
40
60
U∞ = 12 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-60
-40
-20
0
20
40
60
U∞ = 16 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-60
-40
-20
0
20
40
60
U∞ = 18 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0
-60
-40
-20
0
20
40
60
U∞ = 22 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0
-60
-40
-20
0
20
40
60
U∞ = 26 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0-6
-4
-2
0
2
4
6
t [s]h
[mm
]0 0.1 0.2 0.3 0
-60
-40
-20
0
20
40
60
U∞ = 28 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-8
-6
-4
-2
0
2
4
6
8
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-80
-60
-40
-20
0
20
40
60
80
U∞ = 32 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
t [s]
α
0 0.1 0.2 0.3 0.4 0.5 0-8
-6
-4
-2
0
2
4
6
8
t [s]h
[mm
]0 0.1 0.2 0.3 0.4 0.5 0
-80
-60
-40
-20
0
20
40
60
80
U∞ = 34 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
U∞ = 40 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
U∞ = 40 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic simulations - laminar
U∞ = 45 m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
flexibly supported airfoil NACA 0012
RANS + Spallart-Almaras turbulence model
NASTRAN computation with the STRIP model criticalspeed U∞ = 37.7m/s
frequencies and damping comparison
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 7.5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 10m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 12.5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 15m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 17.5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 20m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 22.5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 25m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 27.5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 30m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 32.5m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 35m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 36m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
Solution of the coupled aeroelastic model (h,α), U = 37m/s
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Aeroelastic model - Turbulent Flow
U=37 m/s
velocity isolines, AVI format
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Comparison with NASTRAN computation
Frequencies comparison
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Fluid FlowApproximation
FluidApproximationover MovingStructure
AeroelasticSimulations
Conclusions
Comparison with NASTRAN computation
Damping
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Summary
NS solver numerical results were compared toexperimental/computational data.
NS solver and RANS solver results for aeroelastic problemwere compared each to other.
Performence: RANS x NS (?)
Conclusion
RANS and NS shows good agreement with NASTRANcomputations.
We must provide: careful mesh design, time step value, ...
How to increase reliability?
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Summary
NS solver numerical results were compared toexperimental/computational data.
NS solver and RANS solver results for aeroelastic problemwere compared each to other.
Performence: RANS x NS (?)
Conclusion
RANS and NS shows good agreement with NASTRANcomputations.
We must provide: careful mesh design, time step value, ...
How to increase reliability?
On reliabilityof FEM in FSI
Svacek, P.
Outline
Introduction
MathematicalModel
NumericalApproximation
NumericalResults
Conclusions
Summary
NS solver numerical results were compared toexperimental/computational data.
NS solver and RANS solver results for aeroelastic problemwere compared each to other.
Performence: RANS x NS (?)
Conclusion
RANS and NS shows good agreement with NASTRANcomputations.
We must provide: careful mesh design, time step value, ...
How to increase reliability?
On reliabilityof FEM in FSI
Svacek, P.
Can the grid motion “pollute” the solution?
On reliabilityof FEM in FSI
Svacek, P.
Can the grid motion “pollute” the solution?
AVI format Test Problem: constant fluid velocity v = (1, 0)on rectangle.
On reliabilityof FEM in FSI
Svacek, P.
Use anallytical derivative of ALE mapping ...
wg =∂x
∂t
Approximately 10 % error (!) AVI format
On reliabilityof FEM in FSI
Svacek, P.
Use finite difference formula ...
wg ≈3x (n+1) − x(n) + x(n−1)
2∆t
Grid Motion Influence on Solution (finite differenceapproximation of grid velocity)AVI format
On reliabilityof FEM in FSI
Svacek, P.
The ALE derivative approximation
use vn−1 - defined on Ωn−1
vn - defined on Ωn
and vn+1 - defined on Ωn+1
DAv
Dt≈ 3vn+1 − 4vn + vn−1
2∆t
where vn and vn−1 lives on Ωn+1
On reliabilityof FEM in FSI
Svacek, P.
The ALE derivative approximation
use vn−1 - defined on Ωn−1
vn - defined on Ωn
and vn+1 - defined on Ωn+1
DAv
Dt≈ 3vn+1 − 4vn + vn−1
2∆t
where vn and vn−1 lives on Ωn+1
On reliabilityof FEM in FSI
Svacek, P.
The ALE derivative approximation
use vn−1 - defined on Ωn−1
vn - defined on Ωn
and vn+1 - defined on Ωn+1
DAv
Dt≈ 3vn+1 − 4vn + vn−1
2∆t
where vn and vn−1 lives on Ωn+1
On reliabilityof FEM in FSI
Svacek, P.
Spatial discretization
Stabilization ψ = (w · ∇)ϕ+∇q
L(U,V ) =∑K
δK
(3
2τv − ν4v + (w · ∇) v +∇p, ψ
)K
,
F(V ) =∑K
δK
(4vn − vn−1
2τ, ψ
)K
,
Stabilized problem
Gelhard, T., Lube, G., Olshanskii, M. A., 2004. Stabilizedfinite element schemes with BB-stable elements forincompressible flows. Journal of Computational andApplied Mathematics (accepted).
Gelhard, T., Lube, G., Olshanskii, M. A., 2004. Stabilizedfinite element schemes with LBB-stable elements forincompressible flows. Journal of Computational andApplied Mathematics .
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