Computational modelling

A numerical study is conducted to confirm the experimental results at least qualitatively and to assess the effects of various parameters on FSI and impact loading. For the numerical study, 2-D modeling and analysis is conducted. Even though the experiments are 3-D, it is not believed the qualitative behavior would differ between 2-D and 3-D cases. The 2-D analysis is computationally much less time-consuming, especially with the FSI model containing a large domain of fluid.

The numerical model has a composite beam which has density 2000 Kg/ m3 and elastic modulus 50 GPa. The beam is 400 mm long, 20 mm wide, and 2 mm thick. Therefore, the composite has approximately twice density of water and is 40% lighter than aluminum and one quarter of steel. On the other hand, the elastic modulus is about 70% and 25% of those of aluminum and steel, common structural metals. As a parametric study, both density and elastic modulus of the beam are changed.

The computational model consists of both structure and fluid. Finite element formulations are developed for the FSI study. The beam is modeled using the Euler-Bernoulli beam theory. Because the beam is assumed to be very thin, the transverse shear deformation energy is negligible. The finite element matrix equation for the beam as well as for any general structure is expressed below:

[ Ms ]{d }+[Cs ]{d }+[Ks ]{d} = {Fe}+{Ffs} (2)

where [Ms], [Cs], and [Ks] are the mass, damping and stiffness matrices of the structure, {d} is the nodal displacement vector, {Fe} and {Ffs} are the force vectors from external loading and the fluid loading, respectively, In other words, the fluid loading comes from the FSI. Superimposed dot denotes the temporal derivative.

in which c2 = B / p0 and c is the speed of sound. In order to apply Eq. (10) to FSI problems, the velocity at the FSI interface boundary is computed from the structural dynamics, i. e. Eq. (2), and it is applied to the wave equation through Eq. (7). On the other hand, the fluid pressure is computed from the wave equation using

The acoustic wave equation is processed using the Galerkin method to formulate the finite element matrix equation. The resulting matrix equation for the wave equation is expressed as

-1

^[Mf ]{ }+[Kf ]{0} = {Ff} (12)

where

[ Mf [ H ]T [H ]dQ (13)

[Kf ] ^ [V H]T [V H]dQ (14)

{Ff} =Jrf [ H ] undr (15)

Here, [H] the a vector of shape functions. For the present analysis, four-node quadrilateral elements are used for the fluid acoustic model. In addition, Qf and rf are the fluid domain and boundary, respectively, and un is the fluid velocity normal to the boundary.

For an FSI application, Eqs. (2) and (12) are solved in a staggered matter. For example, the structural analysis is conducted using Eq. (2). Then, the structural velocity at the FSI boundary is computed. From the velocity compatibility condition, both structural and fluid velocities must be the same at the FSI boundary. For an inviscid flow, only the normal velocity components are considered for the compatibility condition. As a result, Eq. (15) is computed from the fluid velocity at the FSI interface and the fluid analysis is performed from Eq. (12). From the fluid analysis, fluid pressure is computed at the FSI interface. From the force equilibrium, the fluid pressure is used as an applied force to the structure. This completes one cycle and the whole process continues as the time increases. For an efficient computation, the explicit time integration technique was used for both structural and fluid analyses. For example, time integration of fluid analysis is conducted as below and similarly for structural analysis.

 [Kf ]{0} = [Mf ]-1 {{Ff}t -[Kf ]{0}f} (16) At At {} + 2 ={} 2 +At{0 }t (17) – At {0}t + ={0}t +At{0} 2 (18)