Proceedings of the International Symposium on Engineering Lessons Learned fromthe 2011 Great East Japan Earthquake, March 1-4, 2012, Tokyo, JapanTSUNAMI IMPACT ON COASTALSTRUCTURES: PARTITIONED SOLUTIONPROCEDURE FOR FLUID–STRUCTURE
INTERACTION WITH FREE–SURFACE FLOW
A. Ibrahimbegovic1 , C. Kassiotis2 and H. G. Matthies3
January 16, 2012
1

LMT-Cachan, ENS Cachan/UPMC/CNRS/PRES UniverSud Paris, France
ai@lmt.ens-cachan.fr
2
Saint-Venant Laboratory for Hydraulics, Université Paris-Est (Joint Research Unit EDF R&D,
CETMEF, École des Ponts ParisTech)
6 quai Watier, BP 49, 78401 Chatou, France
christophe.kassiotis@enpc.fr
3
Institut for Scientific Computing, TU-Braunschweig, Germany
wire@tu-bs.de
Abstract
In this work we discuss a way to compute the interaction between free-surface flows and nonlinear structures. The approach chosen rely on a partitioned strategy that allows to solve strongly
coupled fluid-structure interaction problem. The software coupling is ensured in an efficient way
using the Communication Template Library (CTL). Numerical examples presented herein concern
2D validations case but also 3D problems with a large number of equations to be solved.

INTRODUCTION
In the 2011 Tohoku Disaster, the huge tsunami, caused by an ocean-ridge earthquake of magnitude 9.0,
was the primary source of the destruction and death toll. Moreover, this disaster is characterized by
the vast affected region and by the chain of events where one disaster (earthquake) triggered another(tsunami). Such chains of events pose many challenges to overcome, and many lessons to learn fromthe 2011 Tohoku Disaster. In particular, the unprecedented risks brought about by combining andsuperimposing the disaster threats requires a new approach to understanding the way of mitigating theconsequences, which is best spelled by the word resiliency that is currently being used in earthquake
engineering community.
In most of the current applications, the fluid loading is applied statically on the construction, taking
into account its dynamics and the interaction effects with only security coefficients. Obviously, the problem is coupled (the more a coastal protection resists, the higher the fluid level, and therefore the loadingis), and there is a need to compute both the resistance of structures and the flow main characteristics,
especially when the failure occurs.
In this work we focus on fluid-structure interaction problems, with free-surface flows. The free-surface
flow problem is described by the Navier-Stokes equations with two phases (water and air), set in an ALE
framework [Farhat et al. , 2001, Demirdžić & Perić, 1988, Hirt et al. , 1997] and discretized with a VOF
strategy [Dutykh & Mitsotakis, 2009, Ubbink & Issa, 1999, Sobey, 1998, Stoker, 1992] For the structure
part, it is natural to follow material point motion in a Lagrangian formulation and a discretization with
FEM [Ibrahimbegović, 2009].
For the coupled problem, the monolithic approach is abandoned in favor of the partitioned approach[Farhat & Lesoinne, 2000], [Le Tallec & Mouro, 2001], [Matthies & Steindorf, 2003], [Causin et al. , 2005],[Fernández & Moubachir, 2005], [Matthies et al. , 2006], [Deparis et al. , 2006], [Förster et al. , 2007] and546

[Fernández et al. , 2008]. The latter is preferred for its modularity and the possibility of re-using existing
software. The partitioned approach used here is based on a simple fixed-point strategy based on the Block
Gauss-Seidel algorithm (DFMT-BGS) with an adaptive relaxation parameter [Küttler & Wall, 2008] thatshows sufficient performances for the example proposed. The properties as well as stability of the implicit
coupling DFMT-BGS algorithms used herein are presented in detail in [Kassiotis et al. , 2010a].
In this work, a general fluid-structure interaction framework based on existing software was used. This
framework was built using the middleware Communication Template Library (CTL) [Niekamp et al. , 2009]
which offers good performances, and can therefore be used for scientific computing of large systems. An
important feature is the possibility to couple types of software product that were initially not programmed to be coupled (here FEAP for the structure and OpenFOAM for the fluid), even if they arebased on different discretization techniques (respectively FV for the fluid and FE for the structure)
and were programmed in different languages, C++ and Fortran. For more details on the implementation,see [Kassiotis et al. , 2010b]
The outline of this paper is as follow: in the subsequent section we present the chosen formulations
for the structure and fluid sub-problem. In Section 3, we describe the coupling between the fluid and thestructure sub-problems. In Section 4, we give and comment the results of illustrative numerical examples
dealing with free-surface flow impacting a structure in two and three-dimensions as well as comparison
with existing works. The concluding remarks are given in the last section.

DESCRIPTION OF THE FREE SURFACE FLOWS AND THE
STRUCTURES
We will not here gives the detailled PDE for the structure and the fluid, but give directly the semi-discrete
form of the problem. For the structure, it can be set in a matrix notation by using the real valued vectors
us ∈ Rnd−o−f that give a discrete approximation of the 3D displacement vector field u s :extRs (us ; λ) := Ms üs + f ints (us ) − f s (λ) = 0

(1)

ext
where M is the mass matrix, f int
the consistent nodal
s with a geometrically nonlinear problem, and f sforces. Here the λ represents the boundary forces computed from the fluid flow problem and imposed on
the fluid-structure interface. In order to complete the discretization process, the time integration of thestructure problem can be carried out by using standard time-stepping schemes [Ibrahimbegović, 2009].
In particular, the Generalized HHT-α method is used herein.
The semi-discrete form of the discretized fluid problem can be written in a matrix form as follows.
The discrete fluid mesh motion considers that um is imposed by the motion of the interface u:

Rm (um ; u) := Km um − Dm u = 0(2)

where Dm is a projection/restriction operator and Km governs the extension of the boundary displacement. The (discrete) volume fraction ι, the 3 components of velocity v and pressure p are coupled througha set of non-linear equations. Written in a matrix forms, it gives the following semi-discrete problem:Rf (ι, vf , pf ; um )Mι ι̇ + Nι (vf − u̇m )ι
 Mf (ι)v̇f + Nf (ι, vf − u̇m )v + . . . := . . . Kf (ι)vf + Bf pf − f f (ι) 
BTf vf=0(3)

where Mι and Nι are the matrices associated to the advection problem of the fluid volume fraction, Mf
is a positive definite mass matrix, Nf is an unsymmetric advection matrix, Kf is the conduction matrix
describing the diffusion terms, and Bf is for the gradient matrix, whereas f f is the discretized nodal loads
on the flow. This matrix form also takes into account the boundary conditions; special care has to betaken concerning the discretization of boundary conditions – and especially normal flux – when using the
Finite Volume Method [Ghidaglia & Pascal, 2005].
One way to solve the flow problem is to consider a monolithic solver handling all equations simultaneously. Another way is to consider a split between the mesh motion, the volume fraction advection,
the momentum and the continuity equations, and to use an operator split-like procedure often referred547

to the segregated approach [Patankar, 1980]. This approach is favored for its computational efficiency
compared to the monolithic scheme. Indeed, even with a simple fixed point iteration strategy its cost is
less important than that of the monolithic approach for large size problems [Ferziger & Perić, 2002]. In
the work presented herein, the segregated approach will be used because of its efficiency.

COUPLING STRATEGY AND IMPLEMENTATION
By enforcing the continuity of primal variables at the interface we can eliminate the energy errors thatcharacterize the explicit interface matching. When coupling incompressible flow with structure, theimplicit interface matching is required for stability reason, as proved in [Kassiotis et al. , 2010a]. Thisought to be done by iterating on the following residual to reduce its value below the chosen tolerance:r N +1 := u s,N +1 − u f ,N +1 ' 0

≤ TOL

(4)

In this way we obtain an implicit algorithm requiring more than one iteration to enforce the interface
matching condition. The chosen order of iterations, corresponds to the Block-Gauß-Seidel algorithm
for fluid-structure interaction problem [Matthies et al. , 2006]. Let us note that not only the value atsynchronization points Tn or Tn+1 , but also the interpolated evolution of variables have to be exchangedin the entire time-interval t ∈ [Tn , Tn+1 ] when the time steps are not matching between fluid and structure
sub-problems.
Contrary to explicit algorithms which generate spurious energy at the interface, the present implicit
interface matching algorithm enforce the same evolution of the primal variables at the fluid-structureinterface. In other words, an iterative solution for primal (displacements) continuity as well as the dual
(forces) equilibrium equations at the interface is performed by using the Picard iteration:


(k+1)
(k)
u N +1 = G u N +1 ;G = Ss −1 ◦ −Sf(5)where Sf and Ss are the Steklov-Poincaré operators for fluid and structure defined as traditionallydefined in [Deparis et al. , 2006]. These operators can be formulated using transfer operators and equations (1), (3) and (2):Ss = Tsλ ◦ Rs ◦ Tsu ;Sf = Tfλ ◦ Rf ◦ Rm ◦ Tsu(6)
where the transfer of structure displacement to fluid-structure interface displacement is Tsu , the transfer
of fluid displacement to interface displacement Tsλ , the transfer of structure stresses to the interface Tfu
and the transfer of fluid stresses to the interface Tfλ .
The Picard iterations will continue until convergence of interface residual is achieved:


(k)
(k)
(k)
(k)
(k)
r N +1 = u s,N +1 − u f ,N +1 = G u N +1 − u N +1
(7)
It is clear that this fixed-point algorithm based on Picard iterations has the main drawback that the
search directions for u and λ variables at the interface do not exploit any information from the fixedpoint function G nor the Steklov-Poincaré operators Sf and Ss . Therefore, quite a few iterations may beneeded to reach the convergence.
The stability of such a coupling algorithm is studied in [Kassiotis et al. , 2010a]. We give a formal
proof of potential numerical instability due to the added-mass effect also observed in [Förster et al. , 2007,
Le Tallec & Mouro, 2001]. In order to improve the convergence of the DFMT-BGS method, we can use
a relaxed update:
(k+1)
(k)
(k)
u N +1 = u N +1 + ω (k) r N +1
(8)
Our favorite choice for constructing ω (k) is using a secant methods which can keep the cost of each
iteration as low as possible. The Aitken’s relaxation strategy has been extensively used in fluid-structureinteraction [Deparis et al. , 2006, Küttler & Wall, 2008], and shown sufficient performances to be used in
the following.
The use of different different solvers, for the fluid and the structure part, do not provide in general amatching mesh at the interface. Furthermore, even for matching meshes, as the geometries of the domains
are not the same on both sides of the interface, an optimal numbering of the nodes can lead to different
orders for the interface nodes. In the examples proposed herein, only this latter point is of interest. Last

548

292
g
292
Ωf
146

,2

Ωf

80

80

Ωs

,1

292
146

146

140

12

286

Figure 1: Three-dimensional water column impacting an obstacle: geometry (given in mm) and boundary
conditions
but not least, different discretization techniques (Finite Element versus Finite Volume) or different order
p of the polynomials can be used for constructing solution to fluid-structure interaction problem. In thedomain of FE applied to mechanical engineering, extensive literature can be found on how to build a consistent interpolation for both subproblems at the interface [Felippa & Park, 2004]. For the fluid-structureinteraction problems, an interesting review can be found in [de Boer et al. , 2007]. In our framework, it
was decided not to favor any mesh-based representation of the interface, since, in the most general case,the fluid problem can also be solved by a meshfree-based method [Dalrymple & Rogers, 2006]. Namely,
an interpolation strategy relying on radial basis function is here chosen. This method has already been
employed for FSI in [Monaghan, 1992, Beckert & Wendland, 2001].
The algorithm presented here is simple to implement. We use for this work the Communication
Template Library (CTL, see [Markovič et al. , 2005, Niekamp et al. , 2009]) that allows to re-use existingcodes in a generic way, either called as libraries on the same computer, or as remote executables throughnetwork. With the CTL, we are able to couple existing stand-alone software, in a quite straightforward
way, even if they are programmed in different langages (Fortran for the structure part, C++ for the fluidpart), and to conserve the inner parallelism of each component. For more details on the implementation,
the reader is invited to see [Kassiotis et al. , 2010b].

NUMERICAL SIMULATION
The problem solved is a 3D generalization of dam-breaking event that brings about a sloshing wave
impact on a rigid [Bullock et al. , 2007, Ubbink & Issa, 1999] or flexible structure as presented in 2D
in [Walhorn et al. , 2005, Baudille & Biancolini, 2005]. At initial time t = 0s, a the three-dimensional
water column starts falling down under the gravity loading and eventually hits the obstacle placed in
the way. The flexible obstacle is a slender plate-like body made of elastic material that can undergo
large deformation. The chosen dimension of the problem, as well as the boundary conditions are given
in Fig. 1. Let us note that we propose to use open boundary conditions far from the obstacle in order toavoid the water bounces-back and hits again the structure after breaking off the walls. For that reason,
only the left and bottom planes of the fluid domain are defined as non-slipping walls, while the others
are defined with boundary condition of atmospheric pressure.
The material properties are chosen as follows: the density and the kinematic viscosity are ρf ,1 =1 × 103 kg.m−3 and νf ,1 = 1 × 105 m.s−1 for the high density fluid (water in the reservoir), versus
ρf ,2 = 1 × 103 kg.m−3 and νf ,2 = 1 × 106 m.s−1 for the low density fluid (air in the remaining part ofthe domain). The mesh motion problem is solved by using a Laplacian smoothing material where thediffusion coefficient is a quadratic inverse function of the distance to the interface between solid and fluid.
The results are computed for two meshes with the chosen discretization and the number of cells given
in Tab. 1. For the finest grid, around 64, 60 and 40 cells are used in e x , e y and e z direction. The mesh549

is refined gradually refined around the structure, and initially the cell dimensions are between 8.9 × 10−8
and 3.2 × 10−5 . For this finest grid, the maximum skewness of the mesh observed is 2.947, that does not
generate too large errors. The fluid is handled by second order space discretization and a Van Leer limiteris used for the advection terms. The time integration scheme employed in this computation is implicit
Euler. For such a scale of modeling it is not required to consider surface tension between the two fluids.
For this problem the fluid computation is parallelized, but reduction of the CPU time is obtained byusing a Generalized Algebraic-MultiGrid (GAMG) linear solver. Note that small time steps are required
for the explicit solution of the phase function indicator equation, as well as the half-implicit nature of
the coupling between the momentum predictor and the pressure corrector.
For the structure part, we propose here to use three-dimensional elements with quadratic shapefunctions, where each element has 27 nodes. The material properties used for the solid are: a neoHookean elastic material with Young’s modulus Es = 1 × 106 P a and Poisson’s ratio νs = 0 and a density
ρs = 2500kg · m−3 , which can represent finite deformation. The time integration is carried out by aGeneralized-α scheme with the same parameters as the one used for the 2D case.

Coarse
Fine

fluid
d-o-f
63 × 103
520 × 103

solid
d-o-f
1.1 × 103
6.6 × 103

number oftime steps
1 × 105
1 × 105

Table 1: Number of d-o-f for coarse and fine discretization of the three-dimensional dam-breaking problem
The computation of the coupled problem (with total number of d-o-f given in Tab. 1), is carried out
by an implicit iterative scheme. The results of fluid and solid computations are matched for a time step of
1 × 10−4 for the coarse and 2 × 10−5 for the fine discretization. The coupling scheme used is DFMT-BGS
with Aitken’s relaxation. The initial parameter is ω = 0.25 and the predictor is of order 1. The absolute
tolerance considered is:(k)kr N k ≤ 1 × 10−6
(9)The number of iterations required to reach the convergence criteria is given in Fig. 2(a). Note that
there is no iteration required before the water hits the structure since the effect of air flow can almost
be deemed negligible with respect to the structure. The number of iterations required for the coarse andfine mesh is small. This propertie is observed for most of the simulation carried out with free-surface flow
when the convergence of the pressure solver is easily reached. For the finest grid, the total time required
to perform the whole coupled simulation on a single 3.0GHz Intel processor is 279 × 103 s.
In Fig. 3, the high density fluid domain is represented, as well as some part of the fluid mesh andthe structure displacement. The first 0.1s of the simulations, the water column falls under the gravity
loading. There is no effect whatsoever on the structure until the high density flow reaches its bottom.
The maximum amplitude of the motion is obtained at t = 0.25s, before the solid comes back to its initial
position and oscillates after the shock.
In Fig. 2(b) the motion of the extremity of the solid obstacle is plotted. Contrary of the twodimensional example, small drops of high density fluid are not interacting with the obstacle after themain shock. Therefore, the motion of the flexible structure remains fairly smoothed and it is rather welldescribed with the coarsest grid. We present for comparison the results obtain with 2D models using
either a monolithic stabilized-FEM strategy [Walhorn et al. , 2005] or tight coupling strategy between
a FV and FEM solver [Baudille & Biancolini, 2005]. The use of 2D models ad well as the difference
in strategy explain the differences observed (our results are closer for a 2D simulation, as presented
in [Kassiotis et al. , 2010c]).

CONCLUSION
The proposed solution method allows to perform coupled simulations and obtain reliable solution tocomplex fluid-structure interaction by using the existing codes, that were initially developed to support
either fluid or structure motion computation. This is achieved thanks to the use of the component technology [Kassiotis et al. , 2010b, Niekamp et al. , 2009] providing the coupling between existing software
products. Therefore, the proposed solution method for fluid-structure interaction can utilize very different discretization strategies to obtain the optimal accuracy; The case in point concerns FE for the

550

5

Iteration number

4
3
2
1
0

0

0.2

0.4

0.6

0.8

1

Time (s)(a) Number of iterations in order to make the DFMT-BGS algorithm converge for the threedimensional dam-breaking problem

5b4

b b b b
b

b

bb

b
b

bb

b
b

b
bb

Displacement (cm)coarse meshfine mesh[Walhorn et al. , 2005][Baudille & Biancolini, 2005]

b

bb
b

b
b
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bb

3

b

bb
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b

b b b b bb

bbbbb bbb
bb

b

bbb b

2

bbbb
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b b
b b bb
bb

b bb

1

b
bbb

bbb b
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0

b b bb b bb

bb

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b
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bbb

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bbb

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b b b b b b b b bb b b b b b b b b b b b b b b b
b
b
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b b

b
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-1

bb
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0

0.2

0.4

0.6

b

b

0.8

1

Time (s)(b) Three-dimensional dam break example: obstacle displacement in e x direction of center
point of the top face (40; 6; 80) and comparison with 2D models [Baudille & Biancolini, 2005,
Walhorn et al. , 2005]

Figure 2: Results of three-dimensional simulations

551

Time t = 0.3s
Time t = 0.6s

Time t = 0.2s
Time t = 0.5s

Figure 3: Tri-dimensional dam break problem. Evolution of the free surface and motion of the structure.

552

structure and FV for the fluid. The use of these popular methods for the fluid and solid parts allows tobenefit from the advanced features of the two families of methods, each developed by the experts from thecorresponding domain. Accordingly, on the fluid side, it is possible to use a very efficient semi-implicit
solver for incompressible flow (PISO), inverse techniques (Algebraic Multigrid) or advanced models forfree-surface flows. A very good performance of the proposed technology for fluid-structure interaction is
illustrated with 2D and 3D models for dam breaking examples, which also involve flexible obstacles.
Even if the convergence is quite easy to achieve, the use of the recently proposed semi-implicit couplingstrategy [Astorino et al. , 2009] based on the predictor-corrector work traditionnaly used for pressure
correction could by a good way to decrease the computational cost without loosing accuracy. An otherpoint of interest is the extension of this work to more complex behavior, for both the fluid (turbulence,
non Newtonian flows, debris flows. . . ) and the structure (non-linear behavior, cracks and failure).

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