Fully Coupled Fluid2Structure Interaction Model ...

China Ocean Engineering , Vol. 21 , No. 3 , pp . 439 - 450

Ζ 2007 China Ocean Press , ISSN 089025487

Fully Coupled Fluid2Structure Interaction Model Based on Distributed Lagrange Multiplier/ Fictitious Domain Method 3 J I Chun2ning ( 及春宁) a , DONG Xiao2qiang ( 董晓强) b , ZHAO Chong2jiu ( 赵冲久) c and WANG Yuan2zhan ( 王元战) a ,1 a b

School of Civil Engineering , Tianjin University , Tianjin 300072 , China

College of Architecture and Civil Engineering , Taiyuan University of Technology , Taiyuan 030024 , China

a

Laboratory of Coastal Engineering Structures , Tianjin Research Institute of Water Transport Engineering , Tanggu 300456 , China

( Received 19 October 2006 ; accepted 15 June 2007)

ABSTRACT This paper , with a finite element method , studies the interaction of a coupled incompressible fluid2rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model , the rigid structure is tak2 en as“fictitious”fluid with zero strain rate. Both fluid and structure are described by velocity and pressure. The whole domain , including fluid region and structure region , is modeled by the incompressible Navier2Stokes equations which are discretized with fixed Eulerian mesh. However , to keep the structure’s rigid body shape and behavior , a rigid body con2 straint is enforced on the“fictitious”fluid domain by use of the Distributed Lagrange Multiplier/ Fictitious Domain (DLM/ FD) method which is originally introduced to solve particulate flow problems by Glowinski et al . For the verification of the model presented herein , a 2D numerical wave tank is established to simulate small amplitude wave propagations , and then numerical results are compared with analytical solutions. Finally , a 2D example of fluid2structure interaction under wave dynamic forces provides convincing evidences for the method excellent solution quality and fidelity. Key words : fluid2structure interaction ; f ully coupled model ; distributed Lagrange multiplier/ fictitious domain method ; numerical wave tank

1. Introduction Fluid2Structure Interaction ( FSI) is an engineering problem with many practical aspects such as the swash of seabed , the stagnation of contamination , the settlement of sand around coastal structures , and the resonance of structures under external wave excitations. However , the coupling between two essentially different domains often leads to highly nonlinear behaviors and the development of robust and efficient solution techniques for such problems present one of the great challenges in computational mechanics , as summarized by Ohayon and Felippa ( 2001) . Analyses of the complex Fluid2Structure Interaction can be performed by using partitioned (Lghner ( monolithic” ) ( Hü et al . , 1995 ; Felippa et al . , 2001) or fully coupled “ bner et al . , 2004 ; Heil , 2004 ; Hü bner and Dinkler , 2005 ; Walhorn et al . , 2005) schemes. Partitioned methods utilize sepa2 3 This study is supported by the National Natural Science Foundation of China ( Grant No. 50579046) and the Science Foundation of Tianjin Municipal Commission of Science and Technology ( Grant No. 043114711) 1 Correspondence author. E2mail : yzwang @tju. edu. cn

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JI Chun2ning et al . / China Ocean Engineering , 21 ( 3) , 439 - 450

rated solvers for fluid and structure domains and attempt to obtain a coupled solution via fixed2point it2 erations. This approach is relatively easy to implement and is convenient in when specialized solvers for fluid and solid problems are readily available. However , partition methods also have serious draw2 backs. Fixed2point iterations tend to converge slowly ( Heil , 1998) and , in the presence of strong flu2 id2structure coupling , the iteration can diverge even if a good initial guess for the solution is available. In a fully coupled approach , fluid and structure equations are discretized and solved simultaneously. In the case of large structural displacements ,“monolithic”solution procedures may be preferable in order to ensure stability and convergence of the coupled solution. Borrowing the idea from the Fictitious Domain ( FD) method ( Glowinski et al . , 2001 ) in the multi2phase flow areas , a set of fully coupled Fluid2Structure Interaction governing equations is pre2 sented. In this“monolithic”approach , the structure is taken as“fictitious”fluid with zero strain rate and the whole computational domain is modeled by the Navier2Stokes equations. However , to keep the rigid body shape and behaviors of the“fictitious”fluid , the Distributed Lagrange Multiplier ( DLM) method is applied on this domain. The whole field , including fluid region and structure region , is de2 scribed by velocity and pressure and the entire set of model equations is discretized with fixed Eulerian mesh. Three major advantages of the present formulation include : ( i ) The governing equations unitized both for fluid and structure help capturing the predominant physics of interaction phenomena ; ( ii ) The interfacial force/ displacement between fluid and structure are internal actions for the overall system. Therefore , the stress/ velocity consistency conditions on the fluid2structure interface are automatically satisfied in this fully coupled model ; ( iii ) For the use of fixed Eulerian mesh , it is not necessary to remesh the computational domain , and thus free from mesh distortions. The content of the paper is structured as follows. First , details of the fully coupled Fluid2Struc2 ture Interaction model based on the DLM/ FD method are given in Section 2. Then , the operator2split2 ting approach using the FEM is described in Section 3. In Section 4 , the VOF method based on un2 structured triangular mesh is briefly presented. Finally , the reliability and capability of the proposed method are shown in Section 5 by a 2D small amplitude wave propagation problem and a 2D Fluid2 Structure Interaction problem with a free2surface subjected to external wave excitations.

2. Model Equations 2. 1 　Governing Equations for Fluid and Structure

Let Ω ∈Rd ( d = 2 , 3) be a space region ; supposedly Ω is filled with an incompressible viscous fluid of density ρf and viscosity μ , and it also contains a moving rigid body B of density ρs , as depict2 ed in Fig. 1. Here , Γ indicates the outer boundary of the whole computational domain and 5 B de2 notes the fluid2structure interface. The fluid is modeled by the Navier2Stokes equations

ρf 5 u + ( u ・g) u = ρf g + g ・σ in Ω \ B ( t ) , 5t g ・u = 0 in Ω \ B ( t ) ,

( 1) ( 2)

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u ( x , 0) = u 0 ( x) , Π x ∈Ω \ B ( 0) , with g ・u0 = 0 ,

441

( 3)

to be completed by u = g0 on Γ , with

∫g ・ndΓ = 0 , Γ

0

( 4)

and the following no2slip boundary condition on 5 B , u ( x , t ) = V ( t ) + ω( t ) × G ( t ) x , Π x ∈5 B ( t ) ,

( 5)

　Fig. 1. An example of 22D flow region with one rigid body.

where , V and ω denote the translation velocity and angular velocity of the centroid G of the rigid structure , respectively. The stress2tensor σ verifies

σ = - pI + μ( g u + g u t ) . ( 6) 　　The motion of the rigid structure is controlled by the Newton2Euler equations dV i e ( 7) M = Mg + F + F ( G , Θ) , dt dω ( 8) I - Iω ×ω = Ti + Te ( G , Θ) , dt dG ( 9) = V, dt dΘ ( 10) = ω, dt and to be completed by the following initial conditions ( 11) G ( 0) = G0 , 　Θ ( 0) = Θ0 , 　V ( 0) = V0 , 　ω( 0) = ω0 . In Eqs. ( 7) ～ ( 11) , M and I are the mass and inertia tensor of the rigid body , respectively. Θ is the rotation angle of the rigid body. F i and Ti are the hydrodynamic force and torque at G acting on the rigid body by fluid , respectively. They are formulated by

∫σn d x , =∫Gx ×σnd x . i

F =-

Ti e

e

5B

5B

( 12) ( 13)

F and T are the external force and torque at G acting on the rigid body by outer fields , which are

dependent on specific situations.

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2. 2 　Variational Formulations for the Governing Equations

Similar to the method presented by Glowinski et al . ( 2001) , the variational formulations of the Fluid2Structure Interaction are obtained : define the following functional spaces 1 d W g0 ( t ) = { v | v ∈ ( H ( Ω) ) , 　v = g0 ( t ) on Г },

∫

( 14)

2 2 L 0 ( Ω) = { q | q ∈ L ( Ω) , 　 qd x = 0}

( 15)

Λ( t ) = ( H1 ( B ( t ) ) ) d .

( 16)

Ω

　　The fictitious domain formulation of fluid2structure interaction governing equations with distributed Lagrange multipliers λ ∈Λ( t ) are for a. e. t > 0 , find { u ( t ) , p ( t ) , G ( t ) , Θ ( t ) , V ( t ) , ω( t ) , λ( t ) } such that 2 d 3 u ( t ) ∈ W g0 ( t ) , p ( t ) ∈ L 0 ( Ω) , G ( t ) ∈ R , Θ ( t ) ∈ R , d 3 V ( t ) ∈ R , ω( t ) ∈R , λ( t ) ∈Λ( t ) ,

( 17)

and

5u

∫( 5 t + ( u ・g) u) ・vd x + 2μ∫D ( u) : D ( v) d x 　　 ∫p g ・vd x - (λ, v - Y - θ × Gx) 　ρf

Ω

Ω

Λ( t)

Ω

ρf dV dω e e 　　 + ( 1 - ρ ) ( M ・Y + ( I - Iω ×ω) ・θ - F ・Y - T ・θ d t dt s ρf

∫g ・vd x + (1 - ρ ) Mg ・Y,

= ρf

Ω

　Π v ∈ (

s

H10 ( Ω)

) , 　Π Y ∈ R d, 　Πθ ∈ R 3, d

( 18)

∫

　 q g ・u d x = 0 , 　Π q ∈ L 2 ( Ω) ,

( 19)

Ω

　

dG = V, dt

( 20)

　

dΘ = ω, dt

( 21)

　(μ, u ( t ) - V ( t ) - ω( t ) × G ( t ) x) Λ( t) = 0 , Πμ ∈Λ( t ) ,

( 22)

　u ( x , 0) = u0 ( x) , Π x ∈Ω \ B ( 0) , 　u ( x , 0) = V0 + ω0 × G0 x , Π x ∈ B ( 0) ,

( 23)

( 24) 　G ( 0) = G , Θ ( 0) = Θ , V ( 0) = V , ω( 0) = ω . 　　Obviously , the distributed Lagrange multiplier λ acts as the pressure p and Eq. ( 22) represents 0

0

0

0

the rigid body constraint imposed on the“fictitious”fluid region to maintain the rigid body shape and ) Λ( t) are behavior of the“fictitious”fluid. The two most nature choices for ( ・,・ (μ , v) Λ( t) =

∫ (μ ・v + δ gμ ∶g v) d x , 　Πμ, v ∈Λ( t) , 2

B ( t)

( 25)

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(μ, v) Λ( t) =

443

∫ [μ ・v + δ D (μ) ∶D ( v) ]d x , 　Πμ, v ∈Λ( t) , 2

( 26)

B ( t)

with δ as a characteristic length ( the maximum length of B , for example) . For two dimensional fluid2 structure interaction problems discussed in this paper , we have d = 2 , Θ = ( 0 , 0 , Θ) , ω = ( 0 , 0 ,ω) in Eqs. ( 17) ～ ( 24 ) , where , Θ ∈R , ω ∈R , and the nonlinear term Iω ×ω vanishes in Eq. ( 18) .

3. Time Discretization by Operator Splitting The above fully coupled fluid2structure interaction model contains four numerical difficulties , each of which can be associated with a specific operator , namely , ( a) The nonlinear convection term ; ( b) The incompressibility condition and the related unknown pressure p ; ( c) The rigid body constraint of B and the related distributed Lagrange multiplier λ; ( d) The external forces F e and torque Te at G . Applying the operator splitting scheme to problem Eqs. ( 17) ～ ( 24) , we obtain : 0 0 0 0 0 0 0 u , p ,λ , G , Θ , V , ω are given.

( 27)

For n Ε 0 , knowing u , p , λ , G , Θ , V , ω ,we firstly compute u n

∫

u

n +1/ 4

n

n

n

- u ・vd x + - Δt

Ω

n

∫( u

n

n

n +1/ 4

Ω

n

n + 1/ 4

via the solution of

・g) u n +1/ 4 ・vd x = 0 , Π v ∈ ( H10 ( Ω) ) 2 ,

u n +1/ 4 ∈ W ng+0 1 .

( 28)

Secondly , we compute u n + 2/ 4 , pn + 2/ 4 via the solution of n +2/ 4

n +1/ 4

・vd x + 2μ D ( u ∫ ∫ = ρ g ・vd x , 　Π v ∈ ( H ( Ω) ) , ∫ 　 q g ・u d x , 　Π q ∈ L , ∫ 　ρf

f

u

Ω

- u Δt

1 0

Ω

n +2/ 4

Ω

n +2/ 4

) : D ( v) d x -

∫p

n +2/ 4

Ω

g ・vd x

2

2

Ω

　u n +2/ 4 ∈ W ng+0 1 , 　pn +2/ 4 ∈ L 20 .

( 29)

Thirdly , predict the position , rotation angle , translation velocity and angular velocity of the rigid body as follows. Take Gn + 2/ 4 , 0 = Gn , Θn + 2/ 4 , 0 = Θn , V n + 2/ 4 , 0 = V n , ωn + 2/ 4 , 0 = ωn , then update the variables via the following subcycling technique : For k = 1 , …, N , compute n +2/ 4 , k

V ^

= V

n +2/ 4 , k - 1

+ (Δt/ N ) g + (Δt/ 2 N ) ( 1 - ρf / ρs )

-1

M

-1

e n +2/ 4 , k - 1 n +2/ 4 , k - 1 ) , F (G ,Θ

( 30)

ω ^

n +2/ 4 , k

=ω

n +2/ 4 , k

= G

G ^

Θ ^

n +2/ 4 , k

V

n +2/ 4 , k

n +2/ 4 , k - 1

+ (Δt/ 2 N ) ( 1 - ρf / ρs )

n +2/ 4 , k - 1

n +2/ 4 , k n +2/ 4 , k - 1 ) , + (Δt/ 4 N ) ( V ^ + V

=Θ = V

-1

I

e

T (G

n +2/ 4 , k - 1

n +2/ 4 , k - 1

+ (Δt/ 4 N ) (ω ^

n +2/ 4 , k - 1

+ (Δt/ N ) g + (Δt/ 4 N ) ( 1 - ρf / ρs )

n +2/ 4 , k

+ω

-1

n +2/ 4 , k - 1

,Θ

n +2/ 4 , k - 1

) ,

( 31) ( 32)

) ,

( 33) -1

M

-1

e

F (G ^

n +2/ 4 , k

,Θ ^

n +2/ 4 , k

)

JI Chun2ning et al . / China Ocean Engineering , 21 ( 3) , 439 - 450

444

　 + (Δt/ 4 N ) ( 1 - ρf / ρs ) - 1 M - 1 F e ( Gn +2/ 4 , k - 1 , Θn +2/ 4 , k - 1 ) ,

( 34)

n +2/ 4 , k n +2/ 4 , k ωn +2/ 4 , k = ωn +2/ 4 , k - 1 + (Δt/ 4 N ) ( 1 - ρf / ρs ) - 1 I - 1 Te ( G ) ^ ,Θ ^

　 + (Δt/ 4 N ) ( 1 - ρf / ρs ) - 1 I - 1 Te ( Gn +2/ 4 , k - 1 , Θn +2/ 4 , k - 1 ) , G

n +2/ 4 , k

Θ

n +2/ 4 , k

( 35)

= G

n +2/ 4 , k n +2/ 4 , k - 1 ) , + (Δt/ 4 N ) ( V + V

( 36)

=Θ

+ (Δt/ 4 N ) (ω

( 37)

n +2/ 4 , k - 1

n +2/ 4 , k - 1

+ω

n +2/ 4 , k

n +2/ 4 , k - 1

) ,

　　end do ; and let Gn + 2/ 4 = Gn + 2/ 4 , N , Θn + 2/ 4 = Θn + 2/ 4 , N , V n + 2/ 4 = V n + 2/ 4 , N , ωn + 2/ 4 = ωn + 2/ 4 , N . Fourthly , we compute u n + 3/ 4 , λn + 3/ 4 , V n + 3/ 4 , ωn + 3/ 4 , via the solution of

ρf ρf ωn +3/ 4 - ωn +2/ 4 u n +3/ 4 - u n +2/ 4 V n +3/ 4 - V n +2/ 4 )M ) θ ・ vd x + ( 1 ・ Y + (1 Δt ρs Δt ρs I Δt

∫

　ρf

Ω

n +3/ 4 n +2/ 4 = (λ , v - Y - θ ×G x)

n +2/ 4 B

, Π v ∈ ( H0 ( Ω) ) , Π Y ∈ R , 　Πθ ∈ R , 1

2

2

　(μ, u n +3/ 4 - V n +3/ 4 - ωn +3/ 4 × Gn +2/ 4 x) B n +2/ 4 = 0 , 　Πμ ∈Λn +2/ 4 , 　u n +3/ 4 ∈ W ng+0 1 ,λn +3/ 4 ∈Λn +2/ 4 , V n +3/ 4 ∈ R 2, ωn +3/ 4 ∈ R .

( 38)

Fifthly , correct the position , rotation angle , translation velocity and angular velocity of the rigid body as follows. Take Gn + 1 , 0 = Gn + 2/ 4 , Θn + 1 , 0 = Θn + 2/ 4 , V n + 1 , 0 = V n + 3/ 4 , ωn + 1 , 0 = ωn + 3/ 4 , then up2 date the variables via the following subcycling technique : For k = 1 , …, N , compute V ^ n +1 , k = V n +1 , k - 1 + (Δt/ 2 N ) ( 1 - ρf / ρs )

-1

M - 1 F e ( Gn +1 , k - 1 , Θn +1 , k - 1 ) ,

n +1 , k n +1 , k - 1 -1 -1 e n +1 , k - 1 n +1 , k - 1 ) , ω ^ =ω + (Δt/ 2 N ) ( 1 - ρf / ρs ) I T ( G ,Θ

G ^

n +1 , k

= G

n +1 , k - 1

+ (Δt/ 4 N ) ( V ^

n +1 , k

+ V

n +1 , k - 1

) ,

V

= V

n +1 , k - 1

+ (Δt/ 4 N ) ( 1 - ρf / ρs )

　 + (Δt/ 4 N ) ( 1 - ρf / ρs )

-1

M

-1

e

F (G

-1

M

n +1 , k - 1

-1

e

F (G ^

,Θ

n +1 , k - 1

( 40) ( 41)

Θ ^ n +1 , k = Θn +1 , k - 1 + (Δt/ 4 N ) (ω ^ n +1 , k + ωn +1 , k - 1 ) , n +1 , k

( 39)

( 42) n +1 , k

,Θ ^

n +1 , k

) ,

) ( 43)

n +1 , k n +1 , k ωn +1 , k = ωn +1 , k - 1 + (Δt/ 4 N ) ( 1 - ρf / ρs ) - 1 I - 1 Te ( G ) ^ ,Θ ^

　 + (Δt/ 4 N ) ( 1 - ρf / ρs ) - 1 I - 1 Te ( Gn +1 , k - 1 , Θn +1 , k - 1 ) , G

n +1 , k

Θ

n +1 , k

= G

n +1 , k - 1

=Θ

n +1 , k - 1

( 44)

n +1 , k n +1 , k - 1 ) , + (Δt/ 4 N ) ( V + V

( 45)

+ (Δt/ 4 N ) (ω

( 46)

n +1 , k

+ω

n +1 , k - 1

) ,

　　end do ; and let Gn + 1 = Gn + 1 , N , Θn + 1 = Θn + 1 , N , V n + 1 = V n + 1 , N , ωn + 1 = ωn + 1 , N . Finally , let u n + 1 = u n + 3/ 4 , ρn + 1 = ρn + 2/ 4 , λn + 1 = λn + 3/ 4 . Subproblem Eq. ( 28) is a pure convection problem and can be efficiently solved by the charac2 teristic method presented in Minev and Ethier ( 1999) . Subproblems Eq. ( 29) and Eq. ( 38) are fi2 nite dimensional linear problems with the structure t

Ax + B y = b , Bx = c ,

( 47)

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445

where , matrix A is symmetric. Problems such as Eq. ( 47 ) are known as saddle2point systems in mathematic scopes and their iterative solution by Uzawa/ conjugate gradient algorithms have been exten2 sively discussed. Subproblems Eqs. ( 30) ～ ( 37) and Eqs. ( 39) ～ ( 46) are iterative algebra schemes and can be easily calculated without any difficulty.

4. Free2Surface Tracking Algorithm The algorithm used in tracking the free2surface is the VOF method based on unstructured triangu2 lar mesh , which is introduced by Ji et al . ( 2005) . In this method , the computational domain is dis2 cretized with unstructured triangular mesh which can fit arbitrary complex curve boundaries accurately and directly without introducing any complicated boundary treatment and artificial boundary diffusion , as long as the mesh size is small enough. This method is a combination of the Piecewise Linear Inter2 face Calculation ( PLIC) method which is adopted to obtain a second2order reconstructed interface ap2 proximation , and the Modified Lagrangian2Eulerian Re2map (MLER) method (Ji et al . , 2005) which is applied in advecting fluid volumes on unstructured triangular mesh. For the sake of conciseness , the method is omitted from this paper and one can refer the reference (Ji et al . , 2005) for details.

5. Numerical Experiments 5. 1 　Numerical Model Verification

To verify the method presented herein , a numerical test is carried out trying to certify the accura2 cy of the method. In this test , a 2D Numerical Wave Tank (NWT) of 0. 08 m high and 1. 0 m long is established , as demonstrated in Fig. 2. An absorbable numerical wave generator ( Troch and Rouck , 1999) is installed on the left boundary of the NWT to generate incident waves and a sponge layer (Larsen and Dancy , 1983) started at x = 0. 8 m is placed before the right boundary for wave energy absorption.

Fig. 2. The layout and boundary conditions of the NWT.

In this test , the incident wave is taken as a small amplitude wave with wave height Hw = 0 . 01 m , water depth hw = 0 . 05 m and wave period Tw = 0 . 3 s. As depicted in Figs. 3 and 4 , wave height and fluid velocity contour are presented respectively.

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JI Chun2ning et al . / China Ocean Engineering , 21 ( 3) , 439 - 450

In Fig. 5 , wave surfaces at four time distinct t = nTw , ( n + 1/ 4 ) Tw , ( n + 2/ 4 ) Tw , ( n + 3/ 4 ) Tw within a wave period are compared with the analytic solutions. In this figure , the numerical results are represented by continued lines and the analytical solutions are denoted by dashed line. Obviously , the numerical results are coincident quite well with the analytic solutions except small differences observed in the front and rear parts of the NWT. Moreover , the performance of the sponge layer in absorbing wave energy is also outstanding.

Fig. 3. Wave surface process of the NWT.

Fig. 4. Fluid velocity contour of the NWT.

Fig. 5. Wave surface process at four time distinct within a wave period.

5. 2 　Fluid2Structure Interaction Problem

For the assessment of the capability of the method presented here in simulating complicated FSI problems , another numerical test is performed to provide an objective measurement of the method’s performance. In this test , a 2D Numerical Wave Tank (NWT) of 0. 08 m high and 0. 7 m long is set up , as depicted in Fig. 6 , and in which a structure of 0. 025 m wide and 0. 035 m high is embedded with its centroid initially located at ( 0. 3 , 0. 0225) m. An absorbable numerical wave generator is in2

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447

stalled on the left boundary of the NWT to generate incident waves and a sponge layer started at x = 0. 6 m is placed before the right boundary to absorb the wave energy. The structure is simplified as the mass2spring2friction mechanical model demonstrated in Fig. 7 , where , G is the gravity ; Fcl and Fcr are the left and right spring contact forces , respectively ; Ff l and Ff r are the left and right friction forces , respectively , as defined by - Fwh Ff l =

Vx - μf Fcl , 　| V x | Ε ε | Vx | - Fwh

Ff r =

Fcl , 　| V x | < ε Fcl + Fcr

Fcr , 　| V x | < ε Fcl + Fcr

Vx - μf Fcr , 　| V x | Ε ε | Vx |

( 48)

( 49)

where , Fwh is the horizontal wave force on the structure , which can be computed by integrating the pressure along the structure ’s boundary , μf is the friction coefficient between the structure and the seabed , V x is the structure’s horizontal velocity , εis a small positive. As the fluid dynamic forces on the structure are internal actions , they are omitted in this mechanical model .

Fig. 6. The layout and boundary conditions of the NWT.

　　Fig. 7. The sketch and mechanical model of the structure.

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448

Based on the mechanical model presented above , the external forces in Eq. ( 18) can be formu2 lated as : e

Fx = Ff l + Ff r ,

( 50)

Fey = Fcl + Fcr + G ,

( 51)

T = ( - Ff l sinΘ - Fcl cosΘ) B / 2 + ( - Ff l cosΘ + Fcl sinΘ) H/ 2 e

　 　 + ( Ff r sinΘ + FcrcosΘ) B / 2 + ( - Ff rcosΘ + Fcr sinΘ) H/ 2 ,

( 52)

where , B and H are the structure’s width and height , respectively. In this test , the incident wave is taken as a small amplitude wave with wave height Hw = 0 . 01 m , water depth hw = 0 . 05 m and wave period Tw = 0 . 3 s. The spring’s coefficient is ks = 200 N/ m , the friction coefficient is μf = 0 . 1 and the density ratio is ρs / ρf = 2 . 0. The wave surface process and fluid velocity vectorgraph at four time distinct within a wave period near the structure are presented. Obviously , the wave surface breaks near the structure for the small water depth above the structure , and a small rigid body velocity of the“fictitious”fluid is observed. As illustrated in Fig. 8 , the horizontal velocity of the structure increases continuously in the first two wave periods and then oscillates at zero when the wave propagation is stable. The vertical velocity initially fluctuates with a higher frequency and a smaller and decreasing amplitude , as the result of self2vibration of the mass2spring2friction system , and then oscillates steadily when the wave field is sta2 ble. However , a perceivable secondary oscillation is detected in the structure’s vertical velocity curve , which is mainly caused by wave breaking near the structure. Time histories of the structure’s horizontal and vertical displacements are depicted in Fig. 9. The wave drag and dynamic lift forces , which indi2 cate the wave forces on the structure in horizontal and vertical directions respectively , are not syn2 chronous and a quarter of wave period difference is checked , as demonstrated in Fig. 10. In Fig. 11 , the wave surface process at four time distinct within a wave period is presented. Ob2 viously , the wave propagation shows the characteristics of a partial standing wave before the structure for the maximum of wave height changed staggeredly along x2axis , of a breaking wave near the struc2 ture for the wave height decreases dramatically , and of a traveling wave after the structure for the maxi2 mum of wave height almost reach steady along x2axis. For the determination of the influence of the fluid2structure interaction on the wave forces , a test with all the same conditions but a fixed structure is performed. As shown in Fig. 12 , the drag force amplitude of the FSI case decreases about 15 % as compared with the fixed structure case , mainly due to the lower relative velocity between the fluid and the structure. However , the lift force amplitude of the FSI case increases about 25 % as depicted in Fig. 13 , due to the higher fluid velocity near the wave surface to which the vertical fluctuating structure is much closer in the FSI case than the fixed structure in the fixed2structure case. On the whole , the regularity of numerical results is obvious and becomes relatively steady after a few wave periods. However , from time histories of the structure’s displacement , velocity , and wave drag and lift forces , small oscillations can be detected , which are mainly caused by the irregular dy2

JI Chun2ning et al . / China Ocean Engineering , 21 ( 3) , 439 - 450

Fig. 8. Time history of the structure’s horizontal

Fig. 9. Time history of the structure’s horizontal

and vertical velocity.

Fig. 10. Time history of the wave drag and dynamic

and vertical displacement.

Fig. 11. Wave surface process at four time distinct

lift force on the structure.

Fig. 12. Comparison of the wave drag force between

449

within a wave period.

Fig. 13. Comparison of the wave lift force between

the FSI case and the fixed2structure case.

the FSI case and the fixed2structure case.

namic behaviors of breaking waves. Moreover , the strong coupled Fluid2Structure Interaction process also results in the instability of numerical solutions. For example , for the complicated movement of the structure caused by breaking waves , the wave forces on the structure at the time distinct nTw + t are not exactly equal to those at the time distinct ( n - 1) Tw + t .

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6. Conclusion A fully coupled Fluid2Structure Interaction finite element model based on Distributed Lagrange Multiplier/ Fictitious Domain method is presented. In this“monolithic”method , the structure is taken as“fictitious”fluid with zero strain rate and the whole computational domain , including fluid and structure , is governed by the Navier2Stokes equations. To keep the structure’s rigid body shape and behavior , a rigid body constraint is enforced on this“fictitious”fluid domain by using the DLM/ FD method. The unitized governing equations help capturing the predominant physics of interaction phe2 nomena and the stress/ velocity consistence conditions on the fluid2structure interface are automatically satisfied. Moreover , with the use of fixed Eulerian mesh , it is not necessary to remesh the computa2 tional domain , and thus free from mesh distortions. A 2D NWT problem and a numerical test on the Fluid2Structure Interaction system with a free2surface subjected to external wave forces have verified the method’s excellent solution quality and fidelity. References Felippa , C. A. , Park , K. C. and Farhat , C. , 2001. Partitioned Analysis of Coupled Mechanical Systems , Comput . Methods Appl . Mech. Eng. , 190 , 3247～3270. Glowinski , R. , Pan , T. W. , Hesla , T. I. , Joseph , D. D. and Periaux , J . , 2001. A Fictitious Domain Approach to the Direct Numerical Simulation of Incompressible Viscous Flow past Moving Rigid Bodies : Application to Particu2 late Flow , J . Comput . Phys. , 169 , 363～426. Heil , M. , 1998. Stokes Flow in an Elastic Tube a Large2Displacement Fluid2Structure Interaction Problem , Int . J . Numer. Methods Fluids , 28 , 243～265. Heil , M. , 2004. An Efficient Solver for the Fully Coupled Solution of Large2Displacement Fluid Structure Interaction Problems , Comput . Methods Appl . Mech. Eng. , 193 , 1～23. Hü bner , B. , Walhorn , E. and Dinkler , D. , 2004. A Monolithic Approach to Fluid2Structure Interaction using Space2 Time Finite Elements , Comput . Methods Appl . Mech. Eng. , 193 , 2087～2104. Hü bner , B. and Dinkler , D. , 2005. A Simultaneous Solution Procedure for Strong Interactions of Generalized Newtonian Fluids and Viscoelastic Solids at Large Strains , Int . J . Numer. Methods Eng. , 64 , 920～939. Ji , C. N. , Wang , Y. Z. and Wang , J . F. , 2005. A Novel VOF2Type Volume2Tracking Method for Free2Surface Flows based on Unstructured Triangular Mesh , China Ocean Eng. , 19 (4) : 529～538. Larsen , J . and Dancy , H. , 1983. Open boundaries in short wave simulations2a new approach , Coastal Eng. , 7 ( 3) : 285～297. Lghner , R. , Yang , C. , Cebral , J . , Baum , J . D. , Luo , H. , Pelessone , D. and Charman , C. , 1995. Fluid2Struc2 ture Interaction using a Loose Coupling Algorithm and Adaptive Unstructured Grids , Computational Fluid Dynamics Review , John Wiley , 755～776. Minev , P. D. and Ethier , C. R. , 1999. A Characteristic/ Finite Element Algorithm for the 32D Navier2Stokes Equa2 tions using Unstructured Grids , Comput . Methods Appl . Mech. Eng. , 178 , 39～50. Ohayon , R. and Felippa , C. , 2001. Special Issue : Advances in Computational Methods for Fluid2Structure Interaction and Coupled Problems , Comput . Methods Appl . Mech. Eng. , 190 , 24～25. Troch , P. and Rouck , J . D. , 1999. An active wave generating2absorbing boundary condition for VOF type numerical model , Coastal Eng. , 38 (4) : , 223～247. Walhorn , E. , Kglke , A. , Hü bner , B. and Dinkler , D. , 2005. Fluid2Structure Coupling within a Monolithic Model Involving Free Surface Flows , Computers and Structures , 83 , 2100～2111.