Asian Research Journal of Mathematics

Fluid-structure interaction (FSI) couples fluid motion with structural deformation and is central to problems in aeroelasticity, marine and civil engineering, rotating machinery, and biomechanics. The mathematical difficulty arises because the fluid and solid occupy interacting domains, exchange traction and velocity at a moving interface, and may require a strongly coupled numerical solution when feedback between the two fields is significant. This paper develops a self-contained finite-element framework for an incompressible Newtonian fluid interacting with an elastic structure. The formulation is expressed in an arbitrary Lagrangian-Eulerian description for the fluid and a Lagrangian description for the solid. The governing Navier-Stokes and structural momentum equations, interface compatibility conditions, weak forms, nondimensional groups, mesh-motion requirements, and partitioned and monolithic coupling options are specified in a form suitable for implementation. Particular attention is given to numerical stability, including the added-mass difficulty in partitioned FSI, and to verification and validation procedures that separate discretisation error from agreement with physical or benchmark data. A benchmark matrix is proposed for time-dependent flow past a cylinder, vortex-induced vibration of an elastically mounted cylinder, and flexible-plate flutter. Error measures are defined without presenting ungenerated numerical results. The framework is intended as a reproducible methodological basis for subsequent computational studies and for extensions to turbulence, reduced-order modelling, and multiphysics FSI.