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Development of a coupling strategy between Smoothed Particle Hydrodynamics and Finite Element Method for
violent fluid-structure interaction problems
C. Hermange, David Le Touzé, G. Oger
To cite this version:
C. Hermange, David Le Touzé, G. Oger. Development of a coupling strategy between Smoothed Particle Hydrodynamics and Finite Element Method for violent fluid-structure interaction problems.
3rd International Conference on Violent Flows 2016, Mar 2016, Osaka, Japan. �hal-01658067�
Development of a coupling strategy between Smoothed Particle Hydrodynamics and Finite Element Method for violent fluid-structure interaction problems
Corentin Hermange, David Le Touzé and Guillaume Oger
LHEEA Lab. (ECN/CNRS), Ecole Centrale Nantes, Nantes, FranceAbstract
This paper focuses on a fluid-structure coupling applied to free surface flows through the use of a Lagrangian particle method and a classical meshed structure. We focus here on a coupling strategy between Smoothed Particle Hydrodynamics (SPH) and Finite Element Method (FEM). On the fluid side, the SPH method is well adapted to model complex free surface flows. On the solid side, the classical FEM method is used for its accuracy and stability properties.
In this work a weak coupling strategy is presented. The scheme can be used with any kind of SPH or FEM method and is relatively easy to implement. This paper highlights the use of implicit schemes for structure to preserve coupling stability and computational time.
Interesting stability and computational efficiency properties will be also shown. Finally, coupling validations are performed and discussed on various 2D analytical and experimental test cases.
Keywords: Smoothed Particle Hydrodynamics; Finite-Element Method; Fluid-structure interaction; Coupling; Stability
1 Introduction
Fluid-structure interaction problems arise every time when the motions of a fluid and of a solid are somehow coupled. Fluid-structure interaction effects need to be pre- dicted and eventually mitigated in a large domain of appli- cations: aerospace engineering, nuclear engineering, ocean engineering, biomechanics, car industry and even food processing. As a consequence, different methods have been developed in order to model these coupled phenomena that cannot be neglected due to strong reciprocal effects.
For the last decades, the numerical works have been dominated by mesh based methods. In the fluid domain, several methods can be used such as Finite Volume Meth- od (FVM) or FEM. For structure modelling, FEM is clas- sically adopted [4]. Fluid meshes follow the solid defor- mations and need to adapt to the solid domain. Neverthe- less violent fluid-structure interaction problems might in- duce large deformations of the fluid domain which lead to the need of re-meshing tools. However, in order to handle two meshes under large deformations, mesh adaptation usually requires large CPU costs due to its complexity. The computational cost is therefore greatly increased.
This work focuses on SPH-FEM coupling strategy to model violent fluid-structure interactions involving com- plex free-surface flows with deformable structures. On the solid side, a classical FEM method is used, since it has been proven in terms of accuracy and stability. On the fluid side, a SPH method is used in order to easily consider complex free surface flows. This method has been intro-
duced at the end of the 70’s by Gingold and Monaghan [13] and Lucy [20]. As a Lagrangian particle meshless method, SPH method naturally avoids the problem of mesh distortion at fluid-structure interface: neither contact algo- rithms between fluid and solid, nor free surface tracking algorithms are needed. This feature significantly reduces the complexity of fluid-structure interface treatment. With this coupling strategy both the two subdomains are mod- elled with a purely Lagrangian formulation. Moreover, if desired compressible effects can be taken into account which is particularly useful for high velocity impact prob- lems.
Various SPH-FEM couplings have already been per- formed since the 90’s by Johnson [16] and Attaway et al.
[3]. Most popular strategies are based on a master-slave coupling, in which contact forces are computed to prevent the penetration of SPH particles into FEM elements [5][14][31]. Each of them has their own specificities, FEM nodes can be considered as SPH particles like in [8]. Then Fourey et al. [11][12], Yang [30] and more recently by Li et al. [17][18][19] proved SPH-FEM coupling capacity to model complex FSI without contact algorithms.
Here, an analysis of a SPH-FEM coupling method from the stability point of view is carried out. According to Fahart et al.[10] and Piperno et al.[23][24] energy conser- vation and information transfers are primordial for cou- pling performances, accuracy and stability. Coupling method thus needs to be improved, to build more robust strategies regarding solver parameters together with their combinations. In their work, Li et al. [17][18][19] proposed Proceedings of 3rd International Conference on Violent Flows (VF-2016)
9-11, March 2016, Osaka, Japan
a coupling strategy enforcing the conservation of energy at the interface. However, this technique requires the knowledge of the FEM model, and thus needs to be adapted to the structure solver and to the specificity of the problem to be considered. Here we propose a coupling strategy independent of the FEM solver as the one used by Yang [30]. From the coupling strategy point of view the structural software is used as a black box, receiving fluid loads and returning solid positions and velocities. Yang analyzed the coupling behaviour according to spatial reso- lutions. In this work we propose to introduce a sufficient dissipation of high frequency responses in the structure solution in order to stabilize the coupling. Indeed high fre- quency fluid loads tend to harm the coupling stability [11].
This paper is organized as follows : Section 2 introduc- es the SPH and FEM models. In Section 3 a SPH-FEM coupling strategy is proposed with its different parameters.
Finally in the last section, academic test cases from Scolan [26] and from Antoci [1] are proposed in order to analyze the stability and CPU properties of the coupling.
2 Governing equations
2.1 SPH method
Here we consider a weakly compressible fluid ap- proach. As we focus on fast dynamics flows, viscosity do not have a large influence on the physics. In the SPH for- malism, derivatives are approximated using a kernel func- tion [13], W. Here a Wendland kernel [29] function is used.
The fluid field is described by a set of particles; each of them carries the discrete fields values within the fluid volume. In the rest of this study, we use the Riemann-based formulation introduced by Vila [28]. Our Riemann scheme is defined by the following discrete system for a particle i:
j
j ij i i j i
i v v W
dt d
, ).
(
2
(1)
j
j i ij i ij e e i
i v v x W
dt d
, .
) (
2
(2)
( ( ))
. ,
2
j
j i ij i d e ij e e e i
i
i v v v x PI W
dt v
d
(3)
where x , v
, ρ, P are respectively the position, velocity, density and pressure in the fluid, respectively. Pe and
ve
are the exact Riemann problem solutions between parti- cle i and j. ω is the volume of a particle and
2 / ) ( )
(xij vi vj
v
. For closing the set of equations, Tait equation is used as the equation of state:
. 1 ) ( 7
7
0 2 0 0
i i
c
P (4)
A weakly compressible approach is used in this study for the fluid. The time advance is performed explicitly us- ing the 4th order Runge-Kutta scheme. The time step should therefore respect the CFL condition:
, C0
CFL R t
(5)
where R is the kernel radius and CFL is the Courant num- ber taken as CFL=0.375. This feature leads classically to very small time steps.
2.2 FEM method
FEM solves differential equations characterizing the physics of the structure in a discrete way. The great ad- vantage of this method compared to others relies on accu- racy and stability properties and also on its capability to represent complex geometric structures, materials, etc... As long as the structure undergoes large deformations without distortion of the mesh, the dynamics can be translated into a Lagrangian description of motion. In this section particu- lar attention is paid to the temporal integration schemes.
For more informations about the Finite Elements theory, the reader may refer to the papers such as Bathe [4].
The choice of the time integration scheme is a key point in the Finite Element method. Several models can be used:
implicit or explicit schemes. For fast dynamics, explicit scheme are classically preferred to implicit ones. Actually, the time integration scheme not only acts on the structure only but also on the coupling stability. As a consequence we propose to test different time integration schemes in the structure solver in order to obtain the most stable and ro- bust coupling strategy. Here we focus on two classical im- plicit schemes, Newmark and Hilber-Hughes-Taylor (HHT).
Using Newmark implicit scheme, the dynamical equi- librium at instant tn1can be expressed at each mesh node as:
1.
1 2
1 2
n extn
n
F u
K dt
u M d
(6) Supposing that all the variables are known at instant tn, the new equilibrium solution is obtained by using the Newmark scheme:
, )
( 2
1 2 2 2
2 2 2
1 1
dt u d t dt
u d t dt
u d t u u
n n
n n
n
(7)
, )
1
( 2
1 2
2 2 1
dt u d t dt
u d t dt
u d dt u
d n n n n
(8)
The scheme stability depends on the values of parame- ters β and γ. The scheme is unconditionally stable with γ ≥ 0.5 and β ≥ 41(21 )2. A second order convergence is expected with γ = 21 , without numerical dissipation in the solid. When γ > 21 a numerical dissipation is introduced, and it is actually needed to stabilize the solution if the structure is excitated by high frequency loads. This is par- ticularly useful to couple with SPH method which usually presents high frequency acoustic noise. However, this dif- fusion also affect the low frequency domain, which may harm the solution accuracy.
In order to reduce high frequency oscillations without affecting the low frequency domain the Hilber-Hughes- Taylor scheme have been proposed by Hilber et al. [15].
This approach introduces an offset within the equilibrium equation:
, )
1 ( )
1
( 1 1
2 1 2
n ext n ext n
n n
F F u
K u K dt
u d M
(9)
with α ϵ]-1/3,0], related to γ =0.5-α and β = 0.25(1-α)2. This offset increases the diffusion on the high frequen- cy part but reduces the dissipation in the low frequency part while ensuring a second order convergence. It can be no- ticed that for α = 0, the HHT scheme corresponds exactly to the Newmark one with γ = 21 and β = 41 .
3 Coupling strategies
We aim at developing a coupling strategy allowing to deals with complex fluid dynamics with a solid part which can encase unsteady loads. As a result we use classical par- titioned approaches. In addition, such strategy needs to be robust and compatible with any kind of FEM method and software. Here the deformable body behaviour is modelled by the FEM theory using an open-source generalist soft- ware, Code_Aster develop by EDF (www.code-aster.org).
The fluid part is modelled using SPH-Flow code jointly developed by Ecole Centrale Nantes and HydrOcean (www.sphflow.com).
3.1 Treatment of deformable body
From the SPH algorithmic point of view, at instant t a deformable body is considered as rigid, but with specific local deformation velocities (and node positions) provided by the FEM solver. It is possible to choose any boundary conditions, ghost particle or Normal Flux Method [7]. No special treatment is prescribed for the deformable body interface. As a result, no contact algorithm dedicated to avoid material interpenetration is needed. The pressure loads supplied to the FEM structure is computed using the SPH fluid pressure solution extracted from the near-boundary area. This pressure loading Ppanel is calcu- lated as an average of all particles seen by the wet body panel (Fig.1) within a distance R from the panel, where R is the kernel radius:
N
Aa i N i
panel P
P 1 , (10)
where N is the number of particles within the averaging area, Aa. The choice of the distance R can be explained by his relation with the fluid resolution, indeed at particle spacing convergence we will get an average pressure on local particles which will become closer to the wall. It is important to achieve a proper average pressure to smooth irregularities near the wet body panel according to Oger et al.[21], as such irregularities may be responsible for cou- pling instabilities. It is necessary to have a sufficient num- ber of particles for each panel, but the size of each panel must also be adapted in order to capture load variations.
The influence of the number of particles per panel on cou- pling stability will not be treated in this paper but could be the purpose of a further work.
Fig 1 Averaged pressure calculation on a wet body panel j
3.2 Coupling algorithms
In this study, a Conventional Parallel Staggered (CPS) procedure will be used for parallel algorithm (Fig. 2). The time step is the same for the fluid and solid solver. As the fluid time step is expected to be smaller than the solid one, SPH-Flow imposes its time step to Code_Aster. In practice the fluid solver sends the time step value and pressure loads to FEM solver and receives node positions and velocities.
Then both software works simultaneously to the next time step.
Fig 2 CPS coupling algorithm
The scheme is described in Farhat et al. [10]. This algo- rithm does not fit exactly with the implicit nature of the solid resolution but reduces CPU costs. Indeed, the calcula- tion time of fluid and solid solvers are hidden by a syn- chronous processing. Nevertheless the coupling time step is assumed small enough so that the error is acceptable. It is then important to establish if it corresponds to the best compromise in terms of stability/CPU time.
Recently Li et al. [18] proposed a specific coupling strategy with energy conservation treatment at the interface in order to preserve numerical stability. This technique is based on the FEM model, and therefore need to be adjusted to each structural solver and problem. Here we have a weak-coupling strategy, which is very flexible, easy to im- plement and compatible with any kind of SPH schemes and boundary conditions. Moreover, the solid solver is consid-
ered as a black box from the coupling strategy receiving fluid loads and returning solid positions and velocities, thus any FEM software can be used. No additional CPU time is introduced as no interface system treatment is required.
The quality of our results will be analysed with respect to Li et al. [18] and Yang [30].
4 Numerical results
Characteristics of our coupling strategy were detailed in the previous sections. We present here some applications of this coupling strategy to specific test cases in order to vali- date our coupling strategy. Here we propose to proceed by increasing physical complexity. Two bi-dimensional test cases are considered in order to highlight the stability and robustness of our coupling strategy.
4.1 Deformable beam impact
In this section we now consider a test case with a very high dynamic perfectly suited to the SPH method with a conventional aluminum structure easily modelled by FEM:
the free surface impact of a deformable beam structure [26]
with high velocity (see Fig. 3).
Fig 3 Deformable beam impact configuration The beam is made of aluminum as summarized in Tab.
1. The structure impacts the free surface which is initially at rest with an inclination of 10 degrees. Motions are im- posed on the two vertical boundaries with a vertical veloc- ity of 30 m/s. The structure mesh is composed of 4 width-wise and 40 length-wise elements. The tank is suffi- ciently large to avoid acoustic wave reflexion onto the de- formable body. In order to limit the number of particle in- volved in this test case variable space resolution (varia- ble-h) is used by concentrating the finest resolution in the impact area as proposed by Oger et al. [21] (Fig. 4).
Table 1 Physical and numerical parameters for the de- formable beam impact
L e
Young Modulus E Poisson Coefficient ν ρbeam
ρwater C0
CFL number R/∆xSPH
0.6 m 0.04 m 67.5 Gpa
0.34 2700 kg/m3 1000 kg/m3 1500 m/s
0.375 3.0
Fig 4 Pressure field at t = 0s
Fig. 5 gives the midpoint displacement of the beam for different fluid resolutions. For ∆xSPH = 1 mm, results are close to the semi-analytical solution proposed by Scolan [26] which combined a hydrodynamic Wagner model to a linear model of elasticity for thin shells. The deformation of the plate increases in the first instance, in accordance with the increase in pressure. It nevertheless showed that the deformation of the plate at its midpoint finally appears slightly lower in the case of SPH-FEM coupling than in the analytic case. This difference can be explained by the non-consideration of the jet in the analytical solution, which is the reason why it last ends when the bottom of the structure is completely wet. Moreover, we have also a good agreement of the vertical force applied on the deformable structure for ∆xSPH = 1 mm, Fig. 6.
Fig 5 Time history of center point relative displacement of the deformable beam for different fluid resolution, New-
mark α = -0.3
Fig 6 Time history of vertical force on the deformable beam for different fluid resolution, Newmark α= -0.3
Then by varying the parameter α, we observed that in- creasing the diffusion has no influence on the solution for α
< -0.05 for Newmark scheme. Nevertheless for the HHT scheme α <-0.1 is needed for coupling stability (Fig. 7).
Increasing the diffusion through FEM time integration schemes helps to return to a stable coupling. CPS proce- dure needs diffusion to maintain the stability. It appears that the use of HHT scheme with low diffusion leads to destabilization. However, the use of Newmark scheme helps counter this problem.
Fig 7 Time history of vertical force on the deformable beam for different damping coefficient, ghost particle
method and ∆xSPH = 1 mm
Here the advantage of using an implicit integration scheme (and not explicit) lies in the fact that the coupling time step is not obliged to respect structure CFL condition.
In the work from Li et al. [19], the coupling time step is very small (t = 1,5 E-7 s) contrary to here (t = 7.0 E-7 s) with a smaller ∆xSPH. Indeed, the CFL condition is highly restrictive in light of the speed of sound in the material.
Note that the speed of sound in the material is linked to the material properties so that it cannot be decreased, contrary to water for which the weakly-compressible feature can be used. The CPU time, which is already important due to SPH method would therefore be significantly increased.
Fig. 8 and Fig. 9 show the pressure field for Newmark and HHT schemes with α=-0.05. Using HHT scheme, the introduced numerical diffusion becomes insufficient. As a result the high frequency displacement of the deformable body interface generates acoustic waves in the fluid pres- sure field. It highlights the superiority of Newmark proce- dure for coupling stability. This is especially noticeable after t = 2ms.
Fig 8 Pressure field at t=2 ms, Newmark α = -0.05 and
∆xSPH = 1 mm
Fig 9 Pressure field at t=2 ms, HHT α = -0.05 and
∆xSPH = 1 mm
We have seen that through the use of diffusive time in- tegration scheme that it is possible to obtain a stable cou- pling strategy without use an interface energy treatment, Li et al. [19]. A relatively large computation time can thus be saved using CPS algorithm. No influence of the increase of the diffusion is ascertained. Thus, even for the most im- portant introduced dissipation and corresponding to the use of the Newmark scheme with α = -0.3, the frequencies of interest are not affected. There is no loss of information.
Indeed once it is stable almost all these curves are similar.
We consider this case test as a quite easy problem for our coupling strategy. Indeed, rigid structures impact prob- lem do not leads to major difficulties in SPH method. The explicit Lagrangian feature without mesh used to perfectly deals with this problem where the dynamics is predomi- nant. Furthermore, the structure is made of a relatively hard material, deformations remain low in spite of a significant pressure loading.
4.2 Dam break flow through an elastic gate
Now we propose to perform a case where the flow dy- namics is slower. The fluid is contained in a rigid tank which has an elastic gate on the left. Initially, the water is at rest and the gate made of rubber is not deformed. At the instant t = 0s the elastic gate is released to let the water escape. This test case has been introduced by Antoci [1], according to the configuration presented in Fig. 10 and Tab. 2. In their study, Antoci et al. used a SPH-SPH cou- pling model to simulate this FSI problem and compared it to their experimental data [2].
Fig 10 Dam-break configuration
Table 2 Physical and numerical parameters for the dam-break through an elastic gate
L H e
Poisson Coefficient ν ρrubber
ρwater
∆xFEM
∆yFEM
C0
CFL number R/∆xSPH
79 mm 0.14 m 5 mm
≈0.5 1100 kg/m3 1000 kg/m3 1.975 mm
1.25 mm 30 m/s
0.375 4.0
Fluid particles are distributed uniformly to a finite depth throughout the right of the gate, Fig. 11. Finite ele- ments are used to model the elastic gate. The mesh is composed of 4 width-wise and 40 length-wise elements.
Rubber behaviour is considered as incompressible with a Poisson coefficient close to 0.5.
Fig 11 Pressure field at t = 0s
A non-linear deformation law obtained by experimenta- tions is used to model the rubber gate behaviour (Fig. 12).
Fig 12 Deformation law
Fig. 13 and 14 compares the experiment and numerical time history of the displacements of the gate tip for a diffu- sion value α = -0.3 in the Newmark scheme. Numerical results are in good agreement with experiments.
Good agreements are also observed for the coarsest res- olution (∆xSPH = 1 mm). In particular, the global trend of the gate deformation time history is captured. This offset decreases by the use of finer resolutions. The abrupt de- crease of the displacement observed experimentally from instant t = 0.32s is not reproduced by our coupling, but can be attributed to the leakage of fluid in the non-observable dimension modelled on photographs from the experiments.
Shapes and positions of the gate and free surface position show good agreement with experimental data, Fig. 15. The shapes of the curves obtained numerically are consistent with those recorded experimentally. We still found an un- derestimation of the water level which persists even at convergence.
Fig 13Time history of the horizontal (top) and vertical (bottom) displacements of the gate tip for different fluid
resolutions, Newmark α = -0.3 scheme
Fig 14 Time history of the horizontal (top) and vertical (bottom) displacements of the gate tip for different fluid
resolutions, Newmark α = -0.3 scheme
Fig 15 Time history of the water height for different fluid resolutions behind the gate, Newmark α = -0.3
Fig. 16 shows the displacement of gate tip for different schemes and diffusion coefficient values. Note that using the CPS algorithm with a HHT scheme α = -0.1 the cou- pling is not stable. On the contrary coupling solutions using Newmark scheme is stable and identical for all α values tested.
Fig 16 Time history of the horizontal (top) and vertical (bottom) displacements of the gate tip for different solid
time integration schemes and ∆xS PH = 0.25mm
Once more, CPS procedure has difficulties to maintain coupling stability. Fig. 17 .and Fig.18 show pressure fields at with a low diffusive scheme for both implicit scheme, highlighting one more time the stability feature of New- mark procedure. Using HHT scheme, the deformable body displacement generates acoustic waves in the fluid pressure field.
Fig 17 Pressure field at t=0.24s, Newmark α = -0.05 and
∆xSPH = 0.25mm
Fig 18 Pressure field at t=0.24s, HHT α = -0.05 and
∆xSPH = 0.25mm
It was also shown that a gain in CPU time is possible for this problem by simultaneously performing the SPH and FEM calculations. The use of diffusive time integration scheme leads to counter the loss of precision inherent to CPS algorithm. Vibration modes of the sought structure, essentially the first, are not affected by the use of more dissipative temporal integration schemes. This result is not surprising according to scheme characteristics discussed in the previous chapter. Indeed, the time step used is small compared to the period of vibration of the first mode of the structure, they return similar solutions at low frequencies.
For the problem studied in this section, only low fre- quency vibration modes are sought, particularly the first one. Therefore, the diffusion introduced through the time integration schemes to maintain a stable coupling does not affect the solution. Interesting convergence and stability properties were shown in terms of displacement of the structure and free surface position. Furthermore, a good agreement was found between the solution provided by our model and the experimental results, closer than those ob- tained by Yang [30]. Our weak coupling strategy provide similar results to those presented by Li et al. in [19] with- out any energy treatment at the interface. It is then possible to maintain result accuracy and to reduce the computation time using larger time step through the solid implicit fea- ture and without additional coupling procedure.
5 Conclusion
An analysis of a coupling between SPH method for the fluid and FEM for the solid has been provided from a sta- bility point of view. A weak coupling strategy has been chosen to couple these two methods. The proposed cou- pling method can be used with any kind of SPH or FEM method, relatively easy to implement. It appears that the present method is robust to SPH and FEM parameters, showing that it is compatible with implicit structural algo-
rithms allowing large time step values. Interesting proper- ties have also been shown for the CPS algorithms regarding the compromise stability/CPU time. In all validations the coupling strategy presents good agreements with analytical and experimental results, and also with the literature. SPH is clearly well adapted to model water jets on deformable body. An extension of the present study for more practical applications is still needed for further understanding of this coupling strategy.
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