a strong coupling partitioned approach for fluid-structure interaction with free surfaces

Computers & Fluids 36 (2007) 169–183

www.elsevier.com/locate/compfluid

A strong coupling partitioned approach forfluid–structure interaction with free surfaces

Wolfgang A. Wall a,*, Steffen Genkinger b, Ekkehard Ramm b

a Chair of Computational Mechanics, Technical University of Munich, Boltzmannstr. 15, 85747 Garching, Germanyb Institute of Structural Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany

Received 12 June 2002; accepted 4 August 2005Available online 13 December 2005

Abstract

Fluid–structure interaction (FSI) problems are of great relevance to many fields in engineering and applied sciences. One widespread and complex FSI-subclass is the category that studies the instationary behavior of incompressible viscous flows and thin-walled structures exhibiting large deformations. Free surfaces often present an essential additional challenge for this class of prob-lems. Prominent application areas are fluid sloshing in tanks and numerable problems in offshore engineering and naval architecture.Especially when partitioned strong coupling schemes are used in order to solve the coupled FSI problem the design of an appro-priate overall computational approach including free surface effects is not trivial. In this paper a new so-called partitioned implicitfree surface approach is introduced and embedded into a strong coupling FSI solver. For complex problem classes this approach iscombined with the general elevation equation that is closed through a dimensionally reduced pseudo-structural approach. Thepresented approach shows the same stability properties as a full implicit approach but is by far more efficient—especially in thepartitioned coupled case.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, free surface flow and fluid–structureinteraction problems have received considerable atten-tion from the computational mechanics community.Accurate prediction of the fluid–structure interactioneffects of a flowing liquid including free surfaces is aproblem of great relevance in civil and offshore engineer-ing and naval architecture among many other fields. Itwas only recently that advances in computer and algo-rithmic power allowed to tackle such problem classes.Such problems are particularly difficult because theyconsist of nonlinear boundary conditions imposed onmoving boundaries where the position is part of the

0045-7930/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.compfluid.2005.08.007

* Corresponding author. Tel.: +49 89 28915300; fax: +49 8928915301.

E-mail address: [email protected] (W.A. Wall).

problems� solution. The requirement of accurate androbust prediction of the domain deformations, togetherwith proper representation of the boundary conditionsalong the free surface and the fluid–structure interface,are some of the difficult issues encountered in FSI prob-lems including free surface effects.

In previous papers [1,2] FSI-problems have beencharacterized as hybrid surface and volume coupledthree field problems. Our presented approach is basedon a semidiscretization strategy with a pure finite ele-ment spatial discretization of all fields involved—fluid,structure and mesh. For the flow field stabilized finiteelements along with a fully coupled solver have beendeveloped. This incompressible Navier–Stokes solverhas been extended to time dependent domains throughan ALE-approach. The mesh dynamics needed in thisrespect is captured through a pseudo-structural formu-lation. Finally, the structural field exhibiting largedeformations is described using a discretization with

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170 W.A. Wall et al. / Computers & Fluids 36 (2007) 169–183

advanced finite shell or wall elements based on multi-field principles. For the solution of the coupled problemboth loose and strong coupling schemes have beendeveloped.

The numerical solution of viscous free surface prob-lems was pioneered by Harlow and Welch [3,4] whodeveloped the MAC method. A variant of this approachis the VOF method by Hirt and Nichols [5]. Hirt et al. [6]and many other authors (e.g. Ramaswamy and Kawa-hara [7], Radovitzky and Ortiz [8]) have also used theLagrangian method for solving fluid problems with freesurfaces. Arbitrary Lagrangian–Eulerian (ALE) type ofmethods (see [9]) have also been implemented by Hirtet al. [10] and various other authors (e.g. Onate andGarcıa [11], Navti et al. [12], Soulaimani and Saad[13]). Another popular method for treating movingboundaries and interfaces is the Deforming SpatialDomain/Stabilized Space–Time (DSD/ST) method, thathas been introduced by Tezduyar et al. in the early 1990s[14,15]. From the continuum mechanical point of view itcan also be seen as an ALE-method smartly formulatedin a space–time concept. It has been very successfullyapplied both to free surface flows and also to FSI-prob-lems (first-time in [16,17]).

In the area of fluid–structure interaction many mod-els and computational approaches of different complex-ity have been developed. Partitioned solution strategiesrange from a loose coupling procedure of Felippaet al. [18] to the strong coupling algorithms of Kalroand Tezduyar [19], Le Tallec and Mouro [20] and manyother authors. An approach to fluid–structure interac-tion with surface waves can be found in [11,21]. Anotherinteresting approach for fluid–object and fluid–structureinteraction with free surfaces and two-fluid flows thatcombines attractive features of interface tracking andinterface capturing schemes is the Mixed Interface-Tracking/Interface-Capturing Technique introduced byTezduyar [22].

The driving force behind the study presented in thispaper was to extend the robust strong coupling schemesfor FSI problems that we developed in the past to casesthat include free fluid surfaces. Main requirements forthe new overall approach were to maintain the robust-ness of the original partitioned strong couplingapproach as well as the straightforward applicabilityof developed acceleration schemes and also to achievean overall efficiency as high as possible. All of theserequirements were met through a so-called partitionedimplicit free surface approach. This paper introducesthis approach and also its embedment into a stronglycoupled FSI solver with very efficient accelerationschemes. In addition it is also shown how the generalelevation equation, which is needed for complex prob-lem classes, can very conveniently and generally beclosed through a dimensionally reduced pseudo-struc-tural approach.

An outline of the paper is as follows. In Section 2, thegoverning equations for free surface flows on a movingmesh are given along with the boundary conditions onthe free surface and their algorithmic treatment. In Sec-tion 3, a new partitioned implicit approach is developedand the generalized free surface description is intro-duced. In Section 4, a combination of a partitionedstrongly coupled three field FSI solver with the parti-tioned implicit approach for free surface flows of Section3 is presented. Section 5 presents the conclusions ofour investigations. To demonstrate the numerical for-mulation, several examples are presented for pure freesurface problems and for FSI problems including freesurfaces.

2. Fluid field with free surface

2.1. Governing equations

We consider a linear-viscous, isothermal and isotro-pic fluid on the domain Xf, governed by the unsteadyincompressible Navier–Stokes equations. Free surfaceflows are characterized by a constantly changingsolution domain. Several approaches to free surfacemodelling have been attempted. The arbitrary Lagrang-ian–Eulerian formulation has proved to be very suitablefor simulating free boundary problems, especially whenlarge deformations of the surface but no topologicalchanges are present.

The Navier–Stokes equations in an ALE-frame-work are given in terms of the velocity u and the pres-sure p:

ou

ot

��xm

þ c � ru� 2mfr � �ðuÞ þ rp ¼ b ð1Þ

r � u ¼ 0 ð2Þ

with the stress tensor

rf ¼ �pIþ 2mf �ðuÞ ð3Þ

and the velocity deformation gradient

�ðuÞ ¼ 1

2ðruþ ðruÞTÞ ð4Þ

plus appropriate boundary and initial conditions:

u ¼ g on Cg ð5Þrf � n ¼ h on Ch ð6Þu � n ¼ uG � n on CI ð7Þu ¼ u0 on Xf with r � u0 ¼ 0 for t ¼ 0 ð8Þ

The boundary oXf subdivides into the Dirichlet bound-ary Cg, the Neumann boundary Ch and the interfaceboundary CI (either free surface CFS or fluid–structureinterface CFSI):

W.A. Wall et al. / Computers & Fluids 36 (2007) 169–183 171

oXf ¼ Cf ¼ Cg [ Ch [ CI ð9ÞCg \ Ch \ CI ¼ 0 ð10Þ

The ALE-convective velocity

c ¼ u� uG ð11Þ

is defined as the difference between the particle velocity u

and the grid velocity uG.

2.2. Mesh movement

The mesh movement, necessary for defining themovement of the reference domain within an ALE-typeapproach, is determined by a pseudo-structuralapproach on the domain Xm. The mesh is introducedas a separate field and its displacements are obtainedby solving the elastostatic system of equations

Kmr ¼ fmðrCÞ ð12Þwhere the right-hand side vector fm(rC) results from theDirichlet boundary conditions on the interface bound-ary CI. In order to increase the robustness of this ap-proach for large boundary movements additionalmeasures have to be taken like those introduced in [23]or [24]. More details on mesh moving schemes will beprovided in a forthcoming paper.

In addition to above set of equations dynamic andkinematic boundary conditions have to be specified atthe free surface CFS.

2.3. Free surface—dynamic boundary conditions

The continuity of forces on the free surface impliesthe standard dynamic boundary condition (7). In gen-eral, the interfacial behavior is very complex anddepends on the physiochemical properties of the fluidsand the molecular structure of the interface. In a com-mon simplification, the prescribed surface stress isassumed to consist of two parts, namely a normal stresscomponent originating from the atmospheric pressureand the surface tension on a curved surface and the tan-gential stress component in the case of a surface tensiongradient occurring along the surface:

rf � n ¼ �ðpa þ cfrs � nÞn�rscf ð13Þ

where $s Æ n denotes the curvature j of the surface and$s the surface gradient operator [25].

In this work it is assumed without loss of generalitythat the tangential stress component and the atmo-spheric pressure vanishes, so it is possible to simplifythe continuity of forces on the free surface to one termdepending on the local curvature of the free surfaceand a single material parameter cf, the isotropic and iso-thermal surface tension coefficient:

rf � n ¼ h ¼ cf jn on CFS ð14Þ

2.4. Free surface—kinematic boundary condition

In the particular case of free surfaces and moving no-flux interfaces, the only requirement for the velocity ofthe domain boundary, i.e. also the mesh boundary, is

u � n ¼ uG � n on CFS ð15Þi.e. the normal components of the grid and the fluidparticle must match. The tangential component of uG

remains arbitrary. Evaluating the scalar products ofEq. (15) leads to the general elevation equation:

u1n1 þ u2n2 þ u3n3 � uG1 n1 � uG

2 n2 � uG3 n3 ¼ 0 ð16Þ

with

uGi ¼

ori

otð17Þ

Obviously we are facing a closure problem at this point.For the three unknowns ui and ri, respectively, only oneequation is available up to now. So further assumptionshave to be made to close the problem.

As will become obvious later, both the presented par-titioned implicit free surface approach and the overallstrongly coupled FSI free surface scheme are more orless independent of the treatment of the kinematicboundary conditions. Hence, for the sake of simplicityof the presentation, two simple standard approachesfor treating kinematic boundary conditions, namelythe local Lagrangian and the height function approach,are mentioned in the following. Finally, in Section 3.3 amore general closure approach applicable to complexthree-dimensional cases will be presented.

2.4.1. Local Lagrangian approach

In a local Lagrangian framework the mesh isassumed to be attached to the fluid particles on the freesurface. Thus the grid moves into the same direction asthe fluid particle and so Eq. (15) reduces to

u� uG ¼ 0 ð18ÞThe grid velocity at the free surface of Eq. (18) is used toincrement its position

rnþ1 ¼ rn þ un;GDt on CFS ð19ÞThe simple implementation and its applicability togeneral geometries are the main advantages of thisapproach. However, it suffers from great mesh distor-tions especially for fluid flows exhibiting significantvelocities tangential to the free surface, like in openchannels. There the mesh degenerates quickly and re-meshing with the known difficulties and disadvantagesis unavoidable.

2.4.2. Height function approach

To overcome the disadvantages of the local Lagrang-ian approach the description of the free surface via a


height function is a very common and widely used alter-native. In this case it is assumed that the grid at the freesurface moves along a prescribed direction only, i.e.r = /e. In the case of vertical nodal displacement(r1 = r2 = 0) e is chosen as the unit vector in vertical,i.e. x3-direction, and then /(x1, x2, t) is a regular scalarvalued function representing the position of the free sur-face. For this specific case the normal vector in Eq. (16)can be expressed by

n ¼ 1

k~nk~n with ~n ¼ o/

ox1

o/ox2

1h iT

ð20Þ

and finally the free surface position can be determinedby solving

o/ot¼ �u1

o/ox1

� u2

o/ox2

þ u3 on S ð21Þ

where the domain S is the lower-dimensional regionover which / is determined, typically obtained from Xf

by projection.

FS

x1

x2 intΩ ΩuG or r

Fig. 1. Decomposition of fluid domain.

2.5. Explicit vs. implicit treatment of the free surface

The movement of the mesh at the free surface can becomputed both in an explicit as well as in an implicitway. Within the explicit formulation the free surfaceposition is predicted at the beginning of a time stepand the mesh motion in the domain is solved via apseudo-structural problem. With this predefined defor-mation of the reference domain the ALE fluid problemis solved. This formulation is rather cheap and it is ableto give good results for many applications. But obvi-ously this approach will show stability problems withlarger time steps. To overcome these stability problems,in a lot of situations implicit formulations are manda-tory. Standard implicit schemes for free surface flowsare realized either through a monolithic or through apartitioned iterative staggered solution of the coupledtwo field problem—fluid and mesh. In the first casethe fluid and the mesh problem are solved within onesystem of equations. This is rather expensive becauseof the solution costs of this coupled set of equationsand especially in combination with strongly coupledFSI approaches (see Section 4.3) it leads to a substantialincrease of calls to the computational mesh dynamics(CMD) module. Partitioned iterative staggered schemessolve the fluid and the mesh problem alternatively withinone time step until the position of the free surface hasconverged. Again the main drawback of this procedureappears in combination with strongly coupling FSIapproaches. The additional nested iteration complicatesa convergence analysis and may even destroy the con-vergence properties of the whole problem. And again,as before, one has to face a substantial increase ofCMD calls.

3. Partitioned implicit free surface approach

3.1. Basic idea

To overcome the discussed problems of a pure expli-cit or implicit free surface approach in combination withpartitioned strongly coupled FSI solution algorithms apartitioned implicit algorithm has been developed forfree surface flows. For this purpose the entire integra-tion domain Xf is divided (or partitioned) into the inter-nal domain Xint and a boundary domain close to the freesurface XFS, as displayed in Fig. 1:

Xf ¼ Xint [ XFS ð22ÞThe basic idea then is to include only the free surface po-sition r or the mesh velocities uG on the boundary asadditional unknowns in the fluid solver and solve themin a monolithic way. By this we inherit all the benefitsfrom a fully implicit approach but only add very littlecosts to the overall solution procedures as comparedto an explicit treatment.

A question that remains is how to choose XFS. In thesimplest case it only consists of one layer of elementsadjacent to CFS. Then Xint consists of all elements insidethe fluid domain according to Eq. (22). Of course, whenthe surface deformations are rather large or fast it mightbe necessary to enlarge the domain XFS. In such a casethe size of XFS can be determined by an estimation basedon the element size and the expected maximum gridvelocity at the boundary CFS, and the time incrementDt. Then a structured boundary mesh is used with sim-ple constraints reaching from CFS to the interfacebetween XFS and Xint. Hence, also in this case the num-ber of unknowns and costs remain small as before.

This approach is in principle independent of the freesurface formulation. It can be combined with a localLagrangian approach, with a height function approachor with the generalized elevation equation.

3.2. Finite element formulation

The variational form of the stabilized ALE FE for-mulation for the Navier–Stokes equations e.g. with the

(i) Initialization(ii) Time loop tn+1 = tn + Dt

(iii) Transfer grid velocity uG,n on CFS fromfluid to mesh field

(iv) Computational Mesh Dynamics (CMD)(v) Predict mesh displacement:

rnþ1 ¼ rþ uG;nDt on CFS

(vi) Solve mesh with prescribed displace-ments:

Kmrnþ1 ¼ fm

(vii) Compute new mesh velocity:uG;n!nþ1 ¼ rnþ1�rn

Dt

(viii) Transfer new grid velocity uG,n!n+1 onXint from mesh to fluid field

(ix) Computational Fluid Dynamics (CFD)(x) Compute ALE-velocity: cn = un �

uG,n!n+1

(xi) Nonlinear iteration scheme:(xii) Compute cnþ1

iþ1 ¼ unþ1iþ1 � uG;n!nþ1

on Xint

(xiii) Compute cnþ1iþ1 ¼ unþ1

iþ1 � uG;n!nþ1

on XFS

(xiv) Solve

Mf _unþ1iþ1 þNf ðcnþ1

i ; uG;nþ1i Þ

funþ1iþ1 ; u

G;nþ1iþ1 g þGf pnþ1

iþ1 ¼ ff

(xv) Correct mesh position

rnþ1iþ1 ¼ rn þ uG;nþ1

iþ1 Dt on CFS

(xvi) Check convergence: if not convergedgo to (xi)

(xvii) Check time: if end of simulation not reached, goto (ii)

(xiv) The system of equations has to be solved forfunþ1

iþ1 ;/nþ1iþ1 g.

(xv) The new mesh position on CFS determines to

rnþ1iþ1 ¼ /nþ1

iþ1 � /0. In addition, the mesh velocity

uG,n!n+1 on XFS has to be corrected based on (vii).


local Lagrangian kinematic boundary condition as addi-tional equation for the free surface may now be writtenas follows:

Find uh 2Vhg, uh;G 2Wh and p 2 Ph such that

Bðuh; uG;h; ph; vh;wh; qhÞ ¼ F ðvh; qhÞ8ðv;w; qÞ 2Vh �Wh �Ph ð23Þ

where

Bðuh; uG;h; ph; vh;wh; qhÞ

¼ ouh

ot

��xm

; vh

!þ ðch � ruh; vhÞXint

þ ðuh � ruh; vhÞXFS

� ðuG;h � ruh; vhÞXFSþ ð2mf �ðuh; �ðvhÞÞ � ðr � vh; phÞ

� ðr � uh; qhÞ þ ðuh � uG;h;whÞCFSþ ST ð24Þ

and

F ðvh; qhÞ ¼ ðb; vhÞ þ ðh; vhÞ þ ðcf jn; vhÞCFSþ ST ð25Þ

with the finite-dimensional subspaces

Vhg ¼ fu 2 H1ðXf Þ : u ¼ g on Cgg ð26Þ

Vh ¼ fu 2 H1ðXf Þ : u ¼ 0 on Cgg ð27ÞWh ¼ fuG 2 H1ðXFSÞg ð28ÞPh ¼ fp 2 L2ðXf Þg ð29Þ

The inner products (Æ , Æ) represent the integrals over thewhole fluid domain Xf while the indices of the other in-ner products indicate the respective integration domain.

In case of the height function approach as kine-matic boundary condition only the inner product(uh � uG,h, wh)CFS

in Eq. (24) has to be replaced by theweak form of Eq. (21). Due to the well-known numericalproblems, e.g. spurious oscillations in the solution, thatoccur when these equations are solved using standardFEM, stabilization terms �ST�, based on a GLS-formula-tion, are added to the standard variational form (see[26]). The semidiscrete matrix equations are given as

Mf _uþNf ðc; uGÞfu; uGg þGf p ¼ ff ð30Þand

Mf _uþNf ðc;/Þfu;/g þGf p ¼ ff ð31Þrespectively. Both are integrated in time using directtime integration schemes (like a one-step-H or aBDF2-scheme). The nonlinear term Nf(c, Æ ) is linearizedand the system of equations is solved in a fixed point-like iteration scheme. The approach is realized in aparallel setting via domain decomposition along withparallel iterative solvers (e.g. BiCGstab with ILUpreconditioning).

The overall partitioned implicit solution algorithmfor the two-field coupled free surface problem proceedsas follows:

It should be noted that through our partitionedimplicit approach we are able to get a fully implicit algo-rithm with only one CMD-call per time step in theabove algorithm.

In case of the height function approach only a fewparts of the algorithm have to be modified:

3.3. Generalized free surface description

In many cases of interest, the height function Eq. (21)cannot be applied, since the prescribed direction of themesh at the free surface is not vertical. For example,


sloping reservoir or channel walls would dictate that theadjacent surface nodes move along the wall surface (seee.g. [27]). This is especially crucial when the walls aredeformable themselves. Remember that the free surfacekinematic boundary condition, like the general elevationequation introduced earlier, is a scalar equation andhence only prescribes one of three components in the3D case. This means that in order to handle complexproblem classes like mentioned above more generalassumptions for closing the set of equations at the freesurface have to be introduced. The key point is thatthe free surface nodes should be able to distribute moreor less freely at the free surface without following anysimple and a priori prescribed direction. One way toachieve this is the Line-Tracked Interface Update Tech-nique, which was introduced by Tezduyar in [28] andwhich calculates the motion of each surface node alonga locally selected direction (typically in the surface-nor-mal direction).

The key idea here is to use a dimensionally reducedpseudo-structural approach. A descriptive interpreta-tion of this would be to attach a shell (or better a mem-brane) to the free surface in order to govern thetangential motion (i.e. two components in 3D) of thesurface grid nodes.

Obviously there are several ways to realize this idea.An elegant and straightforward way would be to simplycombine the general elevation Eq. (16) with a membraneequation on the free surface. For convenience we sug-gest another rather simple way here, i.e. to pose it in away that is directly based on the domain decompositionintroduced for the partitioned implicit free surfaceapproach. In other words we, for simplicity, somehow‘‘extend the membrane into the boundary domain’’.Only on the already defined domain XFS (Section 3.1)mesh displacements r are introduced as additionalunknowns. These mesh displacements are assumed tobe governed by a quasi-static momentum equation

r � S ¼ 0 ð32Þwhere S is the Cauchy stress tensor. This idea has alreadybeen applied for determining the mesh displacements onXm. However, this newly introduced pseudo-structureonly exists within the boundary area at the free surfaceand its displacements are determined together with thefluid unknowns during the nonlinear iteration scheme.We then apply Eq. (16) as a boundary condition actingon the equations for the mesh displacements of XFS.

The boundary value problem for the unknowns rk atnode k can be expressed by the weighted residual of thequasi-static momentum equation:

Rkb ¼

ZXFS

rðwkebÞ : SdXFS ¼ 0 ð33Þ

The vector equation for the residual has been convertedto three scalar components by dot products with the

co-ordinate basis vectors eb. Thus we have one scalarequation governing the unknown nodal displacementfor each of the three directions. Since one does not wantto prescribe the distribution of the nodes at the free sur-face, but only the position of this free boundary, thekinematic boundary condition is introduced as a so-called distinguishing condition. The effectiveness of thisprocedure has been demonstrated by Sackinger et al.[29] for different two-dimensional applications. For eachnode k on the free surface boundary CFS one of the threemesh equations defined by Eq. (33) is replaced by theweak form of the kinematic condition

Rkb ¼

ZCFS

wkebðu� uGÞ � ndCFS ¼ 0 ð34Þ

After discretizing Eq. (34) in time, e.g. using a one-step-H-scheme

uG;nþ1 ¼ rnþ1 � rn

HDtþ 1�H

HuG;n ð35Þ

it is possible to solve for the unknown free surfacedisplacements r.

Putting the approach this way, one question remains:which one of the three mesh displacements has to bereplaced? For simple problems the choice may be easybecause the surface lines up with a co-ordinate plane.E.g. if the free surface is facing in the x3-direction andone does not expect this to change during the entire sim-ulation, the kinematic boundary condition wouldreplace the x3-component of the mesh equations. Suchsimple rules break down for general cases, especially ifthe orientation of the free surface rotates during thecomputation. In such cases it may happen that the pro-jection of the distinguishing condition is non-zero result-ing in spurious mesh stresses. And obviously the answerto our question lies in the original membrane idea, i.e. inthe tangential plane. For this we rotate the componentsof the governing equations locally into normal and tan-gential components. Then the distinguishing condition ischosen to constrain the normal motion of the free sur-face. The remaining components of Eq. (33) allow ashear-free redistribution of the nodes in tangential direc-tion. Cairncross et al. [30] successfully applied thismethod to three-dimensional stationary fluid problems.

A further method to reduce possible mesh distortionscoming along with large deformations of the free surfaceis to scale the Youngs modulus of the pseudo structureused for calculating the tangential stiffness values. As ascaling factor the inverse of the Jacobian determinantof the element edge (not of the whole element) is used.This locally increases the stiffness of free surface ele-ments exhibiting a tendency to distort the tangentialnodal redistribution.

Finally a short remark on the computation of theboundary normals. For a standard finite element imple-mentation using linear quadrilateral or hexahedral ele-


ments the normal at a boundary node is not defineduniquely. There are a number of possible ways in whichone may compute this nodal normal. Possible choicesare the uniquely defined normal at the Gauss point onthe element edge or an averaged normal from purelygeometric considerations. Numerical experiments usingthe first possibility came along with a relatively greaterror for the mass conservation which resulted in anadditional undesirable numerical damping of thesystem. Best results when monitoring the mass conserva-tion have been achieved by using the standard mass-con-sistent normal. A detailed description in which the unitnormal at node k is derived from the discrete weak con-tinuity equation for incompressible flow can be found in[31]. After some algebra this yields the following expres-sion for the normal component in j-direction at node k

nkj ¼

1

k~nkk~nk

j with ~nkj ¼

ZXf

oN k

oxjdXf ð36Þ

where Nk is the finite element velocity basis function atnode k.

3.4. Examples

To show the suitability of the presented method, theformulation is applied to three different problemsdescribed in the literature. The first one is a problemof wave propagation. A numerical simulation of thisproblem has been carried out by Ramaswamy [33],where the results are compared with the analytical solu-tion (see [32]). The second example involves a collapsingwater column, which is described among others byRamaswamy and Kawahara [7] as the broken damproblem. In order to be able to compare results, surfacetension effects have been neglected for the first twoexamples, just as in the given references. In the lastexample a water rod was modeled in order to validatethe computation of surface tension effects.

Another purpose of the three examples was to exam-ine the mass conservation of the applied method. In fact,the mass is conserved to within 6.0 · 10�5% of the origi-nal fluid volume at any time during the entire simula-tion. For all three examples it was possible to chooseXFS as one layer of elements adjacent to the free surface.

3.4.1. Solitary wave propagation

The phenomenon of a solitary wave traveling in arectangular channel of uniform depth was first reportedby John Scott Russell in 1834. He defined the solitarywave as a single elevation above the surrounding undis-turbed water level producing a definite transport in thedirection of wave propagation only. The wave travelswithout change of shape and with essentially constantvelocity throughout the observable time of travel.Laitone [32] provides in his analytical study for this

problem an approximate solution for velocity, pressureand free surface elevation, which can be written in theforms

u1 ¼ffiffiffiffiffiffigd

p Hd

sech2f ð37Þ

u2 ¼ffiffiffiffiffiffiffiffi3gd

p Hd

� �3 x2

d

� �� sech2f � tanh f ð38Þ

g ¼ d þ H sech2f ð39Þp ¼ .gðg� x2Þ ð40Þ

where

f ¼ffiffiffiffiffiffiffiffiffi3

4

H

d3

rðx1 � ctÞ and c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigd 1þ H

d

� �sð41Þ

in which H and d are the initial wave height and stillwater depth, respectively. Although Laitone�s formulaholds for an infinitely long channel only, it is possibleto limit the computational domain by two vertical walls,located 8d away from the initial crest [33], as illustratedin Fig. 2, without disturbing the flow characteristicssignificantly.

For the numerical computation the still fluid depth dwas chosen to 10 cm, the wave height H is 2 cm and thehorizontal length of the channel is 16d = 160 cm. Grav-ity is acting downwards with magnitude 10 cm/s2. Thedensity and the kinematic viscosity of the fluid are.f = 1.0 g/cm3 and mf = 1.0 cm2/s respectively. Thedomain is discretized with 160 · 10 Q1Q1 stabilized fluidelements and time increment Dt = 0.05 s is used. Start-ing from the initial conditions (37)–(41) at t = 0.0 sand using a height function approach for the free sur-face, the behavior of the solitary wave was computed.Exemplary the velocity field on the pressure result aftert = 7.0 s is displayed in Fig. 2(b). Laitone�s approxima-tion gives a solution for the run-up height R of the waveon the right vertical wall:

Rd¼ 2

Hd

� �þ 1

2

Hd

� �2

ð42Þ

With H/d = 0.2, R is 4.2 cm, and our computed result isR = 4.22 cm. This clearly shows that the present methodagrees reasonably well.

3.4.2. Collapsing water column

In this example, a rectangular water column in hydro-static equilibrium is confined between two vertical walls,as shown in Fig. 3. The water column is 3.5 cm wideand 7.0 cm high. Gravity is acting downwards with980.0 cm/s2. Material parameters are given by the den-sity .f = 1.0 g/cm3 and the kinematic viscosity mf =0.01 cm2/s. The right wall of the reservoir is removedinstantaneously at time t = 0.0 s and the liquid moves

Fig. 3. Collapsing water column: problem definition.

Fig. 2. Solitary wave propagation: problem definition (a) and computed results at t = 7.0 s (velocity vectors on pressure solution) (b).


under the force of gravity. The fluid domain was discret-ized with 20 · 40 Q1Q1 stabilized fluid elements result-ing in 2641 fluid DOFs. The time step increment waschosen to Dt = 10�4 s. Fig. 4 shows a typical sequenceof the velocity vectors on the pressure results. Experi-mental results [34] and reference values from the litera-ture (e.g. [7]) have been reported for the position xF

versus time of the leading edge of the water as it flowsto the right. The comparison of the computed resultswith those from Ramaswamy and Kawahara [7] plotted

Fig. 4. Collapsing water column: computed resu

in Fig. 4 shows close agreements in all respects. Theappearance of both a vertical and a horizontal free sur-face, however, provides a check on the capability of thepresented local Lagrangian approach to treat freesurfaces that are not single-valued with respect to oneco-ordinate axes.

3.4.3. Non-equilibrium rodWhen a rod or cylindrical drop is deformed, capillary

waves are induced that cause the drop surface to oscil-late about its equilibrium shape. This behavior isobserved in a numerical calculation when an initiallysquare drop responds to unbalanced surface tensionforces. The results were computed for a drop with width1.0 cm, on an unstructured mesh with 10,363 Q1Q1 sta-bilized fluid elements. The free surface position wasdetermined by the local Lagrangian description. Mate-rial parameters were chosen to .f = 1.0 g/cm3, mf =1.0 cm2/s and cf = 73.0 g/s2. The initially square likeshape of the drop results in very strong surface forcesat the high-curvature corners, setting the drop into oscil-lation, as shown in Fig. 5. At a sequence of timest = 0.00, 0.01, 0.02, 0.03, 0.04 and 0.05 s, the oscillationsof the drop are apparent. The present viscosity leads to astationary circular solution. At t = 0.05 s, the drop isalready nearly circular.

lts (velocity vectors on pressure solution).

Fig. 5. Non-equilibrium rod in zero gravity.

1

1

2

tn tn+1

Ωf Ωf

ΩsΩs

Fig. 7. Iterative staggered analysis schemes.


4. Fluid–structure interaction including free surfaces

4.1. Fluid–structure interaction environment

Our partitioned fluid–structure interaction environ-ment is described in detail in [35] and is presented herein a comprising overview in Fig. 6. In this approach anon-overlapping partitioning is employed, where thephysical fields fluid and structure are coupled togetherat the interface, i.e. the wetted structural surface CFS.The third field is the already mentioned pseudo-struc-tural mesh. The single fields are solved by semidiscreti-zation strategies with finite elements and implicit timestepping algorithms.

Desirable key requirement for the coupling schemeswould be to fulfill two coupling conditions: the kine-matic and the dynamic continuity across the interfaceat all times. Kinematic continuity requires that the posi-tion of structure and fluid mesh as well as structural,fluid and fluid mesh velocities are equal at the interface,while dynamic continuity means that all tractions at theinterface have to be in equilibrium. Due to their inherentexplicit character sequential staggered analysis schemes(or loose coupling schemes) fulfill only the dynamicbut not the kinematic continuity requirements along

Fig. 6. Three field FS

the interface CFSI. This results in limited stability ofthe partitioned solution algorithm. At the beginning ofa numerical analysis the partitioned solution might evenbe perfectly stable. However, after some time, exponen-tially increasing perturbations can appear making thesolution useless.

In [1,2,35–37] we developed and realized iterativestaggered, strong coupling, implicit analysis schemeswith several acceleration approaches that guarantee(fast) convergence of the iteration over the fields. Thebasic idea of these schemes is as follows (Fig. 7): In eachtime step [tn! tn+1] an approximate solution at the endof the step tn+1 is determined by any sequential stag-gered scheme. With the updated coupling boundary

I environment.

(iv) Set i = 0(v) Iteration over the fields

(vi) Transfer dnþ1C;i on CFSI from structure

to mesh field and rnþ1i from fluid to

mesh field(vii) Computational Mesh Dynamics (CMD)

(viii) Solve mesh with prescribeddisplacements: Kmrnþ1

iþ1 ¼ fm

(ix) Compute new mesh velocity:uG;n!nþ1

iþ1 in compliance with GCL

(x) Transfer new grid velocity uG,n!n+1

from mesh to fluid field(xi) Computational Fluid Dynamics (CFD)

(xii) Solve fluid partition on deformingmesh for fluid velocity unþ1

iþ1 pres-

sure field pnþ1iþ1 and grid velocity

uG;nþ1iþ1 / grid displacement rnþ1

iþ1 or

height function value /nþ1iþ1 and

correct mesh position on CFS

(xiii) Transfer fluid stresses rf ¼ �pnþ1iþ1 Iþ

2mf �ðunþ1iþ1 Þ on CFSI from the fluid to the

structure field as Neumann b.c.(xiv) Computational Structural Dynamics

(CSD)(xv) Convert rf on CFSI into a consis-

tent nodal coupling vector fnþ1C;iþ1

(xvi) Solve structure partition for new

displacements dnþ1iþ1 ¼ ½~d

nþ1

C;iþ1dnþ1I;iþ1�

T

(xvii) Compute optimal relaxation parameterxi via Aitken iteration or Gradientmethod

(xviii) Relaxation of predicted interface posi-

tion: dnþ1C;iþ1 ¼ ð1� xiÞdnþ1

C;i þ xi~d

nþ1

C;iþ1 onCFSI

(xix) Check convergence. If converged goto (xxi)

(xx) Set i! i + 1, go to (vi)(xxi) Check time: if end of simulation not reached, go

to (ii)


conditions a new solution is calculated for the same timestep. This procedure is repeated with the latest, iterativeimproved coupling information until convergence overthe three fields is reached. Afterwards computation isproceeded for the next time step. Since both the kine-matic and the dynamic continuity requirements arefulfilled exactly a strongly coupling algorithm givingstable and reliable results, is achieved. The overallstrategy can be interpreted as an iterative Dirichlet–Neumann substructuring scheme. In its basic version,and without the novel acceleration schemes, thisapproach is similar to the procedure presented by LeTallec and Mouro [20].

4.2. Embedding the partitioned implicit free surface

approach into the fluid–structure interaction solver

Embedding a free surface description into thedescribed iterative Dirichlet–Neumann substructuringscheme for fluid–structure interaction represents anadditional challenge. Inclusion of the simple explicit freesurface approach is not very advisable here, because itsexplicit fraction would cause stability limit for the over-all approach, although all the rest of the coupled solu-tion scheme is fully implicit which is also paid for. Thedrawbacks of the monolithic and iterative staggered freesurface approaches, respectively, in conjunction with astrong coupling approach of above type are the increas-ing numerical costs due to a substantial increase of callsto CMD and the possible deterioration of convergenceproperties of the whole problem. Hence it is especiallythis class of problems where the partitioned implicit freesurface approach shows its suitability and quality.

4.3. Algorithmic framework

Combining the partitioned iterative staggered analy-sis schemes under consideration with synchronoustime discretizations in the fluid and structural part anda partitioned implicit treatment of the free surfaceis straightforward. The whole procedure can be cast ina unified algorithmic framework. In every time interval[tn! tn+1] the following steps have to be run throughin order to compute the new coupled solution attn+1, starting from a known state of motion before andat tn:

(i) Initialization(ii) Time loop tn+1 = tn + Dt

(iii) Compute predictor:

dnþ1C;0 ¼ dn

C þ Dt3

2_d

n

C �1

2_d

n�1

C

� �on CFSI

rnþ10 ¼ rn þ uG;n!nþ1Dt on CFS

The given scheme is an extension of the algorithmpresented by Mok [35] for pure FSI problems. In thiscontext the partitioned implicit algorithm for the treat-ment of the free surface has particularly proven to bevery advantageous, since only two additions to thealready existing iterative Dirichlet–Neumann substruc-turing scheme are necessary: In addition to the struc-tural predictor at the fluid–structure interface,displacements at the free surface are also predicted atthe beginning of a time step (see (iii)). The solution ofthe fluid field simply follows the procedure describedin Section 3 for a pure fluid free surface flow with thegrid velocity, grid displacement or height function valueas additional unknowns on CFS (see (xii)). Especially

(xviii.a) Compute Aitken factor:

lnþ1i ¼ lnþ1

i�1 � ðlnþ1i�1 � 1Þ

�ðDdnþ1

C;i � Ddnþ1C;iþ1Þ

T � Ddnþ1C;iþ1

ðDdnþ1C;i � Ddnþ1

C;iþ1Þ2

whereas Ddnþ1C;i :¼ dnþ1

C;i�1 � ~dnþ1

C;i and lnþ10 ¼ ln

imax

(xviii.b) Compute new optimal relaxation parameterxi ¼ 1� lnþ1

i


compared with an iterative determination of the freesurface position the advantages of the chosen procedureare evident. In such a case an iteration over the interfacewould have to be integrated into the iteration over thefluid–structure interface. The resulting nested iterationwould lead to considerable additional computation timeand to a problematic convergence analysis.

The iteration described in the above algorithmicframework ensures and accelerates convergence forappropriate relaxation parameters to the simultaneoussolution, exactly fulfilling the required coupling condi-tions. However, a key question remains: How to chooseoptimal relaxation parameters?

We have developed several techniques for choosingthe relaxation parameter for pure FSI problems [1,35–37] and thanks to the partitioned implicit free surfaceapproach these techniques can directly be applied tocases when free surfaces are involved. The two mainapproaches are sketched in the following. They are bothrobust in the sense that they have problem-independentacceleration properties even for nonlinear systems, anduser-friendly in the sense that no problem-dependentparameters are needed.

The first technique is an acceleration based on thetransformation of the basic idea of the gradient method(method of steepest descent) to the iterative substructur-ing scheme. In every iteration a relaxation parameter

xi ¼gT

i gi

gTi S�1

S ðSf þ SSÞgi

ð43Þ

is computed without explicitly computing and storingthe Schur complements of Eq. (43). In this case theabove algorithm has to be extended at position (xviii)by the following steps:

(xviii.a) Compute residual gnþ1C;i ¼ ~d

nþ1

C;iþ1 � dnþ1C;i on CFSI

and transfer it to mesh field(xviii.b) Solve fluid mesh with prescribed displacements

rC ¼ gnþ1C;i

(xviii.c) Compute grid velocity uG in compliance withthe geometric conservation law (GCL)

(xviii.d) Transfer mesh deformation r and mesh veloc-ity uG to fluid field

(xviii.e) Solve fluid partition with r and uG (withoutexternal loads and history terms): ð 1

Dt MfþHNf ðcÞÞuþHGf p ¼ 0

(xviii.f) Transfer fluid stress rf ¼ �pIþ 2mf �ðuÞ onCFSI from the fluid to the structure field asNeumann b.c.

(xviii.g) Convert rf on CFSI into a consistent nodalcoupling load vector fC

(xviii.h) Solve structure partition for dC ¼ dC dI

h iT

(xviii.i) Compute new optimal relaxation parameter

xi ¼ðgnþ1

iþ1ÞT�ðgnþ1

iþ1Þ

ðgnþ1iþ1Þ�ð�dCþgnþ1

iþ1Þ

The computed xi is locally optimal with respect to theactual search direction. The gradient method guaranteesconvergence, but additional computational costs are notnegligible.

A second technique is based on Aitken�s accelera-tion scheme for vector sequences according to Ironsand Tuck [38]. In this case step (xviii) of the FSI-algorithm has only to be extended by two additionaloperations:

Additional computational costs for this technique aremarginal since the computation of the Aitken factorconsists only of vector differences and scalar products.However, obviously no convergence analysis for thevector case exists.

4.4. Examples

4.4.1. Basin with collapsing arch

The first numerical example for testing the FSI-solverincluding the generalized free surface descriptioninvolves a two-dimensional basin as depicted in Fig. 8.The lateral walls and parts of the bottom are rigid.The center-part of the bottom is closed by an elasticarch. The initial fluid depth is 20.0 cm. The basin is filledthrough two lateral channels with u2 ¼ 5:0 cm=s.

The elastic arch has a thickness of 0.1 cm. The struc-tural material parameters were chosen to .s = 500g/cm3, Es = 9.0 · 108 N/cm2 and ms = 0.3. The fluid den-sity is .f = 1.0 g/cm3 and the kinematic viscosity ismf = 9.0 cm2/s. At both lateral inclined walls slipboundary conditions were applied. The fluid domainwas discretized with 3200 Q1Q1 stabilized fluid ele-ments. The time increment was chosen to Dt = 0.025 s.For the structural domain 100 wall elements have beenused.

Due to the increasing fluid depth the pressure on theelastic arch reaches a critical value and the structure col-lapses. As shown in Fig. 9, the presented approach isable to reproduce this highly transient coupled bucklingprocess exhibiting large structural and free surfacedeformations. Due to the damping of the fluid the sys-tem reaches a new steady equilibrium position afterapproximately t = 72.0 s.

Fig. 9. Basin with elastic arch: collapsing arch with computed pressure solution.

Fig. 8. Basin with elastic arch: initial configuration.


4.4.2. Tank with flexible bottom plate

The final example is just given to show the operabi-lity for three-dimensional problems of the presented

approach. Fig. 10 shows a filled liquid storage tank withrigid walls and a flexible membrane at its bottom. Thetank has a quadratic cross Section with a length of

– d

–

aa

p(t)

g(t)

Fig. 10. Filled liquid storage tank with flexible bottom plate.


a = 10.0 cm. The fluid depth in the tank is d = 4.0 cmThe bottom plate has a thickness of 0.2 cm, the densityis .s = 500 g/cm3 and Young�s modulus is Es = 2.1 · 103

N/cm2. Poisson�s ratio is ms = 0.0. For the fluid, materialparameters are comparable to those of a viscose oil: Thedensity is .f = 0.92 g/cm3 and the kinematic viscosity ismf = 9.0 cm2/s. Surface tension effects have beenneglected. At the vertical rigid walls slip boundary con-ditions were applied. The fluid domain was discretizedwith 20 · 20 · 8 Q1Q1 stabilized fluid elements. The freesurface position was determined via the local Lagrang-ian approach. For the structure field 20 · 20 four-nodedshell elements were used. The time increment was chosento Dt = 0.01 s.

In order to avoid modeling of the filling process thetank was taken to be already filled up to the above givenheight from the beginning and an additional starting

3

0

980

0 2 4 6

pg(t)

t

Fig. 11. Time dependence of g

Fig. 12. Oscillations of the flexible bot

procedure was used in the numerical computation.Applying full self-weight of the fluid at time t = 0.0 son the membrane would result in a highly dynamic snapthrough of the bottom plate which requires a very smalltime step size Dt in the numerical computation. Hence,the filling process is simulated by a time dependent grav-ity vector (Fig. 11). During the respective time intervalthe bottom plate was considered to be rigid. The rigidplate is modeled through a surface load applied to themembrane being in equilibrium with the self-weight ofthe fluid. At t = 2.1 s the surface load is removed withinDt = 0.3 s (Fig. 11) and the system starts to oscillateexhibiting large deformations as shown in Fig. 12.

Comparison of the obtained results with alternativeapproaches, like monolithic or standard partitionedschemes, are obsolete here since they obviously haveto match more or less (only differences with respect tomesh deformations) completely. The only real differenceis the algorithmic treatment and hence the costs. For thegiven example the iteration over the fields convergedafter three steps and the fluid solver needed six to eightiteration steps to reach convergence. Thus, thanks to thepresented approach it was possible to save about 20 callsto CMD per time step compared to a monolithic or iter-ative staggered free surface approach. For a localLagrangian kinematic description of the free surfacethe number of unknowns of the fluid problem has only

0

6

(t)

0 2 4 6 t

ravity and surface load.

tom plate with pressure solution.


increased from 13,881 to 15,120 unknowns due to theadditional degrees of freedom at the free surface. Thisdifference is marginal compared to a monolithic schemeand the increase only depends on the number of nodes atthe free surface and not on the whole problem size. Andit is worth mentioning that the relative increase will evenbe much smaller in most real cases since usually the ratioof free surface to internal DOFs will be much smaller.

5. Conclusions

The main goal of the present paper was the develop-ment and embedment of a suitable free surface approachinto a recently developed strongly coupled parti-tioned fluid–structure interaction environment, withoutdestroying the robustness and the efficiency of theoriginal approach. After outlining the main difficultiesassociated with the boundary conditions at the freesurface and discussing the advantages and disadvan-tages of a pure explicit or implicit treatment of thefree surface boundary conditions, a new partitionedimplicit approach was developed and an algorithmicframework for solving incompressible viscous free sur-face flows based on an ALE FE formulation was given.The presented approach is in principle independent ofthe free surface formulation, i.e. treatment of kinematicboundary conditions. The new approach is presentedalong with a local Lagrangian, a height functionapproach and a generalized formulation. For the lattercase, which is especially needed in complex situations, anew closure based on a dimensionally reduced pseudo-structural approach was presented. The proposed algo-rithm was demonstrated with selected numericalexamples.

Afterwards this new incompressible Navier–Stokessolver for free surface flows was embedded into an exist-ing unified algorithmic FSI-framework, based on aniterative Dirichlet–Neumann substructuring schemewith two robust and user-friendly convergence accelera-tion methods. The resulting new method was given andapplied to two numerical examples.

The advantages of the presented idea of a partitionedimplicit treatment of the free surface within a stronglycoupled partitioned FSI framework can be summarizedas follows:

• Embedment into existing partitioned strong FSI cou-pling algorithms is straightforward and does notdestroy their robustness and efficiency.

• Successful convergence acceleration schemes of theoriginal iterative Dirichlet–Neumann schemes forpure FSI problems are still valid for the cases includ-ing free surfaces.

• Calls to the CMD-solver are strongly decreased com-pared to standard monolithic or iterative solution

approaches for free surface flows (especially in FSIcases).

• The size of the resulting fluid equation system ishardly increased. The increase depends only on thenumber of nodes at the free surface and not on thewhole problem size.

• It is independent of the formulation of the kinematicboundary condition and an extension to a generalapproach that combines a local pseudo-structuralformulation with the general elevation Eq. (16) is pos-sible and straightforward.

Acknowledgement

The present study is supported by a grant of the�Deutsche Forschungsgemeinschaft� (DFG) under pro-ject B4 of the collaborative research center SFB 404�Multifield Problems in Continuum Mechanics�. Thissupport is gratefully acknowledged.

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a strong coupling partitioned approach for fluid-structure interaction with free surfaces

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