Keywords: fluid-structure interactions, transport equations, moving mesh.

Improvements in the fidelity of coupled physics models have been driven by the demand for increased predictive capability, and are allied with performance improvements in computer hardware. An important example is fluid-structure interaction (FSI) which, despite significant development, still lacks a generic, robust and accurate framework suitable for industrially relevant problems. Regardless of whether the governing equations are synchronous, staggered, monolithic, or partitioned, several challenges arise from solving the fluid and solid equations in a coupled framework. While the partitioning of a solution algorithm into (say) a finite volume method and a finite element method produces independently efficient schemes, their coupling can be suboptimal and prone to instability. The matching of moving-interface conditions can be problematic in an Eulerian reference frame, leading to numerical problems in the fluids domain, and non-matching meshes at the fluid-solid interface add to the problem of accurate data transfer. Although several algorithms have been introduced to address these issues, work in this field continues with the aim of developing methods to ensure conservation and stability.

Numerical analysis of FSI problems necessarily involves a spatial model that represents the fluid and solid, a common interface, a time-stepping scheme, and additional algorithms. This paper is concerned with a review of some of the existing numerical methods. More specifically, different techniques are examined in the light of their computational efficiency, numerical stability, potential for dealing with moving interfaces, and ability to handle non-matching, independently evolving, mesh partitions. The methods considered include mesh moving strategies as well as those founded on an Eulerian description. In addition, meshless methods purposely developed to overcome some of the problems with mesh-based techniques are examined.

The paper also discusses a recently developed method that aims formally to unify the fluid and solid domains and so reduce the challenges that exist in tackling FSI problems. The proposed method is a monolithic formulation of the fluid and solid equations based on weighted transport forms of the governing equations. The approach makes use of discontinuous elements to provide a finite-volume like structure that is applicable to both solid and fluid alike. All transport equations are established on a moving control volume to provide the necessary flexibility required for handling moving interfaces. A subdivision strategy is employed to deal with gaps and overlays at the common fluid-structure interface. Overall, the fluid and solid equations are treated identically, and the differences arise when the nature of the materials are taken into account in the constitutive equations. The feasibility of the weighted formulation is demonstrated via its application to a simple example.

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