Introduction

Fluid-structure interaction problems are numerous in nature and engineering. Just think of the blood flow in the arteries and veins, the beating of the wing of the birds, and the rolling motion of the Wright Brothers Flyer obtained through wing-warping. In fluid mechanics, the Arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations is unquestionably the most used to describe these problems. The Navier-Stokes equations and the elasticity equations are solved together to predict the forces on the solid and the dynamic of the flow.

The domain of application often dictates the level of interoperability between the different solvers involved in the simulation. For example, modeling the interaction between a submerged buoy and the flow of water requires monolithic calculation of the fluid's degrees of freedom, the solid's distortions, and the displacements of the mesh's nodes to ensure the calculations stability and convergence. Another similar application involves modeling the interaction of submerged pipelines from the well to the offshore platform. The usual weakly coupled approach, in which the fluid's degrees of freedom and the structure are solved sequentially, can not solve this type of problem.

All ALE formulations have either algebraic or partial differential equations to update the mesh to the motion of the solid. In these formulations, the convective acceleration of the flow is a function of the fluid and the mesh velocities. While the pseudo-solid method is certainly the most prevalent and has been successfully applied to monolithic and loosely couples simulations of fluid-structure interaction problems, algebraic methods are easier to implement and are widely used in aeroelasticity or when the fluid-structure interaction problem can be solved sequentially. 

Contribution of this study

In order to further develop the algebraic methods and test their properties and limitations, we present a new algebraic approach for updating fluid-structure interaction meshes. Like any other algebraic methods in this category, it relies on an interpolation method to propagate the deformation of the fluid-solid interface into the fluid computational domain. Specifically, for this purpose we chose the transfinite mean value interpolation (TMI) method, developed in the early 2000s, under the leadership of Professor Michael S. Floater and his colleagues at the University of Oslo. 

This interpolation method possesses the following interesting intrinsic properties:

- It is linearly exact, similar to some instantiations of the radial basis function (RBF) method. It can therefore represent all rigid motions accurately (translationt and rotation) without deforming the mesh, whereas the inverse distance-weighted (IDW) method can only represent exactly rigid displacements. 

- It is explicit and matrix-free like the IDW method, but, unlike the RBF method, it does not require the solution of an algebraic system to obtain the interpolation coefficients.

In this paper, we intend to focus on the mathematical relationship between the RBF method, the IDW method and the TMI method. They will then be meticulously compared to determine their properties, their fields of application and their respective limitations. Indeed, by applying different transformations (translation, rotation or deformation) to simple meshes, we will get the most out of each method.

Preliminary results

Through our mesh updating process, we note that the connectivity is unchanged and that we further allow the nodes to move freely on the cylinder surface. Thus, the ALE formulation of the Navier-Stokes equations allows us to describe the dynamics of the flow at any points of the computational domain under deformation or discretely at the nodes of the mesh of this domain. Furthermore, the role played by the kinematic relationship in updating the time-dependent position of the nodes at t^{n+1} from the previous time step at t^n, as it usually appears during the simulation.