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

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Violent Fluid-Structure Interaction simulations using a coupled SPH/FEM method

G Fourey1, G Oger2, D Le Touzé1 and B Alessandrini1

1

2 Fluid Mechanics Lab., Ecole Centrale Nantes / CNRS, Nantes, France HydrOcean, Nantes, France

E-mail: guillaume.fourey@ec-nantes.fr

Abstract. The Smoothed Particle Hydrodynamics (SPH) method presents different key assets

for modelling violent Fluid-Structure Interactions (FSI). First, this method is a meshless

method, which drastically reduces the complexity of handling the fluid-structure interface

when using SPH to model the fluid and coupling it with a Finite Element Method (FEM) for

the solid. Second, the method is Lagrangian and large deformations of the fluid domain can

thus be followed, which is especially interesting for simulating violent interactions in presence

of a free surface, or which induce large deformations, rotations, and translations of the solid.

Third, the SPH method being explicit, the time scale of the SPH resolution in the fluid domain

is naturally adapted to the FEM resolution in the solid. Free-surface FSIs can also be simulated

without including the air phase when it does not play a significative role. For violent

interactions where the fluid compressibility matters, it is also intrinsically modelled by the SPH

method. The paper details the SPH method used and the coupling. The FEM solver is a

standard open source solver for solid mechanics. Validation test cases are then presented in

detail. They include the hydrodynamic impact of elastic wedges at high speed, where local

pressures and wedge deformations are compared to experimental data.

1. Introduction

Nowadays, thanks to the growth of computing power numerical simulation is widely used to study complex physical problems involving fluids or solids. The modeling of coupled phenomena such as fluid-structure interactions can be carried out as well even though their simulation needs important numerical resources. In literature different methods have been developed in order to predict the deformations of a structure due to non-stationary loads caused by fluid flow, itself modified by the structure evolution. The complexity of the coupling and its simulation is much increased when both the fluid and the structure are largely deformed due to their interaction. We concentrate here on these complex fluid-structure interactions.

In a general way Lagrangian formulations based on the Finite Element Method (FEM) are employed to study the structure behaviour. The mesh follows solid deformations. However, different methods which allow following the moving interface between fluid and solid are used to simulate the fluid evolution when dealing with fluid-structure interaction problems. We can cite mesh-based methods such as Lagrangian or Arbitrary Lagrangian Eulerian (ALE) formulations of Navier-Stokes equations, often using the Finite Volume Method (FVM) or FEM [1], where the fluid mesh is deformed to adapt to the solid domain deformation. This adaptation is costly and of complex implementation. Further, large fluid domain deformations cannot be handled without using remeshing

methods which significantly increase CPU costs of simulations. Another way to solve this problem is to use Volume of Fluid method (VOF) or Level-Set methods which permit to track interfaces on fixed grids [2]. With all these methods, a contact algorithm is used in order to prevent materials interpenetration. Fluid structure simulations have also been performed using the Boundary Element Method (BEM) [3]. In the latter case, only boundaries need to be meshed. However, this is limited to simplified problems in terms of geometry (of the free surface especially) and physics (no vorticity nor viscosity).

Particle-based methods have also been used to model fluid evolution in the context of Fluid-Structure Interactions. Is for instance the case with the Material Point Method in [4], or with the Particle Finite Element Method (PFEM) in [5]. However these methods are not fully meshless.

The method proposed here to simulate fluid-structure interactions relies on the Smoothed Particle Hydrodynamics method (SPH) [7][8][9] resolution of the fluid problem. The latter method being Lagrangian and meshless, no free surface tracking techniques and contact algorithm between fluid and structure domains are needed. In addition, compressibility effects are taken into account when they matter. In this paper, we first present an introduction to the SPH formalism. Then, the way we couple this method to a standard FEM solver for the solid is described. Validation is made on the hydrodynamic impact of an elastic wedge in comparison to the analytical solution. An illustrative test case is then presented where a water column impacts an elastic plate resulting on a strong interaction with a very complex flow pattern.

2. SPH solver

In 1977, the SPH method was proposed by Monaghan, Gingold and Lucy [10][11]. The aim was to simulate complex astrophysical phenomena such as star formations. Later, Monaghan used this method to simulate free surface flows [12].

2.1. Governing equations

As we focus on fast dynamics cases (regarding the fluid, the solid answer being of slow dynamics), viscosity does not play a significant role so that we model the Euler equations which read in Lagrangian formulation

rdxr=v (1) dtrdvr P=g (2) ρdtdρr= ρ .v (3) dt

Since the fluid is assumed compressible, an equation of state is needed in order to close the system of equations above. The fluid is considered to follow the barotropic Tait’s equation of state 72 ρ ρ0c0 1 (4) P P0=7 ρ0

in which ρ0, P0 and c0 respectively denote the nominal density of the fluid, the pressure of reference and the nominal speed of sound.

2.2. Functions interpolation

SPH is a mesh-free method; no connectivity list is needed to perform simulations. Approximation of functions and their derivatives is done by using a kernel function. In this study we used the cubic

spline kernel proposed by Monaghan [13]. We note r the distance between two particles and h the smoothing length. r

2213 ssif 0s1 +≤< 3 2r r 1 W s= =C (2 s)3 if 1≤s<2 (5) h 6 0 else

Constant C ensures kernel normalization, it is equal to 15/7πh2 for two dimensional simulations and to 3/2πh3 for three dimensional ones. The smoothing length is a measure of the spatial resolution. The approximation of values taken by a function is obtained according to the following formula, where D is the compact support of the kernel function rrrr f(

y)=f(x)W(y x)dV (6) ∫D

2.3. Discrete approximation

If we note ωi the volume of a calculation point i, called ‘particle’, and N the number of particles that are included in the domain D, then we get approximations of integrals of functions thanks to the rectangle quadrature formula

D∫Nrrf(x)dV=∑f(xi)ωi (7) i=1

In all the simulations the mean distance between two calculation points is Δx=h/1.23, leading to neighborhoods composed of about 20 particles in 2D, and 50 in 3D.

Noting mi=ρiωi we can now write the discretized scheme which is used to solve the Euler equations

rdxir=vi (8) dtr Pi+Pj dvirrr=g ∑mj Bij W(xi xj) (9) ρiρj dtj mjrrdρrr=ρi∑(vi vj)Bij W(xi xj) (10) dtjρj

Bij=The matrix Bij represents the symmetrized form of the renormalization matrix

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 compressibility effects are present in the problem. We thus perform the fluid-structure interaction simulation with the real velocity of sound in water: c0=1500 m.s-1 at 20°C.

A visualization of this hydro-elastic impact simulation is presented in figure 8. Naturally, when the structure strikes the free surface, strong pressures appear in the flow. The beam undergoes these pressures, resulting in its deformation. In particular, it bends during the jet formation. When the latter

is released, the pressure applied to the structure quickly decreases and the structure gets back.

Figure 5. Displacement of Figure 6. Total vertical force Figure 7. Pressure seen at the

neutral fiber at the middle of the time history applied to the four pressure sensors. beam. beam.

As can be seen in figure 5, 6 and 7, there is globally and locally a great agreement between the results of the coupled SPH/FEM simulation and the analytical solution. A slight undervaluation of the total vertical force can be noticed though. Pressure peaks are correctly reproduced, especially at

locations C and D.

Figure 8. Pressure (fluid) and stress (solid) during the wedge impact.

4.2. Water column impacting an elastic plate

In order to further show the potential of the method, an illustrative case is here presented. It deals with the impact of a water column on a vertical elastic plate. Height and width of the water column are respectively fixed to 1.5 m and 75 cm. Height and thickness of the elastic plate are set to 1 m and 4 mm. We kept the same material properties for the structure as those used for the wedge impact case. The sound speed is fixed to 120 m.s-1 which ensures that the fluid is simulated as quasi-incompressible. Space resolution is defined by a constant smoothing length of 3mm. Hooke’s law is still used to relate strain to stresses. The fluid-structure interaction evolution is presented on figure 9. After 0.75s water reaches the structure which slightly bends under the action of the fluid and water runs out on the beam along its

direction. Then, the vertical velocity of the fluid decreases and the beam keeps bending under the

weight of water leaning on it. When part of the jet is released from the beam to go to impact the lower and the right walls of the tank, one can observe the back motion of the elastic structure, throwing away water that was remaining on its left part. Finally, the beam vibrates until finding back its initial state. One can note the large complexity of the flow and its free surface, underlining the assets of the SPH method to simulate such violent fluid-structure interactions.

Figure 9. Evolution of the fluid-structure interaction between a water column and an elastic plate.

5. Conclusion

The present paper presents a fluid-structure coupling between an SPH method for the fluid and a FEM for the structure. This coupling is very efficient when dealing with fluid-structure interaction problems in presence of a free-surface, and of rather easy implementation. The ability of SPH to fragment and reconnect interfaces presents a great interest when modelling impacts of solids on fluids, and vice versa. As exhibited in the example of validation test presented, a wedge water entry, good agreement is found with the analytical solution. SPH has no difficulty to model water jets and no contact algorithm is needed to avoid material interpenetration. The capabilities of this coupling for simulating violent fluid-structure interactions are further illustrated on a demonstrative case where the flow and the structure display very large deformation.

Further work will be to validate the method for three dimensional test cases, and to apply it to realistic complex cases.

Acknowledgments

Authors wish to acknowledge DGA (Direction Générale de l’Armement) from French ministry of Defense for financial support of this study. The research leading to these results has also received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement 225967 ‘‘NextMuSE’’.

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