This verification problem illustrates the co-simulation feature used to couple ABAQUS/Standard with FLUENT to perform a fluid-structure interaction (FSI) simulation. The problem consists of a slender cantilever beam placed inside a channel with steady, incompressible, laminar flow. For this case a one-way coupling is considered in which fluid pressure along the cantilever beam is computed by FLUENT and is imported into ABAQUS. The problem is simple such that comparison between the numerical and analytical results can be made.

The model consists of a slender cantilever beam inside a channel, as illustrated in Figure 3.17.1–1. The beam length is 1 m, and the thickness is 0.01 m. The depth is considered sufficiently large so that end effects can be neglected and the flow can be considered independent of the z position. A 0.1 m slice of the beam and channel is chosen for this model. The FLUENT model contains two fluid domains that are distinct at one end and merge at the opposite end of the beam: the top channel height is 0.02 m, and the bottom channel height is 0.04 m. The channel cross-section is uniform along the beam.

Mesh:

A two-dimensional model is used. The mesh consists of incompatible mode plane stress elements: 100 elements along the length, and 4 elements stacked in the thickness direction. No mesh parameter studies were performed on the structural mesh. The fluid-structure interface is defined through a surface definition.

The fluid mesh consists of 200 quadrilateral cells along the channel length and 8 cells and 16 cells stacked in the top and bottom channels, respectively. Quadrilateral fluid cells were used since these generally provide better pressure results than triangular fluid cells at the faces.

The structural model uses linear elastic properties with Young's modulus of 1.09 GPa and a Poisson's ratio of 0.3.

The fluid model assumes incompressible flow with a fluid density of 1000 kg/m³ and a dynamic viscosity of 0.001 kg/ms.

Boundary conditions:

The structure is fixed on the inlet end of the channel and free at the outlet end. The velocity inlet flow corresponds to a Reynolds number of 250 in the upper channel and 354 in the lower channel. A pressure outlet with a zero gauge pressure is specified at the outlet, implying that the fluid of the top and bottom channel merge and have the same pressure condition. A fully developed flow is assumed and is specified through the FLUENT user-defined function fsi_channel_2d.c.

Loading:

The fluid flow induces both normal pressure and viscous shear forces on the cantilever. The viscous shear forces are relatively small. The cantilever deforms due to the pressure difference in the top and bottom channels.

Analytical results:

A fully developed flow is assumed through the uniform cross-section channel with an incompressible fluid. Thus, the y-velocity component () and the gradient of the x-velocity component () are zero everywhere; and the governing Navier-Stokes equation for the fluid flow is

Substituting the boundary condition and integrating the Navier-Stokes equation leads to the flow solution for each channel:

The deformation of a cantilever beam subjected to a triangular distributed pressure is given by

The tip deflection due to the flow in each channel is

Units:

The SI unit system is used. ABAQUS does not require that the analysis be run with a particular unit system as long as all properties are specified in a consistent manner. However, the unit system used by ABAQUS must coincide with those used by the third-party analysis code.

The two-dimensional model is identical to the model used for the one-way solution transfer. A three-dimensional model is included and described under this section. In addition, two-dimensional and three-dimensional models with nodal transformations specified at the fluid-structure interface are included.

Mesh:

The three-dimensional structural mesh consists of continuum elements: 100 elements along the length, and 4 elements stacked in the thickness direction. No mesh parameter studies were performed on the structural mesh.

The fluid mesh for the three-dimensional model consists of 200 hexahedron cells along the channel length and 8 cells and 16 cells stacked in the top channel and bottom channels, respectively. Quadrilateral fluid cells were used since these generally provide better surface pressures than prismatic fluid cells.

Boundary conditions:

The boundary conditions are identical to the boundary conditions specified for the one-way solution transfer.

Loading:

The fluid flow over the channel induces both normal pressure and viscous shear forces on the cantilever. The viscous shear forces are relatively small. The cantilever deforms in response to the pressure differential between the flow in the top and bottom channels. The deformations are transferred back to FLUENT, and a new flow solution is obtained. This process is repeated until a steady-state condition is established; specifically, until minor changes in deformation and pressure are observed between consecutive coupling steps.

Analytical results:

The formulation derived under the one-way solution transfer holds only if there is no significant cross-flow; i.e., no flow perpendicular to the cantilever. As the deflection of the cantilever increases, the cross-flow becomes more dominant and, thus, the numerical results deviate from the analytical results.

The problem is identical to the two-dimensional channel problem discussed in the previous sections, with the exception of the time stepping scheme. The rendezvousing scheme is defined through the MpCCI GUI. Specifying a target time period allows ABAQUS to subcycle based on its own time stepping scheme while maintaining exchanges with the third-party code at a fixed frequency. ABAQUS/Standard interpolates the imported loads between the previous coupling step and the target values.

Specifying a rendezvousing time period is typically used for transient simulations. It is being employed for this steady-state simulation purely for illustrative purposes. The rendezvous time period is set at 1.0.