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Coupling strategy

This section proposes a coupling strategy that combines the fluid dynamics and solid dynamics solvers implemented in the FSI framework developed for this work. The solution sequence for each model is presented in Figure 4, along with the main communication lines between models and communicated variables. The sequence is summarized in the list below:

  • 1. Mesh motion model: The fluid mesh is deformed through linear elastostatics, driven by the displacement of the fluid-structure boundaries Tfns which is retrieved from the multibody dynamics solver from the previous time-step. The model returns the mesh velocity um for the ALE formulation to correct advection fluxes in subsequent models that use the deformed fluid mesh.
  • 2. Navier-Stokes model: The model calculates fluid velocities and pressures by solving the Navier-Stokes equations, on the updated mesh topology and using corrected fluxes.
  • 3. Free surface model: For two-phase flow, the new position of the free surface is obtained, based on the updated mesh topology, mesh velocity, and fluid velocities which were calculated by the previous models within the same time-step. Fluid densities and viscosities are also updated according to the new position of the free surface.
  • 4. Added mass model: When instabilities due to added mass effect are present, the added mass is estimated in order to later stabilise the partitioned coupling scheme. The estimation of the added mass increment is based on the method proposed in Soding (2001) and details of the implementation with the context of a CSS coupling scheme are included in de Lataillade (2019).
  • 5. Multibody dynamics model: The new position of bodies are calculated with the hydrodynamic forces and, if included, the added mass estimation. Mooring dynamics are also calculated using a strong coupling with rigid bodies linked to the cables, such as fairleads and anchors. Hydrodynamic forces acting on the cable are taken into account using the fluid velocity solution from the Navier-Stokes model, and collision detection is enabled along the cable.

The coupling interfaces are presented in more details as follows: fluid-structure coupling in Section 4.1, fluid-mooring coupling in Section 4.2, and mooring-structure coupling in Section 4.3.

Workflow of numerical models with current coupling approach (adapted from de Lataillade (2019))

Figure 4: Workflow of numerical models with current coupling approach (adapted from de Lataillade (2019)).

Fluid-structure coupling

4.1.1 Fluid-structure interface

When coupling a fluid problem with a structure problem, new conditions are applied to both problems in order to ensure a well-defined fluid-structure interface Гfns: the continuity of velocity and the continuity of stresses. The continuity of velocity is expressed as:

where U/- and us are the velocity of the fluid and the velocity of the solid at the interface, respectively. The continuity of stresses is expressed as:

where <7/ and cr* are the fluid and solid stresses, respectively, and n is the normal vector to Гfns- For the work presented here, and as is commonly the case for FSI problems, the fluid- structure interface is coupled where the Dirichlet-Neumann method where the continuity of velocity, Equation (31), is imposed as a Dirichlet condition on the fluid velocity, while the continuity of stresses, Equation (32), is imposed as a Neumann condition on the solid. Hydrodynamic forces ff and moments my acting on the structure are integrated along the fluid-structure interface Гfns as:

where x is a point on rfns and n is the normal vector to Ffns.

While a variety of coupling schemes are possible (see Section 3), the partitioned, fully explicit scheme known as the CSS scheme is used for the work presented here. This scheme allows for great modularity, making it possible to use two distinct and highly specialised frameworks: Proteus® for describing the fluid dynamics and Chrono® for solving the structural problem.

4.1.2 Moving boundaries and mesh motion

When mesh-conforming structures are moving within the fluid mesh, the latter must be deformed in order to accommodate this motion while retaining good overall quality and prevent mesh entanglement. This mesh deformation must also be taken into account by all unsteady PDE models that are spatially discretised with the fluid mesh. Internal and external mesh-conforming boundary motion is achieved here by means of the ALE method (Hirt et ah, 1974). According to this method, three different domains-spatial, material and reference—are mapped to each other. These domains correspond to the Eulerian (spatial), Langragian (material) and ALE (reference) frames of reference (see Figure 5). The practical differences with respect to, e.g., temporal evolution of processes are summarised in the following equation, showing the rate of change (time derivative) of a flow variable /:

where um is the mesh velocity, i.e., the rate of mesh node displacement. The mesh velocity is then introduced in the Navier-Stokes equations and other transport models so that advective fluxes take um into account.

The deformation of mesh elements itself is handled here by the method of linear elasto- statics (Dwight and Dwight, 2009):

where f is the body force, and m is the stress tensor that can be expressed in terms of Lame parameters as <7m = Atr (ё) I + 2//ё where I is the identity tensor and ё is the strain tensor expressed as:

where h is the displacement vector, and //. and Л are the Lame parameters: Schematic representation of Lagrangian, Eulerian, and ALE domains on a ID mesh

Figure 5: Schematic representation of Lagrangian, Eulerian, and ALE domains on a ID mesh.

According to this method, the mesh is allowed to deform as a (fictitious) linearly elastic material. Variable stiffness can be used within the mesh according to defined metrics. In the implementation presented here, the Young’s modulus is scaled according to the mesh element volume:

where Eq is a user-defined constant, and Js is the Jacobian of the mesh element £ whose determinant is proportional to the volume of the element. Therefore, in highly refined regions, the elements are stiffer whilst in coarser regions the elements are more flexible. This results in a tendency of the method to maintain the size and shape of more refined elements, which are typically more critical for the quality of the solution, whilst allowing coarser elements to absorb most of the mesh deformation.

While this method maintains acceptable mesh quality for typical engineering FSI that are reasonably short, it was found in de LataiUade (2019) that for long simulations that, e.g., may correspond to a full duration of an extreme storm, mesh quality can deteriorate over time due to multiple nonlinear wave cycles that drive the motion of the floating body. The use of a mesh quality monitor process will very likely address these issues. The application of a mesh monitor function model that deforms the mesh according to target element volumes was explored in (de Lataillade, 2019), based on the techniques proposed by Grajewski et al. (2009). In this work, only the linear elastostatics method is used, as the mesh quality was found to remain acceptable for the whole duration of the simulations presented herein.

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