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Simulation of incompressible fluid flow-elastic structure interactions is targeted by using fully-Lagrangian mesh-free computational methods.A projection-based fluid model (moving particle semi-implicit (MPS)) is coupled with either a Newtonian or a Hamiltonian Lagrangian structure model (MPS or HMPS) in a mathematically-physically consistent manner.The fluid model is founded on the solution of Navier-Stokes and continuity equations.The structure models are configured either in the framework of Newtonian mechanics on the basis of conservation of linear and angular momenta,or Hamiltonian mechanics on the basis of variational principle for incompressible elastodynamics.A set of enhanced schemes are incorporated for projection-based fluid model (Enhanced MPS),thus,the developed coupled solvers for fluid structure interaction (FSI) are referred to as Enhanced MPS-MPS and Enhanced MPS-HMPS.Besides,two smoothed particle hydrodynamics (SPH)-based FSI solvers,being developed by the authors,are considered and their potential applicability and comparable performance are briefly discussed in comparison with MPS-based FSI solvers.The SPH-based FSI solvers are established through coupling of projection-based incompressible SPH (ISPH) fluid model and SPH-based Newtonian/Hamiltonian structure models,leading to Enhanced ISPH-SPH and Enhanced ISPH-HSPH.A comparative study is carried out on the performances of the FSI solvers through a set of benchmark tests,including hydrostatic water column on an elastic plate,high speed impact of an elastic aluminum beam,hydroelastic slamming of a marine panel and dam break with elastic gate.