Violent Fluid-Structure Interaction simulations using a coupled SPH-FEM method(6)

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.

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