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Assessment of Fictitious Domain method for Linear Stability Analysis - - PowerPoint PPT Presentation

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Assessment of Fictitious Domain method for Linear Stability Analysis of Fluid-Structure systems J. Moulin, J-L. Pfister, M.Carini, O.Marquet Office National dEtudes et de Recherches Arospatiales, Dpartement Arodynamique Fondamentale et

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Assessment of Fictitious Domain method for Linear Stability Analysis of Fluid-Structure systems

  • J. Moulin, J-L. Pfister, M.Carini, O.Marquet

Office National d’Etudes et de Recherches Aérospatiales, Département Aérodynamique Fondamentale et Expérimentale Funded by ERC Starting Grant ERCOFTAC-SIG33, Siena, 19-21 June 2017

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SLIDE 2

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Linear Stability Analysis & FSI

Zig-Zag mode (J.Tchoufag, JFM 2014) Navier-Stokes + rigid-solid Fluttering flag (C.Eloy, JFM 2008) Potential flow + 1D elastic solid Spring-Mounted airfoil Quasi-Steady theory + damped oscillator

Fluid modelling complexity Solid modelling complexity

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Linear Stability Analysis & FSI

Zig-Zag mode (J.Tchoufag, JFM 2014) Navier-Stokes + rigid-solid Fluttering flag (C.Eloy, JFM 2008) Potential flow + 1D elastic solid Spring-Mounted airfoil Quasi-Steady theory + damped oscillator

Fluid modelling complexity Solid modelling complexity

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J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Linear Stability Analysis & FSI

Zig-Zag mode (J.Tchoufag, JFM 2014) Navier-Stokes + rigid-solid Fluttering flag (C.Eloy, JFM 2008) Potential flow + 1D elastic solid Spring-Mounted airfoil Quasi-Steady theory + damped oscillator

Fluid modelling complexity Solid modelling complexity

Objective

Assess the use of Fictitious Domain approach for Stability Analysis of FSI systems

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Plan

1

Introduction

2

Conforming vs. Non-Conforming

3

App.1 : VIV on rigid cylinder

4

  • App. 2 : cylinder with flexible appendice

5

Conclusion

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Numerical frameworks for FSI

Description of the separate problems

Fluid : Eulerian description in domain Ωf(t) Solid : Lagrangian description in domain Ωs

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ξs u Ωf(t) Ωs(t)

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Numerical frameworks for FSI

Description of the separate problems

Fluid : Eulerian description in domain Ωf(t) Solid : Lagrangian description in domain Ωs

Description of coupled FSI problems

Conforming methods

Arbitrary Lagrangian-Eulerian method (ALE) (J.Donea, Enc. Comp. Mech. 2004) ➥ Artificial unknowns : mesh displacement in the fluid region ➥ Added equation : fluid mesh movement equation

Non-Conforming methods

Fictitious Domain method (FD) (R.Glowinsky, Int. J. Numer. Meth. Fluid 1999) ➥ Artificial unknowns : fluid velocity in the solid region ➥ Added equation : constraint equation to impose the solid presence

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J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

ξs u Ωf(t) Ωs(t)

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Numerical frameworks for FSI

Description of the separate problems

Fluid : Eulerian description in domain Ωf(t) Solid : Lagrangian description in domain Ωs

Description of coupled FSI problems

Conforming methods (Ref.)

Arbitrary Lagrangian-Eulerian method (ALE) (J.Donea, Enc. Comp. Mech. 2004) ➥ Artificial unknowns : mesh displacement in the fluid region ➥ Added equation : fluid mesh movement equation

Non-Conforming methods

Fictitious Domain method (FD) (R.Glowinsky, Int. J. Numer. Meth. Fluid 1999) ➥ Artificial unknowns : fluid velocity in the solid region ➥ Added equation : constraint equation to impose the solid presence

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ξs u Ωf(t) Ωs(t)

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

An illustrative example

Heat equation in a moving domain Ωf(ξs(t)) : ∆T = 0 in Ωf(ξs(t))

T = 1 T = 1 T = 1 T = 1 T = 0 ξs(t) Ωf

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SLIDE 10

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

An illustrative example

Heat equation in a moving domain Ωf(ξs(t)) : ∆T = 0 in Ωf(ξs(t))

T = 1 T = 1 T = 1 T = 1 T = 0 ξs(t) Ωf

How is the fluid-structure coupling handled in ALE vs. Fictitious Domain frameworks ? 7/22

J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

ALE formalism : an illustrative example

Current configuration x Ωf(ξs)

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

ALE formalism : an illustrative example

Current configuration x Ωf(ξs)

  • Ωf(ξs)

∇T · ∇v dx = 0

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

ALE formalism : an illustrative example

Current configuration x Ωf(ξs) Reference configuration xr Ωf x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ωf(ξs)

∇T · ∇v dx = 0

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SLIDE 14

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

ALE formalism : an illustrative example

Current configuration x Ωf(ξs) Reference configuration xr Ωf x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ωf(ξs)

∇T · ∇v dx = 0

  • Ωf

(F(ξs)−T ∇T)·(F(ξs)−T ∇v) J(ξs)dxr = 0

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

ALE formalism : an illustrative example

Current configuration x Ωf(ξs) Reference configuration xr Ωf x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ωf(ξs)

∇T · ∇v dx = 0

  • Ωf

(F(ξs)−T ∇T)·(F(ξs)−T ∇v) J(ξs)dxr = 0 The geometrical non-linearity can be put either : in the fluid integration domain or in the fluid integrand

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

ALE formalism : an illustrative example

Current configuration x Ωf(ξs) Reference configuration xr Ωf x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ωf(ξs)

∇T · ∇v dx = 0

  • Ωf

(F(ξs)−T ∇T)·(F(ξs)−T ∇v) J(ξs)dxr = 0 The geometrical non-linearity can be put either : in the fluid integration domain or in the fluid integrand Imagine what will happen with full Navier-Stokes ...

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration Ω

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration x Ω Ωs(ξs)

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration x Ω Ωs(ξs)

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs(ξs)

(T − 0) µ, dx

  • constraint

+

  • Ωs(ξs)

λv dx

  • Lagrange multiplier

= 0

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SLIDE 20

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration x Ω Ωs(ξs) Reference configuration xr Ω Ωs x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs(ξs)

(T − 0) µ, dx

  • constraint

+

  • Ωs(ξs)

λv dx

  • Lagrange multiplier

= 0

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration x Ω Ωs(ξs) Reference configuration xr Ω Ωs x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs(ξs)

(T − 0) µ, dx

  • constraint

+

  • Ωs(ξs)

λv dx

  • Lagrange multiplier

= 0

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs
  • T(xr + ξs) − 0
  • µ(xr + ξs) J(ξs)dxr
  • constraint

+

  • Ωs

λ(xr + ξs)v(xr + ξs) J(ξs)dxr

  • Lagrange multiplier

= 0

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SLIDE 22

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration x Ω Ωs(ξs) Reference configuration xr Ω Ωs x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs(ξs)

(T − 0) µ, dx

  • constraint

+

  • Ωs(ξs)

λv dx

  • Lagrange multiplier

= 0

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs
  • T(xr + ξs) − 0
  • µ(xr + ξs) J(ξs)dxr
  • constraint

+

  • Ωs

λ(xr + ξs)v(xr + ξs) J(ξs)dxr

  • Lagrange multiplier

= 0 The geometrical non-linearity can be put either : in the solid integration domain or in the solid integrand

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SLIDE 23

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Fictitious Domain formalism : an illustrative example

Current configuration x Ω Ωs(ξs) Reference configuration xr Ω Ωs x = xr + ξ(xr, t) F = ∂x ∂xr J = det(F)

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs(ξs)

(T − 0) µ, dx

  • constraint

+

  • Ωs(ξs)

λv dx

  • Lagrange multiplier

= 0

  • Ω

∇T · ∇v dx

  • physical equation on total domain

+

  • Ωs
  • T(xr + ξs) − 0
  • µ(xr + ξs) J(ξs)dxr
  • constraint

+

  • Ωs

λ(xr + ξs)v(xr + ξs) J(ξs)dxr

  • Lagrange multiplier

= 0 The geometrical non-linearity can be put either : in the solid integration domain or in the solid integrand Nothing different with the full Navier-Stokes ...

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Problem set-up

u · n = 0 free outflow u · n = 0 U∞, ρf, ν k, c, ρs D

Modelisation

Fluid model : Navier-Stokes Solid model : damped linear

  • scillator

Non-dimensional numbers :

Re = ρfU∞D ν = 40 ω0 =

  • k

m D U∞ = 0.8, ζ = c 2 √ km = 0.01 ˜ m = ρs ρf = 10

Numerics :

Finite elements modelisation in Freefem++(Taylor-Hood elements) Newton’s method for baseflows calculations Shift and invert method for eigenvalue computations with ARPACK

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SLIDE 25

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

Fictitious Domain

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SLIDE 26

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

Fictitious Domain ALE FIGURE – Baseflows : x-velocity

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SLIDE 27

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

Fictitious Domain ALE FIGURE – Baseflows : x-velocity

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

Fictitious Domain ALE FIGURE – Baseflows : x-velocity

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

50 100 150 200 1.56 1.58 1.6

Ncyl CD

ALE FD

FD converges towards ALE value when refining on the solid walls Excellent comparison of converged CD values

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Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Eigenvalue problem : comparison of spectrums

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.8 −0.6 −0.4 −0.2

ω λ

ALE FD FIGURE – Spectrums FD and ALE spectrums are similar

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SLIDE 31

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of eigenmodes

Fictitious Domain

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SLIDE 32

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of eigenmodes

Fictitious Domain ALE FIGURE – Eigenmodes : x-velocity

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SLIDE 33

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of eigenmodes

Fictitious Domain

v 1 0.5

ALE

v 1 0.5

FIGURE – Eigenmodes : x-velocity

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SLIDE 34

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of eigenmodes

Fictitious Domain

v 1 0.5

) ALE

v 1 0.5

FIGURE – Eigenmodes : y-velocity

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SLIDE 35

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Problem set-up

Geometry

x y

D l h

U∞, ρf, ν E, ρs, νs

Modelisation

Fluid model : Navier-Stokes Solid model : linear elasticity

Non-dimensional numbers :

Re = ρfU∞D νf = 100 Kb = Eh3 12ρfU2

∞l3 = 0.04, νs = 0.35,

h l = 0.03 ˜ m = ρs ρf = 85

Numerics :

Finite elements modelisation in Freefem++(Taylor-Hood elements) Newton’s method for baseflows calculations Shift and invert method for eigenvalue computations with ARPACK

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SLIDE 36

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

FIGURE – Stationary solution : ALE vs FD

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SLIDE 37

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of stationary fields

FIGURE – Stationary solution : ALE vs FD

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SLIDE 38

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Eigenvalue problem : comparison of spectrums

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.6 −0.4 −0.2 Mode A Mode B Mode C

ω λ

ALE FD FD and ALE spectrums are similar ...

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SLIDE 39

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Eigenvalue problem : comparison of spectrums

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.6 −0.4 −0.2 Mode A Mode B Mode C

ω λ

ALE FD FD and ALE spectrums are similar ... ... Except from the unstable steady mode C

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SLIDE 40

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Comparison of eigenmodes

Unstable unsteady modes

FIGURE – Mode A : pressure distribution

➥ VIV instability

FIGURE – Mode B : pressure distribution

➥ Mix flutter-VIV instability

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SLIDE 41

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

What about the steady mode ?

FIGURE – Mode C : pressure distribution

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SLIDE 42

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

What about the steady mode ?

FIGURE – Mode C : pressure distribution

➥ Aeroelastic divergence instability

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SLIDE 43

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

What about the steady mode ?

FIGURE – Mode C : pressure distribution

➥ Aeroelastic divergence instability

added stiffness due to the fluid forces 21/22

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SLIDE 44

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder

  • App. 2 : cylinder with flexible appendice

Conclusion

Conclusion

Present work

We disscussed the use of a Fictitious Domain framework for linear stability analysis of FSI problems The Fictitious Domain predicts accurate baseflows and eigenvalue spectrums in most of our test cases at lower computationnal cost (7-10 times faster) Its precision is still in question for some types of instabilities where the interface stresses need particularly accurate evaluation

Perspectives

Other types of FSI instabilities should be tested (wing flutter, galloping) The use of more recent non-conformal methods might be considered

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