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An overview of fluid-structure coupling schemes

Coupling schemes used for modelling fluid-structure interaction include two main types of schemes that handle the exchange of information between the fluid and the solid phase: i) monolithic schemes and ii) partitioned schemes, which can be implicit or explicit. These coupling schemes can be combined with any numerical method for representing moving solids in a fluid such as ALE mesh, immersed boundary method or cut-cell method. The main traits of these coupling schemes, as well as their subcategories, are summarised below.

Monolithic schemes

Monolithic schemes are the optimal choice for numerical stability and accuracy. In these schemes, the calculation of the fluid pressure and solid acceleration are intertwined, leading to an interdependent solution that is found simultaneously for the fluid and the solid. The idea can be described at a fundamental level by the following equation:

where Г^1 is the position of the fluid-structure interface at timestep n + 1, Vp is the fluid pressure gradient representing the local fluid acceleration, p f is the fluid density, and Xrrns is the acceleration of the boundary Гfns- The total acceleration acting on the boundary includes gravity and hydrodynamic forces and any other stresses imposed by additional considerations (e.g., flexible structures) or external restrictions (e.g., moorings and springs). The solid boundary acceleration Xrlns is a function of the hydrodynamic pressure itself (among other variables), hence Equation (20) results in an implicit system. This equation is eventually fed into the pressure solver which is arguably the most critical part of the numerical solution for incompressible flows, as it is the condition that ensures mass conservation. Finding the numerical solution for the fluid pressure is also considered the most time-consuming part for incompressible solvers.

In the special case of a rigid and free floating body, Xrlns is solely dependent on the weight and the fluid pressure. It is therefore relatively straightforward to develop an algorithm using Equation (20) as an implicit system of equations for the fluid pressure which can be combined with the overall pressure solution for the fluid and thus result in a truly monolithic scheme for, e.g., floating structures. This approach was followed in Chen et al. (2016) in the context of a Particle-In-Cell (PIC) model and the cut-cell method for modelling floating structures. This work has shown the merits of using monolithic schemes as there were no numerical stability issues reported. The most significant drawback of monolithic schemes is nevertheless its lack of modularity. By including additional forces such as mooring, springs or solid stresses, the global system of equations for the pressure at tn+1 must be redesigned to take into account the additional forces. For example, including mooring with linear springs and damping would require to express the forces in Equation (20) at t = fn+1, without using data from tn, as doing otherwise will compromise the monolithic character of the scheme. This task can be proven particularly challenging for implementing a simple spring mooring system. When coupling the CFD model with external and highly sophisticated software libraries for, e.g., introducing cable or chain moorings (or even other types of solid deformable structures), maintaining a true monolithic scheme may prove an impossible task, as an analytical expression to express the forces at tn+1 is not available. In this case, some level of partitioning must be introduced, which compromises the monolithic character of the scheme.

Partitioned schemes

Partitioned schemes solve the fluid and solid kinematics separately, and can be divided in three main categories: explicit, implicit, and semi-implicit schemes. All of them allow for greater modularity as each solver can be treated as a black-box, using available data from previous time-step or iteration to proceed. While this enables the use of highly-specialised software for each of the coupled problems, it is important to note that partitioned schemes can be subject to numerical instabilities or time consuming calculations, depending on the chosen scheme, as discussed further below.

3.2.1 Explicit schemes

In explicit schemes, calculations for the solid and fluid kinematics are separated and only performed once per time step. A representative sequence of calculations can be expressed as:

where Uf is the fluid velocity, JF and S are the fluid and solid discrete time advancement operators (explicit or implicit), respectively, Sils is the motion increment of the solid domain, and flf,n+1 is a prediction of the position of the solid domain at tn+1. Equation (21) can be varied by further breaking down steps and by employing different time advancement schemes for T and T, or using subcycling (e.g., use smaller times steps for the solid within a global time step). If = fl", then this is known as the Conventional Serial Staggered

(CSS) scheme. It is also possible to calculate pn +1 from a predicted solid domain position

Qn+i.P = Qn + SQn+i,P with.

where Qs is the velocity of the solid domain, and qo and eti are known coefficients controlling the order of time-accuracy of the prediction. Using predictions from Equation (24) has been introduced as the Generalised Serial Staggered (GSS) in Piperno et al. (1995). Second-order time-accuracy of the coupling scheme can be achieved with the Improved Serial Staggered (ISS) scheme (Lesoinne and Farhat, 1998) using the midpoint rule with prediction:

Note that using Equation (25) actually leads an asynchronous method where the fluid solution is technically found for the predicted position of the solid at tn+? and used to advance the solid solution from t" to tn+1.

Explicit partitioned schemes are the most flexible approach, in terms of implementing multiple physical processes relevant to the solid body motion/response. The calculation step that involves the update of the position of the body is not intertwined with the hydrodynamic equation, as in the monolithic schemes, and this allows for the introduction of external forces or solid stresses without any particular complications. It is also evident that this scheme is at the lowest possible threshold in terms of computational resource requirements. This nevertheless comes with the most severe stability issues among any other alternatives, due to the lag between the solut ions of fluid and solid kinematics, which will be discussed further below.

3.2.2 Implicit schemes

Implicit partitioned schemes were developed in order to mitigate some of the issues experienced in explicit schemes, whilst maintaining the flexibility of adding multiple physical processes relevant to the solid body response and motion. The idea is to iterate through Equation (21) until convergence is achieved, up to a user defined tolerance. Common approaches to converge to a solution are Block Gauss-Seidel (BGS), block Jacobi, and block Newton. Using fixed-point formulation, the BGS approach can be compactly defined as minimising the following residual:

where к is the iteration number, S represents the solid solver, and T represents the fluid solver. Because multiple iterations are usually required for convergence of the solution, these schemes are likely to become prohibitively expensive in the context of modelling complex FSI applications. The bottleneck is usually the iteration for the fluid step, which can be very demanding computationally when using highly refined meshes. Note that a single iteration of Equation (26) is equivalent to a fully explicit scheme.

These schemes are less prone to numerical stability issues than explicit schemes as relaxation factors can be used when iterating between the fluid and solid solver (e.g., see Devolder et al. (2015)), but they remain less robust than monolithic schemes.

3.2.3 Semi-implicit schemes

It. is worth mentioning semi-implicit schemes, which are more computationally efficient than implicit schemes, and can be made more stable than explicit schemes. The main constraint of these schemes is that they require segregation of the fluid velocity and pressure equations, which can be achieved using a projection scheme such as the Chorin-Temam scheme (Chorin, 1968; Temam, 1969a,b). This allows to first solve once per time step for the fluid velocity and then to iterate between the fluid pressure and solid solutions. It is essentially equivalent to replacing the full fluid solver T of Equation (26) by a pressure equation. Although these schemes are faster than implicit schemes, they still require significant additional computational cost when compared to explicit schemes due to iterations with the pressure and solid solver, especially for incompressible flows, where the solution of a Poisson equation for the pressure is a bottleneck in terms of computational time. Furthermore, updating the pressure equation but not the velocity field may pose flow continuity issues if large deformations are involved, and may require additional treatment to ensure mass conservation.

Coupling instabilities

As mentioned in the previous section, while monolithic schemes are inherently stable, partitioned schemes may experience stability issues. Within the framework of moving/floating structures, these stability issues are caused by the added mass effect, which is briefly described below. The added mass of a solid submerged in a fluid—resulting from the moving solid accelerating the fluid in its vicinity—depends on the density of the surrounding fluid and on the acceleration of the solid within this fluid. Introducing the added mass matrix, Equation (9) for a single rigid body can be written as:

where M and A are the mass and added mass matrix, respectively, s is the generalised coordinate array, fe is the generalised external force array (e.g., gravity), f/ is the generalised fluid force array, and f f is the generalised fluid force array exempt from added mass contributions. Because CFD models directly calculate f/, the added mass contribution is inherently taken into account. This means A is unknown, leading to a time discretised formulation for explicit partitioned schemes of the form:

where the added mass matrix A cannot be retrieved separately, leading to an added mass force term calculated in t". When using an explicit partitioned scheme, the fluid forces are integrated at a specific position of the solid domain in time-Q”-but when the solid evolves from tn to tn+1, the effect of the change in acceleration of the solid (sn+1 ф s'1) within the fluid is ignored. Ignoring the effect of the added mass while stepping for the solid produces a force term acting opposite to the previous acceleration of the solid. This effect can lead to a diverging solution, due to strong oscillations in the acceleration of the solid. Such instability can clearly be visualised when solving Equation (28) for a few time steps while setting f f and fe as constants over time and with A > M. This instability also applies to implicit or semi-implicit schemes (when no relaxation is used), by replacing the time dependence n of s in Equation (28) by the iteration number k. Note that if this effect is significant, it renders the coupling unconditionally unstable, i.e., increasing the spatial or temporal refinement does not recover stability, as reducing numerical diffusion errors actually destabilises the system even more.

Stabilising simulations prone to the added mass effect has been an active field of research for the past decades. For implicit schemes, relaxation factors can be used to penalise oscillations as in Soding (2001); Yvin et al. (2013, 2014); Devolder et al. (2015). For explicit schemes, Robin- Neumann boundary conditions can be used (Fernandez et al., 2013) or, alternatively, Robin and Dirichlet-Nitsche boundary conditions with a penalty term on the pressure from the fluid solver (Burman and Fernandez, 2007, 2009), or prediction-correction steps (Banks et al., 2017a,b).

In the work presented here, Dirichlet Neumann coupling is used with no prediction- correction steps. Numerical instabilities related to the added mass effect are countered by introducing an accurate estimation of the added mass (following the method presented by Soding (2001)) and using it to take into account the change of acceleration of the solid within the fluid, leading to:

where A is the estimation of the added mass. It is clear that using a good estimation of the added mass—i.e., A « А-leads to having Equation (29) equivalent to the time discretised version of Equation (42), which is unconditionally stable. More details on this technique can be found in de Lataillade (2019).

 
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