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ECCM '99

European Conference on Computational Mechanics August 31 { September 3 Munchen, Germany

NUMERICAL METHODS FOR FLUID-STRUCTURE INTERACTIONS OF SLENDER STRUCTURES Trond Kvamsdal

Department of Numerical Simulation SINTEF Applied Mathematics, Trondheim, Norway e{mail: [email protected]

Jrn Amundsen

Center for computer services at NTNU (ITEA) Norwegian University of Science and Technology, Trondheim, Norway e{mail: [email protected]

Carl B. Jenssen

SINTEF Research Center, Trondheim, Norway e{mail: [email protected]

Knut M. Okstad

Department of Numerical Simulation SINTEF Applied Mathematics, Trondheim, Norway e{mail: [email protected]

Key words: Fluid-Structure Interactions, Parallel Computing 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. Herein we give an overview of results obtained within FluidStructure 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.

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad

1 Introduction An important eld in computational 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 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 show how a 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, and alternative numerical schemes are discussed. In Section 3 we study the scalability of the total FSI-system through the simulation of an benchmark case. Finally, in Section 4, we study a wind engineering application example (the Great Belt bridge), where important modeling considerations for such types of problems together with numerical results for utter speed computations are given. 2

ECCM '99, Munchen, Germany

2 Computational approach 2.1 Architecture of FSI-system

In [1] it is shown how the variational formulation of the complete FSI-problem may be reformulated into three coupled subproblems: (1) Fluid problem, (2) Structural problem and (3) Mesh movement. The developed FSI-system is based on an iterative solution (enhanced version of the \staggered solution procedure" [2, 3]) of the three coupled subproblems. 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 Computational Fluid Dynamics, CFD, and Computational Structural Dynamics, CSD, equations) with only minor adjustments to each code. The main programming effort 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. 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 consistent Coupler nodal forces

CFD

new mesh coordinates

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Figure 1: Overview of the FSI system 3

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad Dirichlet conditions along ? for solving the uid problem, and the corresponding extension of the displacements along the FSI-interface needed in the Arbitrarian Lagrangian Euler, ALE, formulation are provided for the CFD-code, respectively. As described below (see also [4]), to ensure computational eciency and parallel scalability, only a small part of mesh movement computations is carried out by the coupler, while most of the computations are carried out within the CFD-code. 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 [3].

2.2 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. However, for FSI-problems with large structural displacements the conventional staggering procedure often provides inaccurate solutions and numerical instabilities may occur. Alternatively, we may use an inter- eld staggering approach as illustrated in Figure 3. Fluid Structure

W - - 6 6    3. 1.     = 4. -= - 

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Figure 2: The conventional staggering procedure 4

ECCM '99, Munchen, Germany W (n)



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Figure 3: The basic inter- eld procedure

 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 .  The structure is advanced using the uid surface load at t n (n)

( )

With this procedure, the CFD and CSD solvers can run in parallel during the time interval [t n ; t n ]. 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, and numerical instabilities is to be expected for FSI-problems involving large structural displacements. The process of advancing the solution from t n to t n consists normally of a set of non-linear iteration steps. A means of increasing the accuracy and numerical stability of the overall procedure, is 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 Schwartz 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. We therefore advocate the use of the following iterative solution procedure for problem involving large displacements of slender structures: ( )

( +1)

( )

( +1)

For each time step t n do: ( )

1. Predict the displacement and the velocity of the structure at the time t n 2. Repeat until convergence (a) Update the uid grid to conform with the new structural boundary. (b) Advance the ow using the new boundary conditions. (c) Update the surface load on the structure based on the new ow eld. (d) Advance the structure using the new uid surface load.

( +1)

5

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad

2.3 Mesh movement strategy In coupled analysis of FSI-problems the displacements of the structure and the use of ALE formulations for discretization of the uid domain raise the need for moving the nodes in the uid domain. 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. However, it is 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 , such an approach would require the order of N operations on a single processor as well as having to send 3N values over the network. Since in the CFD code, the computational cost and communication costs are of the order N and N 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 relevant FSI-problems 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 3

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ECCM '99, Munchen, Germany

Elliptic FEM Host Proc.

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Figure 4: Mesh movement strategy. number of blocks, and not mesh points. Parallelization 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 [4], 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. 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 (nearly) remains as in the initial grid, and if the corner nodes of the block remain unchanged, the mesh within the block also stays unchanged. The present meshmovement algorithm may also be combined with mesh adaption by means of adaptive relocation of nodes, see [4]. 7

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad

3 Benchmark results related to scalability In order to verify the scalability of the chosen approach we here report results obtained for a benchmark test. The uid domain of the benchmark test case consists of 43000 grid points in CBJ divided into 28 computational blocks. The geometry of these blocks is illustrated in Figure 5. There are two blocks in the length direction of the bridge. The structure is modeled in USFOS using a single beam element which is attached to a set of springs in each end to simulate the global stiness of a bridge. The test runs are performed across the NTNU-SINTEF Internet with a physical distance of approximately 5 km between the CFD and coupler hosts. The coupler host is attached to an Ethernet and the CFD host to a FDDI. Between the two systems, PVM messages are transported in four network hops. The CSD application runs on the same host as the coupler, a 2-CPU SGI IRIX system with 150 MHz R4400 processors, whereas the CFD host is a CRAY T3E. In the time step loop, the coupler sends 26380 bytes to the CFD application and receives 33548 bytes in return, in average approximately 30 Kbytes. The round-trip time, as measured with 30  30 Kbyte trace-route packets is 75.8 ms on the average. The timings listed in Table 1 give the CPU time used by the CFD code and the rest of the FSI system based on the two-level mesh-movement solver described above. The table also report the overhead represented by the FSI system and the speed-up for the complete system as a function of the number of processors used by the CFD code. The CPU time for the additional FSI parts, T , includes all CPU time associated with FSI. fsi

Figure 5: Computational blocks of the benchmark example. 8

ECCM '99, Munchen, Germany Table 1: CFD application scalability with the two-level mesh movement solver. Ncpu

2 4 8 12 16

Tcfd

[sec.] 294.9 129.9 65.1 44.4 38.4

Tfsi

[sec.] 12.3 12.3 11.1 12.0 12.0

Ttot

[sec.] 307.2 142.2 76.2 56.4 50.4

Overhead Speedup [%] 4.0 1.0 8.65 2.2 14.57 4.0 21.28 5.4 23.81 6.1

Table 2: CFD application scalability with a one-level mesh movement solver. Ncpu

2 4 8 12

Tcfd

[sec.] 216.6 102.6 54.6 37.2

Tfsi

[sec.] 280.2 280.8 280.2 285.6

Ttot

[sec.] 496.8 383.4 334.8 322.8

Overhead Speedup [%] 56.4 1.0 73.2 1.3 83.7 1.48 88.5 1.54

As we can see, acceptable performance is obtained for up to 12 processors. The CPU time for the FSI parts remain almost constant as expected, while the CFD code shows very good scalability. This benchmark is overly pessimistic in terms of the scalability of the complete system, as the FSI part appears as a sequential bottleneck that causes a low parallel eciency for 16 processors or more. However, if the accuracy of the simulation is to be increased, the need for higher resolution would be on CFD side, calling for a many fold increase of the number of grid points. Thus, the communication time and sequential CPU time would remain the same, and assuming linear speed-up of the CFD code, the number of processors could grow proportional to the number of grid points without increasing the FSI overhead. As a contrast, Table 2 contains time-measurements obtained with a one-level meshmovement procedure. This procedure is not run in parallel and we see that it totally destroys the scalability of the whole system. The timings in Tables 1 and 2 were obtained with a FSI communication library which still contains much instrumentation code. In addition, spikes in campus network load will result in varying communication times. However, we believe this is a second-order contribution compared to the mesh-movement overhead.

9

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad

4 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 FSI-Bridge system we refer to the presentation given in [5].

4.1 Modeling 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 [6], 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 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 FSIsimulation where each block is treated as a linear-elastic continuum element [4]. 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. 10

ECCM '99, Munchen, Germany 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 B   p (1) Re

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.,  , 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 modeled, 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. 50

4.2 The Great Belt bridge The bridge pro le studied in [7] is considered. A block grid and computational grid for this model, generated based on the directions discussed above is shown in Figure 6. In [7], 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 7.

a)

b)

Figure 6: Computational model for the Great Belt bridge. a) The block grid. b) Close-up look on the computational grid. 11

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad 5.0 0.0 0.0 H2

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Figure 7: Aerodynamic derivatives based on simulations and wind tunnel experiments. 12

ECCM '99, Munchen, Germany The comparison shows that in almost all cases, the discrepancies between the 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 (denoted \CFD forced oscil." in Figure 7) and the free motion (denoted \CFD" in Figure 7), respectively. This is in good agreement with previous estimates based on wind tunnel experiments [8] 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 [7]. 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, see the numerical results shown in [7]. 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.

5 Conclusions We have herein presented numerical methods for solving uid-structure interaction (FSI) problems involving slender structures discretized by beam elements. The approach is characterized by a modular coupling of existing codes for Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) through a separate module denoted FSI-Coupler. The numerical procedure for solving the coupled sub-problems obey the governing conservation laws. This is achieved by means of iterations similar to the non-overlapping alternating Schwartz algorithm with variational consistent transfer of loads and velocities along the uid-structure interface. The developed methodology for two-level mesh-movement for modular FSI-simulations accounts for large structural displacements. The parallel eciency of the overall system is ensured through the splitting of the mesh movement algorithm into a global part computed sequentially by the FSI-Coupler, and a local part computed in parallel within the CFD code. By using the developed simulation tool we have been able to do realistic numerical prediction of critical utter speed for wind induced vibrations of aerodynamic bridges. Comparison between our numerical results with performed wind-tunnel tests of the Great Belt Bridge in Denmark shows less than 10% deviations. 13

Trond Kvamsdal, Jrn Amundsen, Carl B. Jenssen, Knut M. Okstad

Acknowledgments 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] T. Kvamsdal, C. B. Jenssen, K. M. Okstad, and J. Amundsen. Fluid{structure interaction for structural design. In T. Kvamsdal et al., editors, Proceedings of the International Symposium on Computational Methods for Fluid-Structure Interaction (FSI'99), pages 211{238, Trondheim, Norway, February 1999. Tapir Publishers, Trondheim, Norway. [2] C. A. Felippa and K. C. Park. Staggered Transient Analysis Procedures for Coupled Mechanical Systems: Formulation. Computer Methods in Applied Mechanics and Engineering, 24:61{111, 1980. [3] P. Pegon and K. Mehr. Report and algorithm for the coupling module. Technical Report I.97.77, EC/JRC, Ispra, Italy, 1997. [4] T. Kvamsdal, K. M. Okstad, K. Srli, and P. Pegon. Two-level adaptive mesh movement algorithms for FSI-computations. In T. Kvamsdal et al., editors, Proceedings of the International Symposium on Computational Methods for Fluid-Structure Interaction (FSI'99), pages 121{132, Trondheim, Norway, February 1999. Tapir Publishers, Trondheim, Norway. [5] S. O. Hansen, I. Enevoldsen, C. Pedersen, and L. T. Torbek. Practical design perspectives in FSI-simulations of wind load on large bridges. In T. Kvamsdal et al., editors, Proceedings of the International Symposium on Computational Methods for Fluid-Structure Interaction (FSI'99), pages 239{260, Trondheim, Norway, February 1999. Tapir Publishers, Trondheim, Norway. [6] C. B. Jenssen. The Development and Implementation of an Implicit Multi Block Navier{Stokes Solver. Dr. Ing. dissertation, Department of Mechanical Engineering, The Norwegian Institute of Technology, Trondheim, Norway, 1992. [7] C. B. Jenssen and T. Kvamsdal. Computational methods for FSI-simulation of slender bridges on high performance computers. In T. Kvamsdal et al., editors, Proceedings of the International Symposium on Computational Methods for Fluid-Structure Interaction (FSI'99), pages 31{40, Trondheim, Norway, February 1999. Tapir Publishers, Trondheim, Norway. [8] T. A. Reinhold, M. Brinch, and A. Damsgaard. Aerodynamic design of the Great Belt east bridge. In A. Larsen, editor, Aerodynamics of Large Bridges, pages 269{284. Balkema, Rotterdam, 1992. 14