An Eulerian-Lagrangian approach for fluid-structure coupled systems using X-FEM Antoine Legay / CNAM-Paris

Transcription

1 An Eulerian-Lagrangian approach for fluid-structure coupled systems using X-FEM Antoine Legay / CNAM-Paris Collaborations: A. Tralli, P. Gaudenzi, Università di Roma La Sapienza, Italia 8th USNCCM, 25-27th July 2005, Austin

2 Introduction Position of the problem Fluid-structure interaction Transient dynamics ΩS ΩS Γ Γ Closed structure Open structure t t t ΩF ΩF ΩF 8th USNCCM, 25-27th July 2005, Austin 2/ 26

3 Introduction Basic ideas Fluid: Eulerian description, fied mesh Structure: Lagragian description Incompatible meshes Enrichment of the fluid fields using X-FEM [Moës, Belytschko 99] 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

4 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

5 τ i j Description of the problem Strong form Fluid: incompressible, viscous, Eulerian description ρ F v i v i i ρfviv j 0 in ΩF τ i j pδi j 2µei j j g i in Ω F Structure: thin, Lagrangian description ρ S ü i σi j j g i in Ω S u ugi i on uωs σ i j n S Fg j i on F Ω S Fluid-structure interface v i τ i j σi j n j 0 on Γ u 0 i on Γ penalty term: β 2 δ Γ v i u i 2 dγ 8th USNCCM, 25-27th July 2005, Austin 5/ 26

6 Description of the problem Weak form Fluid: Find v i such that δvi Ω F δv i ρ F v i vi v j j dω F Ω F δv i jτ i j dω F Ω F δv ig i dω F Ω F δv iτ i j n F j ds F β Γ δv i v i u i dγ 0 penalty term Structure: Find u i such that δui Ω S δu i ρ S ü i dω S Ω S δu i jσ i j dω S Ω S δu ig i dω S F Ω S δu if g i dss u Ω S δu iσ i j n S jds S β Γ δu i v i u i dγ 0 penalty term This leads to the strong form [Legay, Chessa and Belytschko 2005] 8th USNCCM, 25-27th July 2005, Austin 6/ 26

7 Pressure: N4 i Discretization Space discretization: finite element method Fluid: mied formulation 9/4-node element: Velocity: N 9 i Structure: 1D beam Incompatible meshes: Enrichment of the fluid (X-FEM) 8th USNCCM, 25-27th July 2005, Austin 7/ 26

8 Discretization Time discretization: fractional time step method Semi-implicit scheme, 3-steps process: t n t n 1 t U n U n 1 P n P n 1 Collaboration with A. Tralli and P. Gaudenzi, Università 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

9 Discretization Localization of the interface The position of the structure is described by a level-set function φ t Γ t : φ t 0 φ φ t t 0 on the interface Γ 0 on one side of the φ t structure φ t 0 on the other side φ t 0 φ t 0 Discretization of φ t : n φ t N9 I φi Normal on Γ: n Gradφ t φ: signed distance to the interface 8th USNCCM, 25-27th July 2005, Austin 9/ 26

10 Enrichment Partition of unity Set of functions f i defined on ΩPU such that ΩPU i f i 1 Obviously, i f i ψ ψ New approimation of g in Ω PU : g j N j G j regular part i f i ψ Ag i enriched part note that : G j 0et A g i 1 g ψ 8th USNCCM, 25-27th July 2005, Austin 10/ 26

11 Enrichment Choice of enrichment Velocity continuous discontinuous derivative Enrichment: φ Pressure discontinuous Enrichment: sign φ φ sign φ v p 8th USNCCM, 25-27th July 2005, Austin 11/ 26

12 Enrichment Velocity enrichment v i N 9 i Vi j N 4 j φ Av j 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

13 Enrichment Pressure enrichment p i N 4 i Pi j N 4 j sign φ p Ap j Regular part Partition of unity Enrichment: discontinuous Note: This enrichment does not introduce etra-terms in blending elements 8th USNCCM, 25-27th July 2005, Austin 13/ 26

14 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

15 Applications Driven Cavity, streamlines Compatible mesh, no enrichment. Non-compatible mesh, velocity and pressure enrichment. 8th USNCCM, 25-27th July 2005, Austin 15/ 26

16 Applications Driven Cavity, velocity and pressure Comparison of velocities Comparison of pressure compatible mesh non-comp. mesh, with no enrichment non-comp. mesh, with enrichment compatible mesh non-comp. mesh, with no enrichment non-comp. mesh, with enrichment 8th USNCCM, 25-27th July 2005, Austin 16/ 26

17 Applications Driven Cavity, zoom around structure Comparison of velocities Comparison of pressure 0 04m.s 1 compatible mesh non-comp. mesh, with no enrichment non-comp. mesh, with enrichment compatible mesh non-comp. mesh, with no enrichment non-comp. mesh, with enrichment 8th USNCCM, 25-27th July 2005, Austin 17/ 26

18 Applications Fied structure in a driven cavity The structure is fied 1m.s 1 1 4m Fluid mesh: deg 2m Reynolds number: m 2 6m 4m 8th USNCCM, 25-27th July 2005, Austin 18/ 26

19 Applications Fied structure in a driven cavity, streamlines 8th USNCCM, 25-27th July 2005, Austin 19/ 26

20 Applications Translating straight structure 4m Fluid mesh: Reynolds number: 10 Structure velocity: 1m.s m time=0s time=2s 2m 1m 2m 8th USNCCM, 25-27th July 2005, Austin 20/ 26

21 Applications Translating structure 0.03 s 0.7 s 1.4 s 2.0 s 8th USNCCM, 25-27th July 2005, Austin 21/ 26

22 Applications Rotating structure 0 7m Fluid mesh: The structure is a rotating straight line Angular velocity: Ω 1rad.s 1 2m 2m 8th USNCCM, 25-27th July 2005, Austin 22/ 26

23 Applications Rotating structure 8th USNCCM, 25-27th July 2005, Austin 23/ 26

24 Applications Two Rotating structures 0 7m Fluid mesh: Two rotating straight lines Angular velocities: Ω 1 0 5rad.s 1 Ω 1 2 1rad.s Ω1 0 8m Ω 2 2m 4m 8th USNCCM, 25-27th July 2005, Austin 24/ 26

25 Applications Two Rotating structures 8th USNCCM, 25-27th July 2005, Austin 25/ 26

26 Conclusions and future work Conclusions Eulerian-Lagrangian fluid-structure approach take advantage of the eisting 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, fleible structures 8th USNCCM, 25-27th July 2005, Austin 26/ 26

27 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, A. Tralli and P. Gaudenzi. Simulation of unsteady incompressible flows by a fractional-step FEM. International Journal for Numerical Methods in Engineering, submitted, Contacts antoine.legay@cnam.fr aldo.tralli@uniroma1.it 8th USNCCM, 25-27th July 2005, Austin