fluid structure interaction based upon a stabilied (ale) finite element method

COMPUTATIONAL MECHANICSNew Trends and Applications

S. Idelsohn, E. Onate and E. Dvorkin (Eds.)c©CIMNE, Barcelona, Spain 1998

FLUID-STRUCTURE INTERACTION BASED UPON ASTABILIZED (ALE) FINITE ELEMENT METHOD

Wolfgang A. Wall and Ekkehard Ramm

Institute of Structural Mechanics, University of StuttgartPfaffenwaldring 7, D-70550 Stuttgart, Germanye-mail: {wall, ramm}@statik.uni-stuttgart.de

web page: http://www.uni-stuttgart.de/ibs/{wall, ramm}.html

Key words: fluid-structure interaction, stabilized finite element methods, arbitraryLagrangean-Eulerian (ALE) formulation, incompressible Navier-Stokes, geometrically non-linear structural dynamics, staggered time integration

Abstract.The computational procedure presented in this study is focussed on the time-dependent solution of two-dimensional coupled motions of geometrically nonlinear struc-tures and viscous incompressible Newtonian fluids. For the fluid part a stabilized finiteelement method based upon an arbitrary Lagrangean-Eulerian (ALE) approach has beendeveloped. For both the fluid and the structural systems direct time integration schemes,along with finite element spatial discretization, are employed. In order to enable large de-formations of the flexible structures geometrically nonlinear effects are taken into account.The overall numerical model is treated as a three-field problem, including the moving meshas an own system, and solved through partitioned procedures.

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1 INTRODUCTION

Interaction effects between structures and internal or external flows play an important role ina variety of physical systems. Very different engineering disciplines, like aerospace–, civil– orbioengineering are concerned with such kind of coupled problems. Thus the investigation offluid–structure interaction problems has already a long tradition. It appears that a lot of ap-proaches described in the literature contain rather severe restrictions with respect to the complex-ity of the applied models. This concerns either the level of approximation for the fluid models(potential flows, acoustics, Euler flows ...) or for the structural model (rigid bodies, linear elastic-ity through modal analysis ...). Formulations with a high level of physical complexity for bothfields are still very rare. A classification of recent efforts in the simulation of fluid–structure inter-action problems with respect to physical and numerical complexity of the involved models canbe found in Cebral4.

Quite a number of different approaches that emerged in recent years were concerned with theinteraction of a rigid body with compressible or incompressible, viscous or inviscid fluids14, 20,

21, 28. Formulations involving flexible structures and advanced flow models have recently beenpublished by the groups of Farhat8, Tezduyar25, Löhner4, the ISPRA–group3 and others1, 19 –due to the wide spreading character of the fluid–structure interaction research field this ‘flash’on recent contributions is necessarily incomplete. A more elaborate discussion on existing ap-proaches and recent literature on this topic is given in reference 27.

The present study presents a formulation which incorporates the complete incompressibleNavier–Stokes equations and a geometrically nonlinear description of the structure along witha finite element discretization for both domains and a semidiscrete representation in time. There-fore, a fully stabilized finite element method for the fluid part is extended to time dependent do-mains and is embedded in an algorithmic framework, dealing with the involved nonlinearitiesand the time dimension. This extension to time dependent domains is based on an ALE–formula-tion. The structural, i.e. the nonlinear elastodynamic, part is based on plane–stress elements inconnection with the generalized–� time integration method. Viewing the moving mesh, neededin the ALE–framework, as an own system leads to the interpretation of fluid–structure interactionas a three–field coupled problem, which will be solved through partitioned procedures. All de-scribed and developed concepts have been realized within CARAT, the Computer Aided ResearchAnalysis Tool developed at the Institute of Structural Mechanics at the University of Stuttgart.

The outline of the paper is as follows. In section 2 the problem of concern is stated along withthe basic equations for the fluid and for the structural models and a review of the ALE approach.Sections 3 to 5 sketch the developed solvers for the three involved fields – fluid, structure andmesh. In section 6 the derived computational procedure for the coupled problem is described.Section 7 presents some numerical examples, followed by some concluding remarks in section8. A more detailed treatise of the different topics and developed procedures can be found in refer-ence 27.

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2 STATEMENT OF PROBLEM

2.1 Fluid model

The fluid part is modelled as an incompressible, viscous, isothermal, isotropic Newtonianfluid which is fully described by the instationary, incompressible Navier–Stokes equations. Ex-pressed in primitive variables, velocity u and kinematic pressure p, momentum and continuityequations in an Eulerian description can be given as

(1)�u�t � u⋅�u � 2��⋅�(u) ��p � b in �f � (0,T)

�⋅u � 0 in �f � (0,T)

with initial and boundary conditions according to

(2)u � g n⋅� � h

u � u0 in �f for t � 0

on �g � (0,T); on �h � (0,T)

Here � represents the kinematic viscosity, i.e. dynamic viscosity � divided by the fluid density�, and � the stress tensor defined as � � –pI � 2��(u) with the symmetric part of the velocitygradient �(u) � 1�2 �u � (�u)T. The whole set of equations is defined on a bounded domain�f with boundary �� split into its complementary subsets denoted as Dirichlet boundary �g andNeumann boundary �h, and the time interval (0,T). In operator notation this set of equations canbe denoted by

�C(u) � �contu � 0

�M(p, u) � �u��t � �advu � �viscu � �presp � b (3)

with �adv, �visc, �pres and �cont defining the advective, viscous, pressure and continuity differen-tial operators, respectively.

2.2 Structural model

For the structural part a (total) Lagrangean description is employed. The governing equationfor the structural motion/deformation describes the momentum conservation and is in this contextalso referred to as (Cauchy) equation of motion

(4)� d..� � � S� � b in �s� (0,T)

with � denoting the structural density, d, d. and d

.. displacement, velocity and acceleration of a

material point, S the second Piola–Kirchhoff stress tensor – related to the Cauchy stress tensor� via a ‘pull–back’ operation employing the material deformation gradient – and b the body force.Boundary and initial conditions are written as

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(5)d � d

~N⋅S� S

~

d � d0; in �s for t � 0

on �d � (0,T); on �s� (0,T)

d.� d

.

0

with the Dirichlet or displacement boundary �d and Neumann or traction boundary �s of thestructural domain �s. Assuming a linear–elastic material model of St.–Venant–Kirchhoff type(small strains), the constitutive equation for the structure leads to the following relation betweenthe second Piola–Kirchhoff stress tensor S and the Green–Lagrange strain tensor E

(6)S� C : E

with C denoting the constitutive tensor. The range of large displacements is captured if the fullynonlinear kinematic relations are applied

(7)E � 12�FT � F�I

Here F is the material deformation gradient.

2.3 Arbitrary Lagrangean Eulerian (ALE) formulation

As given in the previous sections, fluid equations are traditionally described in an Eulerianformulation whereas a Lagrangean description is used for the structural part. This is due to therespective advantages of either formulation: easy capturing of large deformations and proper def-inition of the boundary. In order to close the gap between the fluid and structural domains an Arbi-trary Lagrangean Eulerian (ALE) formulation is employed for parts of the fluid domain.

ALE techniques were originally developed along with finite differences (e.g. Hirt et al.13) andwere later on also derived in connection with finite elements (e.g. Hughes et al.17, Donea7, Huertaand Liu15). The basic idea is the introduction of an additional reference domain �x with referencecoordinates x. Material domain / coordinates and spatial domain / coordinates are denoted by �z

/ z and �y / y, respectively. This reference domain is allowed to move arbitrarily independent ofspatial or material points. Continuum mechanical derivations are now made for points definedby a fixed position in this reference domain. For the formulation of conservation laws the materialtime derivative of some quantity f,

(8)DfDt

��f�t

(z, t)�z�

�f�t

(x, t)�x� ci

�f�yi

(y, t)

also referred to as ALE fundamental equation, is needed. Here, c denotes the ALE convectivevelocity, given as difference between particle and mesh velocity. It is important to note that ALEis not a question of the specifically chosen coordinate system but rather a question of the pointof reference for the continuum mechanical view. For the ease of implementation a formulationwritten in spatial coordinates y while keeping the time derivative in reference coordinates x is

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employed. Exemplified for this case conservation of momentum, already given for an Euleriandescription in (1), transforms to

(9)�ui�t�x� cj

�ui�yj

��ij

�yj� bi in �y

with � denoting the Cauchy stress tensor.In the finite element context the ’arbitrarily’ moving finite element mesh serves as representa-

tion of the reference domain as sketched in Fig. 1. As can be seen from this figure in special casesof an ALE description certain points, of course, can represent pure Eulerian (E) or Lagrangean(L ) character, respectively.

t

continuum

... ”Lagrange node”

... ”Euler node”E

y1

y2

L L

E

reference domainspatial domain

material domain

Figure 1: Arbitrary Lagrangean Eulerian description

Since ALE is a continuum mechanical formulation rather than a specific technique, in the au-thors’ opinion, one should also denote space–time concepts with deforming domains, like theDSD/ST (deforming spatial domain/space time) method of Tezduyar et al.26 or Hansbos12 ap-proach among others, as ALE formulation.

3 COMPUTATIONAL FLUID DYNAMICS (CFD) SOLVER

One concept to deal with the numerical problems associated with the finite element solutionof the incompressible Navier–Stokes equations stated in section 2.1, like incompressibility, pres-sure–coupling, hyperbolic character, in a unified way is the concept of stabilized finite elementmethods. On the one hand this concept originates from the streamline upwind Petrov–Galerkin(SUPG) method, introduced by Brooks and Hughes2 for advection dominated flows, and on theother hand on the ’circumventing Babuska–Brezzi’ (CBB) method, introduced by Hughes et al.16

for the Stokes problem. Numerous papers have been published on this subject since then.Applying the method of weighted residuals and a Bubnov–Galerkin type approximation to

equations (1), the general variational formulation for a stabilized finite element method can bewritten as

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(10)

Find uh �hg and ph �

h : � �vh, qh� �h � �

h

��uh

�t� uh⋅�uh, vh� � �2��(uh), �(vh)� � ��⋅vh, ph� � ��⋅uh, qh� � ST

� �b, vh� � �h, vh��h

with �hg and �h denoting appropriate finite–dimensional function spaces and vh and qh denoting

the respective test functions for velocities and pressures. Using the operator notation of (3) thestabilization term ST, introduced in (10), is given as

(11)ST� �K�h

��M�ph, uh�, �M L�qh, vh��K� ��C

�uh�, �C �cont�vh��K

where �M and �C represent the residuals of the momentum and continuity equations, respec-tively and L is a subset of �M of (3). �M and �C are referred to as stabilization parameters. As canbe seen from (11), the stabilization terms are mesh dependent terms, evaluated elementwise – thesum over all elements K of the triangulation �h – and are functions of the residuals of the Euler–Lagrange equations. Thus stability is enhanced, still preserving consistency and accuracy. Differ-ent methods emanate, depending on which parts of the differential operator �M are included inthe definition for L�qh, vh� in (11). Therefore methods like the streamline upwind Petrov–Galer-kin method (SUPG), the pressure stabilizing Petrov–Galerkin method (PSPG), the ‘pure’ or ‘un-usual’ Galerkin least–squares method (GLS), are included in this definition.

In this study a fully stabilized finite element method for incompressible, viscous flows ontime–dependent domains is developed, based on the fixed domain method of Franca and Frey11

and the ALE approach discussed in section 2.3. ‘Fully’ in this context refers to the inclusion ofall terms (except the time derivative) of the differential operator �M in the definition forL�qh, vh�. The method can be written as

(12)Find uh �

hg and ph �

h :

B�uh, ph, vh, qh� � F�vh, qh�� vh, qh� �

h � �h

with the appropriate function spaces given as

(13)

�h � vh H1

0��f�N �vh�

K Rk(K)N, K �h�

�hg � vh H1��f

�N �vh�K Rk(K)N, K �h, vh � g on �g�

�h � ph C0��f

� L20��f�� ph�

K Rk(K), K �h�

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Rk(K)N denotes an interpolation of order k on element K in the N–dimensional space and H10 de-

notes the first order Sobolev space with compact support. In full length the left hand side of (12)takes the form

(14)

� �K�h

��uh

�tx� ch⋅�uh ��ph � 2� �⋅�(uh),

B�uh, ph, vh, qh� � ��uh

�tx� ch⋅�uh, vh�� 2��(uh), �(vh)� � ��⋅vh, ph� � ��⋅uh, qh�

� ��⋅uh, �cont�⋅vh�K

�mom�y, ReK(y)� �ch⋅�vh ��qh–2� �⋅�(vh)��K

whereas the right hand side can be written as

(15)

F�vh, qh� � �f, wh� � �h, vh��h

� �K�h

�f, �mom�y, ReK(y)� �ch⋅�vh ��qh–2� �⋅�(vh)��K

The stabilization parameters could be defined as

(16)�mom�hK

2ch(y)2��ReK(y)�; �cont� ch(y)

2hK��ReK(y)�

with � being defined as the function

(17)��ReK(y)� � ReK(y),1,

0 � ReK(y) � 1,ReK(y) � 1

of the modified element Reynolds number

(18)ReK(y) �mK

ch(y)phK

4�(y)

hK is a measure for the element size and mK a measure for the order of interpolation, respectively.An important feature of this definition appears in the case when, through application of the ALEapproach, local Lagrangean regions appear, where the Navier–Stokes equations turn to time de-pendent Stokes problems. The above definition ensures that in this case the stability parameterstake the correct order with respect to the element length hK.

When the trial functions are expanded in terms of their finite element basis or shape functions,the semi–discrete matrix equation emanates

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(19)Mfu. � Nf(u)u � Gfp � Ff

For time discretization of the first order equations as well as for the treatment of the involvednonlinearities different methods have been adopted. This includes methods like the fractional–step–� scheme, semi–implicit one– and two–step methods, semi–explicit methods and fixed–point as well as Newton iterative schemes. For the ease of presentation only the simplest scheme,the one–step–� scheme, is discussed leading to a system of equations of the form

(20)�Mf(u) ��tNf(un�1) �un�1 ��tGfpn�1

� �Mf(u)–(1–�)�tNf(un)�un–(1–�)�tGfpn ��t Ffn�1� (1–�)�t Ffn

When choosing � � 0 an implicit scheme results. Often this nonlinear system of equationsneeds not to be iterated until convergence in every time step, i.e. only one iteration step of a fixed–point like scheme could be performed resulting in a kind of semi–implicit method. This approachcould be interpreted as ’linearization through extrapolation in time’. As discussed earlier, the do-main on which integration has to be performed is deforming with time. Formation of the elementmatrices therefore means integration over the element domain at a certain time instance governedby the time integration parameters.

4 COMPUTATIONAL STRUCTURAL DYNAMICS (CSD) ANALYZER

Spatial discretization of the structural model presented in chapter 2.2 is done through differenttypes of plane–stress elements for the two–dimensional problems under investigation. On the onehand fully integrated 9–node and reduced integrated 8–node displacement elements are used. Onthe other hand hybrid–mixed elements, like enhanced assumed strain (EAS) or hybrid stress ele-ments, are utilized. In the surface coupled problem context the second class of elements can alsobe employed, as pure displacement elements since involved additional quantities, stemming fromthe underlying multi–field functionals, are eliminated on the element level. Therefore in theglobal system matrices only displacement degrees of freedom remain – a crucial point with re-spect to load and motion transfer at the fluid–structure interface.

Once the spatial discretization is done, the nonlinear semidiscrete equation of motion is

(21)Msd..� �Csd

.�� Ns(d) � Fs

where Ms is the structural mass matrix, Ns is the vector of internal forces and Fs denotes the exter-nal forces. Damping of the structure could be included in the model via the damping matrix Cs.This however is negligible in many cases since the major source of damping for the structure re-sults from the surrounding fluid.

In the present approach the system given in (21) is solved using the ’Generalized–� Method’of Chung and Hulbert5 along with consistent linearization and a Newton–Raphson iterative

8


scheme. The ’Generalized–� Method’ is an implicit, one–step time integration scheme based onNewmark approximations in the time domain and contains schemes like the Hilber–Hughes–Taylor method, the Bossak–� method or the Newmark method (with its sub–methods trapezoidalor midpoint, linear acceleration, Fox–Goodwin, central difference, etc.) as special cases. There-fore a modified form of the equation of motion

(22)Ms d..

�� Ns�(d) � Fs

�

is introduced. Subscripts � denote evaluation of the respective quantities within the time intervalaccording to time integration parameters �m and �f which may be individually selected.

(23)d..

� � (1� �m) d..n�1

� �m d..n

d.

� � �1� �f� d

. n�1� �f d

. n

Ns�(d) � �1–�f� Ns�dn�1� � �f Ns(dn)

Fs� � �1� �f� Fs�dn�1� � �f Fs(dn)

Herein Newmark approximation in the time domain, i.e. a linear approximation of the accelera-tion within the time interval with a modification through the Newmark parameters � and �, is usedwhich is given by

(24)

d.� �

��t�dn�1 � dn� � ��

�d. n�

�� 2�2�

�td..n

d..� 1

��t2�dn�1 � dn� � 1

��td. n�� 1

2�� 1�d..n

The resulting scheme is second order accurate and an appropriate selection of the involvedtime integration and Newmark parameters governs its numerical characteristics, i.e. its numericaldissipation of low and high frequency modes and its stability.

5 COMPUTATIONAL MESH DYNAMICS (CMD) SOLVER

The flexibility gained through employing an ALE concept for moving domain flow problemsof course has to be paid for, since the ’arbitrary’ movement of the reference domain, i.e. the mesh,has to be fixed – asking for some kind of mesh moving scheme. Possibilities for the definitionof the mesh movement are the use of user–prescribed functions (applicable when dealing withsimple geometries) or some kind of mesh smoothing algorithms. A very powerful approach isto view the mesh as a pseudo–structural system. This could be done through a kind of spring/masstype idealization (see e.g. Farhat8) or by solving directly the elasticity equations (see e.g. Johnsonand Tezduyar18).

In the present study a pseudo–structural approach is adopted. In this context the mesh isviewed as an own system and therefore the fluid–structure interaction problem will be formulatedas a coupled three–field rather than a two–field problem. Considering the mesh as an elastic body

9


leads to the solution of an equation similar to (21) within the presented CMD solver. In this con-text, however, geometrically nonlinear effects are neglected leading to a linear system of equa-tions. Time–dependent effects are skipped in order to avoid numerical difficulties which couldarise through artificially ‘oscillating meshes’. Thus the CMD solver consists in the solution ofan elastostatic system of equations

(25)Km q � Fm with q � d on �FS and q � 0 on � \ �FS

per structural time step. The mesh deformations q are driven through the structural displacementsd on the fluid–structure interface �FS. These displacement boundary conditions are incorporatedin (25) through the right hand side term Fm.

Additional concepts realized in the presented computational procedure in order to extend theversatility or to reduce the computational costs in connection with our pseudo–structural ap-proach are:

� assigning different element stiffnesses to small and large elements (just by neglectingthe Jacobian determinant during element integration) or to elements close or far fromthe structure

� applying constraints related to the movement of the structure to certain mesh areas inorder to increase the ‘domain of influence’ of the applied prescribed deformations

� restricting the moving mesh domain only to a certain part of the whole flow domain‘close’ to the structure

� when using direct solvers – reusing the factorized stiffness matrix for several steps andjust back–substituting with the new right hand side vector emanating from the respec-tive structural deformations

� applying only a few iteration steps when using iterative solvers, instead of iteratinguntil convergence

6 COUPLED PROCEDURE

Two different approaches are available for the solution of the resulting three–field coupledproblem (see Fig. 2) – a fully coupled monolithic scheme on the one hand and solution throughpartitioned analysis or staggered procedures introduced by Park and Felippa22 on the other hand.The two different approaches are also sometimes referred to as strong and loose coupling, respec-tively (e.g. Cebral4). Due to obvious appealing features of the second approach, especially whenaiming at quite general situations of fluid–structure coupled problems, a loose coupling proce-dure is adopted in this study. A survey on the application of partitioned analysis to coupled sys-tems is given in Felippa et al.10.

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Loose coupling:

Mfu. � Nf(u)u � Gfp � Ff

Computational Fluid Dynamics – CFD:

Computational Mesh Dynamics (ALE) – CMD:

Computational Structural Dynamics – CSD:

Kmq � Fm

Msd..� �Csd

.��Ns(d) � Fs

Strong coupling:

CSDCMDCFD

CSDCMDCFD

Figure 2: Overview of three–field coupled problem and principle coupling strategies

The overall partitioned computational procedure, consisting of the implicit solvers describedin the previous chapters, used in this study can roughly be described through the followingscheme:

1. Initialize CSD, CMD and CFD solvers2. Begin global time loop3. Global time step tn � tn+14. CSD – advance structural system n � n+15. TRANSFER structural displacements at fluid–structure interface as kinematic bound-

ary conditions to CMD solver6. CMD – elastostatic solution for mesh movement within timestep7. TRANSFER mesh deformations and mesh velocities to CFD solver8. CFD – advance fluid system n � n+19. TRANSFER pressure (and viscous stresses) at fluid–structure interface as external

(fluid) loads to CSD solver10. Set n = n+1 and go back to step 311. End global time loop

It could appear in many situations that the involved time scales for the structural and the fluidsystem and/or the optimal time step sizes for the CSD and CFD solvers are quite different. In thesecases one solver step of the smaller time step system in the above algorithm is replaced with alocal time loop of the respective solver, in order to advance the subcycled system one global timestep. In most of these cases fluid flow requires the smaller temporal resolution. Iteration (untilconvergence is achieved) within each time step, i.e. steps 4 to 9 in the above algorithm, turns thisstaggered approach over to a kind of full or strong coupling procedure. Recent presentations ofa variety of staggered algorithms along with discussions with respect to accuracy, stability, sub-cycling and parallel processing for aeroelastic problems can be found in Farhat8 and Farhat etal.9.

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In many cases load transfer at the interface can be restricted to pressure forces only and viscousforces may be neglected. Nevertheless also in these examples viscosity plays an important rolein so far that the overall flow characteristics are governed and certain boundary conditions areenabled.

7 NUMERICAL EXAMPLES

In order to demonstrate the performance of the overall computational procedure two examplesare given. The first example is presented for verification purposes for the moving domain fluidalgorithm. The second example demonstrates the ability of the scheme to deal not only with com-plex flow phenomena but also with large structural deformations.

7.1 Channel flow with moving indentation

To verify the presented procedure for flow simulations on time dependent domains calcula-tions for a channel flow with a moving indentation are performed. There has been great interestin such flow examples, e.g. for studying the early stages of atherosclerosis, and therefore exper-imental as well as numerical results are available (see e.g. Pedley and Stephanoff23, Ralph andPedley24, Demirdzic and Peric6).

y

x

b

h(t)

hmax� � b

x1

x3x2 � 0.5(x1 � x3)

l ol i

Figure 3: Geometry of fluid domain (not to scale)

The geometry of the fluid domain (see Fig. 3) and the analytical functions approximating thechannel shape used in the experiments

(26)y(x) � 0.5h �1–tanh�ax–x2��

0

h for 0 � x � x1

for x1 � x � x3

for x � x3

are taken from Pedley and Stephanoff23, with x1 � 4.0 b and x3 � 6.5 b. The channel heightb is chosen to be 1.0 cm and the distances from the center of the symmetrical indentation(x � 0.0) to the inflow and outflow boundaries are l i � 8.0 cm and l o � 18.0 cm, respectively.The movement of the indentation develops as a harmonic function in time

(27)h(t) � 0.5 � 1� cos2�t^��; t^ �t � t0

T

12


with T denoting the oscillation period. As governing, dimensionless parameters for the flow phe-nomena the Reynolds and Strouhal numbers are defined as

(28)Re�u0 b� ; St� b

u0 T

based on the bulk velocity u0 = 1.0 cm/s.At the channel walls no slip boundary conditions were specified. All calculations started from

fully developed channel flow at t � t0. The computational domain has been discretized with 18x 208 x 2 stabilized P1P1 elements. The time increment �t was approximately T/175. As meshupdate strategy a simple linear interpolation scheme was employed (see Fig. 4).

Figure 4: Snapshots of moving mesh (zoom around downstream end of indentation)

In the experiments23 two different flow regimes were observed depending on the Strouhalnumber being less or greater than a critical value of St� 0.005, independent of the Reynoldsnumber in the observed range 360� Re� 1260. Two calculations are presented here, one foreach flow regime, both with 38% maximum indentation (i.e. � � 0.38) to study if the algorithmalso correctly captures these two flow phenomena.

In case the Strouhal number is less than the critical value a quasi–steady flow can be observedwith one eddy forming downstream of the indentation. It develops and decays in phase with thewall oscillations. The same simple flow pattern is observed in our numerical results (see Fig. 5).

time

t^ � 0.00

t^ � 0.23

t^ � 0.50

t^ � 0.76

t^ � 1.00

Figure 5: Velocity field ux and instantaneous streamlines for flow regime ISt� 0.0033, Re� 507, � � 0.38; (y–scaling factor = 2.0)

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If St� 0.005 the flow is no longer quasi–steady. Instead, a complex scenario of flow phenom-ena develops. During each cycle a propagating train of waves appears in the core flow down-stream of the indentation and closed eddies form in the separated flow regions beneath their crestsand above their troughs. One of the most remarkable phenomena appearing in such flows is theso called ‘eddy–doubling’, i.e. eddies break up and second corotating eddies develop in the sameseparated flow region upstream of the primary eddies. And although the flow becomes markedlydisturbed later in the cycle, all disturbances are swept downstream when the indentation is fullyretracted. This also means that flow of one cycle does not interact with waves and eddies of thenext cycle allowing us to study all mentioned flow phenomena during a numerical investigationof just one cycle.

t^ � 0.44

t^ � 0.58

t^ � 0.65

t^ � 0.72

t^ � 0.82

t^ � 1.06

t^ � 0.27

Figure 6: Velocity field ux / instantaneous streamlines (left) and pressure contours (right) for flow regime IISt� 0.038, Re� 610, � � 0.38; (y–scaling factor = 2.0)

The flow structure observed in our prediction on a rather coarse grid with large time step,shown in Fig. 6, is in good agreement with both the experimental study of Pedley and Stephan-off23 and the numerical results presented in Ralph and Pedley24 and in Demirdzic and Peric6.Checking was done with respect to the appearing flow phenomena and with respect to the timeswhen the primary and secondary vortices appear. It should be noted that both spatial and temporaldiscretization are significantly coarser as the ones used in reference 24 (along with an explicitfinite difference scheme) and also coarser as in reference 6 (applying an implicit finite volumemethod). Using a ’linearization through extrapolation in time’, instead of typically 40 iterationsper time step6, additionally increases computational efficiency of the applied procedure.

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7.2 Vortex–induced oscillations of flexible structure in the wake of a bluff body

This example has been chosen to demonstrate the ability of the procedure to deal with complexflow – flexible structure interaction problems exhibiting large deformations.

y

x

Fluid:

5.5 14.0

4.01.0

1.0 0.06 12

.0

slip–bc

slip–bc

U

� � 1.18� 10–3

� � 1.82� 10–4

U � 51.3

Figure 7: Flexible structure behind fixed, rigid bluff body (not to scale)

As sketched in Fig. 7 a slender flexible structure is fixed at the downstream end of a bluff bodyand developing vortices induce structural oscillations. Fluid properties are chosen according tothe transverse oscillating cylinder example of Nomura and Hughes21 and are given in Fig. 7.Boundary conditions were chosen as: no–slip along the body and the structure, slip boundaryconditions at the top and bottom wall and ’do nothing’ outflow boundary conditions. All com-putations were started from a stationary flow regime obtained with a totally fixed structure.

Only the mesh surrounding the structure was chosen as ALE mesh, far from the structure(closer to inflow and outflow boundary) a pure Eulerian approach was chosen. Mesh deforma-tions were obtained by employing the pseudo–structural mesh moving scheme according to sec-tion 5. Thus remeshing could be totally avoided throughout the whole simulation. TremendousCPU–time savings were obtained through reuse of the factorized mesh stiffness matrix. In bothcalculations presented in Fig. 8 the mesh stiffness matrix has been assembled and factorized onlyonce.

The fluid domain was discretized with 6340 stabilized Q1Q1 elements and the structure with20 nine–node plane stress elements. The total number of degrees of freedom was roughly 19000for the fluid domain, 8500 for the mesh and 240 for the structural field, respectively.

Figure 8 shows the response for two structural models. Displacement plots of the center andthe tip of the structure, respectively, indicate that structure 1 oscillates mainly in the first modewhereas in the second structural model also higher modes clearly contribute to the structural de-formation. The difference between these two models consists in a small change of Young’s modu-lus and a rather drastic modification of the structural density as indicated in Fig. 8, at a constantPoisson’s ratio of 0.35. Applying the same modifications of structural parameters for a linearstructural model the first eigenfrequency would be shifted from 19.1 to 3.8 or in other terms, theoscillation period associated with the first mode would be shifted from 0.33 to 1.65.

15


E � 2.0� 106

� � 2.0

Structure 2:

–1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0–1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

time time

tip

center center

tipE � 2.5� 106

� � 0.1

Structure 1:

Figure 8: Time history for vertical displacements of structural tip/centerfor two different structural models

A part of the vertical deformations time history of the second structural model is presentedin Fig. 9 along with a series of scaled plots of the actual structural deformations. In this figurealso the time instances for the flow snapshots, presented in Fig. 10, are indicated.

structural tip

0.0–1.25 1.25

3.00

3.50

4.00

4.50

Vertical displacements

structural center

time

Flow snapshot A

Flow snapshot B

Flow snapshot C

Flow snapshot DFlow snapshot E

Flow snapshot F

Figure 9: Structural deformations at various time instances (square bodies not to scale)

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The partially complex flow characteristics are indicated through a series of flow snapshots inFig. 10. The flow velocity vector plots reveal a series of vortices developing in the vicinity ofthe deformed structure. Background colors give the pressure distribution ranging from blue forlow (=negative) pressure, i.e. suction, to red for high compression. The figures are scaled withthe grey squares indicating the real size of the bluff body.

Figure 10: Flow snapshots – pressure and velocity vectors,A (t=3.44), B (t=3.64), C (t=3.78), D (t=4.02), E (t=4.08), F (t=4.18)

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8 CONCLUDING REMARKS

A computational concept for fluid–structure interaction problems, capable of dealing with vis-cous, incompressible flows and large structural deformations, has been presented. The approachis based on finite element discretization of both the fluid and the structural domain and on directtime integration schemes. The presented stabilized finite element method for incompressibleNavier–Stokes equations on time dependent domains, based on an ALE continuum mechanicaldescription, has been shown to correctly capture the complex characteristics of moving domainflow problems. The overall staggered three–field solution procedure has been described and ap-plied to an unsteady two–dimensional example.

Future work will, among others, focus on efficiency aspects, the three–dimensional extensionof the presented approach and an in–depth study of different partitioned procedures as well asthe application of various load and motion transfer concepts at the interface.

9 ACKNOWLEDGEMENTS

The present study is supported by grants of the German National Science Foundation (DFG),within the graduate collegium ’Modelling and Discretization Methods for Continua and Flows’and project B4 of the collaborative research center SFB 404 ’Multifield Problems in ContinuumMechanics’. This support is gratefully acknowledged.

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fluid structure interaction based upon a stabilied (ale) finite element method

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