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Proceedings of OMAE2009 28th International Conference on Offshore Mechanics and Arctic Engineering

May 31- June 5, 2009, Honolulu, Hawaii

OMAE2009-79804

FULLY COUPLED FLUID-STRUCTURE INTERACTION FOR OFFSHORE APPLICATIONS

Rajeev K. Jaiman

ACUSIM Software, Inc. Mountain View, CA 94043

Farzin Shakib ACUSIM Software, Inc.

Mountain View, CA 94043

Owen H. Oakley, Jr. Chevron Energy Technology Company

Yiannis Constantinides Chevron Energy Technology Company

ABSTRACT CAD integrated tools are accelerating product development by incorporating various aspects of physics through coupling with computational aided engineering (CAE) packages. One of the most challenging areas is fluid-structure interaction (FSI) of low mass bodies such as flexible marine risers/cables with vortex-induced vibrations. The focus of this work is on the application of a new Multi-Iterative Coupling (MIC) procedure to couple AcuSolve (fluid solver) with Abaqus (structural solver). The proposed new scheme has superior stability and convergence properties as compared to conventional explicit staggered schemes, especially for low mass-density ratios of structure to fluid. Demonstrations and validation of the scheme are presented and discussed along with additional challenges associated with FSI in production environments. The addition of an FEA solver enables the modeling of the nonlinear aspects of flexible riser VIV, namely, contacts with gaps, multi-body dynamics, seabed interaction, geometric and/or material nonlinearities.

INTRODUCTION The nonlinear dynamic analysis of fluid-structure interactions has been given increased attention during recent years in offshore applications. This is largely because computational

methods and resources have become very powerful and they can be used at reasonable costs with great benefits in numerous offshore engineering problems [1-8], e.g., drilling and production risers, mooring lines, marine cables and offshore platforms. From both the design and operational standpoint, it is important to be able to predict the hydrodynamic forces and motion of such structures caused by fluid-structure coupling. Due to the complexity of the hydroelastic problem, theoretical and semi-empirical models remain incomplete and problem specific. Typically, the simplified models rely on the force input as well as the added mass coefficient and geometric correlation parameters. Until recently, prediction of hydroelastic interactions and vortex-induced vibrations were primarily based on such semi-empirical methods. Fully coupled FSI analysis represents the class of multiphysics problems in which it is important to study the effects of fluid flow on flexible structures and their subsequent interactions. There are many problems in offshore where a direct fully-coupled analysis is needed to model the physics of the fluid-structure problem accurately. This particularly becomes a need for a structure with geometric and material nonlinearities that undergoes large deformation while interacting with turbulent flow. For example, in the simulation of long marine risers, the contact conditions of the riser with support guides or centralizers are nonlinear. Similarly, there are nonlinear soil structure interactions at the touchdown point for steel catenary risers. Although material nonlinearities are not important for typically steel risers, new concepts with composite material and various end-termination components have nonlinear

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material properties. The presence of these effects means that a nonlinear structural model must eventually be incorporated in the solution of riser problems and also that large mesh motions must somehow be accommodated in the fluid flow solution.

In such studies, it is important to develop an efficient and general approach towards solving fluid flows coupled with structures. The flows can be modeled as fully incompressible or slightly compressible and can include turbulence effects. The structural components can comprise solids, shells, beams and other elements including contact surfaces, gaps etc. Our objective is to present some developments to solve fluid flows with structural interactions. In particular, we are interested in realistic simulations of offshore applications involving low mass bodies such as flexible marine risers/cables with vortex-induced vibrations. We present two methods with an aid of direct-coupled FSI framework: Conventional Sequential Staggered (CSS) scheme [9] with explicit integration and multi-iterative coupling procedure based on implicit sub-iterations. The framework combines AcuSolve, a general-purpose fluid dynamics solver distributed by ACUSIM Software, Inc. [10] and the structural solver Abaqus distributed by Simulia [11]. In the following sections we describe the direct-coupling FSI method of combining 3D fluid flow solver with the structural solver. We then briefly discuss explicit sequential staggered and new multi-iterative coupling schemes. Benchmark test cases of flexible baffles and short riser sections are presented to analyze the stability and robustness of the multi-iterative coupling scheme. Finally, a full scale riser model is simulated using the new scheme.

NOMENCLATURE

A = motion amplitude [m] D = riser diameter [m] L = riser length [m]

, ,x y z = coordinates [m] U = current velocity; maximum for sheared profile [m/s] Re = UD /ν = Reynolds number [-] t = time [s]

DIRECT-COUPING FSI

The coupled fluid-structure interaction equations comprise the initial-boundary value problems of the fluid and the structure, complemented by the traction (dynamic) and displacement (kinematic) boundary conditions at the fluid-structure interface. The Acusolve Direct-Coupling technique can be used to solve complex FSI analyses by coupling to the external structural solver Abaqus in the partitioned manner as shown in Fig. 1. In the partitioned approach, the decomposed domains of the fluid and solid share a common interface boundary. The structural equations are conventionally

formulated in Lagrangian coordinates on a mesh that moves along with the material, while the fluid equations are formulated in Eulerian coordinates, where the mesh serves as a fixed reference for the fluid motion. For the coupling of these media, the fluid solutions are accommodated with the structural motion by an arbitrary-Lagrangian-Eulerian formulation [12]. Particular care must be taken to properly couple the fluid and the structure along the interface between the media. Figure 1– Direct-coupling FSI simulation cycle. The coupling of the fluid and structural response can be achieved numerically in different ways, but the interface conditions of displacement continuity (motion transfer) and traction equilibrium (momentum transfer) along the fluid-structure surfaces must be satisfied:

:

:

d d

P P

Displacement continuity

Traction equilibrium

on

on

= Γ

= Γ

f s fsi

f s fsi

where df and ds are the displacements, Pf and Ps are the

tractions of the fluid and structure, respectively, and Γfsi is the interface of the fluid and structural domains. These conditions must be imposed efficiently in the coupled numerical scheme. While the ALE fluid system computes the traction field Pf on the element surfaces of the interface, the structural system is solved based on a set of forces at the nodes on the interface. Assume that the fluid and structural domains have been meshed independently and the nodal positions do not match (see Figure 2). The projection and interpolation of tractions and displacements are achieved across the non-matching discrete interface. The projected fluid tractions are imposed as concentrated forces onto the structures, and the structural

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Motion Transfer

AcuSolve

Momentum Transfer

Interface Tractions

Abaqus

Put Forces

Solve ALE Motion

Get Displacements

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displacements are applied as motion change onto the ALE fluid domain.

Figure 2– Interface conditions transfer across non-matching spatial resolutions.

EXPLICIT COUPLING SCHEMES At each time step, explicit staggered techniques are generally used to satisfy the continuity of velocity and traction conditions in a sequential manner, often referred to as the conventional sequential staggered (CSS) procedure [9]. The explicit CSS scheme appears attractive from a computational viewpoint as the solutions of the fluid and the structures are computed once per time step. However, it may suffer significantly from destabilizing effects introduced due to a lag between the fluid and structure solutions [13, 14]. Depending on the direction of the interface acceleration, high under- or over-prediction of pressure may occur due to the lack of energy equilibrium across the fluid-structure interface. In other words, any small error in the interface displacements imposed onto the fluid by the structure will result in large errors in fluid pressure. The numerical instability depends upon the material properties of fluid and structure and also on the relative geometric scales of the domain and the compressibility of the fluid. Notably, the sequential staggering introduces an explicit nature into the coupling even if both domains are solved implicitly. For incompressible fluid interacting with low mass structures, reducing the time step size does not cure the instability regardless of whether the Courant condition for the individual domain is satisfied. This implies the instability is inherent in the sequential staggered scheme due to the strongly nonlinear nature of fluid-structure coupling. A variety of structural predictors and force corrections (added-mass) can be employed at each time step to enhance the coupled numerical stability, for example: Generalized Sequential Staggered (GSS) [15] and Modified Combined-Interface Bundary Conditions (M-CIBC) [16]. At each time step, the prediction of displacement dP

f and the correction of

forces PCs are formed in a sequential manner along the fluid-

structure interface:

:

:*

*

d d + d

P P + P

Displacement prediction

Traction correction

on

on

δ

δ

= Γ

= Γ

Pf s fsi

Cs f fsi

where displacement prediction *dδ and the force correction

*Pδ terms are explicit functions of spatial and temporal solutions (velocity, accelerations, traction gradients etc) of fluid and structure along the interface. These interface prediction and correction terms are only applied to the right hand side of fluid and structure equations. The above predictor-corrector procedure generally determines the order of temporal accuracy and provides an interface based relaxation to the coupled numerical instability. However, such interface predictors and force correction terms may not provide sufficient stability and robustness for the light-weight structures interacting with highly incompressible fluid. In addition, the explicit correction terms involve coupling parameter and coefficients which depend on the underlying spatial discretization and the time integration schemes. In the next section, we turn our attention to the multi-iterative coupling scheme based on the implicit sub-iterations.

MULTI-ITERATIVE COUPLING SCHEME In the multi-iterative coupling scheme, the full partitioned system is treated with an implicit integration operator similar to the monolithic integration. This implicit integration is performed by the predictor-corrector iterations while maintaining the explicit “loosely-coupled” nature of the information transfer similar to the CSS. To begin, the resulting coupled algebraic system of equations to be solved for each time step can be written in abstract form:

ff fs f f

sf ss s s

A A q RA A q R

RA q

Coupled Nonlinear System

∆ = − ∆

∆

where fq and sq denote ALE fluid and structural solution

fields, respectively, and fR and sR represent equation systems for the fluid and structure model, respectively. The derivatives of fluid and solid equations (Jacobian matrices) are given by the left-hand side matrix A . The off-diagonal terms, namely, fsA and sfA are not explicitly formed in the MIC scheme. Instead, the scheme proceeds in a similar fashion as the above predictor-corrector schemes by constructing a force correction.

(2)

(3)

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The interface force correction terms are formed by iterating between the solutions of fluid and structure. Sub-iteration is done by cyclic substitutions at the same time step. For example, the force correction approximation *Pδ k at the kth

sub-iteration can be constructed in the following manner:

-1fs f

:

*

P = A R

Iterative traction correction

onδ Γkfsi

where -1fsA represents inverse of successive approximation of

fsA (without explicitly forming fsA and its inverse) , and

fR denotes the approximated linearized fluid equation effects along the fluid-structure interface. This force correction procedure is based on a nonlinear feedback monitoring, as we go back and forth between the two physics. In summary, the fluid and structure equations are solved sequentially (similar to the explicit coupling schemes) to pass latest interfacial solutions from one model to the other in the MIC. For both Abaqus and AcuSolve, the nonlinear sub-iterations are continued until the coupled iterative forces have been solved up to residual tolerance. The multi-iterative force corrections allow to achieve asymptotic convergence of forces along the fluid-structure interface for a range of mass density and frequency ratios. To illustrate the stability of the schemes, we conduct numerical experiments on prototypical fluid-structure interaction problems in the next section.

TEST CASES

The objective of this section is to present test simulations that briefly demonstrate the numerical capabilities described above. In all simulations, the variants of Newmark approximation [18, 19] in time will be applied for the fluid, the structure and the moving ALE mesh equations. Of interest in this work is to characterize the influence of mass-density ratio on coupled FSI problems involving large displacements. The relative characteristic mass ratio µr is defined as

:

s

f

Mass density ratiomm

µ =r

where ms and mf are the characteristic masses of structure and fluid, respectively. We remark that the mass ratio determines which physics dominates the coupled dynamics of fluid-structure system. In particular, for

1 ( 1) µ µ<< >>r r the coupled dynamics is dominated by the fluid inertia (structure).

Flexible Baffle Problem To assess the effects of mass-density ratio, we first consider the hydroelastitc deformations of a flexible baffle in a channel

with inflow speed of 10 m/s (see Fig. 4a). No-slip boundary conditions are applied at the top and bottom of the channel. The fluid domain is discretized by 1856 elements and the structure is modeled by 64 elements of geometrically nonlinear continuum type. The top end of baffle has zero displacement condition. A matching coupling time step of ∆tc =0.005 sec is used which allows a high resolution of the coupled dynamics. The selected fluid flow and solid material properties lead to a substantial deflection of the structure as shown in Fig. 4b. In the figure, a snapshot of the highly transient dynamics of the coupled problem is given where velocity vectors are plotted on the absolute velocity. It highlights the significant influence of the structural motion on the flow field and vice versa.

Figure 3– Flexible baffle problem: (a) geometry and material data; (b) representative solution of the coupled problem. Table 1– Summary of test cases for varying density ratios

Test Case1 Case2 Case3 Case4 Case5 Mass density

ratio µr 9.0 6.0 3.0 1.5 0.1

The influence of the structural density is compared for the sequential staggered and multi-iterative schemes on the

Inflow

No-slip wall

Baffle

Outflow

Baffle: height = 2.0 m, width = 0.5m, thickness= 0.5 m

Fluid: Structure:F 3density ρ =1000 kg/m

material properties

Young's modulus E = 1.0e9 Paviscosity ν =0.001 Pa.s Poisson's ratio = 0.33

(a)

(b)

(4)

(5)

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motion of the flexible baffle. Oscillations in the integrated baffle displacements indicate numerical instability. The test cases corresponding to decreasing mass density ratio (lowering of structural density) are summarized in Table 1. We observe the behavior of the schemes in the time interval [0, Tmax = 2 sec]. For the sequential staggered scheme, the normalized stream- wise and cross-stream displacements of the first three cases are plotted in Figs. 4(a)-(b). The onset of instability is predicted roughly at the mass density ratio of 6.0 (Case2) and the coupled simulation becomes unstable at the mass density of 3.0 (Case3). Unfortunately, decreasing the time-step size and increasing the number of nonlinear sub-iterations does not eliminate the onset of instability. Moreover, the fluid pressure is highly sensitive with respect to the correctly determined fluid-structure interface position and small over-prediction/under-prediction along the interface yields large coupling force errors. Thus implicit coupling of the pressure is unavoidable, which leads to the need for multi-iterative coupling.

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Figure 4- Temporal evolution of baffle displacements for sequential staggered procedure Figures 5(a)-(b) present the displacement solutions of the five cases for the multi-iterative coupling. It can be seen that the scheme remains stable and convergent for all the cases up to the mass density of 0.1.

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Figure 5- Temporal evolution of baffle displacements multi-iterative coupling procedure: (a) streamwise (b) cross-stream The stability limits with respect to the mass density ratio is further displayed in Table 2. The results confirm that the explicit coupling schemes for the baffle problem suffer stringent lower limit (i.e., mass density ratio µr > 1.0) for the structural surface density than the multi-iterative coupling scheme. Table 2– Estimated lower limit of stable mass density ratio

Method CSS GSS M-CIBC MIC Lower Limit of Stable µr 6.0 3.5 1.75 0.05

While the sub-iterations based force corrections provide sufficient stability to the fluid-structure system, the multi-iterative scheme requires at least two iterations of fluid and structure field solutions per time step. For very thin and light structures, the required sub-iterations per time step may rise up to 6-8; therefore solving the fully-coupled solution may require large hardware clusters for practical engineering problems1. For further effectiveness, efficient formulations are desirable for this special class of coupled problems. Next we 1 The explicit coupling schemes cannot even solve this problem.

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present a more general test case related to offshore applications using the multi-iterative coupling scheme.

Short Span Riser Model This test case is designed to simulate the behavior of flow around a short span riser with L/D = 50 with mass density ratio of 1.5µ =r . The present example has been designed for two purposes. Firstly it demonstrates the capability of the fully-coupled scheme to simulate the vortex induced vibrations at low-mass density ratio. Secondly, it is also used to compare a simulation computed from beam structural elements to one performed with shell structural elements. The geometry of the initial problem is depicted in Fig. 6. The fluid has the material data of sea water. At the inflow, the water enters at the fluid domain with a horizontal velocity of 1.2 m/s. This speed corresponds to the subcritical Reynolds number of ReD=81640. Symmetry boundary conditions are applied at the top, bottom and side walls of the fluid domain. The structural material properties are: density=3791 kg/m3, Young’s modulus E =1.5E9 Pa and Poisson’s ratio=0.42. The pinned-pinned boundary conditions are imposed at both end points of the structural model. The fluid domain is discretized by a total of 4619981 tetrahedral elements and 805105 nodes and the riser structural surface is modeled by 15000 quad elements (reduced-order integration) for the thin shell case and 100 line elements for the beam model. These spatial discretizations have been selected by a grid independent study of the standalone fluid and structure solutions. Since this problem is highly transient in nature, we have used a small matching coupling time step of ∆tc =0.005 sec. Figure 6- Geometry and cross-sectional mesh distribution of short span riser problem Even if the fluid mass density is roughly the same as the structural mass density, the sequentially staggered scheme is unstable for this riser problem due to the greater geometric scale of the problem. Therefore, the multi-iterative scheme

with 2 sub-iterations between AcuSolve and Abaqus has been used to obtain the stable solutions for both the beam and shell element cases. In Fig. 7, the evolution of velocity field at t=1 sec (top) and t=2 sec (bottom) of the coupled problem is shown. The results are obtained using the beam (left) and shell (right) structural elements. The results suggest that the coupled dynamic behavior for the two discretizations is quite similar. In particular, the above plots can be used to contrast the overall deflection of riser and the vortex shedding in the fluid domain. The similarity is also obvious from the streamwise and cross-stream displacements of the riser as depicted in Fig. 8. At least before 1.5 sec, the two simulations predict similar behavior. After 2.0 sec, the discretization by shell elements provide lower streamwise displacement than the beam elements, but not much apparent difference is seen in the solution of cross-stream displacements. The boundary conditions for the fluid domain and the structural models are identical for both the cases and a detailed grid convergence studies are carried out for individual domains for both cases.

Figure 7- Coupled solutions of short span riser problem for beam (left) and shell (right) structural elements. Color map represents the fluid velocity magnitude It is obvious that the computational effort required with the shell element case (surface elements) is greater than the beam element (line elements) case for the standalone structural

t = 1 sec

t = 2 sec

Inflow Outflow

D

L

Pinned

Pinned

7

solver. In the current DC-FSI implementation of AcuSolve and Abaqus coupling, the shell element case was also found to be about 2 times more expensive for the overall coupled solution than the counterpart beam element case. These observations are important with respect the computational efficiency of the fully-coupled FSI simulations, and they suggest utilizing the low-cost beam elements for riser type of applications to obtain a similar structural dynamic response. Figure 8- Temporal evolution beam and thin shell elements: (a) stream-wise displacements, (b) cross-stream displacements.

SIMULATION OF MEDIUM L/D FLEXIBLE RISER The final application for fully-coupled FSI consists of the riser model with L/D=1407 tested by the Norwegian Deepwater Programme (NDP) and reported in [17]. The riser has a length of 38 meter length and 0.027 meter diameter and the model was towed with the linear sheared currents at a subcritical Reynolds number. The numerical approach taken here assumes that the fluid flow over the riser must be described using a three dimensional flow solution. The numerical modeling has three parts: the resolution of the boundary layer, the resolution of the near wake and the resolution of the far field. A variety of mesh solutions have been used here in an attempt to find the most economical mesh. To further reduce the initial grid nodes, a small rectangular domain around the riser was modeled. The inflow and outflow planes of the riser domain are moved along with the riser structural motion.

The riser fluid domain was modeled with an initial mesh of 9.9M nodes (which was near the limit of the 24 CPU computer used). A unidirectional steady shear current of 2 m/s was applied. The structure part was modeled as an elastic beam of hollow circular section divided into segments. The nonlinear and dynamic fluid-structure coupling was achieved by considering a complete description of the hydrodynamic forces and instantaneous displacements.

Flow visualization A sample flow pattern is presented in Fig. 9 for the linear shear case at 2.0 m/s. As the vibration moves down the riser vortices are shed in the wake. The vortices break down and split. Vortex shedding modes such as 2P and 2S have been identified in the isolated cases examined. The adaptive motion of inflow and outflow planes can also be seen in the visualization. The linear shear current vectors are shown at the moving inflow plane.

Figure 9- Vorticity contours at deformed shape of riser model.

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Dynamic response Using the direct-coupling FSI approach, the crossflow and in-line responses for the riser were analyzed in an effort to understand the complex cable dynamics. Figures 10(a)-(b) show the calculated cross-stream and inline displacements of the riser at two different locations. Also, a transient response is shown by the corresponding trajectories of crossflow and in-line displacements in Figs. 10(c)-(d).

Figure 10- Comparison of cross-flow (CF) and inline (IL) displacements at two different 2 different spots and the corresponding trajectories. The complex response shown in Fig. 11 reveals a strong traveling wave pattern with standing waves at the ends of the riser, due to the end conditions. The riser response is in fact in nonlinear equilibrium between the flow induced excitation forces and the structural dynamics, and is characterized by varying amplitudes and phases along the riser length. The excitation is caused near the top of the riser at the high current region and the wave propagates to both ends. This region, referred to as power-in, was observed to shift in time. In the cases examined, the power-in region was located in the high current region between 0.6 and 0.95 z/L. The shift to areas of higher and lower velocity may be caused by the inline motion of the riser that drifts back and forth due to the slow change in drag force, or due to the mode or frequency shift.

Figure 11- Responses in shear currents at 2.0 m/s for L/D= 1407 riser model: (a) crossflow (b) in-line

CONCLUSIONS A full three dimensional CFD model was coupled with a structural model to predict complex fluid-structure interactions. We considered quite general flow fields and very general structural conditions, as simulated with the direct-coupling FSI technique of AcuSolve and Abaqus. For the sequential staggered schemes, the instability is highly influenced by the ratio of structural to fluid mass density. Whenever the structure and fluid mass densities are comparable, the numerical instability is almost unavoidable for the physically stable solution. Only in the cases where the fluid density is much lighter than the structure (e.g., air with aluminum) is a conditionally stable solution possible from the

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(b)

(c) (d)

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explicit staggered schemes. The interface structural predictions and the explicit force correction operators postpone the onset of instability while being unable to prevent it for a very low mass density ratio. To overcome these deficiencies we have developed a sub-iteration based multi-iterative coupling procedure. The current implementation of the procedure can be used for complex fluid flow problems with structure interactions, and it retains the modularity of both codes. We see considerable potential to further increase its robustness and effectiveness. To improve the computational efficiency of the multi-iterative coupling scheme, we need to employ accurate and consistent prediction techniques in order to accelerate the sub-iteration convergence per time-step. We demonstrated the versatility of the multi-iterative coupling scheme for increasing complexity of problems. The response of a long flexible marine riser in a shear current was investigated with details using the scheme.

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[5] Constantinides Y, and Oakley O. H, "Numerical prediction of VIV comparison with field experiments", OMAE2008-57215, 2008

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[8] Herfjord, K., Drange, S.O., Kvamsdal T. Assessment of vortex-induced vibrations on deepwater risers by considering fluid-structure interaction, J.OMAE, 121 1999

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[10] AcuSolve Command Reference Manual, Version 1.7e, 2008.

[11] Abaqus Analysis User’s Manual, Version 6.7, 2007. [12] Donea, J., “Arbitrary Lagrangian-Eulerian finite element methods”, Computational Methods for Transient Analysis, Elseiver Science Publisher, pp. 474-516, 1983. [13] Forster, Ch., Wall W.A.,and Ramm, E., “Artificial added mass instabilities in sequential staged coupling of nonlinear structures and incompressible viscous flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 1278-1293,, 2007. [14 Causin, P., Gerbeau, J-F, and Nobile, F. “Added-mass effect in the design of partitioned algorithms for fluid-structure problems”, Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 4506-4527, 2005. [15] Farhat, C., van der Zee, K.G., and Geuzaine, “Provably Second-Order Accurate Time-Accurate Loosely-Coupled Solution Algorithms for for Transient Nonlinear Computational Aeroelasticity”, Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 1973-2001, 2006. [16] Jaiman, R., Geubelle, P. , Loth, E. and Jiao, X. “Stable and Accurate Loosely-Coupled Scheme for Unsteady Fluid-Structure Interaction”, AIAA Paper, 07-334, 2007.

[17] Trim, A. D., Braaten, H., Lie, H. and M. A. Tognarelli, “Experimental Investigation of Vortex-Induced Vibration of Long Marin Risers,” J. Fluids and Structures, Sept. 2005

[18] Newmark, N.M., “A method of computation for structural dynamics,” Journal of the Engineering Mechanics Division (ASCE), Vol 85, 67-94, 1959.

[19] Chung, J. and Hulbert, G.M., “A time integration algorithm for structural dynamics with improved numerical dissipation,” Journal of Applied Mechanics Division, Vol 60, 371-375, 1993.