An Eulerian-Lagrangian approach for fluid-structure coupled systems using X-FEM Antoine Legay / CNAM-Paris Collaborations: A. Tralli, P. Gaudenzi, Universita` di Roma ”La Sapienza”, Italia
8th USNCCM, 25-27th July 2005, Austin
Introduction
Position of the problem
2/ 26 8th USNCCM, 25-27th July 2005, Austin
ΩF
Open structure Closed structure
t ΩF
ΩF
ΩS ΩS
∆t Γ t Γ
Transient dynamics Fluid-structure interaction
Introduction
Basic ideas Fluid: Eulerian description, fixed mesh Structure: Lagragian description Incompatible meshes ¨ Belytschko 99] Enrichment of the fluid fields using X-FEM [Moes,
Previous work: Compressible fluid, thin structure Fluid on one side of the structure No enrichment [Legay, Chessa and Belytschko 2005]
8th USNCCM, 25-27th July 2005, Austin
3/ 26
Introduction
Other approaches Classical ALE-approach: need to update the fluid mesh mesh distortions: can not deal with large relative structure displacement Similar methods: Immersed boundaries [Peskin 89, 02] Fictitious domain [Glowinski 94] [Bertrand 97] [Baaijens 01] Immersed FEM [Wang 04] [Zhang 04] Do not enrich the fluid field around the structure
8th USNCCM, 25-27th July 2005, Austin
4/ 26
Description of the problem
Strong form Fluid: incompressible, viscous, Eulerian description
ρF v˙i ρF vi v j τi j vi i 0 in ΩF τi j pδi j 2µei j
gi in ΩF
j
Structure: thin, Lagrangian description
ρS u¨i σi j j gi in ΩS ui ugi on ∂u ΩS σi j nSj Fig on ∂F ΩS
Fluid-structure interface 2
u˙i
dΓ
Γ
vi
β δ penalty term: 2
0 on Γ
u˙i
0 on Γ
vi
σi j n j
τi j
8th USNCCM, 25-27th July 2005, Austin
5/ 26
Description of the problem
Weak form Fluid: Find vi such that δvi
u˙i dΓ
0
Γ
δvi vi
β
δvi j τi j dΩF
ΩF
δvi τi j nFj dSF
∂ΩF
dΩF
j
ΩF
δvi gi dΩF
vi v j
ΩF
δvi ρF v˙i
penalty term Structure: Find ui such that δui
ΩS
δui gi dΩS
u˙i dΓ
0
Γ
δui vi
β
∂u
ΩS
δui σi j nSj dSS
∂F
ΩS
ΩS
δui j σi j dΩS
g
δui Fi dSS
ΩS
δui ρS u¨i dΩS
penalty term This leads to the strong form [Legay, Chessa and Belytschko 2005] 8th USNCCM, 25-27th July 2005, Austin
6/ 26
Discretization
Space discretization: finite element method Fluid: mixed formulation 9/4-node element:
Pressure: Ni4 x
Velocity: Ni9 x
Structure: 1D beam Incompatible meshes: Enrichment of the fluid (X-FEM)
8th USNCCM, 25-27th July 2005, Austin
7/ 26
Discretization
Time discretization: fractional time step method Semi-implicit scheme, 3-steps process: tn
tn
Un 1
2
3
Pn
Pn
t
Un
1
1 1
Collaboration with A. Tralli and P. Gaudenzi, Universit a` di Roma, Italia [Tralli and Gaudenzi, 3rd MIT conf., 2005] [Tralli and Gaudenzi, submitted to IJNME, 2005]
8th USNCCM, 25-27th July 2005, Austin
8/ 26
Discretization
Localization of the interface The position of the structure is described by a level-set function
0 on one side of the
φxt
structure
φxt
0
Discretization of φ x t :
0
0 on the other side
φxt
x
φxt
φxt
0 on the interface Γ
φxt
0
Γt :φxt
φxt
n
Normal on Γ: n
8th USNCCM, 25-27th July 2005, Austin
Grad φ
NI9 x φI t
φxt
φ: signed distance to the interface
9/ 26
Enrichment
Partition of unity
Set of functions f i x defined on ΩPU such that
x
1
∑ fi
ΩPU
x
i
Obviously,
ψx
xψx
∑ fi i
New approximation of g x in ΩPU :
j
g f x ψ x A ∑i i
∑Nj x Gj
gx
i
regular part
gx
1
ψx
8th USNCCM, 25-27th July 2005, Austin
0 et Agi
note that : G j
enriched part
10/ 26
Enrichment
Choice of enrichment Velocity
Pressure
continuous discontinuous derivative
8th USNCCM, 25-27th July 2005, Austin
sign φ x
φx
v
Enrichment: sign φ x
Enrichment: φ x
discontinuous
p
11/ 26
Enrichment
Velocity enrichment φ x Avj
∑ j N 4j x
v
∑i Ni9 x Vi
v
Enrichment: Regular part
Partition of unity
continuous, discontinuous derivative
Note: This choice of P.u. avoids problems in blending elements [Legay, Wang and Belytschko, 2005] 8th USNCCM, 25-27th July 2005, Austin
12/ 26
Enrichment
Pressure enrichment
sign φ x
∑ j N 4j x
A pj
p
∑i Ni4 x Pi
p
Partition of unity Regular part
Enrichment: discontinuous
Note: This enrichment does not introduce extra-terms in blending elements
8th USNCCM, 25-27th July 2005, Austin
13/ 26
Applications
Driven Cavity The fluid domain is divided into 2 parts: the left part is a driven cavity the right part should not move The structure is a rigid wall Reynolds number: 10
8th USNCCM, 25-27th July 2005, Austin
14/ 26
Applications
Driven Cavity, streamlines
Compatible mesh, no enrichment.
Non-compatible mesh, velocity and pressure enrichment.
8th USNCCM, 25-27th July 2005, Austin
15/ 26
Applications
Driven Cavity, velocity and pressure Comparison of velocities
Comparison of pressure
— compatible mesh
— compatible mesh
— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment
— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment
8th USNCCM, 25-27th July 2005, Austin
16/ 26
Applications
Driven Cavity, zoom around structure Comparison of velocities
0 04m.s
Comparison of pressure
1
— compatible mesh
— compatible mesh
— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment
— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment
8th USNCCM, 25-27th July 2005, Austin
17/ 26
Applications
Fixed structure in a driven cavity 1 4m
1m.s
1
The structure is fixed Fluid mesh: 24
12
Reynolds number: 10
31deg 0 5m
2m
2 6m
4m
8th USNCCM, 25-27th July 2005, Austin
18/ 26
Applications
Fixed structure in a driven cavity, streamlines
8th USNCCM, 25-27th July 2005, Austin
19/ 26
Applications
Translating straight structure 4m
Fluid mesh: 36
18
Reynolds number: 10
time=0s
Structure velocity: 1m.s 1
2m
time=2s
0 88m 1m
8th USNCCM, 25-27th July 2005, Austin
2m
20/ 26
Applications
Translating structure
8th USNCCM, 25-27th July 2005, Austin
0.03 s
0.7 s
1.4 s
2.0 s 21/ 26
Applications
Rotating structure 0 7m
Fluid mesh: 12
12
The structure is a rotating straight
2m
line
1rad.s
Angular velocity: Ω
1
2m 8th USNCCM, 25-27th July 2005, Austin
22/ 26
Applications
Rotating structure
8th USNCCM, 25-27th July 2005, Austin
23/ 26
Applications
Two Rotating structures 0 7m Fluid mesh: 24
12
Two rotating straight lines Angular velocities:
1rad.s
Ω2
0 5rad.s
Ω1
1
2m
0 8m
1
Ω2
Ω1
4m
8th USNCCM, 25-27th July 2005, Austin
24/ 26
Applications
Two Rotating structures
8th USNCCM, 25-27th July 2005, Austin
25/ 26
Conclusions and future work
Conclusions Eulerian-Lagrangian fluid-structure approach take advantage of the existing formulations for both fluid and structure Incompatible meshes no fluid mesh updating no mesh distortion Enrichment of fluid field around the interface using X-FEM improve accuracy around the interface
Future work Further validation Improve the enrichment at the structure tip Rigid structures with inertia terms, flexible structures 8th USNCCM, 25-27th July 2005, Austin
26/ 26
References A. Legay, J. Chessa and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, in press, 2005. A. Tralli and P. Gaudenzi. Simulation of unsteady incompressible flows by a fractional-step FEM. International Journal for Numerical Methods in Engineering, submitted, 2005.
Contacts
[email protected]
8th USNCCM, 25-27th July 2005, Austin
[email protected]