For computation of the FSI (Fluid-Structure Interaction) problems, the combined formulation (Hesla, 1991) is adopted which incorporates both the fluid and structure equations of motion into a single coupled variational equation so that it is not necessary to calculate the fluid force on the surface of structure explicitly when solving the equations of motion of the structure. Before tackling complex FSI problems, laminar flow around a freely falling cylinder is considered. The Navier-Stokes equations are solved using a P2P1 Galerkin finite element formulation with ALE (Arbitrary Lagrangian-Eulerian) algorithm and Newton’s equations of motion for cylinder are solved. The adaptive mesh refinement technique is also adopted which uses stress error as a posteriori error estimator together with an efficient variable reordering and element-reordering method for unstructured finite element meshes. The newly reordered global matrix has a much narrower bandwidth than the original one, making the MILU (Nam et al., 2002) preconditioner perform better. The cylinder falls oscillating in the transverse direction and rotating about the center. It rotates in the positive rotational direction while it moves to the positive transverse direction and vice versa. The Strouhal number for a freely falling cylinder is lower than that for a fixed cylinder at the same Reynolds number and this is mainly due to the transverse oscillation. As a second FSI problem, laminar channel flow divided by a thin plate is considered and the dynamic response of the plate under the influence of channel flows is studied. For simplicity, we assume a 2-D laminar flow so that the plate can be modeled by a Bernoulli-Euler beam. The numerical simulation results are compared with Wang (1999). As a third FSI problem, oscillations of a vertical plate in resting fluid are studied and the results are compared with Glu¨ck et al. (2001). Finally, an impinging jet flow on a flexible plate is considered. There exists clearly an interaction between the impinging jet and the plate; the plate is deflected due to the impinging jet and the deflected plate then affects the flow field. This fluid-structure interaction continues until the damping by fluid viscosity terminates the vibration of the plate. The frequency response of the plate is different from that in free vibration case because the vibration of the plate is damped by fluid viscosity.

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