I am not an expert in fluid structure interaction, but I think it is fair to say that the main difficulties in this kind of problem are represented by the coupling of the fluid and structure time discretizations. As a consequence, as long as the method employed for space discretization is able to handle complex meshes (and all those you mention are able to do so, obviously with different specific features) probably the main difficulties will come from the time discretization, in particular the choice between different splitting approaches or monolithic approaches. A review of these issues can be found, among many others, in these papers

Hübner, B., Walhorn, E., & Dinkler, D. (2004). A monolithic approach to fluid–structure interaction using space–time finite elements. Computer methods in applied mechanics and engineering, 193(23-26), 2087-2104.

Hou, G., Wang, J., & Layton, A. (2012). Numerical methods for fluid-structure interaction—a review. Communications in Computational Physics, 12(2), 337-377.

Heil, M., Hazel, A. L., & Boyle, J. (2008). Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches. Computational Mechanics, 43(1), 91-101.

@ Lucas Bonaventura Thanks for your detailed insightful response and references. It is well appreciated. Cheers!!!

As correctly pointed out by @Luca Bonaventura, the main challenge is in resolving the fluid-solid coupling.

Monothic schemes are better in terms of stability but they are the most difficult to implement, particularly when using blockbox solvers for fluid and structural problems. So, partitioned approaches where solid and structural problems are solved separately via some data transfer between the two are preferred. Partitioned approaches suffer from instabilities. There are quite a few partitioned schemes with varying degrees of accuracy and stability limits.

The type method for the flow problems one can use (FVM, FDM, FEM, SEM etc.) is secondary, although the implicit or explicit nature of the CFD scheme can have some effects on the FSI scheme.

I have worked on different types of FSI coupling schemes. Would be happy to discuss these FSI schemes. Message me.

As a general comment about the FSI methodologies, at present the trend is the Immersed Boundary Method. The genesis of this formulation is in the historical series of Peskin papers. Different formulations have beenrecently developed and presented in the literature. The IBM method can be implemented in FD, FV, FEM numerical formulations.

@ Chennakesava Kadapa and Filippo Maria Denaro . I really appreciate to insightful comments. Quite elucidating...

Fluid-structure interaction (FSI) problems can be solved in a monolithic or partitioned way. Loosely or strongly coupled partitioned solution procedures solve the fluid problem separately from the structure problem; consequently this approach is popular due to its modularity and simplicity of implementation. In classical partitioned approaches, known as Dirichlet-Neumann (DN) schemes, the fluid problem is solved with a Dirichlet boundary condition (structure velocity) at the fluid-structure interface while the structure problem is solved with a Neumann boundary condition (fluid stress) at the interface. Loosely coupled partitioned DN schemes are stable only if the structure density is much larger than the fluid density. This requirement is easily achieved in some applications like aerodynamics, but not for example in hemodynamics where the density of blood is of the same order of magnitude as the density of arterial walls. In these cases, the energy of the discrete problem in the DN partitioned algorithm does not accurately approximate the energy of the continuous problem, introducing numerical instabilities known as the added mass effect. A possible solution to this problem is to sub-iterate between the fluid and structure sub-problems at each time step until the energy at the fluid-structure interface is balanced. Schemes that require sub-iterations, also known as strongly coupled partitioned schemes, may suffer from slow convergence issues, however this can be mitigated using Aitkens dynamic relaxation method or reduced order models[1].

[1] Zeljko Tukovi´c, Martina Bukaˇc, Philip Cardiff, Hrvoje Jasak, Alojz Ivankovi´c, “Added mass partitioned fluid-structure interaction solver based on a Robin boundary condition for pressure”, May 2018.

@ Ideen Sadrehaghighi Wow! Your comment is highly insightful. Thanks for your invaluable contribution.

Fractional relaxation

high order godunov type for both CFD and CSD with exact Riemann's solver for FSI