Violent Fluid-Structure Interaction simulations using a coup(2)

1Bi+Bj) (11) (2rrrrBi=∑(xj xi) (xi xj)ωj (12)

j

This renormalization permits to locally increase approximations of gradients. The symmetrized form used allows using Riemann solvers to stabilize the explicit centered scheme presented above. Following Vila [14], it is possible to rewrite the SPH formalism in a slightly different manner inspired from the Finite Volumes formalism. The pairs of particle to particle interactions (i,j) in equations (9) and (10) can be seen as the result of a flux acting at the middle of them, leading to rdmivirrrrrr=mig ωi∑ωj2ρEvE vE v0(xij)+pEBij W(xi xj) (13) dtj()

where v0is a transport velocity (Arbitrary Lagrangian Eulerian framework), equal to the fluid velocity rrvin a Lagrangian framework. The terms ρEand vEare the solution of the exact Lagrangian Riemann problem solved at the middle of each pair of particles using a Godunov numerical scheme [14], in which variables are extrapolated thanks to a linear approximation with a limiter through the Monotone Upstream-centered Schemes for Conservations Laws (MUSCL) [15] scheme.

2.4. Temporal integration

An explicit fourth order Runge-Kutta scheme is employed for temporal integration. We note c the velocity of sound in the considered fluid. Thus, the time step has to respect the Courant–Friedrichs–Lewy (CFL) condition rdmirrrr= ωi∑ωj2ρEvE v0(xij)Bij W(xi xj) (14) dtj()Δt<kCFLh (15) c

This leads to very small time steps. However, if compressibility effects are negligible in the fluid, then we can use an artificially decreased sound speed provided we always remain in the weak-compressibility zone (Ma<0.1). In addition, in order to decrease significantly the simulation time the SPH solver we have developed, SPH-flow, is parallel, based on a domain decomposition strategy, the inter-process communications being achieved with the use of MPI (Message Passing Interface) libraries. The explicit feature of SPH provides many advantages concerning the parallelization of the model since each task is relatively independent from the other, resulting in very good parallelization performance. In recent years different SPH solvers have been parallelized and run on massive parallel clusters, see e.g. [16][17][18]. The solver SPH-flow which we have developed exhibits a superlinear scalability between 1 and 32 processors in two dimensions, and good properties in 3D [16]. To reach high scalability figures load balancing procedures have been implemented, see [16]. When applying it to Fluid-Structure Interaction problems, the solution in the solid computed via a FEM method is obtained vey quickly by using one core on only, while the flow evolution is computed on many cores. The global scalability is almost unaffected.

2.5. Boundary conditions

There is no specific treatment to do in order to satisfy free-surface conditions with this SPH scheme

[9]. However, at solid boundaries free-slip and non penetration conditions have to be imposed. To do so we use the ghost particle technique to complete the support kernel of a particle which is close to the boundary [19]. As shown in the next figure, a ghost particle is created for each particle which distance

to the boundary is less than two smoothing lengths.

Figure 2. Ghost particles for SPH solid boundary conditions.

If we note Vi and Vp, respectively, the velocities of particle i and of the boundary, then the rr

velocity of the ghost particle is given by

VGin=2Vpn Vin (16)

in normal and tangential directions.

3. SPH/FEM coupling

To perform our solid mechanics simulations we use a standard FEM open source solver called Code_Aster, developed by Electricité De France (EDF). The coupling algorithm used here is parallel; fluid and structure evolutions are calculated at the same time. Time step is based on the fluid calculation but it is also possible to run several fluid time steps while running only one, bigger, for structure. Similarly, if necessary, a subdivision of time steps of the SPH simulation can be achieved in order to reach convergence for the structure calculation.

In order to couple SPH with a finite element code, we have to be able to get pressure on the solid boundary. Different methods can be employed to do this. Here we simply sample pressures of particles which are closed to the boundary as shown in figure 3. VGit=Vpt (17) Pboundary1=N∑P (18) i

i=1N

The knowledge of this loading permits to perform FEM simulations. Once performed, new positions and velocities of the solid boundaries are known and a new time step can be run for the fluid

solver.

Figure 3. Local pressure evaluation

at the boundary using a sampling.

In practice, several processors are assigned to compute the fluid evolution, and one more processor executes the FEM solver.

4. Validation and illustrative cases

4.1. Violent hydrodynamic impact of an elastic wedge

We consider here the water entry of an aluminium beam with high velocity impact. Comparisons are made with analytical data [20] on deformations of the beam (figure 5), on the vertical force applied to the structure (figure 6) and on pressure peaks evaluated at four pressure sensors (figure 7). The size of these sensors is 5 millimeters. The smoothing length used for the fluid calculation is set to 2 millimeters.

Figure 4. Geometry of the aluminium beam and locations

of pressure sensors.

Relation between strain and stresses follows Hooke’s law. The beam presents the following characteristics: Young’s modulus E=67.5 GPa, Poisson’s ratio υ=0.34 and density ρ=2700 kg.m-3.

When running this simulation with a rigid solid, we observed density variations higher than one percent of initial given density for water (ρ0=1000 kg.m-3). This means that non-negligible …… 此处隐藏：5832字，全部文档内容请下载后查看。喜欢就下载吧 ……