fluid-structure interaction for structural design - CiteSeerX

FLUID-STRUCTURE INTERACTION STRUCTURAL DESIGN FOR Trond Kvamsdal1, Carl B. Jenssen2 , Knut M. Okstad3 and Jrn Amundsen4 1 SINTEF

Applied Mathematics, N-7034 Trondheim, Norway e-mail: [email protected]

2 SINTEF

Applied Mathematics, N-7034 Trondheim, Norway Presently at Statoil, N-7005 Trondheim, Norway e-mail: [email protected]

3 SINTEF 4 Norwegian

Applied Mathematics, N-7034 Trondheim, Norway [email protected]

University of Science and Technology, N-7034 Trondheim, Norway e-mail: [email protected]

Abstract

An important eld in computational science and engineering is the simulation of the coupled problem of uid ow interacting with a moving structure. A common notation for this is uid-structure interaction (FSI) problems, which include both aeroelasticity and hydroelasticity. Such multi-physics systems have received renewed attention as a result of the increase in length of slender structures e.g. suspension bridges and oshore risers, as well as increased simulation capabilities due to faster computers and new numerical methods. Herein we give an overview of results obtained within Fluid-Structure Interaction - Structural Design (FSI-SD) a project that is co-funded under the EC-ESPRIT IV/HPCN Simulation and Design (project number 20111). Our main ambition with the present paper is to disseminate the developed computational framework and give insight into some of the numerical challenges involved in addressing FSI-problems related to slender structures. We show that the improved simulation tool developed herein makes it possible to perform realistic numerical prediction of critical utter speed for wind induced vibrations of aerodynamic bridges.

Key words: uid-structure interaction, modular coupling, high performance computing, aeroelasticity, hydroelasticity

1 Introduction

An important eld in computational science and engineering is the simulation of the coupled problem of uid ow interacting with a moving structure. A common notation for this is uid-structure interaction (FSI) problems, which include both aeroelasticity and hydroelasticity. Such multi-physics systems have received renewed

attention as a result of the increase in length of slender structures e.g. suspension bridges and oshore risers, as well as increased simulation capabilities due to faster computers and new numerical methods. A segment of industry for which uid-structure interaction (FSI) problems are believed to be of increased importance in the near future is the oshore oil industry. This industry is facing exploration of waters down to 1500 meters on the continental shelf of Europe. The engineering concepts of oating oil production will meet new challenges. The importance of resonant dynamic response will increase. The new challenges will mainly be encountered in the design of oil and gas risers and platform mooring systems. The excitation of the dynamics of these systems arise partly from wave action and partly from viscous eects (vortex shedding). The latter eects also constitute a critical eect in limiting the motion response and structural loads. Today, there exist no commercially available uid ow simulation tool which can properly compute such eects. Another industrial problem area characterized by uid-structure interactions, is design of slender and exible structures as bridges. When the Tacoma Narrows bridge collapsed in a gale in November 1940, after barely four months of service, the aerodynamic phenomena that caused the 10 m high oscillations of the road deck and the subsequent failure of the structure were not well understood. Only afterwards did designers become aware of the eect known as vortex shedding and the periodic forces that it generates. Today, all suspension bridges are constructed bearing in mind the delicate interaction between ow eld and structure. The trends towards longer suspended free spans of up to 3000 m calls for new improved design tools facilitating more accurate response predictions. Numerical calculations using FSI will play a major role in these future designs of dynamically sensitive structures. Herein we give an overview of results obtained within Fluid-Structure Interaction - Structural Design (FSI-SD) a project that is co-funded under the EC-ESPRIT IV/HPCN Simulation and Design (project number 20111). Our main ambition with the present paper is to disseminate the developed computational framework and give insight into some of the numerical challenges involved in addressing FSI-problems related to slender structures. This paper is organized as follows: In Section 2 we give the mathematical formulation for the coupled FSI-problem, and how it can be formulated as three decoupled sub-problems. We also outline the nite element formulation of the related uid problem. In particular we address how variationally consistent interface forces may be computed in the chosen described FE-code. In Section 3 we discuss how the partitioned solution algorithm may be implemented using existing codes for the uid and structural part with a coupling module in between. The chosen two-level algorithm for mesh movement are presented. Furthermore we describe how we address very slender structures by means of the strip theory approach. Since the overall FSI-system consists of several program module running together, a graphical userinterface has been designed to control the simulation process. This FSI-manager is discussed in Section 4. In Section 5, the accuracy of the variationally consistent recovered interaction forces are addressed by studying a benchmark CFD problem with known analytical solution. Finally, in Section 6, we study a wind engineering application example (the Great Belt bridge), where important modelling considerations for such types of problems together with numerical results for utter speed computations are given.

2 Mathematical formulation

2.1 Governing equations Basic de nitions Given an open bounded region  IR2 with boundary @ . We assume that is connected and that the boundary is \smooth", i.e., that @ is Lipschitzian [1]. The corresponding unit outer boundary normal, n 2 IN, will then exist \almost everywhere" (a.e.) on @ . The restriction to 2D domains herein is done for notational simplicity, only. The presented theory may be carried over to 3D domains without any mathematical constraints. Consider a body occupying the open domain subjected to applied forces represented by the body force densities f 2 L2( ) and surface force densities t 2 L2(@ t  IN). The total forces acting on by f and t, respectively, are then de ned by

F (f ; ) := F (t; @ t ) :=

Z

Z @ t

f dV

(1)

t dA

(2)

Similarly, the total moments of f and t about a xed point x0 2 IR2 are given by, respectively M (f ; ; x0 ) := M (t; @ t ; x0) :=

Z

Z

(x ? x0 )  f dV

(3)

(x ? x0)  t dA

(4)

@ t

We will now establish the governing equations for the coupled problem of uid

ow interacting with a moving structure. A common notation for this is uidstructure interaction (FSI) problems, which include both aeroelasticity and hydroelasticity. We will start by posing the FSI-problem using conservation laws together with the governing kinematical and constitutive relations. A variational formulation of the mechanical FSI-problem will then be established, and reformulated into three coupled subproblems. The route followed here is similar to what the authors have followed before for establishing variational formulation of problems within one of the disciplines uid or structural mechanics. However, the presentation herein of the variational problem for the coupled uid-structure interaction phenomena, is inspired by the choice of notation and formalism established by Le Tallec and Mani [2]. The system under consideration is therefore assumed to occupy a moving domain

= (t), where we by 0 = (0) understand a xed reference con guration. The deformed uid part of the domain is denoted F and the deformed structural part is denoted S. Let 0F and 0S denote the reference con guration for the uid domain and the structural domain, respectively. The FSI-problem consists in determining the time evolution of the two domains

F and S, the uid ow velocity eld uF, the structural velocity eld uS and the Cauchy stress tensor within the uid, F , and within the structure S.

Our purpose is to address uid-structure interaction involving slender structures. Thus, large structural displacements have to be taken into account. For this reason it is necessary to take into account the changes of the domains F and S when evaluating integrals. Typically, we formulate the uid part of the problem by using Eulerian coordinate system, whereas for structural problems we use the Lagrangian system. For FSI problems, it is customary to introduce a mapping [2]: x :

0  IR ?! (t) (x0 ; t) ?! x(x0; t)

(5)

which maps any point x0 of the xed con guration 0 to its present position x(x0; t) in the deformed con guration (t). We may choose the reference con guration 0 arbitrarily, and since this formulation represents a mixture of Lagrangian and Eulerian coordinate description we use the term Arbitrary Lagrangian Eulerian (ALE) formulation. We may think of this as an Eulerian description with moving coordinate system, and the de nition of the so-called grid velocity ug reads: (6) ug = @ x j

@t

x0

The governing equations for the present mechanical system may be characterized by three conservation laws posed on the current deformation of the domain

(and similarly for current deformation of any subdomain D 2 ) together with kinematical compability and constitutive relations. Conservation of mass Given the mass density  de ned throughout the domain in its deformed con guration , conservation of mass on any sub-domain D  reads

@ Z  dV + Z (u ? ug )
n dA = 0 @t D @D

(7)

By using the following relationship between volume and surface integral of an vector valued function v Z

r
v dV

@D

=

Z

v
n dA

(8)

D

we may alternatively write the conservation of the mass on any sub-domain D 

@ Z  dV + Z @t D D

r
[(u ? u )] dV g

= 0

(9)

Conservation of momentum Assume the existence of a vector eld t 2 L2 (  IN) such that: For any sub-domain D of where the unit outer normal vector n exists a.e. on @D \ @ t (10) F (t; @D \ @ t ) = F (t; @D \ @ t )

Force balance: For any sub-domain D of

F (f ; D) + F (t; @D) =

@ Z u dV + Z  (u
r)(u ? ug ) dV @t D D

(11)

Moment balance: For any sub-domain D of and for any xed point x0 2 IR2 M (f ; D; x0 ) + M (t; @D; x0)

Z Z @ = @t (x ? x0)  u dV (x ? x0)  [ (u
r)(u ? ug )] dV (12) D D

The axioms of force and moment balance are also referred to as the conservation of linear and angular momentum, respectively. These relations are nothing but integral forms of Newton's second law which for a mass particle expresses proportionality between applied forces and the resulting acceleration. Furthermore, notice that no references to the material properties for the uid or structure has been introduced so far. Kinematical compatibility We assume here that the uid is viscous and that the interface between the uid and the structure is a so-called no-slip boundary. This means that we have the following kinematical compatibility condition along the uid-structure interface ? = F \ S: uFj? = uSj? (13) The restriction operator \j?" is to be understood as the trace operator, i.e. the kinematical restriction of a velocity eld de ned either in F or S to the interface ?. In general, we may have that the velocity for the uid and/or the structure is prescribed along parts of the boundary. The corresponding kinematical compatibility conditions reads: (14) uF = uF on @ F;u uS = uS on @ S;u (15) where uF and uS is the prescribed velocity for the uid ow along @ F;u and the movement of the structure along @ S;u, respectively. In order for the mapping introduced in Equation (5) to be continuous the deformation of the uid part and the structural part have to be equal along the interface: x F j ?0 = x S j ?0 (16) Regarding choice of coordinate system we choose the Lagrangian description for the structural part. Thus, we always have that the grid velocity ugS equals the physical velocity uS for the material particles within the structure. However, for the uid part of the domain we use the ALE-formulation. The kinematical constraints on the mapping for the uid part reads [2]: xF = Ext (xF j?0 ) (17) @ xF j = u j (18) F ? @t ?0

where Ext (xF j?0 ) is to be understood as any reasonable extension of the material deformation along the interface ? to the domain F. The relation between the uid velocity uF and the strain rate tensor "F is assumed to be given as  1 (19) "F = r
uF + (r
uF )T 2 The kinematical relation for the structural part is given by the relation between the deformation gradient F S and the Green strain tensor E S  1 (20) E S = F TS
F S ? I 2 Constitutive relation The constitutive relation for the uid, i.e., the relation between the Cauchy stresses F and the strain rates "F is assumed to be linear. For the case with zero strain rates the stress tensor F is assumed to comply with the hydrostatic pressure condition. Hence, the stress tensor reads: F = ?pI + 2"F

(21)

where I is the identity tensor, and  is the viscosity parameter. We assume an isotropic and hyperelastic material response of the structure. Thus, the constitutive relation between the Green strain tensor and the second Piola Kircho stress tensor reads: SS = C S : ES (22) where C S is the fourth order constitutive tensor for the structural material. 2.2 Variational formulation To establish the variational formulation of the mechanical problem de ned above, we introduce the following spaces: n

o

U (D) = v : D ! IR2 ; v 2 H 1(D) : v = v a.e. on @Du n o V (D) = v : D ! IR2 ; v 2 H 1(D) : v = 0 a.e. on @Du Q(D) = fq : D ! IR; q 2 L2(D)g

(23) (24) (25)

where v is assumed to be a prescribed vector eld (herein the prescribed eld may either be position or velocity) along the part @Du of the outer boundary to the domain D of interest. Given a partition of the open domain in two non-overlapping subdomains F and S such that = F [ S. Assume that all the domains have Lipschitzian boundaries. The uid-structure interface is then given as: ? = F \ S and assume that this surface is of nonzero Lebegues measure. Let the domain be subjected to applied forces represented by the body force densities f 2 L2 ( ) and surface force densities t 2 L2 (@ t ). The variational formulation of the uid-structure interaction problem can be written [2] Find the structural deformation xS 2 U ( S), the uid density F in deformed con guration, the uid pressure p 2 Q( F ), the uid velocity uF 2 U ( F ), the Lagrange multiplier  2 W (?), and the uid con guration mapping xF 2 U ( F ).

such that for any fuF; pg 2 V ( F)  Q( F ) and any uS 2 V ( S) the following is satis ed: @ Z 
q dV + Z r
[ (u ? ug )]
q dV = 0 (26) F F F F

@t F

F

@ Z  u
u dV + Z [ (u
r)(u ? ug )]
u dV + Z  : " dV F F F F F F F @t F F F F

F

F + = +

Z

0S

Z

F

Z ?

Z

0 x S
uS dV + S S : E S dV f
uF dV +

Z

0S

@ F;t

Z

Z

S

@ S;t

t
uF dA + f
uS dV +


(uFj? ? uSj?) dA

uFj? = uSj? @ xF j = u j xF = Ext (xFj?0 ) ; ?0 F ?

@t

t
uS dA

(27)

(28) (29)

Here,  2 W (?) = [H 1=2(?)]0 is the Lagrange multiplier related to the kinematical constraint ensuring continuity (in weak sense) in the test function over the

uid-structure interface. Note that [H 1=2 (@D)]0 is the dual space to the space corresponding to the traces on @D of functions in H 1(D). Thus, the last integral in Equation (27) makes sense. The Lagrange multipliers  corresponds to surface tractions, or contact force densities along the uid-structure interface ?. Equation (26) corresponds to the conservation of mass in Equation (7), and Equation (27) corresponds to the conservation of linear and angular momentum speci ed in (11) and (12). Note that the angular momentum is ful lled through the use of symmetric stress tensor. Equations (28) and (29) ensure ful lment (in strong form) of the kinematical compatibility constraints on the uid-structure interface and for the uid con guration map, respectively. 2.3 Variational formulation of coupled subproblems Although the above variational formulation is an very compact mathematical representation of the FSI-problem it inherits certain disadvantages when it comes to computational implementation. Compared to traditional uid and structural problems we have additional unknowns (the Lagrange multiplier ) together with kinematical constraints on the trial functions along ?. However, the above variational formulation of the FSI-problem may be reformulated into three coupled subproblems.

Fluid problem Let uS = 0, and de ne the uid test velocity space VF by:

VF( F ) = fV ( F ) : v j? = 0g

(30)

We then obtain a Navier-Stokes problem written in ALE form on the moving domain

F(t). Along the interface ? we have imposed the Dirichlet boundary conditions uF = uS speci ed by the structural problem. The solution uF to this Navier-Stokes problem corresponds to a force acting on the interface ? de ned through: F (; ?; uj?) =

Z ?


uFj? dA

Z Z @ = @t F uF
uF dV + [F(uF
r)(uF ? ugF)]
uF dV (31)

F

F

+

Z

F

Z

F : "F dV ? f
uF dV ?

F

Z

@ F;t

n?

t
uF dA

Structural problem Let uF = 0, and impose the traction force densities  speci ed by the uid problem, then we obtain the following structural problem: Find the structural deformation xS 2 U ( S ) such that for any uS 2 V ( S ) the following is satis ed: Z

0S

Z

Z

0S

S

0 x S
uS dV + S S : E S dV = +

Z ?

f
uS dV +

Z

@ S;t

t
uS dA (32)


uSj? dA

Mesh movement In order to ful l the kinematical compatibility constraints on the uid-structure interface and for the uid con guration map, we have to solve the following equations: @ xF j = u j x = Ext (x j ) ; (33) F

F ?0

@t

?0

F ?

This may be done either by introducing an explicit extension map (e.g. a prede ned blending function) or by solving an equation system (e.g. an elliptic problem). Note that the coupling to the other two problems are via the condition relating the

uid grid velocity to the value of the physical uid velocity which by compatibility constraints is speci ed by the physical structural velocity. 2.4 Finite element approximation The NAVSIM solver The nite element (FE) method is a numerical method for approximate solution of the variational formulation of boundary value problems. Let M be a partition of

into a collection of nel = nel (M) sub-domains, called elements, e, with boundaries @ e , 1  e  nel such that

1. nel(M) < 1

el ; \ = ; for k 6= l 2. = Sne=1 e k l 3. e are Lipschitzian with piecewise smooth boundaries @ e 4. ?kl = @ k \ @ l ; 1  k; l  nel 5. We set ?0l = @ \ @ l To compute the FE-solution In the present study we use triangular elements with continuous linear interpolation (P1(x) = a0 + a1x + a2 y) of both the velocity and the pressure eld. The FE basis is denoted

n

X h = vh 2 C 0( ) : veh = vhj 2 P1( e) e

o

(34)

The nite dimensional counterpart to the trial- and test spaces de ned above are now given through

U h( F ) = X h( F ) \ U ( F ) V h( F ) = X h( F ) \ V ( F ) Qh( F ) = X h( F ) \ Q( F )

(35) (36) (37)

The FE approximation to the variational form of the uid problem stated above is given by substituting u; u; p; p; U; V; Q with uh; uh; ph; ph; U h; V h; Qh, respectively. The superscript h denotes FE-quantities that are computed with a mesh with characteristic element size equal to h. To compute the FE-solution fuh; phg we use the ow solver NAVSIM developed by Herfjord (1996). The Navier-Stokes equations are in NAVSIM solved by the fractional step method attributed to Chorin [3], who called it a split operator technique. In the implementation of this method, the so-called balancing tensor viscosity [4, 5] is used, and to avoid the need for solving an equation in Step 1 and 3 (see below) mass lumping is applied. The three steps in the fractional step method, which are described in more detail in [6], are: Step 1 The velocity is advanced in time via a momentum step. Step 2 The pressure eld is computed from a Poisson type of problem. Step 3 The velocity is corrected to comply with the computed pressure eld.

The NAVSIM ow solver is using three-noded triangular elements with linear interpolation for both the velocity and the pressure eld. According to the classical compatibility conditions (the Ladyshenskaya-Babuska-Brezzi condition) the use of equal order interpolation may suer from severe spatial oscillatory behaviour in the computed pressure eld ph. However, recent research has shown that the use of the projection technique described above leads to satisfactory results provided that the time step is large enough, with respect to the element size. Computation of interface forces To ensure conservation of forces we use the concept of Variationally Consistent Post-processing (VCP) as described in [7] to compute the total forces and moments acting on the structure from the ow eld at each CFD-plane. The importance of conservative load transfer in uid-structure interaction problems has recently been

addressed by Farhatet al. [8]. Following the notation in [7] we establish the following variational equation at the time step n: Z Z Z uh ? uh (38) teln+1
w dA =  n+1 n
w dV + (uhn
r)(uhn ? ugn)
w dV
t D ? D +

Z

(u ; p h n

h n+1

Z

Z

D

@D

) : "(w) dV ? f n+1
w dV ?

D

n?

teln+1
w dA

The `Finite element boundary tractions' tel is a vector eld de ned on all element boundaries such that both internal and external equilibrium is satis ed. Thus, on an internal element boundary shared between element k and l we have telk + tell = 0. Whereas on the exterior boundary @ t we have tel := t. The velocity eld w is called the `extraction function' and is de ned such that it is zero outside the open subdomain D  . The term @D n ? in the last integral denotes the part of the outer-boundary of D P that is not covered by the surface ?. We now let wk = I 2
? NI (x)ek de ne an extraction function for the kth force component, where
? = fI 2 f1; : : : ; nndg : xI 2 N (M) \ ?g (nnd denotes the total number of nodal points in the mesh M, and N (M) is the corresponding nodal set). Then wk (x) jx2?  ik where k 2 f1; 2g, i1 = [1; 0]T and i2 = [0; 1]T . If we insert wk in Equation (38) the left hand side equals the de nition of the surface force along ?, viz. Z Fk (telk ; ?) := telk dA (39) ?

P

Now, let w = I 2
? NI (x)[?(yI ? y0); (xI ? x0 )]T then w (x) jx2?= [?(yI ? y0); (xI ? x0 )]T and if w is inserted in Equation (38) the left hand side equals the de nition of the moment due to tractions along ?, viz.

M

Z

; x0 ) := [?(y ? y0); (x ? x0)] tel dA

(tel; ?

?

(40)

The quantities computed by means of Equation (39) and Equation (40) are denoted fF1vc; F2vc; M vc g, and are applied to the structure as line force densities at the beam nodes. Note that in order to determine the uid forces acting on the uid-structure interface ?, it is not usually necessary to explicitly calculate the `Finite element boundary tractions' tel. As long as the last integral in Equation (38) is zero we only have to sum the appropriate \nodal forces" for the uid nodes lying on ?. E.g. for cylinders the uid-structure interface ? is an circle (or ellipse). We then have @D n ? = ;, i.e. the last integral is identical to zero. As the nodal forces are computed in the nite element assembly procedure, the use of VCP is very cost eective. The alternative to the above recovery scheme is to use the `kinematical consistent FE tractions' th = (uhF;n; phn+1)
n instead of tel. However, the equality in Equation (38) is not ful lled if we substitute tel with th , i.e. conservation of linear and angular momentum is not guaranteed. Kvamsdal et al. [9] have shown (see also Section ?? herein) that we may expect superior accuracy of the variationally consistent recovered forces compared to those obtained by integration of the th eld. Furthermore, the use of variationally consistent post-processing makes it possible to use `duality' techniques to estimate the spatial discretization error in computed

forces and moments [7, 9]. The contribution from each element to the estimated error is computed, and this information enable the use of adaptive mesh re nement as developed in [7]. Thus, we may obtain a prescribed accuracy in the interface forces with a minimum number of elements.

3 Computational approach

3.1 Architecture of FSI-system The developed FSI-system is based on a \staggered solution" [10, 11] of the three coupled subproblems given in Section 2.3 above. The main principle for a staggered procedure is to solve for the variables of one equation sub-system while keeping the variables of the other sub-system `frozen'. The frozen variables are then applied as boundary conditions or loads on the active equation sub-system. Such a procedure enables us to use existing codes for the two sub-systems (the CFD and CSD equations) with only minor adjustments to each code. The main programming eort in developing the FSI-system is thus related to the third code entering the system; the coupling module. The FSI-system is designed so that each of the three dierent codes can run on completely separate computers, and communicate using Parallel Virtual Machine (PVM) message passing. Typically, the CFD-code would run on a large number of processors on a parallel machine, while the CSD-code would run on a workstation or a single node of the parallel machine. The coupler, which contains a user interface, is designed to run on the user's own workstation. A schematic view of the communication between the dierent codes is shown in Figure 1. As we can see, all communication goes through the coupler. The coupler thus has the task of restricting the uid forces acting on the surface of the structure as discretized by the CFD mesh, to the corresponding nodes connecting the beam elements in the structure model. This corresponds to provide the CSD-code with the necessary Neumann conditions along the uid-structure interface ? for solving the structural problem de ned in Equation (32). Also, the coupler receives the displacements of the structure nodes and computes new mesh coordinates which are passed on to the CFD-code. Thus, the necessary Dirichlet conditions along ? for solving the uid Coupler nodal forces

CFD

new mesh coordinates

ΣF

move mesh

restricted forces Structure FE code

element displacements

Figure 1: Overview of the FSI system

problem stated in Equation (equ:FluidProblem), and the corresponding extension of the displacements along the uid-structure interface (see Equation 33)) needed in the ALE formulation are provided for the CFD-code, respectively. As described in the next section (see also [12]), to ensure computational eciency and parallel scalability, only a small part of mesh movement computations are carried out by the coupler, while most of the computations are carried out within the CFDcode. As the structure is assumed to be discretized with 1D beam elements, contact interface algorithms have been developed that are capable of (a) transforming surface tractions obtained from the CFD-code to variationally equivalent beam nodal forces and (b) determining the movements of the the contact interface corresponding to the computed beam nodal displacements [11]. 3.2 Staggering procedures The conventional staggering procedure commonly used in FSI-simulations is illustrated in Figure 2. It is based on the following steps:

1. Update the uid grid to conform with the structural boundary at time t(n) . 2. Advance the ow using the new boundary conditions. 3. Update the surface load on the structure based on the uid solution at t(n+1) . 4. Advance the structure using the new uid surface load. Due to its simplicity, this procedure is very popular and has been widely used for aeroelastic computations. For a problem constructed from the linearization of the

uid{structure equation around an equilibrium point, a rst order accurate explicit forward Euler scheme for the uid and a second order accurate implicit trapezoidal scheme for the structure, this procedure is rst order in time. Alternatively, we may use an inter- eld staggering approach as illustrated in Figure 3.  The uid grid is updated to conform to the structural boundary at time t(n) and the uid is advanced using the structural boundary conditions at time t(n) .  The structure is advanced using the uid surface load at t(n) Fluid Structure

W (n)

6

2.

W-(n+1)

W-(n+2)

6     3.  1.     = 4. -= - 

x(n) , u(n)

x(n+1) , u(n+1) x(n+2) , u(n+2)

Figure 2: The conventional staggering procedure

Fluid Structure

W (n)

6

W-(n+1)

?

-?

x(n) ,

u(n)

6

x(n+1) ,

u(n+1)

W-(n+2)



-

x(n+2) ,

u(n+2)

Figure 3: The basic inter- eld procedure With this procedure, the CFD and CSD solvers can run in parallel during the time interval [t(n) ; t(n+1) ]. Inter- eld communication and I/O transfer is needed only at the beginning of each time interval. However, when compared with conventional staggering procedures which are inherently sequential, parallelism is achieved at the expense of larger errors in the uid and structure responses. The process of advancing the solution from t(n) to t(n+1) consists normally of a set of non-linear iteration steps. A means of increasing the accuracy of the overall procedure, is then to exchange information between the CFD and CSD code between each iteration, and not only in the beginning of the time step. This will yield a stronger coupling between the uid and structure equations and corresponds to a non-overlapping alternating Schwarz iteration procedure. Since the main bulk of the computations for the FSI-problem addressed herein is consumed within the CFD solver, the additional communications will not increase the overall simulation cost signi cantly, provided the number of time steps and non-linear CFD-iterations remains unchanged. 3.3 Mesh movement strategy In coupled analysis of FSI-problems the displacements of the structure and the use of Arbitrary Lagrangian Euler (ALE) formulations for discretization of the uid domain raises the need for moving the nodes in the uid domain, see Section 2. In case of ow around slender structures (e.g., risers and pipelines) with large length to width ratio, large displacement of the structure can occur. This implies also large displacements of the uid nodes in the vicinity of the structure. Hence, the choice of algorithm for the mesh movements is important for the accuracy of the numerical simulations. Due to the dierence in size and dimensionality of the 3D CFD problem on one side, and the essentially 1D structural problem on the other side, the main bulk of computations are carried out in the CFD code. The parallel eciency of the coupled codes thus rest on the parallel performance of the CFD code, provided that the additional mesh movement and communication through the coupler can be carried out in an ecient manner. The problem of moving the mesh to conform with one or more moving bodies as well as a number of non-moving boundaries is global in nature. It is a problem of book-keeping to keep track of several moving bodies and their relative location, but also a mesh generation problem to create a smoothly moving mesh with as little distortion as possible. The global nature of the problem calls for some sort of elliptic

procedure as commonly used in mesh generation. Both the elliptic problem and the book-keeping challenge makes it tempting to implement the whole mesh movement procedure as single processor code. It is however clear that a system architecture as shown in Figure 1, would not be feasible if at each iteration, all mesh coordinates were to be updated by the coupler and then passed to the CFD code. If the 3D CFD problem has size N 3 , such an approach would require the order of N 3 operations on a single processor as well as having to send 3N 3 values over the network. Since in the CFD code, the computational cost and communication costs are of the order N 3 and N 2 respectively, the coupler would then create a scalar bottleneck and an increase of the communication cost of one order of magnitude. On the other hand, embedding the mesh movement algorithm completely within the parallel CFD code would eliminate both the communication overhead and scalar bottleneck, but would instead introduce complicated book-keeping and require the implementation of an elliptic solver within the CFD code. Our solution to this problem has been to split the mesh movement in two levels; a global and a local level. The method is linked to the use of structured multi-block mesh generation and thus well-suited for parallel computation. A main aspect of multi-block mesh generation techniques is the sub-division of the domain into blocks topologically equivalent to cubes. For problems relevant for FSI-computations the number of blocks are usually a few hundred or less, whereas the number of elements and grid points may be as large as millions. In the global level of the mesh-movement problem an elliptic problem for the location of the block vertices is solved, whereas independent problems for the location of the mesh nodes inside each block are solved afterwards on the local level. Thus, the coupler only has to solve a very coarse elliptic problem, and the communication cost is proportional to the number of blocks, and not mesh points. Parallelisation of the local problem within each block is trivial, as the CFD code is already parallelized over the blocks. The principle of our approach is shown in Figure 4. An elliptic problem where the computational blocks are considered as solid elastic blocks are rst solved by the coupler and the updated coordinates of the block corners are passed to the CFD code. Following [13], the blocks of the multi-block mesh can be considered as linear elastic bodies, characterized by a Young modulus, E , and a Poisson ratio,  and each block is regarded as bi- (or tri-) linear isoparametric element, ei . The mesh movement problem then becomes an elliptic problem with only Dirichlet boundary conditions, and it can be solved using any linear solver. This method has the additional advantage that the Young's modulus can vary form block to block, making it possible to keep blocks in the vicinity of the structure almost unchanged in shape, while most of the distortion is taken up by more distant blocks. Once the global mesh movement problem is solved for the corners of the blocks, the updated values are sent to the CFD code, where Trans Finite Interpolation (TFI) is used to update the interior coordinates of the blocks. Note that we are using TFI on the displacement, or change between each iteration, of the mesh coordinates. Thus, the overall mesh quality remains as in the initial grid, and if the corner nodes of the block remain unchanged, the mesh within the block also stays unchanged. This ensures that the mesh retain the properties such as clustering and orthogonality as imposed by the original mesh generation and do not inherit properties of the TFI mesh generation. The present mesh-movement algorithm may also be combined

Elliptic FEM Host Proc.

TFI

TFI

Proc. 1

Proc. 2

TFI

Proc. 3

TFI

Proc. 4

Figure 4: Mesh movement strategy. with mesh adaption by means of adaptive relocation of nodes, see [12]. 3.4 Strip theory approach Among the objectives for the present development of computational methods for

uid-structure interaction has been to make reliable numerical simulation of vortex induced vibrations (VIV) of full-length oshore risers feasible. In order to achieve this, it has been necessary to simplify the problem by doing the computations in a number of 2D-planes normal to the riser, i.e. a strip theory approach. We present below a partitioned analysis algorithm for the strip theory approach that relies on the so-called conventional staggered solution procedure described above. In our FSI-program we have implemented this approach using the the CFDcode NAVSIM [14] and the CSD program USFOS [15]. They are both linked to the FSI-coupling module, which is administrating the communication and controls all processes and processors involved. The communication is based on the PVM (Parallel Virtual Machine) programming library. After the FSI-coupler has initiated the FSI-simulation the procedure for each time step is as follow (for a case where the number of CFD-planes is equal to np):

1. The ow is advanced one time increment in NAVSIM by performing Step 1 to Step 3 (see Section 2.4) for each of the CFD-planes nF;i subject to the updated boundary conditions along the uid-structure interface: unF+1 = ugF;n on @ nF;i \ @ nS 8i 2 f1; npg

(41)

2. Variationally consistent post-processing (see Equation (38)) is performed in NAVSIM for every CFD-plane nF;i to compute the surface forces and the surface moment [F1vc;n+1 ; F2vc;n+1; Mnvc+1]Ti on the uid-structure interface @ nF;i \ @ nF .

3. Linear interpolation of the uid load between each of the CFD-planes is performed in the FSI-Coupler, i.e. for i = 1 to i = np ? 1 do

qS +1( ) = n

i

(i)[Fnvc+1; Fnvc+1; Mnvc+1]Ti + (1 ? i)[Fnvc+1; Fnvc+1; Mnvc+1]Ti+1

(42)

where i = 1:0 at the CFD-plane i, i = 0:0 at the CFD-plane i + 1, and i varies linearly along the beam axis between the CFD-planes i and i + 1. 4. The new structural position xnS +1, that ful l Equation (32) with the updated forces qnS +1 as the loading along the `wet surface' @ nS;t , is computed in USFOS. 5. A rigid body translation of the uid mesh to restore the compatibility between the CFD-planes (in updated positions) nF;i+1 and the structural domain nF+1 is performed in the FSI-Coupler. Furthermore, the uid grid velocity is updated as follows: @ v (i = 1:0) on n+1 8i 2 f1; npg ug;n+1 = (43) F

@t

F;i

The validity and stability of the above staggering procedure depends on the type of system considered, i.e. the degree of non-linearities and how they are captured. In our case, the time step is determined in order to capture the variation in the ow. The non-linearities in the riser, like variation of the stiness due to the variation of the shape, are captured well by the same time step as the CFD computations, see [16] for numerical investigations. The CSD computation performed in USFOS is very fast relative to the time needed by NAVSIM in order to do the ow calculation. Time consumption needed for communication between the two codes and the FSI-coupler does not in uence the speed of the computation to any large degree. This is due to the fact that only three load components are communicated from each of the CFD-planes to USFOS via the FSI-Coupler. Similarly only three displacements and three velocity components are returned from USFOS via the FSI-coupler to each of the CFD-planes.

4 The simulation process

4.1 Introduction When conducting a numerical FSI-simulation, the end user will often have to deal with a range of computer based tools, ranging from a set of pre-processing tools to construct the computational grid, de ning boundary conditions and setting up initial conditions. The simulation itself may also be carried out using several tools, as is the case with the FSI-simulation system discussed in Section 3.1, which is an integration of two (or more) simulation codes originally written to work as standalone programs. Finally, the post-processing may involve yet another set of tools. Often, a signi cant overhead is imposed on the end user of such a system. A large fraction of this overhead is related to keeping track of les that are used by the various tools. This involves both the le ow itself, i.e., how les produced by one tool are passed as input to other tools, and the time dependency between les, i.e., whether updating one le requires updating other les. The act of invoking a tool may also contribute to the administrative overhead during a simulation session.

Dierent tools may use dierent conventions for providing the input; some tools read an input le that has to be set up by the user, whereas other tools take their input as command line arguments, or there may be a combination. To reduce this overhead imposed on the user, we created the FSI-manager. It is a Tcl/Tk-based graphical user interface (GUI) from where the various activities involved in the modelling and simulation of a FSI problem may be launched. 4.2 Execution control The main window of the FSI-manager is the Execution Control display shown in Figure 5. It contains a matrix of information directly related to the execution of the various activities involved in the simulation of a FSI-problem. Each activity is represented by one row of information and a separate launch push-button. From this display the user can select the host computer and execution directory for each activity, and may also alter the command line arguments when needed. The display also contains indicators telling whether an activity need to be re-executed due to that an input le is more recent than last execution of that activity. Input- and output les of activities that are speci ed as command-line arguments1 are highlighted in the Command column of the execution control display. These are all ASCII les that the user may alter { or just inspect { the contents of, just by pressing the mouse-button on the highlighted le name. Some of these les are typical log- les containing information about the solution process. Such a le is opened in a read-only viewer. Files that are typical input les are instead opened in a `helper' which may be regarded as a high-level special-purpose editor. The helpers enable the user to alter some input parameters to a program without running any risk for destroying the syntactical correctness of the input le when the user is unfamiliar with the syntax. Two examples of such helpers are shown in Figure 6.

Figure 5: The FSI-manager execution control display.

4.3 File ow The second major window of the FSI-manager is the File Flow display shown in Figure 7. It gives a graphical representation of the les owing in and out from the various activities. The activities and les, which both are represented with xed logical names, are connected with directed arrows to indicate input and output relationships. The le ow display also shows the last modi cation time for each le as well as the physical name of the le. The physical le name may be changed by the user, either through a le browser which is attached to each le, or by directly editing the le name in the Name column. The le ow window contains also some les that are not present in the execution control window. These are typically les that pass information from one activity to another and that the user usually not have to inspect or modify directly. The CBJ model and result les fall into this category. The advantage of the le ow window is that the user has to change the name of a particular le in only one place, regardless on how many of the activities that use that le. The FSI-manager then updates the aected launch commands and/or input les automatically. It also veri es that the new le actually exists, and if it does; noti es if activities have to be re-executed. 4.4 Pre-processing activities The rst activity of the execution control display (see Figure 5) launches the preprocessor CAD x [17] for geometry de nition and grid generation. A specialized GUI for FSI-applications has been made on top of CAD x featuring a set of geometry templates for rapid generation of typical wind- and oshore engineering problems (see Figure 8). The FSI GUI also contains translator utilities that creates the necessary model input les for the simulation codes. However, the translators may also be launched directly from the FSI-manager. Before launching a parallel FSI-simulation using the CBJ solver, it is usually necessary to re-partition the user-de ned grid blocks from the preprocessor into a number of computational blocks. This operation is performed by the split block activity in the FSI-manager. The optimal number of computational blocks depends The names given here are usually xed logical names. The associated user-de ned, physical le names are de ned in the File Flow display (see section 4.3). 1

a) b) Figure 6: Input le helpers: a) For the split-block utility. b) For the CBJ solver.

Figure 7: The FSI-manager le ow display. on the number of processors available as it is bene cial to have the same amount of blocks (assuming they are approximately of equal size) on each processor for optimal load balancing. 4.5 Simulation activities There are three dierent simulation activities that may be launched from the FSImanager, depending on which simulation codes are involved. First, we have the pure CFD-simulation which only involve the CBJ solver. When starting on a new model, it is usually necessary to rst run a pure CFD-simulation in order to generate proper starting conditions for the later FSI-simulation. The next category is a simulation with prescribed movements of the structure (in the FSI-manager, we denote this activity for ALE-simulation, see Figure 5). The structure is here considered as a rigid body that moves according to some prede ned function. Such simulations are useful for determining aerodynamic derivatives of the force coecients. Finally, we have the full FSI-simulation that involves both the CFD-solver (CBJ [18]) and the CSD-solver (USFOS [15]), in addition to the coupler module. The ALE- and FSI-simulation activities are divided into a main activity, and two and three sub-activities, respectively (see Figure 5). Each sub-activity correspond to one of the simulation codes and is characterized by the lack of an Execution push-button in the execution control display. However, the user may select the execution host and -directory as well as modify the command-line arguments for the sub-activities in the same manner as for the main activities. When the user launches

The CBJ Geometry Tool

Selecting a bridge section Choose a bridge section from the library of sections • Define the parameters for the chosen section • Press “OK” to define the section in the geometry tool. • If new section definitions are added to the section library then press “Update” to update the list of available sections shown.

Figure 8: The FSI graphical user interface. the main simulation activity, the main activity in turn starts the sub-activities on the speci ed computers.

5 Numerical example involving recovery of surface forces

In order to verify the claimed superiority of the computation of interface forces based on Variationally Consistent Post-processing (VCP) we present here an numerical investigation of a non-trivial Navier-Stokes problem with known analytical solution. It is beyond our scope herein to perform an extensive investigation with the purpose to verify that the we actually may achieve a better accuracy for relevant industrial FSI-problems. However, we hope that the following example demonstrates the qualitative performance of the recovery procedure compared to traditional procedure based on the use of kinematically consistent nite element tractions (see de nitions above). 5.1 The Jeery{Hamel ow The Jeery{Hamel ow is the exact solution to the incompressible Navier{Stokes equations on a wedge-shaped domain with a source or sink at the origin, see Figure 9. The analytical solution to this problem is derived by White [19] (pp. 184{189) which gives the radial velocity and pressure as, respectively

ur = Ur0 f () ;  =  !2    U 0 p = p0 ? 2 r=r f 2  Re1  f 00 0

(44) (45)

where U0 is the radial velocity at a given point (r; ) = (r0; 0),  is the mass density of the uid and p0 is an arbitrary constant. The angular velocity, u is identically zero. The constant p0 is here selected such that p = 0 at the point (r; ) = (r0; 0).

y Line source or sink

r α

θ

x = 1.0

x = 2.0

x

Figure 9: The Jeery{Hamel ow problem. The Reynolds number Re for this problem is Re = U0r0 =. The + sign in front of the last term in (45) is for ow in the positive x-direction and the ? sign for ow in the negative x-direction. The function f () is determined through the following equation

f 000 + 2 Re  ff 0 + 4 2f 0 = 0 (46) with boundary conditions f (0) = 1, f (1) = 0 and f 0(0) = 0. This third order non-

linear equation is solved numerically using MATLAB with our choice of parameters, Re = 10 and  = =4. The Cartesian components of the velocity eld are ux = ur cos  and uy = ur sin , respectively. However, since x = r cos  and y = r sin , we may write the velocity eld in terms of Cartesian coordinates as u(x; y) =

(

ux uy

)

( )

= Ur20 f () xy

(47)

The gradient of u in Cartesian coordinates is now obtained through a partial dierentiation using the substitution of variables r2 = x2 + y2 and  =  tan?1 xy . This results in # # " " 2 2 2 ?2xy U U ? xy x y ? x 0 0 0 ru =  r4 f () ?y2 xy + r4 f () ?2xy x2 ? y2 (48) 5.2 Numerical results In the present study, the quadrilateral region indicated in Figure 9 is analysed with the ow in positive x-direction. The analyses are performed with the FE code NAVSIM [14] which uses linear triangular elements. A series of four uniform meshes are used in this numerical study, with 11  11, 21  21, 41  41 and 81  81 nodal

points, respectively. The largest challenge in a numerical simulation of this ow problem is to specify in ow and out ow conditions. Imposing the full exact solution at these boundaries will violate the nature of the hyperbolic part of the equations. Usually for channel ows, velocity is speci ed at the in ow while pressure is speci ed at the out ow. In the present simulations, however, the exact velocity is imposed as Dirichlet conditions along both the in ow and out ow boundaries. The pressure is then determined up to a constant. Thus, in order to get comparable solutions it is prescribed equal to zero at the point (x; y) = (1:0; 0:0). The results presented in Figure 10 (the left column) clearly indicate the increased level of accuracy for variationally consistent forces F vc (tel) := F vc (uh; ph; tel; ?) recovered along the rigid wall, compared to F (th) := F (th; ?) obtained through integration of kinematically consistent FE tractions th. In order to check the accuracy of the numerical integration scheme both the exact uxes F vc (t) := F vc (u; p; t; ?) and F (t) := F (t; ?) are computed. In the right column of Figure 10 the total net ux through @ is compared. The superior results obtained by using VCP should be noted. For this case the convergence rate for the error in the variationally consistent force F el := F vc (uh; ph; tel; ?) is O(h2) i.e., one order higher than the error in the kinematically consistent force F h := F h(th; ?) which is O(h) (h is the characteristic element size).

6 Wind engineering applications

Here we illustrate the use of the developed computational methods for solving uidstructure interaction problems such as wind induced motion of suspension bridges. Our aim is to provide some insight into the complexities of the computational models, both regarding the size of the element mesh and the CPU-times involved. Some guidelines are given, but for more detailed information about the practical use of the so-called FSI-Bridge we refer to the presentation given in [20]. 6.1 Modelling considerations The success of a FSI-simulation of a wind engineering problem depends heavily on the computational grid. The mesh should be ne enough to resolve important small-scale features of the uid ow near the structure. The shape of the elements must also be reasonably good. On the other hand, the geometry of the structure may vary from simple bridge cross sections to more complex geometries with various appendages corresponding to railings, wind vanes, etc. Due to the computational approach of the CFD solver employed herein [18], we are restricted to the use of structured meshes only. Therefore, any fully automatic mesh generator based on triangulation, which are commonly used in nite element analysis when complex geometries are involved, cannot be used here. It is therefore clear that constructing a suciently good mesh is a great challenge for the engineer conducting the analysis. After having de ned the geometry of the structure, the rst step is to construct a coarse block grid where each block element is a topological cube. As the number of blocks usually are quite low (less than 100), this coarse grid may be established manually by the user. The blocks may correspond to the computational blocks used by the CFD solver, but they do not have to, as the block-partitioning utility may be employed prior to the simulation. The block grid may well be unstructured, and it is used to partition an irregular domain geometry into a set of regular sub-domains

29

-114 F^vc(t^el) F(t^h) F(t)

-116 28 -118 27

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26

F^vc(t^el) F(t^h) F(t)

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a)

23 100

1000

10000

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-132 100

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1000

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-15 F^vc(t^el) F(t^h) F(t)

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-16 -17 -18

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Fx-Fx^el Fy-Fy^el Fx-Fx^h Fy-Fy^h

1 0.1

0.01

0.1

1000 10000 f) 100 1000 10000 e) 100 Figure 10: Results for the uniform mesh sequence. a) Horizontal force Fx along the wall boundary, i.e., ? = f(x; y) 2 @ : x = yg. b) Net horizontal force Fx, i.e., ? = @ . c) Vertical force Fy along the wall boundary. d) Net vertical force Fy . e) Errors in integrated forces F h and variationally consistent forces F el along the wall boundary. f) Net errors. 0.001

0.01

that each may be meshed with a structured grid. When constructing the block grid, there are a few important considerations to keep in mind: The block grid is also used by the global mesh movement solver during the FSI-simulation where each block is treated as a linear-elastic continuum element [12]. Therefore, it is important that the shape of the blocks also are reasonably good. Moreover, one should avoid block boundaries parallel to the cross section surface within the boundary layer as this will degrade the solution considerably. Having de ned the block grid, the grid generation within each block should be automated to the extent which is feasible. The user should have the possibility to specify the number of grid points along each line segment of the polygon(s) de ning the structural cross section. In addition, the degree of clustering along the FSI-surface towards its corners should be user-speci ed. We suggest that for each

corner, the size of two adjacent surface elements is speci ed by the user. The size of the remaining elements should then be automatically computed, e.g. by using some geometric sequence, to ensure a smooth grid. Proper default values for both the number of elements (e.g., at least 200 elements around the cross section) and element size ratios (e.g., a ratio of 1 to 10 between the smallest and the largest element) should be provided to aid the user. A means should also be provided to cluster the grid points in radial direction in order to resolve the boundary layer while at the same time using fairly large grid cells at the far- eld boundary. The far- eld boundary should be located suciently far from the cross section (e.g., at least 10 times the cross section width) and possibly extended even further on the downstream side. The length scale for the boundary layer thickness can be estimated from the formula

  pB Re

(49)

where B denotes the cross section width and Re is the user-speci ed Reynolds number. We suggest that the height of the grid cells closest to the cross section, and the appendages, if any, should be set as default to, e.g., 50 , and that the total number of elements in the radial direction should default to at least 100 elements. The boundary conditions should normally be set to no-slip on the main cross section and any appendages. However, depending on the mesh resolution and the degree of detail to which the appendages have been modelled, all or some of the appendages could be assigned a slip condition instead if the viscous boundary layer on the appendages is not to be resolved. 6.2 The Great Belt bridge The bridge pro le studied in [21] is considered. A block grid and computational grid for this model, generated based on the directions discussed above is shown in Figure 11. In [21], the estimated aerodynamic derivatives based on a full FSI-simulation with free motion of the structure are reported. Similar results, but now obtained using a forced motion of the structure instead are presented in Figure 12. The comparison shows that in almost all cases, the discrepancies between the

a) b) Figure 11: Computational model for the Great Belt bridge. a) The block grid. b) Close-up look on the computational grid.

5.0 0.0 0.0 H2

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Experiments 1 Experiments 2 CFD CFD curve fit CFD forced oscil.

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5 U/(ft*B)

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U/(fz*B)

Figure 12: Estimated aerodynamic derivatives based on simulations compared to estimates based on wind tunnel experiments.

numerical results and the experimental results are within the deviations between the two dierent sets of experimental data. The resulting critical utter speed was found to be 64 m/s and 70 m/s for the forced and the free motion, respectively. This is in good agreement with previous estimates based on wind tunnel experiments [22] giving a value of 74 m/s for the critical utter speed. As the CPU time scales more or less linearly with the number of grid points and number of time steps it is an easy exercise to estimate the relative CPU time between dierent numerical models for a given problem. The total CPU time used for the ve free motion simulation runs was 187.5 hours using a mesh with 40 000 grid points and a total of 230000 time steps [21]. However, for the ten forced motion runs (here we need separate runs for heave and torsion) the number of time steps was only 15500. Thus, the computational cost for obtaining these derivatives are 15 times higher when using free motion compared to the forced motion approach. The vortex shedding frequency is much higher than the oscillation frequency of the bridge pro le, see the numerical results shown in [21]. The time step size is for both free and forced motions, governed by the vortex shedding frequency. For free motion problems, it is necessary to simulate a large number of vortex shedding periods in order to activate the relevant oscillations of the structure. Whereas, for forced motions the structural response are given, and the length of the time series are equal to two or three eigenperiods for the structure. We have also run a pure CFD simulation of this problem with error estimation. Figure 13 shows the computed error distribution in the vicinity of the bridge cross section in terms of the re nement indicators. The re nement indicators have here been computed based on an prescribed error tolerance of 1% error within each block, i.e., in areas where the re nement indicators are greater than 1.0, the element size should have been reduced, whereas in areas where they are less than 1.0, the elements could be made larger. We notice that the areas with highest error is not necessarily at the boundary, but slightly o the boundary. This information may be utilized to determine the grading of the element size in the normal direction to the bridge, as well as providing guidelines for where any extra elements should be added. Note that the slight discontinuity observed across the block boundaries, is due to that the error estimation is performed independently within each block. We want to emphasize that the underlying simulated ow eld does not show such discontinuities.

Acknowledgements

This work was supported by the European Commission through the FSI-SD project (ESPRIT IV contract no. 20111). See also http://tina.sti.jrc.it/FSI.

References

[1] P. G. Ciarlet. Mathematical Elasticity, Volume I: Three-dimensional Elasticity. North{Holland, Amsterdam, Netherlands, 1988. [2] P. Le Tallec and S. Mani. Conservation laws for uid-structure interaction. In T. Kvamsdal et al., editors, Proceedings of the International Symposium on Computational Methods for Fluid-Structure Interaction (FSI'99), Trondheim, Norway, February 1999. Tapir forlag, Trondheim, Norway. [3] J. C. Chorin. Numerical Solution of the Navier{Stokes Equations. Math. Comp. American Mathematical Society, 22:745{762, 1968.

Figure 13: Computed error distribution in terms of element level re nement indicators for the Great Belt bridge [4] A. Kovacs and M. Kawahara. A Finite Element Scheme based on the Velocity Correction Method for the Solution of the Time-Dependent Incompressible Navier{Stokes Equations. International Journal for Numerical Methods in Fluids, 13:403{423, 1991. [5] P. M. Gresho, S. T. Chan, R. L. Lee, and C. D. Upson. A Modi ed Finite Element Method for Solving the Time-Dependent, Incompressible Navier{Stokes Equations. International Journal for Numerical Methods in Fluids, 4:557{598, 1984. [6] K. Herfjord, S. O. Drange, and T. Kvamsdal. Assessment of vortex-induced vibrations on deep water risers by considering uid{structure interaction. In Proceedings of the 17th International Conference on Oshore Mechanics and Arctic Engineering, volume on CD-ROM, Lisbon, Portugal, July 1998. [7] T. Kvamsdal. Variationally Consistent Postprocessing. In Proceedings of the 4th World Congress on Computational Mechanics, volume on CD-ROM, Buenos Aires, Argentina, June/July 1998. [8] C. Farhat, M. Lesoinne, and P. Le Tallec. Load and Motion Transfer Algorithms for Fluid/Structure Interaction Problems with Non-matching Discrete Interfaces. Computer Methods in Applied Mechanics and Engineering, 157:95{ 114, 1998. [9] T. Kvamsdal, K. M. Okstad, and K. Herfjord. Error estimator for recovered surface forces in incompressible Navier{Stokes ow. In Proceedings of the 17th International Conference on Oshore Mechanics and Arctic Engineering, volume on CD-ROM, Lisbon, Portugal, July 1998. [10] C. A. Felippa and K. C. Park. Staggered Transient Analysis Procedures for

[11] [12]

[13] [14] [15] [16]

[17] [18] [19] [20]

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