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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2013; 93 :1–22Published online 27 July 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4370

A new staggered scheme for uid–structure interaction

Wulf G. Dettmer and Djordje Peri c* ,†

College of Engineering, Civil and Computational Research Centre, Swansea University, Singleton Park,Swansea SA2 8PP, UK

SUMMARY

Staggered solution procedures represent the most elementary computational strategy for the simulation of uid–structure interaction problems. They usually consist of a predictor followed by the separate executionof each subdomain solver. Although it is generally possible to maintain the desired order of accuracy of thetime integration, it is difcult to guarantee the stability of the overall computation. In the context of largesolid over uid mass ratios, compressible ows and explicit subsolvers, substantial development has beencarried out by Felippa, Park, Farhat, Löhner and others. In this work, a new staggered scheme is presented.It is shown that, for a linear model problem, the scheme is second-order accurate and unconditionally stable.The dependency of the leading truncation error on the solid over uid mass ratio is investigated. The strategyis applied to two-dimensional and three-dimensional uid–structure interaction problems. It is shown thatthe conclusions derived from the investigation of the model problem apply. The new strategy extends theapplicability of staggered schemes to problems involving relatively small solid over uid mass ratios andincompressible uid ow. It is suggested that the proposed scheme has the same range of applicability as theDirichlet–Neumann or block Gauß–Seidel type strategies. Copyright © 2012 John Wiley & Sons, Ltd.

Received 21 October 2011; Revised 22 April 2012; Accepted 6 May 2012

KEY WORDS : uid–structure interaction; staggered solution procedure; partitioned solution procedure

1. INTRODUCTION

In generic uid–structure interaction problems, any change in the uid domain triggers a response of the structural domain and vice versa. The resolution of this strong coupling poses a major challengein the computer simulation of uid–structure interaction. We speak of strongly coupled solutionstrategies if both kinematic consistency and equilibrium of the traction forces at the interface aresatised in each time increment. In a weakly coupled strategy, we generally allow for a violation of the traction force equilibrium.

A variety of strongly coupled solution strategies has been suggested during the last decade.These include partitioned/monolithic, exact/inexact Newton–Raphson procedures, see, for example,[1–15], and Dirichlet–Neumann iterative strategies with relaxation (also known as block Gauß–Seidel or xed point iteration), see, for example, [16–21]. Both Newton–Raphson andDirichlet–Neumann strategies have been combined with multi-grid technology to improve com-putational efciency, see, for example, [21, 22]. Dirichlet–Neumann strategies inherently fail if thesolid does not possess any inertia, and it is difcult to achieve convergence if the solid over uidmass ratio is small, see, for example, [23].

Weakly coupled strategies have become known as ‘staggered’ schemes and are inherentlypartitioned. They may be organised as serial or parallel schemes, that is, the subsolvers have tobe executed sequentially or may run simultaneously. Staggered schemes are normally expected to

*Correspondence to: Djordje Peri c, College of Engineering, Civil and Computational Research Centre, SwanseaUniversity, Singleton Park, Swansea SA2 8PP, UK.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.



2 W. G. DETTMER AND D. PERI C

be computationally more efcient than strongly coupled solution procedures because they do notrequire iteration within each time step. For many problems, the computational time is one orderof magnitude shorter for staggered than for strongly coupled strategies. Furthermore, staggeredschemes allow for the employment of existing highly specialised subsolver software, which is gener-ally not the case for the strongly coupled Newton–Raphson procedures. The drawback of staggeredschemes is their restricted range of applicability as described next. Substantial research and devel-

opment in the area of staggered schemes have been carried out by Felippa, Park, Farhat, Piperno,Lesoinne, Löhner and co-workers, see, for example, [24–35].A general mathematical setting for staggered strategies as well as the detailed stability analysis of

selected schemes was provided by Felippa and Park in [25,26]. This work was crucial in laying thefoundations for all later development. First-order accurate explicit/implicit serial and parallel stag-gered schemes have been proposed by Piperno, Farhat, Lesoinne and co-workers and applied to thesimulation of aeroelastic problems at high ow velocities, see [27–29]. The stability is governed bythe stability of the explicit compressible uid ow solver. In [27], Piperno and Farhat also proposean implicit/implicit unconditionally stable, rst-order accurate staggered scheme on the basis of thetrapezoidal and the generalised midpoint rule. However, this scheme is only considered on the con-ceptual level. More accurate improved explicit/implicit schemes have been developed by Lesoinneand Farhat [29, 30]. In [31], an implicit/implicit algorithm based on the three point backward dif-ference scheme for the uid ow and the trapezoidal rule for the structural dynamics is proposedby Farhat et al. and proven to be second-order accurate. However, the stability properties are notinvestigated, and the numerical examples are restricted to the area of aerospace and compressibleuid ow. In [32], Farhat et al. propose provably second-order accurate explicit/explicit schemesfor compressible uid–structure interaction problems. Explicit coupling strategies have also beenemployed by Löhner and co-workers for the modelling of free surface ows with oating objectsand for the simulation of explosions, see, for example, [33,34] and references therein. A review of staggered schemes was published by Felippa et al. [35].

To the best knowledge of the authors, none of the staggered schemes proposed in literature hasbeen shown to be suitable for the simulation of incompressible uid–structure interaction. It isgenerally thought that such problems, even if they feature moderate solid over uid mass ratios,require the employment of strongly coupled solution strategies. Staggered schemes are, at best,assumed to be conditionally stable in such situations. The reason for this is the pronounced added

mass effect in problems with incompressible uid ow and moderate or small solid over uid massratios. The added uid mass entrained by the solid structure is not accounted for in the structuraldynamics solver. Thus, one is led to argue that the deformation of the structure, when loaded withthe uid traction forces, is generally too inaccurate to allow for the design of a stable staggeredalgorithm. For the mathematical derivation of critical conditions for selected staggered schemes, werefer to [36,37].

In this work, we propose a new unconditionally stable second-order accurate implicit/implicitserial staggered scheme, which is applicable to problems involving incompressible uid ow andrelatively low solid over uid mass ratios. We investigate the effect of the mass ratio on the per-formance of the coupling strategy and conclude that the proposed scheme has the same range of applicability as the Dirichlet–Neumann or block Gauß–Seidel type strategies. In particular, theproposed scheme possesses a simple control parameter very similar to a relaxation factor, whichallows to overcome the added mass effect to a similar extent as in Dirichlet–Neumann iterationswith relaxation.

The scheme is based on the generalised- ˛ method for the rst-order and second-order differentialequations in time, see [38] and [39], respectively. For linear problems, the generalised- ˛ methodis unconditionally stable and second-order accurate and allows the user to control high frequencydamping through the spectral radius of an innite time step. The generalised- ˛ method is a popu-lar choice for the modelling of uid–structure interaction and related problems, see, for example,[4,13,40–42].

The remainder of this paper is organised as follows. In Section 2, the proposed staggeredscheme is described. The properties of the scheme are assessed on the basis of a linear, one-dimensional model problem in Section 3. Similarities between the proposed staggered scheme and

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2013; 93 :1–22

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A NEW STAGGERED SCHEME FOR FLUID–STRUCTURE INTERACTION 3

the Gauß–Seidel strategy are investigated. Section 4 presents a number of two-dimensional andthree-dimensional numerical examples, and it is investigated to what extent the conclusions drawnfrom the model problem apply to fully resolved incompressible uid–structure interaction problems.Concluding remarks are given in Section 5.

2. THE STAGGERED SCHEME

2.1. Governing equations

After spatial and temporal discretisation, the uid–structure interaction problem may be expressedas (see [4,5, 7] for details)

r f . u f , u i / D0 (uid), (1)

g f . u f , u i / Cg s . u i , d s / D0 (interface), (2)

r s . u i , d s / D0 (solid), (3)

where u f and d s denote, respectively, the degrees of freedom of the uid and the structure whichare not associated with the interface boundary. In the context of incompressible uid ow and niteelement discretisation in space, u f typically comprises the nodal uid velocity and pressure values,whereas d s contains the structural displacements. The degrees of freedom on the interface boundaryof the uid and solid meshes (not including the uid pressure) are exclusively dependent on theinterface degrees of freedom u i . Given non-matching meshes, this dependency may, in the simplestcase, be based on an interpolation strategy. The vectors r f and rs represent the nodal residual forcesin the uid and the solid domains, whereas g f and g s denote the traction forces exerted on theinterface by the uid and the solid, respectively.

A strong solution strategy satises exactly the coupled system of algebraic Equations (1)–(3) ineach time step.

In the context of staggered schemes, it is useful to introduce the interface traction force vector t i

and rewrite (1)–(3) as

r f . u f , u i / D0g f . u f , u i / Dt i ³ (uid), (4)

r s . d s , u i / D0

g s . d s , u i / D t i³ (solid). (5)

We note that, in the context of an arbitrary Lagrangian–Eulerian formulation, the conguration of theuid nite element mesh depends on the interface motion u i . The position of the internal nodes of the uid mesh is adopted to the current interface conguration by means of an appropriate strategy,see, for example, [5,42] and references therein.

2.2. Solution algorithm

In the remainder of this paper, the subscripts 0, 1, 2, : : : , n 1, n , n C1, : : : , N identify the discretetime instants for which we seek to compute the degrees of freedom of the problem under considera-tion. Given the solution at time instants tn and tn 1 , we propose the staggered scheme summarisedin Box 1 for the approximation of the solution of the system (4)–(5) at tn C 1 .

The key ingredients of the scheme are: 1. a traction force predictor; 2. a structural solver, whichaccepts external loads; 3. a uid ow solver, which accepts prescribed motion of the no-slip inter-face boundary; and 4. the averaging of the traction force vectors. We note that the structure hasresponded to the traction force predictor t i

Pn C 1 , whereas the traction force computed from the uid

ow is t i n C 1 . Thus, the difference t iPn C 1 t i n C 1 is a measure of the violation of the system of

Equations (4) and (5). In Step 4, the traction force t i n C 1 is based on a weighted average of t iPn C 1


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and t i n C 1 . The parameter ˇ has to be set by the user, and its role is investigated in Section 3. A nat-ural choice is ˇ D0.5. If ˇ is set to zero, the impact of the uid on the solid is ignored. In case of adaptive time stepping, the predictor becomes

t iPn C 1 D .1 C n C 1 / t i n n C 1 t i n 1 , (6)

where n C 1 D t n C 1 = t n is the ratio of the current to the previous time step sizes.

3. ANALYSIS OF THE SCHEME BASED ON A MODEL PROBLEM

We introduce a one-dimensional linear model problem and employ the generalised- ˛ method to dis-cretise the governing equations in time. In Section 3.2, we consider the strongly coupled solution andinvestigate its stability and accuracy properties. The purpose of this is to introduce the assessmentprocedure and to set the background for the staggered scheme, which is investigated in Section 3.3.In Section 3.4, we identify similarities between the proposed staggered scheme and strongly coupledGauß–Seidel strategies. The analyses presented in Sections 3.2–3.4 were performed with a symbolicsoftware tool.

3.1. The model problem

Consider the model problem consisting of an elastic spring, a dashpot and two point masses asdisplayed in Figure 1. The point masses are connected by a rigid link. The parameters k , c and mdenote the spring stiffness, the viscosity of the dashpot and the total mass, respectively. The ratio of the masses connected to the spring and to the dashpot is ˛=.1 ˛/ . Note that, in comparison with ourearlier article [40], for clarity of presentation, the relative masses ˛ and 1 ˛ have been swapped.

The equations of motion for the point masses may be written as

˛ m Ru Ck u DF (7)

.1 ˛/ m Ru Cc Pu D F , (8)

where u is the displacement of the point masses and F is the force acting in the rigid link betweenthem. After summation, (7) and (8) reduce to the equation of motion of the damped single degree of freedom oscillator

m Ru Cc Pu Ck u D0 . (9)


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Figure 1. One-dimensional linear model problem.

By using ! Dr km , D c

2 p k mand f D

F m

, Equations (7) and (8) become

˛ Ru C! 2 u Df (10)

.1 ˛/ Ru C2 ! Pu D f (11)

or, added together,

Ru C2 ! Pu C! 2 u D 0 . (12)

The introduction of discrete time instants t0 , t1 , t2 , . . . with t D tn C 1 tn results in the exactamplication matrix A exact given as

² un C 1

Pu n C 1 t ³DA exact ² un

Pu n t ³ , A exact D exp 0 1

! 2 t 2 2 ! t . (13)

The eigenvalues of A exact are

1 ,2 D e˙ i p 1 2 ! t

. (14)

In the following and with reference to uid–structure interaction, we use the superscript ‘s’ for allquantities associated with the elastic subsystem and the superscript ‘f’ for all quantities associatedwith the viscous subsystem. Furthermore, we distinguish f s and f f . Thus, we may write

˛ Ru s C! 2 u s Df s (15)

.1 ˛/ Ru f C2 ! Pu f Df f (16)

f s C f f D0 (17)

Pu s D Pu f , (18)

where, because of the initial condition u s0 Du f

0 , Equation (18) ensures that the displacement of the

two point masses is identical at all time instants. Remark 1In the context of incompressible uid–structure interaction, the relevance of the model problemconsidered here is not necessarily obvious. Concerns may arise with respect to the absence of anyconvective ow as well as the incompressibility constraint. Thus, it may even be argued that themodel problem represents structure–structure interaction rather than uid–structure interaction. Inresponse to such concerns, we note the following:

The fact that the uid domain is modelled on the basis of an arbitrary Lagrangian–Eulerianformulation is not relevant for the analysis of the coupling strategy. We also note that most


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uid–structure interaction systems do not allow for any slip of uid particles along the interface,and the interface itself is represented with Lagrangian kinematics. The essential feature, whichmakes the model problem representative of uid–structure interaction, is the coupling of asecond-order differential equation (displacement and acceleration) with a rst-order differentialequation (velocity and acceleration).

Incompressibility imposes a kinematic constraint on the velocity degrees of freedom of

the uid. Any uid pressure variables are merely Lagrangian multipliers used to enforcethis constraint.With respect to the absence of pressure variables from our model, we remark that, in the

context of nite element models for uid ow, the pressure at the domain boundary is accountedfor by the traction forces that correspond to the velocity degrees of freedom, that is, there is noneed to include the pressure variables in the coupling strategy. For more information, we referto [4].

With respect to the absence of an incompressibility constraint from our model, we referto [23], where an extended model problem is considered, which includes internal degrees of freedom in each subproblem. These are subjected to a kinematic constraint. The elimination of the internal degrees of freedom then leads to a problem identical to the one considered here.

Last but not least, the motivation for this particular model problem may be found in the factthat it offers explanations for all major observations made in the application of the proposedstaggered scheme to the multi-dimensional incompressible uid–structure interaction problemsin Section 4.

We also note that this model problem has previously been employed to explain numericaldifculties that arise in computational uid–structure interaction in the context of the block Gauß–Seidel procedure and of time integration strategies (see [23] and [40], respectively).

3.2. Analysis of the strongly coupled solution strategy

We recall the generalised- ˛ method as proposed in [38] for linear second-order differential equationsin time. Application to Equation (15) renders the following system of linear equations

˛ Ru sn C ˛ s

m C! 2 u sn C ˛ s

f D f sn C ˛ sf

(19)

Ru sn C ˛ sm D˛ sm Ru sn C 1 C 1 ˛ sm Ru sn (20)

u sn C ˛ s

f D˛ sf us

n C 1 C 1 ˛ sf u s

n (21)

Pu sn C 1 D Pu s

n C t .1 s / Ru sn C s Ru s

n C 1 (22)

u sn C 1 Du s

n C t Pu sn C t 2 1

2 ˇ s Ru sn Cˇ s Ru s

n C 1 , (23)

where the parameters ˛ sm , ˛ s

f , ˇs and s are related to the spectral radius of an innite time step

s1 by

˛s

m D 2 s

1

1 C s1 , ˛s

f D 1

1 C s1 , ˇs

D 1

4 1 C˛s

m ˛s

f

2

, s

D 1

2 Cs

m ˛s

f . (24)

Similarly, we recall the generalised- ˛ method as proposed in [39] for linear rst-order differentialequations in time. Application to Equation (16) renders

.1 ˛/ Ru f n C ˛ f

m C2 ! Pu f n C ˛ f

f D f f n C ˛ f f

(25)

Ru f n C ˛ f

m D˛ f m Ru f

n C 1 C 1 ˛ f m Ru f

n (26)

Pu f n C ˛ f

f D˛ f f Pu f

n C 1 C 1 ˛ f f Pu f

n (27)


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Pu f n C 1 D Pu f

n C t .1 f / Ru f n C f Ru f

n C 1 , (28)

where

˛ f m D

12

3 f 1

1 C f 1

, ˛ f f D

11 C f

1, f D

12 C˛ f

m ˛ f f . (29)

Kinematic consistency as given by Equation (18) requires

Pu sn C 1 D Pu f

n C 1 . (30)

It has been shown in [40] that the forces f sn C ˛ s

f and f f

n C ˛ f f

need to be interpolated appropriately to

maintain second-order accuracy for the overall problem, that is,

f sn C ˛ s

f D˛ sf f sn C 1 C 1 ˛ s

f f sn (31)

f f n C ˛ f

f D˛ f f f f n C 1 C 1 ˛ f

f f f n . (32)

Thus, equilibrium of the traction forces as given in Equation (17) can be written asf s

n C 1 C f f n C 1 D 0 . (33)

The system of Equations (19)–(23), (25)–(28) and (30)–(33) comprises 13 linear equations and 13unknown displacement, velocity, acceleration and force terms. All quantities associated with timeinstant tn are known. Eliminating all quantities associated with fractional time instants between tnand tn C 1 and dening

u n WDu sn , Pu n WD Pu s

n D Pu f n , f n WDf s

n D f f n (34)

renders

8<ˆ:

u n C 1

Pu n C 1 t

Ru sn C 1 t 2

Ru f n C 1 t 2

f n C 1 t 2

9>>>>>=>>>>>;D2666664

A strong 37777758<ˆ:

u n

Pu n t

Ru sn t 2

Ru f n t 2

f n t 2

9>>>>>=>>>>>;, (35)

where A strong is a ve-dimensional amplication matrix. The coefcients of A strong depend on theproblem parameters ˛ , ! and as well as on the time integration parameters s , f and t .

Stability : The system (35) is unconditionally stable if the spectral radius of A strong does not exceedone for any time step size t . The spectral radius of an amplication matrix of dimension D isdened as

D max .

j1

j,

j2

j, : : : ,

jD

j/ , (36)

where i are the eigenvalues of the amplication matrix. The derivation of a closed form proof of unconditional stability, that is, 6 1 for any t > 0, is difcult. Hence, in [40], the eigenvalues of A strong have been evaluated for various combinations of parameters. In all cases, 6 1, thus sug-gesting unconditional stability of the discretised system. In Figure 2(a1) and (a2), the eigenvaluesof A strong are plotted in the complex plane for different sets of parameters. The gures also showthe associated spectral radii as functions of t=T , where T D 2 =! . It follows from a limit caseanalysis that

1 D limt !1 D max s

1 , f 1 . (37)


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(a1) (b1)

(a2) (b2)

Figure 2. Strongly coupled solution procedure (35): eigenvalues (a1,a2) and spectral radii (b1,b2) for ! D1, D0.1 and different values of ˛ with s

1 D0.2 , f 1 D0.8 (a1,b1) and s

1 D0.8 , f 1 D0.2 (a2,b2).

Accuracy : To assess the accuracy of the system (35), we evaluate the characteristic polynomial of A strong for the eigenvalues of the exact amplication matrix as given in Equation (14). The resultdivided by t 2 corresponds to the local truncation error and is expanded as a Taylor series of t(see, for instance, [38]). The order of the leading term corresponds to the order of accuracy of the

scheme. In the specic case of (35), the leading truncation error is found to be of the order of t 2 .Thus, the second-order accuracy of the generalised- ˛ method in the subsystems is retained for thestrongly coupled solution scheme.

3.3. Analysis of the staggered scheme

To perform the same analysis for the staggered scheme, Equations (31) and (32) are replaced,respectively, by

f sn C ˛ s

f D˛ sf f s

P

n C 1 C 1 ˛ sf f s

n (38)

f f n C ˛ f

f

D˛ f

f f f n C 1

C1 ˛ f

f f f n , (39)

where f s P

n C 1 represents the traction force predictor applied to the elastic subsystem and f f n C 1 is the

traction force obtained from the response of the viscous subsystem. Equation (33) is replaced by

f s P

n C 1 D2 f n f n 1 (40)

f n C 1 D ˇ f f n C 1 C.1 ˇ/ f s P

n C 1 , (41)

where we have introduced the averaged traction forces f n 1 , f n and f n C 1 . The minus sign in front of ˇ f f

n C 1 in Equation (41) is due to the fact that the orientation of the average traction forces is aligned


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with f s P

n C 1 and opposed to f f n C 1 . We note that Equations (40) and (41) correspond, respectively, to

Steps 1 and 4 in the description of the staggered solution algorithm in Section 2.2.By using the system of Equations (19)–(23), (25)–(28), (30) and (38)–(41), all quantities

associated with fractional time instants can be eliminated and one may write

8<ˆ:

u n C 1

Pu n C 1 tRu s

n C 1 t 2

Ru f n C 1 t 2

f n C 1 t 2

f n t 2

9>>>>>>>=>>>>>>>;D266666664

A weak 3777777758<ˆ:

u n

Pu n tRu s

n t 2

Ru f n t 2

f n t 2

f n 1 t 2

9>>>>>>>=>>>>>>>;, (42)

where A weak is a six-dimensional amplication matrix. The entries in the sixth row of A weak are

A weak 61 DA weak 62 DA weak 63 DA weak 64 DA weak 66 D0 and A weak 65 D1 .

All other matrix coefcients depend on the problem parameters ˛ , ! and , on the time integrationparameters s , f and t as well as on the averaging parameter ˇ .

Stability : We recall the denition of the spectral radius as given by (36). The derivation of a closedform assessment of the stability of the solution strategy, that is, 6 1 for any t > 0, is difcult.Thus, we evaluate the eigenvalues of the amplication matrix A weak for a large number of differentproblem and integration parameters. For some selected combinations of parameters, the eigenvaluesand the spectral radii of A weak are displayed in Figure 3. The results of this investigation suggestthat unconditional stability is obtained if ˇ is chosen sufciently small. The critical value of ˇ ismainly determined by the mass ratio ˛ . If ˛ decreases, the critical value of ˇ decreases, too. Inother words, if a smaller proportion of the total mass is associated with the elastic subsystem, then a

(a1) (b1)

(a2) (b2)

Figure 3. Staggered solution procedure (42): eigenvalues (a1,a2) and spectral radii (b1,b2) for ! D 1, D0.1 and different values of ˛ with s

1 D0.2 , f 1 D0.8 (a1,b1) and s

1 D0.8 , f 1 D0.2 (a2,b2); the

parameter ˇ is set to 0.05, 0.3 and 0.7 for ˛ D1=10, ˛ D5=10 and ˛ D9=10, respectively.


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smaller value of ˇ is required to achieve unconditional stability of the proposed staggered solutionstrategy. Furthermore, it follows from the limit case analysis of (42) that

1 D limt !1 D max s

1 , f 1 ,p 1 ˇ . (43)

Thus, in situations where small values of ˛ require the choice of a small value for ˇ , the limit of the

spectral radius for large time steps is p 1 ˇ 1. We note that the user-controlled high frequencydamping associated with the generalised- ˛ method is retained in each subsystem.

Accuracy : The order of accuracy of the proposed staggered scheme is determined in the sameway as for the strongly coupled solution strategy. For (42), the leading term of the local truncationerror is found to be of the order of t 2 , independently of the choice of ˇ . Thus, the second-orderaccuracy of the generalised- ˛ method in the subsystems is retained for the weakly coupled solutionscheme. It is however noted that the leading error term tends to innity as ˛ approaches zero, that is,

lim˛ ! 0

D 1 8 t > 0 . (44)

Thus, the accuracy of the proposed staggered scheme deteriorates if the proportion of the total massthat is associated with the elastic subsystem approaches zero. It may be concluded that this imposesa restriction on the time step size, if reasonable accuracy is to be maintained for systems with smallvalues of ˛ .

Summary : A brief summary of the properties of the proposed staggered scheme as obtained fromthe aforementioned analysis of the model problem reads:

(i) Second-order accuracy is maintained.(ii) Small values of ˛ lead to large truncation errors.

(iii) Small values of ˛ require small values of ˇ to achieve unconditional stability.(iv) High frequency damping of the overall system is reduced if p 1 ˇ > max s

1 , f 1 .

3.4. Similarities with the Gauß–Seidel strategy

Following the procedure described in [23], the application of a Gauß–Seidel strategy with relaxationto the model problem under consideration leads to

f .i C 1/n C 1 D ˇ G S AG S f .i /

n C 1 C bn C .1 ˇ G S / f .i /n C 1 , (45)

where AGS is a scalar valued function of ! , , ˛ , s1 , f

1 and t . The scalar quantity bn con-tains all terms associated with solution data at time instant tn . The factor ˇ GS controls relaxation.Equation (45) may be rewritten as

f .i C 1/n C 1 D bAGS f .i /

n C 1 C ˇ GS bn with bAGS D 1 Cˇ GS .A G S 1/ . (46)

The iteration (46) converges if jbAGS j< 1 . The limit analysis of AGS renders

limt ! 0

AG S D C s1 , f

1 1 ˛

˛ , (47)

where 3=4 6 C s1 , f

1 6 3=2 for 0 6 s1 6 1 and 0 6 f

1 6 1. Thus, AGS increases as ˛decreases. It follows from (46) that small values of ˇ GS can be used to achieve convergence forsmall values of ˛ , that is, for jAGS j> 1 .

If ˇ GS is chosen such that bAGS D 0, then the exact result is obtained after one iteration. Thus,

bAGS D 0 for ˇ GS D ˇ optGS , where ˇ opt

GS is the optimal value of the relaxation parameter. Simi-larly, there exist critical values ˇ crit

GS , which must not be exceeded to guarantee convergence of theiteration. For ˇ GS Dˇ crit

GS , we have bAGS D j1j. For the model problem at hand, it can be shown that

lim˛ ! 0t ! 0

ˇ optGS D lim

˛ ! 0t ! 0

ˇ critGS D 0 . (48)


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We point out that this result is also obtained if (45) is formulated in terms of the interface displace-ment. Because of ˇ GS D0 not being an admissible choice, it is clear that the Gauß–Seidel iterationfails for ˛ D0 (in theory, convergence might be achieved for large time steps, but such solutions, if they can be obtained, are of no use in terms of accuracy). Furthermore, Equation (48) shows that, as˛ decreases, smaller values of ˇ GS are required to achieve convergence. Because of the increasedsensitivity of the method with respect to ˇ GS , we may also expect an increased number of required

iteration steps for small values of ˛ . This is particularly the case for practical problems with morethan one degree of freedom, where a value of ˇ GS that only requires one iteration does not exist.Evidence of this is shown, for instance, in [16] and in Section 4.4.

Thus, the following similarities between the Gauß–Seidel iteration and the proposed staggeredstrategy are established:

The staggered scheme and the Gauß–Seidel iteration fail for ˛ D0. For small non-zero values of ˛ , the parameters ˇ and ˇ GS need to be chosen sufciently smallto achieve stability and convergence, respectively.

For small values of ˛ , the accuracy of the staggered scheme and the convergence rate of theGauß–Seidel scheme deteriorate.

It may be concluded that the Gauß–Seidel scheme and the proposed staggered scheme have a similar

range of applicability.We note that Equations (45)–(48) are equivalent to statements made in [23], if the differentnotation is taken into account. Namely, the mass ratio ˛ m and the relaxation factor ˇ in [23] corre-spond, respectively, to the terms .1 ˛/=˛ and 1 ˇ GS in this paper. Furthermore, the analysis in[23] is based on the trapezoidal rule, which is reproduced by using s

1 D f 1 D1 and C.1 , 1/ D1.

Finally, it is worth mentioning that it has been reported in [43] that the best initial guess for ablock Gauß–Seidel iteration, in terms of minimising the number of iteration steps, is obtained froma low-order traction force predictor. The proposed staggered scheme is essentially based on the rst-order traction force predictor given in Step 1 of Section 2.2. Experimentation with other predictorswas unsuccessful.

4. NUMERICAL EXAMPLES

In this section, a number of numerical examples of two-dimensional and three-dimensional uid–structure interaction problems is presented. All examples involve incompressible Newtonian uidow. The uid ow is modelled with linear stabilised velocity/pressure SUPG/PSPG nite ele-ments. An arbitrary Lagrangian–Eulerian strategy is employed to accommodate the independentmesh motion. The mesh update is based either on the optimisation of the element aspect ratios or ona nonlinear pseudo-elastic strategy. For more details, we refer to [4, 5]. The structures are modelledwith appropriate nite elements as described in the following sections. The time integration is per-formed with the generalised- ˛ method analogously to the model problem in Section 3. We point outthat, unless this is implemented carefully and accurately, the stability of the scheme is easily jeop-ardised. Computations are performed with the proposed staggered scheme and also with a stronglycoupled Newton procedure to allow for the detailed assessment of the staggered scheme. Particularfocus is placed on the question to what extent the analysis of the model problem is relevant for the

nonlinear uid–structure interaction problems with many degrees of freedom.

4.1. Flow induced oscillations of a exible beam

This problem was rst addressed in [44] and has since become a benchmark problem forcomputational uid–structure interaction strategies. An elastic beam is attached to a rigid squareand immersed in uniform uid ow. The vortices in the wake of the rigid square interact with thebeam and excite large amplitude oscillations. The details of the problem are described, for instance,in [4]. Here, we note that the density of the beam is 84.7 times larger than that of the uid and thatits length is 66.7 times larger than its thickness. The beam is modelled with 20 nine-noded quadraticfully integrated continuum elements, and the uid is discretised with 4336 linear velocity/pressure


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Figure 4. Oscillations of a exible beam: detail of the uid mesh.

Figure 5. Oscillations of a exible beam: typical ow pattern; vorticity contour plot.

Figure 6. Oscillations of a exible beam: vertical tip displacement; strongly (SC) and weakly coupled(WC) solutions.

elements. A detail of the mesh is shown in Figure 4. The parameters ˇ , f 1 and s

1 are all set to 0.5.Figure 5 shows a typical ow pattern once the oscillation is fully established. The vertical beam tipdisplacement is displayed versus time in Figure 6. The diagram shows the solutions obtained with

two different time step sizes.For t D0.020 , the weakly and strongly coupled solutions deviate visibly once the oscillation is

established. However, they agree well in terms of amplitude and frequency. Remarkably, the rene-ment of the time step size by factor ve sufces to achieve excellent agreement of the weakly andthe strongly coupled solutions. Note that, for this problem, the weakly coupled scheme does notimpose a restriction on the time step size.

4.2. Membrane roof subjected to wind

This example is concerned with a preliminary study of a membrane roof under wind loading.The geometry and the boundary conditions of the problem are displayed in Figure 7. An elastic


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Figure 7. Membrane roof: geometry and boundary conditions, dimensions not in proportion; membrane,trusses and ropes are marked, respectively, with red, green and blue colour; the pressure is xed at the top

right corner.

Figure 8. Membrane roof: detail of the discretisation.

membrane is suspended between the tip of two pin joint trusses which are held in position by tworopes. The ropes and the membrane carry tensile stresses only. All structural components consist of incompressible Neo-Hooke elastic material. Thus, the axial Cauchy stress is related to the stretch

by

D 4 1

2 . (49)

The shear moduli of the membrane, the ropes and the trusses are, respectively, 160, 1000 and2000 . In the geometry given in Figure 7, the membrane is prestressed such that D1.2. The ropesand the trusses do not interact with the uid ow. The wind prole follows a quadratic parabola.The vertex is located at the upper boundary of the domain. The uid density and viscosity are set,respectively, to 1.0 and 0.002 .

The uid domain is discretised with 20,200 elements, whereas the membrane is represented by25 linear geometrically exact truss elements. Each rope and each pin joint truss are modelled withone element of the same type. A detail of the discretisation is shown in Figure 8. The integrationparameters are set to f

1 D s1 D0.5. Different time step sizes t are considered.

4.2.1. Light wind. For this simulation, the mass of the membrane and the trusses is set to 20 perunit length, whereas the ropes are not given any inertia. The parameter ˇ is set to 0.5. Within 40 timeunits, the inow velocity is increased from zero to Nu D 1 with a sinusoidal ramp. Figure 9 showsthe long-term response as obtained from the staggered scheme and the strongly coupled strategy for

t D4. Figure 10 displays the results obtained with t D4 and t D1 for the rst 500 time units.


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Figure 9. Membrane roof, light wind: vertical displacement of membrane centre point; strongly (SC) andweakly coupled (WC) solutions.

Figure 10. Membrane roof, light wind: vertical displacement of membrane centre point, details; strongly(SC) and weakly coupled (WC) solutions.

Figure 11. Membrane roof, strong wind: vertical displacement of membrane centre point; strongly (SC) andweakly coupled (WC) solutions.

Figure 12. Membrane roof, strong wind: vertical displacement of membrane centre point, details; strongly(SC) and weakly coupled (WC) solutions.

For t D4, the weakly and the strongly coupled solutions deviate visibly, whereas for t D1,they agree extremely well. We note from the comparison with the converged solution that, for

t D4, the strongly coupled procedure does not produce a more accurate solution than the proposedstaggered scheme.

4.2.2. Strong wind. All problem parameters are identical to the example in Section 4.2.1, exceptfor the inow velocity, which is raised to Nu D2 within 100 time units, and the time steps, which areset to either t D2 or t D0.2. Figures 11 and 12 show the response of the structure. A vorticitycontour plot of a typical ow pattern is shown in Figure 13.


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Figure 13. Membrane roof, strong wind: vorticity contour plot of a typical ow pattern.

Figure 14. Membrane roof, small structural mass, strong wind: vertical displacement of membrane centrepoint, large time steps; strongly (SC) and weakly coupled (WC) solutions.

Figure 15. Membrane roof, small structural mass, strong wind: vertical displacement of membrane centrepoint, small time steps; strongly (SC) and weakly coupled (WC) solutions.

We observe that the membrane oscillates more violently. The weakly and strongly coupledsolutions deviate visibly for t D 2 but agree accurately for t D 0.2 during the rst 340 timeunits. Similarly to the oscillations in light wind, neither the strongly nor the weakly coupled solutioncan be said to be of a better quality than the other.

4.2.3. Small structural mass; strong wind. All problem parameters are identical to the example inSection 4.2.2, except for the structural mass, the parameter ˇ and the time step sizes. The mass of the membrane and the trusses is set to 2 per unit length and ˇ is set to 0.1.

We note that, in comparison with Section 4.2.2, ˇ has to be reduced to avoid the developmentof numerical oscillations and the eventual collapse of the computation. This is consistent with theobservation made in Section 3.3. Namely, as the structural mass is decreased relative to the uidmass, a smaller value of ˇ is required to maintain the stability of the simulation.

The long-term response of the structure, as obtained with t D 0.5, is shown in Figure 14.Figure 15 displays the results obtained with t D 0.05 for the rst 350 time units. For t D 0.5,the weakly and the strongly coupled solutions deviate visibly but agree reasonably well in terms of amplitude and frequency. For t D 0.05, the solutions agree accurately during the rst 160 time


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units. We note that in Sections 4.2.1 and 4.2.2, a better agreement of the staggered scheme with thestrongly coupled strategy was obtained. For the reduced structural mass considered here, small timesteps are required to match the strongly and the weakly coupled solutions. This is consistent withthe observation made in Section 3.3: As the structural mass is decreased relative to the uid mass,the proposed staggered scheme becomes less accurate.

4.3. Flutter of an elastic beam

In this example, we consider an elastic beam with rectangular cross-section, which is subjected totransverse uid ow as shown in Figure 16. Both ends of the beam are fully clamped. The inowis uniformly distributed. The beam is composed of Neo-Hooke elastic material. The density, theshear and bulk moduli are, respectively, 70.6, 823.2 and 1783.6. The uid viscosity and densityare set to 0.01 and 1, respectively. The inow velocity is increased with a sinusoidal ramp from0 to 2.5 within the rst 10 time units and then held constant. The uid domain is discretised with146,594 stabilised linear elements, whereas 60 eight-noded linear brick elements are used to repre-sent the beam. Details of the discretisation are shown in Figure 17. Because this study is focussedon the performance of the proposed staggered scheme, we are not concerned about the coarsenessof the nite element model of the beam. The integration parameters f

1 , s1 , ˇ and t are all

set to 0.5.Figure 18 shows the evolution of the vertical displacement of the leading edge centre point in time.

For the rst 500 time units, the solutions obtained with the weakly and strongly coupled strategiesagree excellently. Figure 19 shows typical deformed congurations of the beam. The bending andtwisting motion of the beam corresponds to the well-known phenomenon of utter.

Figure 16. Flutter of an elastic beam: geometry and boundary conditions.

Figure 17. Flutter of an elastic beam: details of the discretisation.


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Figure 18. Flutter of an elastic beam: vertical deection of the leading edge centre point; strongly (SC) andweakly coupled (WC) solutions for t D0.5.

Figure 19. Flutter of an elastic beam: typical deformed congurations; pressure contour plots on uid meshboundary surfaces.

4.4. Flow through a exible tube

In this example, we consider an elastic tube, which is clamped at the inow boundary as shown inFigure 20. The length of the undeformed tube is 5 times larger than the inner radius. The densityof the solid wall and that of the uid are chosen as 1. At the clamped boundary, a parabolicallydistributed inow is prescribed with a maximum velocity u

max at the centre, which is increased in

time in a sinusoidal fashion, but then reduced to zero, as shown in Figure 21. The solid nite ele-ment mesh consists of 720 eight-noded F-Bar elements, whereas the uid is discretised with 7832stabilised linear four-noded elements. Details of the meshes are shown in Figure 22. To minimisethe effect of spatially unresolved high frequencies, we choose f

1 D s1 D0.

Figure 23 shows the results obtained with a strongly coupled Newton strategy. We note that thedeformation of the tube is axisymmetric only during the rst 0.1 time units. It is also observed thatthe solutions obtained with t D 0.005 and with t D 0.001 almost coincide. Some deformedcongurations of the tube at different time instants are displayed in Figure 24.

Numerical experimentation with the proposed staggered scheme shows that the simulation isinherently unstable for ˇ > 0.01. Thus, we choose ˇ D 0.005. Figure 25 displays the solutions


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Figure 20. Flexible tube: geometry and boundary conditions.

Figure 21. Flexible tube: temporal evolution of inow velocity.

Figure 22. Flexible tube: the uid and solid nite element meshes.

Figure 23. Flexible tube: solutions obtained with a strongly coupled Newton procedure; radial coordinatesof two points on the interface located at ´ D0.25 at opposite positions.

obtained for different time step sizes. In general, we note that small changes of t and ˇ causelarge changes in the computed solution. In particular, we make the following observations:

(i) The solutions exhibit strong non-physical oscillations, the onset of which depends on thetime step size.


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Figure 24. Flexible tube: deformed congurations of the uid domain; pressure contour plots on theinterface boundary.

Figure 25. Flexible tube: solutions obtained with the staggered scheme (WC) and the strongly coupledNewton procedure (SC); radial coordinate of a point on the interface at ´ D0.25.

(ii) For t D 0.005 and t D 0.001 , the strongly coupled scheme renders excellent solutions,whereas the staggered scheme, even before the onset of any oscillations, produces largeapproximation errors.

(iii) For t D0.0002 , the solution of the staggered scheme coincides with those of the stronglycoupled strategy for the rst 0.5 time units.

Assuming that the characteristic solid over uid mass ratio of the tube problem is ‘very small’,explanations for these observations can be derived from the analysis of the model problem inSection 3.3:

(i) A very small value of ˇ is required to achieve stability. In the case of the model problem,the spectral radius lies in the region of p 1 ˇ . Thus, the numerical damping of the stag-gered solution strategy is removed almost entirely, and non-physical oscillations are likelyto occur.

(ii) It was shown on the basis of the model problem that, for small solid over uid mass ratios,the staggered scheme produces large truncation errors. This explains the poor performanceof the strategy for t D0.005 and t D0.001 .

(iii) The fact that, with sufciently small time steps and prior to the onset of any oscillations, the

staggered scheme is capable of reproducing the correct solution conrms that the strategy is,in principle, correct, even though not suitable for the simulation of this problem.

Remark 2Further numerical experimentation with the tube problem has shown that the performance of thestaggered scheme improves as the tube radius or the solid mass are increased. On the other hand,the problems become worse for more slender tubes or for smaller solid masses. This is entirely con-sistent with the well-known added mass effect. In [36], Causin et al. derive critical conditions forthe convergence of different staggered or strongly coupled block Gauß–Seidel type iterative solutionstrategies with respect to ow through a exible tube. These conditions relate the radius, the length,the wall thickness and the uid and solid densities.


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Figure 26. Flexible tube: number of iterations required for block Gauß–Seidel strategies to reach a toleranceof 10 6 in the norm of the change of interface traction forces from one iteration to the next.

Thus, the experimentation with the tube problem allows for a clearer denition of the character-istic solid over uid mass ratio: It is identied as the ratio of solid mass over the added uid mass.

The added uid mass represents the amount of uid, which is entrained by the solid. It is typicallylarger in internal than in external ow problems. Hence, if the solid mass is small in comparisonwith the added mass, it is the inertia of the latter that is essential for the computation of a viableapproximation of the interface displacement. However, the added uid mass is not accounted for bythe solid subsolver. As demonstrated in this work, the proposed staggered strategy with its controlparameter ˇ is capable of providing computationally efcient stable solutions in the presence of signicant added mass effects but fails if these become too pronounced.

Remark 3We have also attempted to simulate the tube problem with a block Gauß–Seidel solution strategy. Inparticular, we have used a xed relaxation parameter ˇ GS and, alternatively, adaptive relaxation onthe basis of the Aitken acceleration strategy [16, 45]. We employ a low-order predictor and relax-ation for the interface displacement. We note that this is a perfectly admissible and commonly usedchoice but does not necessarily minimise the number of required iterations.

For ˇ GS D 0.05, the computations fail to converge in the rst time step. For theAitken acceleration, convergence is obtained for a number of time steps before failure occurs(see Figure 26). The combination of small time steps with a small relaxation parameter ˇ GS D0.005does not fail to converge but requires excessive numbers of iterations per time step and is thereforeaborted after 0.03 time units.

The fact that the block Gauß–Seidel strategies are not suited for the simulation of this problem isa conrmation of the assertion made in Section 3.4 that the proposed staggered scheme has a rangeof applicability similar to Gauß–Seidel strategies, while being computationally far more efcient.

5. CONCLUSIONS

This work proposes a new staggered scheme for the simulation of uid–structure interaction.The solution strategy has been applied to a linear one-dimensional model problem. The

conclusions drawn from the analysis can be summarised as follows:

(i) The strategy is second-order accurate in time.(ii) Small ratios of solid over uid mass render large truncation errors.

(iii) Small ratios of solid over uid mass require small values of the control parameter ˇ toachieve unconditional stability.

(iv) Small values of ˇ lead to reduction of high frequency damping of the overall system.(v) Strong similarities between the proposed scheme and the Gauß–Seidel iteration exist.


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Four numerical examples of multi-dimensional uid–structure interaction have been investigated.This includes convergence studies and the comparison with solutions obtained from stronglycoupled strategies. The conclusions may be summarised as follows:

(i) All properties of the scheme as identied on the basis of the model problem also applyto real uid–structure interaction problems. In other words, the model problem is indeedrepresentative of uid–structure interaction problems.

(ii) The characteristic ratio of solid over uid mass is identied as the ratio of solid over addedmass of uid.

(iii) The strategy allows for the simulation of problems which have up to now been believedto require a strongly coupled solution strategy. This includes problems of incompressibleuid–structure interaction and of uid-thin–structure interaction.

(iv) The numerical experimentation performed at this stage suggests that the proposed schemehas a range of applicability very similar to the Gauß–Seidel strategy, while being computa-tionally far more efcient.

Future work should include the adaptation of the proposed staggered scheme to time integrationschemes other than the generalised- ˛ method. This will make the strategy applicable to differentcombinations of existing subsolver software.

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