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Euler–Lagrange coupling with damping eﬀects:Application to slamming problems
N. Aquelet
a,*
, M. Souli
a
, L. Olovsson
b
a
Universite´ des Sciences et Technologies de Lille, Laboratoire de Me´ canique de Lille, UMR CNRS 8107,Bd. Paul Langevin 59655 Villeneuve d’Ascq, France
b
Swedish Defense Research Agency, S-147 25 Tumba, Sweden
Received 17 March 2004; received in revised form 13 January 2005; accepted 13 January 2005
Abstract
During a high velocity impact of a structure on a nearly incompressible ﬂuid, impulse loads with high-pressure peaksoccur.Thisphysicalphenomenoncalled
slamming
isaconcerninshipbuildingindustrybecauseofthepossibilityofhulldamage.ShipbuildingcompanieshavecarriedoutseveralstudiesonslammingmodelingusingFEMsoftwarewithaddedmass techniques to represent ﬂuid eﬀects. In the added mass method inertia eﬀects of the ﬂuid are not taken into accountandareonlyvalidwhenthedeadriseangleissmall.Thispaperpresentsthepredictionofthelocalhighpressureloadonarigid wedge impacting a free surface, where the ﬂuid is represented by solving Navier–Stokes equations with an EulerianorALEformulation.Theﬂuid–structureinteractionissimulatedusingacouplingalgorithm;theﬂuidistreatedonaﬁxedor moving mesh using an ALE formulation and the structure on a deformable mesh using a Lagrangian formulation.A new coupling algorithm is developed in the paper. The coupling algorithm computes the coupling forces at theﬂuid–structure interface. These forces are added to the ﬂuid and structure nodal forces, where ﬂuid and structureare solved using an explicit ﬁnite element formulation. Predicting the local pressure peak on the structure requiresan accurate ﬂuid–structure interaction algorithm. The Euler–Lagrange coupling algorithm presented in this paper usesa penalty based formulation similar to penalty contact in Lagrangian analyses. Both penalty coupling and penalty con-tact can generate high frequency oscillations due to the nearly incompressible nature of the ﬂuid. In this paper, a damp-ing force based on the relative velocity of the ﬂuid and the structure is introduced to smooth out non-physical highfrequency oscillations induced by the penalty springs in the coupling algorithm.
2005 Elsevier B.V. All rights reserved.
Keywords:
Fluid–structure interaction; Euler–Lagrange coupling; ALE formulation; Hydrodynamic impact; Slamming0045-7825/$ - see front matter
2005 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2005.01.010
*
Corresponding author.
E-mail addresses:
nicolas.aquelet@ed.univ-lille1.fr, nicolas_aquelet@yahoo.fr (N. Aquelet), mhamed.souli@univ-lille1.fr (M.
Souli), larolo@foi.se (L. Olovsson).Comput. Methods Appl. Mech. Engrg. xxx (2005) xxx–xxxwww.elsevier.com/locate/cma
ARTICLE IN PRESS
1. Introduction
Themainconcerninﬂuid–structureinteractionproblemsisthecomputationoftheﬂuidforcesthatactona rigid or deformable structure. In shipbuilding industry, empirical solutions and handbook calculationshave been used to estimate forces due to hydrodynamic eﬀects. The hydrodynamic forces are applied asexternal loads on the structural dynamic model to predict failure, stress fatigue, and creep damage to thestructure.Sinceforcecalculationsrelyonanaccurateknowledgeoftheﬂowﬁeld,anaccurateﬂuid–structureanalysis requires skilled engineering talent and careful modeling. The application of ﬂuid–structure interac-tion technology allows us to move beyond handbook design techniques which use generic experimentalcorrelation, to correctly predict the hydrodynamic forces by solving the hydrodynamic equations and usingan appropriate coupling algorithm to communicate forces between ﬂuid and structure for dynamicequilibrium.In the structural design of ship hulls, an estimation of slamming loads is important to avoid substantialdamage on the forebody. For instance, the probability of slamming on the fore part of high-speed vessels inrougher seas is signiﬁcantly high and therefore, wave impacts can damage the bow. Because of the practicalimportance of the slamming study in shipbuilding engineering, several investigations have been carried out.The former work is due to Von Karman (1929) [1] who developed an asymptotic theory for ﬂat impactproblems with linearized free surface and body boundary conditions. The impact load on a sea plane duringlanding was estimated by the force on a two dimensional wedge upon entry into calm water, neglecting thewater surface elevation. But this idealized theory based on momentum conservation underestimates the im-pact load for wedges with small deadrise angle, which represents the initial incidence angle between the freesurface and the structure. The Von Karman formula was modiﬁed by Wagner (1932) [2] to account forpiled-up water on the wedge. This improvement makes it possible to obtain a theory that is generally wellsuited to solve the problem of entry of 2D-wedges into water, if the deadrise angle is small. Wagner
s pres-sure formula is singular on the edge of the expanding plate. To prevent this problem, the eﬀect of the non-linear jet ﬂow at the interface between the wedge and the free surface was included by Watanabee (1986) [3]by matching the solution in the splash region to the expanding plate solution of Wagner. In 1996, at smalldeadrise angle, Zhao et al. [4] developed an analytical solution for the pressure distribution on the body bygeneralizing the work of Wagner [2]. Within the framework of a study carried out by a research laboratoryfor shipbuilding, this paper compares the results obtained by Zhao
s theory [4] with the numerical resultsobtained using a ﬂuid–structure coupling algorithm with damping formulation for high frequency modes.This was developed and implemented in LS-DYNA, an explicit ﬁnite element code for general ﬂuid–struc-ture interaction problems.In the ﬂuid–structure coupling algorithm, two superimposed meshes are considered, a ﬁxed Eulerian orALE mesh for the ﬂuid and a deformable Lagrangian mesh for the structure. Unlike existing algorithmsthat couple two separate codes, a CFD and a structure code, the ﬂuid–structure interaction algorithmdeveloped in the next section is fully coupled.The main purpose of this paper is to describe the ﬂuid–structure coupling algorithm, which computescoupling forces at the ﬂuid structure interface. In this paper, both ﬂuid and structure problems are solvedusing an explicit time integration method, which are suitable for high impact problems. For some problems,the coupling problem generates high frequency oscillations, which disturb the coupling forces. Dampingforces computed from relative velocities of ﬂuid and structure are introduced in a coupling algorithm inorder to damp out high frequency oscillations. Similar damping forces have already been introduced forcontact problems. In contact problems, the slave and master meshes geometrically deﬁne the contact inter-face, whereas in the ﬂuid–structure coupling method developed in this paper, the ﬂuid coupling interface isdeﬁned by the material surface.In the ﬁrst section of the paper, the governing equations for the ﬂuid and structure are presented togetherwith boundary conditions. In Section 2, a detailed description of the Euler–Lagrange coupling algorithm is
2
N. Aquelet et al. / Comput. Methods Appl. Mech. Engrg. xxx (2005) xxx–xxx
ARTICLE IN PRESS
presented together with the description of a regular penalty contact algorithm. The damping formulationimplemented in the coupling is described in the last part of this section. In Section 3, the improvements tothe Euler–Lagrange coupling are validated in a simple test problem, a column of ﬂuid compressed by anelastic piston. The ﬂuid–structure interaction in this example is modeled using two diﬀerent methods, anEuler–Lagrange coupling algorithm and a classical Lagrangian formulation using a tied contact at theﬂuid–structure interface. Since the piston problem is a simple one, the ﬂuid mesh distortion is not severeand so the approach of the problem by a Lagrangian formulation is reliable. The solution of the couplingproblem is compared to the Lagrangian solution, which is considered to be a reference solution. In the sec-ond part of Section 3, the asymptotical matching of the pressure presented by Zhao et al. [4] is described.The analytical pressure from Zhao et al. is compared to the numerical results obtained by the Euler– Lagrange coupling algorithm with damping. Finally, the slamming modeling is described and the numericalresults are compared to existing theoretical results.
2. Description of ﬂuid and structure problems
The ﬂuid is solved by using an Eulerian formulation on a Cartesian grid that overlaps the structure,while the structure is discretised by a Lagrangian approach. For simplicity, the numerical simulations inthis paper have been restricted to an Eulerian formulation for the ﬂuid, although the formulation can beextended to an ALE formulation.The Eulerian formulation is a particular case of the ALE ﬁnite element formulation. Thus a generalALE point of view is ﬁrst described for Navier–Stokes equations. The structure is assumed rigid to allowfor comparisons with Zhao
s theoretical solution, however, the method is not restricted to rigidstructures.
2.1. General ALE description of Navier–Stokes equations
In the ALE description of motion, an arbitrary referential coordinate is introduced in addition to theLagrangian and Eulerian coordinates. The total time derivative of a variable
f
with respect to a referencecoordinate can be described as Eq. (1)d
f
ð
~
X
;
t
Þ
d
t
¼
o
f
ð
~
x
;
t
Þ
o
t
þ ð
~
v
~
w
Þ
~
grad
f
ð
~
x
;
t
Þ
;
ð
1
Þ
where
~
X
is the Lagrangian coordinate,
~
x
is the ALE coordinate,
~
v
is the particle velocity and
~
w
is the velo-city of the reference coordinate, which will represent the grid velocity for the numerical simulation, and thesystem of reference will be later the ALE grid. Thus substituting the relationship between the total timederivative and the reference conﬁguration time derivative derives the ALE equations.Let
X
f
2
R
3
, represent the domain occupied by the ﬂuid particles, and let
o
X
f
denote its boundary (Fig.1). The equations of mass, momentum and energy conservation for a Newtonian ﬂuid in ALE formulationin the reference domain, are given by
o
q
o
t
þ
q
div
ð
~
v
Þ þ ð
~
v
~
w
Þ
grad
ð
q
Þ ¼
0
;
ð
2
Þ
q
o
~
v
o
t
þ
q
ð
~
v
~
w
Þ
grad
ð
~
v
Þ ¼
~
div
ð
r
Þ þ
~
f
;
ð
3
Þ
q
o
e
o
t
þ
q
ð
~
v
~
w
Þ
~
grad
ð
e
Þ ¼
r
:
grad
ð
~
v
Þ þ
~
f
~
v
;
ð
4
Þ
N. Aquelet et al. / Comput. Methods Appl. Mech. Engrg. xxx (2005) xxx–xxx
3
ARTICLE IN PRESS
where
q
is the density and
r
is the total Cauchy stress given by
r
¼
p
Id
þ
l
ð
grad
ð
~
v
Þ þ
grad
ð
~
v
Þ
T
Þ
;
ð
5
Þ
where
p
is the pressure and
l
is the dynamic viscosity. Eqs. (2)–(4) are completed with appropriate bound-ary conditions. The part of the boundary at which the velocity is assumed to be speciﬁed is denoted by
o
X
f 1
.The inﬂow boundary condition is
~
v
¼
~
g
ð
t
Þ
on
o
X
f 1
:
ð
6
Þ
The traction boundary condition associated with Eq. (3) are the conditions on stress components. Theseconditions are assumed to be imposed on the remaining part of the boundary
r
~
n
¼
~
h
ð
t
Þ
on
o
X
f 2
:
ð
7
Þ
One of the major diﬃculties in time integration of the ALE Navier–Stokes equations (2)–(4) is due to thenonlinear term related to the relative velocity (
~
v
~
w
). For some ALE formulations, the mesh velocity canbe solved using a remeshing and smoothing process.In the Eulerian formulation, the mesh velocity
~
w
¼
~
0, this assumption eliminates the remeshing andsmoothing process, but does not simplify the Navier–Stokes equations (2)–(4).To solve equations (2)–(4), the split approach detailed in [5,6] and implemented in most hydrocodes is
adopted in this paper.Operator splitting is a convenient method for breaking complicated problems into series of less compli-cated problems. In this approach, ﬁrst a Lagrangian phase is performed, using an explicit ﬁnite elementmethod, in which the mesh moves with the ﬂuid particle. In the CFD community, this phase is referredto as a linear Stokes problem. In this phase, the changes in velocity, pressure and internal energy due toexternal and internal forces are computed. The equilibrium equations for the Lagrangian phase are
q
d
~
v
d
t
¼
~
div
ð
r
Þ þ
~
f
;
ð
8
Þ
q
d
e
d
t
¼
r
:
grad
ð
~
v
Þ þ
~
f
~
v
:
ð
9
Þ
The mass conservation equation is used in its integrated form Eq. (10) rather than as a partial diﬀerentialequation [7]. Although the continuity equation can be used to obtain the density in a Lagrangian formu-
lation, it is simpler and more accurate to use the integrated form Eq. (10) in order to compute the currentdensity
qq
J
¼
q
0
;
ð
10
Þ
where
q
0
is the initial density and
J
is the volumetric strain given by the Jacobian:
Fig. 1. Fluid domain.4
N. Aquelet et al. / Comput. Methods Appl. Mech. Engrg. xxx (2005) xxx–xxx
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