INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2000; 00:1–6
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Partitioned Vibration Analysis of Coupled Internal
Fluid-Structure Interaction Problems
José A. González1,∗ and K. C. Park2,3
1 Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, E-41092 Seville, Spain.
2Division of Ocean Systems Engineering, School of Mechanical Engineering, KAIST, Daejeon 305-701,
Republic of Korea.
3 Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of
Colorado, Campus Box 429, Boulder, CO 80309, USA
SUMMARY
A partitioned variational formulation for the vibrations of bounded fluid-structure systems proposed
by Park et al. [25] taking into account the combined effects of sloshing, acoustic waves and initial
pressurization, is implemented for its performance evaluation. The systems are modeled using finite
elements and coupled with localized Lagrange multipliers, and the resulting analysis software module
c 2000
has been applied to the problem of a homogeneous liquid in a closed container. Copyright John Wiley & Sons, Ltd.
key words: Internal acoustic and gravity waves, fluid-structure interaction, partitioned analysis,
localized Lagrange multipliers
1. Introduction
Vibrating structures inducing waves in a fluid and the opposite case of acoustic or gravity
waves inducing structural vibrations constitute an active area of research. The two connected
domains, the flexible structure and the enclosed fluid, can be strongly coupled and in that case
the fluid-structure system must be studied using specific techniques.
Fluid structure interaction problems involving only internal acoustics or only sloshing are
relatively well understood. The concurrent interaction of both intense acoustics and sloshing
with flexible containers, however, is not well understood, and presents challenge for the design
of high pressure liquid and gaseous containers.
Normally, in the structural domain the primary variable to describe the motion is the
displacement. For the fluid domain several different approaches can be used but normally
∗ Correspondence
to: José A. González, Escuela Superior de Ingenieros, Camino de los descubrimientos s/n, E
41092 Sevilla, Spain. Tel.: +34 954487326, Fax: +34 954467370. e-mail: japerez@us.es
Contract/grant sponsor: Junta de Andalucı́a, Spain; contract/grant number: P08-TEP-03804
c 2000 John Wiley & Sons, Ltd.
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J. A. GONZÁLEZ AND K. C. PARK
including the pressure as a variable. Including the pressure variable on the fluid requires
additional efforts to satisfy the equilibrium and compatibility conditions between the fluid and
the structure. A displacement formulation for the fluid domain allows to impose directly the
compatibility and equilibrium conditions along the interface, simplifying the treatment of the
interface.
Treatment of the interface is based on the Localized Lagrange Multiplier (LLM) method
[20, 21, 22], a general variational framework used to solve partitioned systems that introduces
a kinematic frame in the fluid-structure interface. Two multipliers fields separately connect
the frame to the fluid on one side and to the structure wet surface on the other. Both
multiplier spaces are discretized as delta functions collocated at the fluid interface and wetsurface structural nodes. These can be physically interpreted as interaction point forces. For
an inviscid fluid only normal-to-the-interface multipliers are needed. Main goal of this interface
treatment is complete decoupling of fluid and structure models, in the sense that those can be
constructed separately by different teams, or different mesh generators. Consequently finite
element meshes will not necessarily match over the interface.
This localized approach, initially dedicated to structural substructuring and contact-impact
problems [26] allows the treatment of non-matching meshes and can be constructed in order
to preserve the constant-stress interface patch test, as demonstrated by Park et al. [24].
The LLM method proposed by Park and Felippa [20, 21, 22] is a general variational
framework used to solve partitioned systems in mechanics that also introduces between the
substructures an intermediate surface called frame, that is endowed with independent degrees
of freedom and treated with a FEM discretization to approximate the interface variables.
Finally, this frame is connected to the coupled systems using classical Lagrange multipliers
defined on the substructure interface nodes, obtaining an approximation that naturally allows
the treatment of non-matching meshes and can be constructed in order to preserve the
constant-stress interface patch test, as demonstrated by Park et al. [24].
In this work we apply this LLM formulation for the simulation of structures experiencing
significant FSI effects with coupled acoustic and gravity waves. Many theoretical and
implementation issues behind the techniques proposed in [25] are reviewed, extended and
explained in detail as appropriate to understand its possibilities. Also additional theory topics
are included, as treatment of fluid quasi-incompressiblity, procedures for interfacing nonmatching meshes in the case of inviscid fluids and treatment of the interfaces using LLMs.
This is especially so as the sloshing fluid should be allowed to move tangentially along the
container walls. A software package has been developed to illustrate the capabilities of the
proposed methodology, demonstrating the generality of the formulation for modeling sloshing,
internal acoustics, and especially combined sloshing and acoustics.
2. Statement of the problem
We consider the problem of determining the small amplitude motions of an inviscid,
homogeneous and compressible liquid with a free surface Σf and contained in a linear elastic
structure, as the case represented in Fig. 1. The internal fluid is considered to be initially at
rest occupying a volume Vf under hydrostatic equilibrium due to the action of the gravity
field g. At the same moment, the elastic structure containing the liquid occupies a volume
Vs and presents a wet surface, common interface with the fluid denoted Γb . The structure
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Figure 1. Container with a liquid in motion. Finite elements are used to describe the motion of the
fluid using the Lagrangian approach.
presents external forces and/or displacements that will produce a coupled oscillation of the
two systems.
A global reference system X(X, Y, Z) is introduced with the vertical direction Z and its unit
vector k measured from the bottom of the tank, defining the undisturbed fluid level inside the
tank as H. In order to describe the motion of the two systems we use the displacement fields
uf and us defined on Vf and Vs respectively, that added to the reference configurations Xf
and Xs provide the current positions xf and xs
xf = Xf + uf
xs = Xs + us
(1)
magnitudes expressed in the global system.
3. Localized Lagrange multiplier method for FSI
The LLM treatment of the problem is schematized in Fig. 2 where the total FSI system is
divided into three subsystems: the fluid, the interface frame and the structure.
To derive the equations of motion for the FSI system we use the variational formulation
proposed by Park and Felippa [20, 21, 22] and later particularized to FSI problems [23, 25].
Under this variational framework, the problem is treated like if all systems were entirely
independent, formulating the virtual work of the system by summing up the contributions
of each subsystem and adding finally the interface contribution obtained multiplying the
constraint equations by the LLMs. The variational functional that represents the stationary
condition of the total energy of the system δΠ is then composed of three terms
δΠ = δΠf + δΠs + δπ int = 0
(2)
where δΠf is the virtual work for the fluid; δΠs is the virtual work for the flexible structure;
and, δπ int is the partition interface constraint that groups contributions from the fluid and
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J. A. GONZÁLEZ AND K. C. PARK
Figure 2. Partitioning of an internal fluid-structure coupled system into three different subsystems:
fluid, interface frame and structure.
the structure. It should be noted that δΠf = 0 yields the governing equations of motion for
the fluid. Similarly, δΠs = 0 yields the equations of motion for the structure. This implies that
the present formulation may utilize two independently developed fluid and flexible structural
dynamics analysis software modules to perform FSI analysis, with the addition of an interface
treatment module whose main function is to enforce the interface constraint δπ int = 0.
In the following sections, the approximations adopted for each one of these three virtual
work expressions are described.
4. Fluid formulation
In this section we derive the approximation used for the fluid.
In the absence of gravity effects, a fluid with no shear stiffness presents rotational zero energy
modes. These free rotational modes in the vertical plane disappear when the gravity effects
are introduced, see [1, 2] for theoretical demonstration.
Because our interest is in fluid-structure interaction, a pure displacement formulation is
preferred, where the absence of the pressure will facilitate the connection with the structure.
4.1. Lagrangian description of fluid motion
Let us consider an inviscid and homogeneous fluid under the action of gravity field g, initially
at rest with density ρ0f and hydrostatic pressure due to gravity p0f = pint + ρ0f g(H − z).
Next, we suppose small deviations from this equilibrium position and introduce a Lagrangian
displacement field uf following the fluid particle to a point M 0 , as represented in Fig. 3.
Conservation of mass equation states
0
ρ0f = ρM
f J
(3)
with J = det F and where F = 5uf is the deformation gradient tensor.
Linearizing the kinematics, the inverse of the dilatation J can be approximated as J −1 ≈
(1 − 5 · uf ) and substituted back in (3) to provide a linearized version of mass conservation
0
0
ρM
f = ρf (1 − 5 · uf )
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M’
uf
0
H
Z
g
X
Figure 3. Initial state of the fluid under the action of gravity. Discretization and Lagrangian description
of the fluid motion.
In the other hand, for general equations of state, the speed of sound (cf ) relates variation
of pressure with variation of density in adiabatic conditions, i.e.,
0
c2f
,
0
pM
f − pf
(5)
0
0
ρM
f − ρf
magnitude that can be considered constant and substitute (4) to express the fluid pressure in
Lagrangian form as
0
0
0 2
pM
(6)
f = pf − ρf c 5 · uf
an equation of state for the fluid that relates Lagrangian pressure with displacements. This
approximation is well known and has been intensively used to model the fluid motion, see e.g.
[6, 4, 5, 18, 1, 2].
Note also that there exists a relation between Lagrangian pressure at point (M 0 ) and Eulerian
pressure in the original position of that point (M ) that can be approximated as
0
M
0
pM
f − pf = −ρf guf ·k
(7)
ρ0f guf ·k
where
is the weight of the water column between M and M’. This well known relation,
see e.g. [17], together with (6) gives an expression for the Eulerian description of pressure
0
0 2
0
pM
f = pf − ρf c 5 · u + ρf gu · k
(8)
that differs from Lagrangian pressure only in the third term.
4.2. Principle of virtual work
We can write the equation for the principle of virtual work in terms of the first Piola-Kirchhoff
tensor as
Z
Z
δΠf =
T : δF dV −
bf · δuf dV
Vf
Vf
Z
−
Z
tf · δuf dA +
∂Vf
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ρ0f üf · δuf dV
(9)
Vf
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J. A. GONZÁLEZ AND K. C. PARK
with internal force bf =ρ0f g due to gravity.
We observe from this equality that the stress tensor conjugate to the deformation gradient
F is the so-called first Piola-Kirchhoff stress tensor given as
T = JσF−T
(10)
note that like F, the first Piola-Kirchhoff stress tensor is an unsymmetric tensor.
For an inviscid fluid, the Cauchy stress tensor σ presents only an hydrostatic component
and can be expressed as
σ = −pf I
(11)
a symmetric tensor in the deformed configuration.
A linear expansion of the deformation gradient tensor gives
J
F
−T
≈ (1 + 5 · uf )
(12)
T
≈ (I − 5 uf )
(13)
Substituting this linearization in (10) gives us an approximation of the first Piola-Kirchhoff
stress tensor
T ≈ −(1 + 5 · uf )(pf I)(I − 5T uf )
(14)
and substituting in the general principle of virtual work (9) gives
Z
Z
δΠf = −
pf (5 · δuf ) dV −
pf (5 · uf )(5 · δuf ) dV
Vf
Z
+
Vf
pf (1 + 5 · uf )5T uf : 5δuf dV −
Vf
ρ0f g · δuf dV
Vf
Z
−
Z
Z
tf · δuf dA +
∂Vf
ρ0f üf · δuf dV
(15)
Vf
Considering that the hydrostatic pressure equation writes
5p0f = ρ0f g
(16)
and that using Green-Gauss theorem the following equality holds
Z
Z
Z
ρ0f g · δuf dV =
p0f nf · δuf dA −
p0f (5 · δuf ) dV
Vf
∂Vf
(17)
Vf
let us now modify the gravitational body force term in (15) to obtain
Z
Z
δΠf = −
(pf − p0f )(5 · δuf ) dV −
pf (5 · uf )(5 · δuf ) dV
Vf
Vf
Z
T
Z
pf (1 + 5 · uf )5 uf : 5δuf dV −
+
Vf
Z
+
(tf + p0f nf ) · δuf dA
∂Vf
ρ0f üf · δuf dV
(18)
Vf
where we can appreciate that the second volume integral and the second term of the sum in
the third volume integral present second order terms.
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Now we use the approximation of Lagrangian pressure derived in (6) and substitute pf = pM
f
to obtain
Z
Z
0 2
δΠf =
(ρf cf − pf )(5 · u)(5 · δu) dV +
pf (1 + 5 · uf )5T uf : 5δuf dV
Vf
0
Vf
Z
(tf + p0f nf ) · δuf dA +
−
∂Vf
Z
ρ0f üf · δuf dV
(19)
Vf
where we can identify the acoustic term first, the initial stress term due to confinement pressure
second and finally the terms due to the external and inertia forces.
After elimination of the second order terms, we rewrite (19) as
Z
Z
δΠf =
κf (5 · u)(5 · δuf ) dV +
p0f 5T uf : 5δuf dV
Vf
V
Z
(tf + p0f nf ) · δuf dA +
−
∂Vf
Zf
ρ0f üf · δuf dV
(20)
Vf
a linearized approximation of the virtual work that incorporates the effects of acoustic, surface
and initial stiffness. Note that in a first order approximation, the volumetric stiffness of the
fluid is κf = ρ0f c2f − p0f .
4.3. Boundary term
The boundary integral of (20) needs additional treatment in order to impose the free-surface
boundary condition. The fluid surface ∂Vf can be divided in two different regions, the free
surface Σf and the surface in contact with the structure Γf , with ∂Vf = Γf ∪ Σf . When
integrating on Σf , the free surface condition pM − p0f = ρ0f g(uf · k) is used to obtain
Z
Z
−
(tf + p0f nf ) · δuf dA =
ρ0f g(uf · k)(nf · δuf ) dA
(21)
Σf
Σf
with a traction on the free surface of the fluid tf = −(p0f + ρ0f g(uf · k))nf .
Substituting back in (20) the final form of the variation for the fluid is obtained
Z
Z
δΠf =
κf (5 · uf )(5 · δuf ) dV +
p0f 5T uf : 5δuf dV
Vf
Vf
Z
+
ρ0f g(uf · k)(nf · δuf ) dA +
Σf
Z
ρ0f üf · δuf dV
(22)
Vf
from where we will derive the discretized equations of motion.
4.4. Mean dilatation method for incompressibility
A standard finite element discretization is unfortunately not applicable to simulations involving
incompressible or quasi-incompressible behavior. It is well known that without further
development such a formulation is kinematically over-constrained, resulting in the overstiff phenomenon known as volumetric locking. Well known solutions to this problem are:
to impose the incompressibility condition by penalization or by using reduced integration
methods [16], to adopt an augmented Lagrangian formulation enforcing quasi-incompresibility
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condition [31, 30] or the use of u-p elements that satisfy the inf-sup condition [3, 8]. Among
all these possibilities, a total displacement formulation with variational treatment of near
incompressibility is preferred in order to facilitate the treatment of the interface without
scarifying theoretical support.
The three-field canonical functional of linear elastostatics identified as the Hu-Washizu
functional in the mechanics literature [11, 32] will be applied to treat nearly-incompressibility
of the fluid. In this functional, three different interior fields: displacements, stresses and strains,
are defined and independently varied.
In our case, the Hu-Washizu (HW) functional for the fluid uses independent kinematic
descriptions for the volumetric and distortional deformation, introducing a new kinematic
¯ representing the dilatation independent of uf and a Lagrange multiplier (p̄f ) to
variable (J)
enforce the condition J¯ = J(uf ).
The HW principle states, see e.g. [12, 7]:
Z
Z
¯
¯ dV
ΠHW
(u
,
J,
p̄
)
=
Ψ̂
(F)
dV
+
Uf (J)
f
f
f
f
Vf
Vf
Z
+
¯ dV − Πext
p̄f (J − J)
f (uf )
(23)
Vf
¯
where Ψ̂f (F) = Ψf (F̂) represents the deviatoric part of the strain energy function and Uf (J)
the volumetric strain energy.
The stationary conditions of (23) are
Z
T : δF dV − δΠext
= 0
(24)
f
Vf
Z
(
Vf
¯
∂Uf (J)
− p̄f )δ J¯ dV
¯
∂J
Z
¯ p̄f dV
(J − J)δ
=
0
(25)
=
0
(26)
Vf
¯ and (δ p̄f , δ J)
¯ are constant over the
Mean dilatation procedure assumes that (p̄f , J)
integration volume Vf . Introducing the volumetric strain energy function as
¯ =
Uf (J)
1
κf (J¯ − 1)2
2
where κf is the volumetric stiffness, equations (25) and (26) reduce to
Z
1
J¯ =
J dV
V f Vf
p̄f = κf (J¯ − 1)
(27)
(28)
(29)
conditions that will be incorporated in our finite element formulation for the fluid.
4.5. Discrete approximation of fluid functional
The mean dilatation approach for a given volume Vf leads to a constant pressure over that
volume, as indicated by equations (28, 29). When this formulation is applied to an individual
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fluid element identified by superscript (e) in a finite element mesh, the pressure becomes
constant over the element volume. In particular, considering (28) in an element with volume
(e)
Vf together with (12) an average of the element dilatation is obtained
Z
1
5 · uf dV
(30)
(5̄ · uf )(e) = (e)
(e)
Vf
Vf
and using the expression of uniform pressure given by equation (29) at the element level, the
uniform element pressure can be expressed
(e)
p̄f = κf (5̄ · uf )(e)
(31)
(e)
where pressure p̄f is constant inside the element.
Continuing with the discretization, recall the linearized virtual work equation (22) which
for an element (e) is
Z
Z
(e)
δΠf
=
κf (5 · uf )(5 · δuf ) dV +
p0f 5T uf : 5δuf dV
(e)
(e)
Vf
Vf
Z
−
(e)
∂Vf
(tf + p0f nf ) · δuf dA +
Z
(e)
Vf
ρ0f üf · δuf dV
(32)
with κf = ρ0f c2f and apply the mean dilatation method. Considering that in the first volume
integral the pressure p̄f = κf (5̄ · uf ) is constant over the element volume, the average
dilatations are introduced for an element (e) as
Z
(e)
(e)
δΠf
= κf Vf (5̄ · uf )(e) (5̄ · δuf )(e) +
p0f 5T uf : 5δuf dV
(e)
Vf
Z
−
(e)
(tf + p0f nf ) · δuf dA +
∂Vf
Z
(e)
ρ0f üf · δuf dV
(33)
Vf
where we observe that the element stiffness matrix is composed of an acoustic term plus an
initial stress stiffness matrix.
(e)
When element displacements are discretized and expressed as uf = Nuf , where Nf collects
e
the element shape functions and uf collects nodal values of the element. Applied forces include
body forces collected in Bf and surface tractions collected in tf . The fluid element matrices
are given by
The stiffness matrix is finally composed of three terms,
K(e)
K(e)
ac
(e)
Kini
(e)
(e)
= K(e)
ac + Kini + Kslo
Z
0 2 Z
ρf cf
T
(5
·
N
)
dV
(5 · Nf ) dV
=
f
(e)
(e)
(e)
Vf
Vf
Vf
Z
=
p0f (5Nf )T (5Nf ) dV
(34)
(35)
(36)
(e)
Vf
(e)
Kslo
Z
=
(e)
Σf
ρ0f g(NTf · n)T (NTf · n) dA
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J. A. GONZÁLEZ AND K. C. PARK
Figure 4. Spectral analysis of the stiffness matrix for a single 3D fluid-element including: initial
stress, sloshing and acoustic terms. Two zero energy modes appear first, corresponding to in-plane
translations. Next eigenvalue (0.41) corresponds to the sloshing mode, a higher eigenvalue (1.36) to
the filling mode and highest eigenvalue (1067.64) to the acoustic mode.
(e)
(e)
where Kac is the element volumetric stiffness matrix, Kini is the element initial hydrostatic(e)
stress stiffness matrix and Kslo is the surface gravity-effect stiffness matrix that will be present
only in elements located at the free-surface Σf .
Spectral analysis of the stiffness matrix (34) for an 8-node, 24 degree-of-freedom, brick
element with a regular (cubic-like) geometry under exact 2 × 2 × 2 integration reveals, see
Fig. (4), that only two in-plane translations remain in the fluid element as free zero-energy
modes. The acoustic stiffness matrix (35) introduces one pure-volumetric mode, the sloshing
stiffness (37) contributes with a new vertical mode and the initial-stress stiffness matrix (36)
is responsible for the remaining 20 non-zero energy modes that include rotation, shear, torsion
and filling modes.
After assembling the finite element matrices, we arrive to the matrix form of the discretized
variational
δΠf = δuTf {Mf üf + Kf uf − f f }
(38)
in which Mf and Kf are the mass and stiffness matrices of the fluid, uf the vector of fluid
node displacements and f f the applied node force vector. As we can see, the second order
ODE representing the fluid equations of motion is only function of displacements and its time
derivatives.
5. Structure formulation
The structure is described by the equation of motion for a continuum body. Because in the
variational formulation previously derived for the fluid, liquid motion is considered to be small,
similar consideration has to be taken for the structure. To this end, we write the the variational
formulation of the structural partition with the following expression for the virtual work
Z
Z
Z
δΠs =
σs : ∇δus dV −
ts · δus dA +
ρs üs · δus dV
(39)
Vs
∂Vs
Vs
where
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Figure 5. Treatment of non-matching interfaces using localized Lagrange multipliers. The fields of
multipliers on the fluid side λf and the structural side λs are discretized using Dirac delta functions
collocated at the nodes. If the fluid is inviscid, only normal displacements of the frame and normal
components of these multipliers are considered.
5.1. Discrete approximation of structure functional
The structure is discretized using the classical FEM approximation, where the assembly of
element contributions by the direct stiffness method leads to the semidiscrete equations of
motion
δΠs = δuTs {Ms üs + Ks us − f s }
(40)
in which Ms and Ks are the mass and stiffness matrices, us the vector of node displacements
and f s the applied node force vector.
6. Interface frame
To formulate the interface problem, instead of considering the direct interaction between the
fluid and the structure, we insert between them a deforming non-physical surface or interface
frame denoted as Γb and reformulate the FSI problem in terms of interaction of the two systems
with this auxiliary surface using LLMs collocated on each side of the frame.
The interaction forces acting on the frame are represented in Fig. 5 where the LLMs
connecting the fluid domain Vf with the frame are named using the vector quantity
λf = {λfn , λft1 , λft2 }T and the multipliers connecting the structure Vs are named λs =
{λsn , λst1 , λst2 }T . These forces are expressed using a local orthonormal base system attached
to the frame; B = [n, t1 , t2 ] that depends of the position occupied by the node on the frame
and that is used to describe λf and λs . This frame local system is defined by convention in
the following way; t1 and t2 are the orthogonal vectors contained in the frame tangent plane
at the considered point ξp and vector n points towards structure Vs .
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6.1. Frame kinematics
The motion of the interface frame is described using a displacement field ub = {ubn , ubt1 , ubt2 }T
and its initial configuration Xb , providing a current position
xb = Xb + ub
(41)
that is restricted to follow the fluid-structure interface permanently. To do so, the frame-fluid
and frame-structure relative positions are forced to be zero, conditions expressed in the frame
local system as
BT (xf − xb ) = 0
(42)
BT (xs − xb ) = 0
and extended to the fluid interface Γf and the structure interface Γs .
If the fluid is inviscid, tangential stress on the wall is zero and tangential displacements of the
fluid and the structure at the interface become independent. In this case, only the kinematic
condition of (42) in the normal direction applies, i.e.,
nT · (xf − xb ) = 0
nT · (xs − xb ) = 0
(43)
reducing the displacement field on the frame to ub = {ub } with only one component in the
normal direction and transforming the LLMs of the fluid side to λf = {λn } and the structural
side to λs = {λs } as normal components.
7. Variationally Based Interface Coupling
Using a variational formulation of the interface equations to obtain treat the fluid-structure
connection possesses the significant advantage of preserving the symmetry of the equations of
motion. In the present section the variational framework proposed in [25] is used to obtain the
coupling equations associated to the interface.
7.1. Interface constraint functional
Consider the fluid-structure boundary Γb shown in Fig. 5 where the interface frame presents an
independent field of displacements (ub ). The fluid boundary displacements (uf ), the structure
boundary displacements (us ), as well as their associated multipliers on the fluid (λf ) and the
structure (λs ) will appear in the variational formulation of the interaction conditions. This
variational is divided in the following way:
δπ int = δπfint + δπsint
(44)
grouping contributions from the fluid and the structure.
Let us obtain first the interface functional for the fluid. It can be expressed by multiplication
of the LLMs and the constraint equation related with the kinematic positioning of the frame
(42) that is enforced in a weak sense using the variational form
Z
δπfint =
δ{λf · [BT (xf − xb )]} dA
(45)
Γb
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that is differentiated by parts to provide the expression for the total variation of the interface
potential at the fluid side
Z
δπfint =
(λf · δ{BT (xf − xb )} + δλf · {BT (xf − xb )}) dA
(46)
Γb
and similarly
δπsint
Z
(λs · δ{BT (xs − xb )} + δλs · {BT (xs − xb )}) dA
=
(47)
Γb
for the structural side.
To deal with the first term of equations (46) and (47) we decompose them in the following
way
λf · δ{BT (xf − xb )} = (δuf − δub ) · {Bλf }
(48)
λs · δ{BT (xs − xb )} = (δus − δub ) · {Bλs }
(49)
a first order approximation where the variations of the normal and tangential frame unitary
vectors are considered to be very small. Substitution into (46) and (47) leads the interface
potential for the fluid
Z
δπfint =
(δλf · {BT (xf − xb )} + (δuf − δub ) · {Bλf }) dA
(50)
Γb
and
δπsint =
Z
(δλs · {BT (xs − xb )} + (δus − δub ) · {Bλs }) dA
(51)
Γb
for the structure.
Grouping contributions from both sides (44) the final expression for the interface variation
is obtained
δπ int (xf , xs , λf , λs , xb ) =
Z
δuf · {Bλf } dA +
δus · {Bλs } dA
Γb
Γb
Z
Z
δλf · {BT (xf − xb )} dA +
δλs · {BT (xs − xb )} dA
Γb
Γ
Zb
−
δub · {B(λf + λs )} dA
Z
(52)
Γb
where independently varied fields are identified in parenthesis on the left-hand side.
Note that in equation (52) the following terms can be identified from the fluid side
Z
Guf (λf ; δuf ) =
δuf · {Bλf } dA
Γb
Z
Gλf (xf , xb ; δλf ) =
(δλf · {BT (xf − xb )}) dA
Γb
Z
Gub (λf ; δub ) = −
δub · {Bλf } dA
(53)
(54)
(55)
Γb
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obtaining similar expressions for the structure; where Guf (λf ; δuf ) represents the virtual work
of interface forces acting on the fluid, Gλf (xf , xb ; δλf ) constitutes the kinematic constraint
between the fluid and the frame and Gub (λf ; δub ) evaluates the virtual work done by the
interface forces on the frame. Once combined these terms with their counterparts from the
structure, the final conditions that will have to be satisfied in the frame are
Gub (λf ; δub ) + Gub (λs ; δub ) = 0
(56)
or frame equilibrium due to forces coming from both sides.
If the fluid model is inviscid, interaction between the fluid and the structure takes place only
in the normal direction to the wet-surface n, conventionally selected to be exterior to the fluid.
In this case, the localized Lagrangian multipliers (λf , λs ) and the frame displacements (ub )
will present only the normal component and the appropriate form of the interface functional
becomes
δπ int (xf , xs , λf , λs , xb ) =
Z
δuf · {λf n} dA +
δus · {λs n} dA
Γb
Γb
Z
Z
+
δλf (xf · n − xb ) dA +
δλs (xs · n − xb ) dA
Γb
Γb
Z
−
(λf + λs )δub dA
Z
(57)
Γb
where λf and λs are localized Lagrange multiplier functions in the normal direction that link
the independently varied normal displacement function of the frame ub with the displacement
fields of the fluid uf and the structure us .
Finally, it is important to mention that the preceding interface constraint functional is a
linear approximation that will be valid for small variations of local system B. For a complete
non-linear formulation of the moving frame, the reader is referred to the works [26, 13, 14].
7.2. Discretization of interface constraint functional
To produce the matrix form of the interface equations, functional (57) is discretized by
assuming shape functions for the independently varied fields: λf , λs and ub .
Approximation for boundary displacements uf and us come from the finite elements used
for the fluid and structure models respectively and are expressed in the global reference system.
The normal displacements ub (ξ1 , ξ2 ) of the interface frame are interpolated using isoparametric
finite elements in the classical form


 ub1 
:
ub (ξ1 , ξ2 ) = Nb (ξ1 , ξ2 )
= Nbi (ξ)ubi
(58)


ubnb
where ξ = (ξ1 , ξ2 ) is the frame coordinate, nb is the number of nodes in the frame element and
the shape functions for Nb will be piecewise-linear, C 0 -continuous [19, 24, 27, 28, 29].
If we assume discretized fields of normal multipliers λf = Nλf λf , λs = Nλs λs , insert these
interpolations in δπ int , and integrate over Γb , the final expression of the discretized interface
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functional is obtained
+δλTf
δπ int = δuTf Bf uf + δuTs Bs us
BTf xf − Lf xb + δλTs BTs xs − Ls xb
−δuTb LTf λf + LTs λs
(59)
in which
Z
Bf =
NTf nNλf dA
(60)
NTs nNλs dA
(61)
NTλf Nb dA
(62)
NTλs Nb dA
(63)
Γb
Z
Bs =
Γb
Z
Lf =
Γb
Z
Ls =
Γb
and where Nf and Ns are the shape functions of the fluid and structure displacement elements,
respectively, evaluated on the discretized interface surface Γb . In some cases, as in curved
geometries, the discretized interface may differ slightly from the original one [25].
The integral evaluations in (60-63) are greatly simplified by assuming that Nλf and Nλs are
Dirac delta functions collocated at the fluid and structure interface nodes, respectively. If so,
Bf and Bs become Boolean matrices that select and normal-project node boundary freedoms
from the complete state vectors uf and us while Lf and Ls simply become the evaluation on
the frame of boundary node positions.
Our choice for multipliers discretization, represented in Figure 6, is to model them as
concentrated forces, that is, support functions of the multipliers are Dirac delta functions
λf (ξ) = Nλf i (ξ)λf i
;
Nλf i (ξ) = δ(ξ − ξp )
(64)
λs (ξ) = Nλs i (ξ)λsi
;
Nλs i (ξ) = δ(ξ − ξp )
(65)
where ξp is the frame coordinate of the fluid or structural interface node projected on the
frame. If the interface is defined in terms of pairs or couples, formed by an interface node and
its projection on the frame, then integrations of the multiplier field over the contact zone are
reduced to summations over the pairs, i.e.
Z
np
X
(p)
λf (ξ) · f (ξ) dΓb =
λf · f (ξp )
(66)
Γb
p=1
with f (ξ) a general function and np the total number of pairs on the fluid side. Expression (66)
is the key in order to maintain the interface treatment generic, leading to modular coupling
software since the frame needs to know very little information about the system it is interfacing.
To manage the discrete variables we have to introduce the interface nodal indicator B, the
well known boolean finite element assembling operator defined in the following way
(p)
(p)
uf = Buf uf
(p)
(p)
us = Bus us
(p)
(p)
ub = Bub ub
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J. A. GONZÁLEZ AND K. C. PARK
Figure 6. Exploded view of the fluid-structure interface discretization. Interface nodes of the fluid and
the structure are projected on the frame constituting pairs or couples formed by the hitting node and
its closest frame element.
and
(p)
(p)
(p)
(p)
λf = Bλf λf
(68)
λs = Bλs λs
(p)
where B is used to extract the variable associated with a boundary node (p) from the global
vector of unknowns with = uf , us , λf , λs , ub . By using this operator together with (66),
matrices (60-63) are simply obtained assembling contributions of each pair (p) in the following
way
np
P
(p)T
(p)
Bf =
Buf n(ξp )Bλf
Bs =
Lf =
Ls =
p=1
np
P
p=1
np
P
p=1
np
P
p=1
(p)T
(p)
Bus n(ξp )Bλs
(69)
(p)T
(p)
(p)T
(p)
Bλf Nb (ξp )Bub
Bλs Nb (ξp )Bub
with sums extended to the np active pairs of the corresponding side: fluid or structure.
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17
Figure 7. Steps in the definition of the frame based on position xn
b : (a) detection of active interface
nodes, (b) computation of nodal forces corresponding to a constant pressure transmission and (c)
construction of a new frame discretization using the ZMR. Separation between fluid and structure is
exaggerated for better visualization.
8. Frame construction
The interface formulation presented so far rely on the assumption that there exists a
discretization for the frame representing the fluid-structure interface. But the problem of
defining a mesh for the interface frame is not unique [29]: it could perfectly valid solutions,
for example, to construct a frame mesh matching with the fluid discretization, to make it
coincident with the structure discretization, or to define a completely new mesh between these
two. However, these arbitrary discretizations in general will not satisfy the patch test through
the interface [33, 10].
For constructing an interface mesh that satisfies the requirement of unicity and passes the
patch test, the node locations of the interface frame are decided using the zero-moment rule
(ZMR) originally proposed for contact-impact problems [26]. The main concept behind the
ZMR is to assure that the LLMs can transmit a constant stress state through the interface
frame passing the patch test. This condition can be satisfied if the interface node locations are
determined by the roots of the frame moment-equilibrium condition [19, 24].
In the case of a viscous fluid, imposing the kinematic condition (42) fixes the relative position
of fluid and structure meshes at the interface, therefore the interface frame mesh is constant.
However, the case of an inviscid fluid requires free relative tangential motion between the
meshes of the fluid and the structure at the interface, as dictated by condition (43), making
the position of the fluid and structure interface nodes on the frame time dependent. This
situation requires the frame mesh to be recalculated every time step using a similar procedure
to that employed by Park et. al. in contact-impact problems [24].
Let us suppose that we have arrived to time step tn and we know the total positions of the
fluid (xnf ), frame (xnb ) and structure (xns ). The procedure used to construct a frame satisfying
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the ZMR for time step tn+1 is graphically explained in Fig. 7 and can be summarized in the
following steps:
1. Define the active boundary for tn+1 . Search for active interface nodes based on the total
position of the fluid xnf relative to the position of the structure xns . Find free nodes and
disconnect them from the interface.
2. Suppose the transmission of a constant normal stress through the active boundary.
Compute the contribution of every interface element to the constant stress by evaluating
its equivalent node forces.
3. Using previous node forces together with their total position (xnf , xns ) apply the ZMR
to obtain the position of the new frame nodes.
4. Construct a frame discretization using isoparametric elements with the previous frame
nodes.
5. Project the current total position of the active nodes on the frame mesh and form pairs.
For each pair calculate the position ξp of the interface node in the frame element, the
normal in that position n(ξp ) and shape functions Nb (ξp ).
Once the frame is defined, the FSI problem is solved for time step tn+1 obtaining new
positions xn+1
, xn+1
and xn+1
. The frame definition algorithm can then be repeated for a
s
f
b
new time step.
9. Partitioned fluid-structure formulation
The coupled partitioned fluid-structure interaction model can now be constructed following
variational statement given in (2) for the discrete form of the total energy functional variation
δΠ. Inserting the discrete variational internal constraint (59), the discrete variational fluid Eq.
(38) and the discrete variational structural Eq. (40) into the foregoing equation, the stationarity
of the resulting expression yields the partitioned equations of motion:
δΠ
= δuTf {Mf üf + Kf uf + Bf λf − f f }
+δuTs {Ms üs + Ks us + Bs λs − f s }
+δλTf {BTf uf − Lf ub − hf } + δλTs {BTs us − Ls ub − hs }
−δuTb {LTf λf + LTs λs }
(70)
with vectors hf = (BTf Xf − Lf Xb ) and hs = (BTs Xs − Ls Xb ) function of the initial
configuration.
The partitioned equations of motion for the frame-based FSI problem are obtained from the
stationary condition of variational (70), that can be expressed as
 



d2
0
Bf
0
0
Kf + Mf dt
uf 
ff 


2





 


d2




0
Ks + Ms dt
0
Bs
0 
2
  us   f s 

T


λf
hf
=
(71)
Bf
0
0
0
−Lf 


 





λ
h




s
s
0
BTs
0
0
−Ls  




 


ub
0
0
0
0
−LTf −LTs
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19
where first two rows are the equilibrium equations of each system with terms Bf λf and Bs λs
representing the interface forces exerted on the fluid and structure expressed in the global
system, the third and fourth equations impose fluid-frame and structure-frame displacement
compatibility respectively and the last equation enforces the frame equilibrium.
9.1. Vibration analysis of FSI systems
Equation (71) can be specialized to vibration formulation by replacing the time-differentiation
d2
2
and making external forces f f = f s = 0, whose substitutions leads to
operator dt
2 with −ω
 



Kf − ω 2 Mf
0
Bf
0
0
uf 
0 







 



0
Ks − ω 2 Ms
0
Bs
0 
 us 
 
 0 



T

Bf
0
0
0
−Lf 
λf
0
=
(72)



 


0
0
−Ls  
0
BTs
 λs 
 
 0 






 

ub
0
0
0
−LTf −LTs
0
leading to an eigenvalue problem from where we can determine natural frequencies and mode
shapes of the coupled system [23].
9.2. Transient analysis of FSI systems
The semidiscrete equations of motion (71) are directly integrated in time using Newmark’s
method, where displacements, velocities and accelerations at current time step tn+1 are related
with known quantities of the previous time-step tn by the expressions
un+1
n+1
u̇
n+1
=
=
un + ∆tu̇n + ∆t2 [( 21 − β)ün + βün+1 ]
n
n
u̇ + ∆t[(1 − γ)ü + γü
n+1
]
(73)
(74)
n
where ∆t = t
−t is the time step and (β, γ) are time integration parameters that determine
stability and accuracy characteristics. These parameters are taken to be the same for both fluid
(u = uf ) and structure (u = us ).
Suppose now that computations have proceeded until tn . On inserting (73,74) into the matrix
equation (71) and moving information from the previous time-step to the right side yields
n+1
n+1 


Mf + β∆t2 Kf
0
Bf
0
0
gf 
üf 













0
Ks + β∆t2 Ms
0
Bs
0 

 gs 

 üs 



BTf
0
0
0
−Lf 
hf
λf
=








0
BTs
0
0
−Ls  
 hs 

 λs 









0
üb
0
0
−LTf −LTs
0
(75)
where the free-term vector groups the dynamic forces acting on the fluid and the structure
gfn+1
gsn+1
= f n+1
− Kf [unf + ∆tu̇nf + ∆t2 ( 21 − β)ünf ]
f
=
f n+1
s
−
Ks {uns
+
∆tu̇ns
+
∆t2 ( 21
−
β)üns }
(76)
(77)
and the dynamic stiffness matrices appear on the diagonal of the system.
It has been demonstrated by Ross et al. [28] that in partitioned systems connected with
localized Lagrangian multipliers, as in monolithic ones, Newmark’s method with (β = 14 , γ = 12 )
provides A-stability and global second-order accuracy. This selection of parameters, known as
the Trapezoidal Rule, is used for time integration in all our numerical applications.
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J. A. GONZÁLEZ AND K. C. PARK
Figure 8. Sketch of a two-dimensional cavity with three rigid walls closed by a flexible steel beam on
the top.
10. Numerical applications
The following applications serve to verify and illustrate the capability of the proposed
formulation in the study of FSI systems. Examples are selected to verify a correct
approximation of the three different effects considered in the fluid, i.e., initial hydrostatic
state, sloshing and acoustic waves, together with strong fluid-structure interaction effects.
10.1. 2D acoustic cavity with a flexible wall
To isolate and study the contribution of the acoustic term of (34), we first consider a twodimensional problem consisting of a rectangular cavity of dimensions Lx = 8m by Lz = 20m
with rigid walls, full of water and closed on the top by a flexible steel structure that is simply
supported at both ends, see Fig. 8.
The properties considered for the fluid (water) are ρf = 1000kg/m3 and cf = 1500m/s and
the material properties of the steel structure are Es = 210GP a and ρf = 50kg/m, with inertia
of the section Iy = 1.59 · 10−4 m4 .
The interior of the rectangular cavity is meshed using 8 × 20 fluid elements, corresponding
to 320 fluid degrees of freedom. Each fluid element is square with a side length of 1m. The
structure on the top is modeled using 20 beam elements with two degrees of freedom per node
(vertical displacement and rotation) and connected to the fluid using localized Lagrangian
multipliers.
The natural frequencies and structural modes that describe the dynamic behavior of the
simply supported beam vibrating in vacuo are also represented in Fig. 10.
10.1.1. Case 1: Infinitely rigid beam In order to compare with the flexible case, we first
consider the limit case Is → ∞ where the top beam acts as another rigid wall, transforming the
problem into an acoustic cavity with rigid walls. This new problem has a well known analytical
solution, it is reasonable to expect resonance at frequencies for which the corresponding
wavelengths are in harmony with the dimensions of the cavity, i.e.,
2
ωl,m
= cf kl,m
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2 1/2
where ωl,m is the resonant frequency and kl,m = (( Llπx )2 + ( mπ
is the wave number.
Lz ) )
Acoustic freq.
Mode (1,0)
Mode (0,1)
Mode (1,1)
Mode (2,1)
Mode (1,2)
Mode (2,2)
Analytical
37.5 Hz
93.7 Hz
100.9 Hz
120.1 Hz
191.2 Hz
210.9 Hz
Computed
37.5 Hz
94.3 Hz
101.3 Hz
120.1 Hz
195.6 Hz
205.1 Hz
Table I. Comparison of analytical and numerical acoustic frequencies obtained for a two-dimensional
acoustic cavity with rigid walls.
Such frequencies are computed analytically using (78) and represented in Table I together
with the numerical results obtained performing a vibration analysis using the technique
described in Section 9.1. These first six resonant frequencies have been selected for contour
plotting in Fig. 9, where the pictures represent horizontal and vertical displacements of the
fluid, together with fluid pressure level that is constant inside each element as required by the
mean dilatation method.
As expected, the approximation error obtained of the acoustic modes in the horizontal
direction is lower than in the vertical direction that presents a coarser mesh. In all cases error
is less than 3% and it can be observed that acoustic waves for these selected frequencies are
correctly represented.
10.1.2. Case 2: Flexible beam Next we study the effect of using a flexible wall with the
dynamic characteristics summarized in Fig. 10 on the top of the cavity. The first mode of
the beam vibrating in vacuo involves more volume variation of the fluid in the cavity, so it is
expected to appear at higher frequencies in the coupled solution.
Fig. (11) shows the three first vibration shapes of the cavity with elastic boundary with their
associated frquencies. The shapes are greatly exaggerated and contour plots of the displacement
fields are also represented. It can be observed that the three first coupled modes are controlled
by the vibration of the flexible beam, however the natural frequencies are lower due to the
added mass effect.
10.2. Sloshing in rigid containers
In this section the sloshing problem with rigid containers is studied for three different
geometries: plane, cubic and cylindrical. The fluid considered is always water, with properties
ρf = 2500kg/m3 and cf = 1500m/s. Sloshing frequencies and mode shapes can be computed
from system (72) eliminating the terms associated with the structure. Equivalent procedure
is used to solve the transient sloshing problems, where system (75) is integrated in time using
Newmark method.
10.2.1. Sloshing in rigid containers: vibration analysis First case considered, represented in
Fig. 15 (left), is a 2D square container of width Lx = 1m filled with water up to a height
H = 1m and open to the atmosphere. The fluid volume is discretized using a structured mesh
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J. A. GONZÁLEZ AND K. C. PARK
Figure 9. Results of vibration analysis for a two-dimensional acoustic cavity with rigid walls. Acoustic
modes for different frequencies.
with 20 × 20 quadrilateral fluid elements and 441 nodes, see Fig. 12, with a mesh refinement in
the vertical direction. This refinement is introduced to capture pressure gradient effects under
the free-surface correctly.
Under the assumptions of incompressible inviscid fluid and irrotational flow, exact analytical
solution for this problem can be obtained by substitution of the free-surface condition into the
velocity potential equation [15] leading to the classical approximation for sloshing natural
frequencies
2
ωl,m
= gkl,m tanh(kl,m H)
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COUPLED INTERNAL FLUID-STRUCTURE INTERACTION PROBLEMS
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Figure 10. Structural natural frequencies of the beam vibrating in vacuo and associated mode shapes.
Figure 11. Coupled vibration modes obtained for a 2D acoustic cavity with flexible beam.
where g is the acceleration of gravity, kl,m the wave number and H the water depth. This
analytical solution is compared with numerical results in Table II for the first four sloshing
frequencies. Computed mode shapes corresponding to these frequencies are presented in Fig.
12 with contour colors representing elevation. It can be observed that correspondence with
analytical frequencies up to the fourth mode is excellent with only 20 elements on the free
surface.
Next we consider a 3D version of the previous problem, with a cubic container of side L = 1m.
The volume of the fluid is discretized using five divisions on each direction with a slight mesh
refinement in the vertical direction, obtaining a finite element mesh of 216 nodes and 5 × 5 × 5
8-node isoparametric fluid elements. Analytical solution is computed using equation (79) and
compared with numerical results in Table III.
Compared with the 2D case, it can be appreciated in Fig. 13, that more complicated
bidimensional sloshing modes appear on the free-surface, corresponding to even and odd
combinations of the wave number. Even with such a crude mesh, the first two modes and
their combinations in two directions are correctly represented with an error under 3% in the
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Sloshing freq.
Mode 1
Mode 2
Mode 3
Mode 4
Analytical
0.88 Hz
1.25 Hz
1.53 Hz
1.77 Hz
Computed
0.88 Hz
1.25 Hz
1.54 Hz
1.79 Hz
Table II. Analytical/numerical comparison of the first four sloshing frequencies of a square container
with rigid walls.
Figure 12. Sloshing modes with natural frequencies in a two dimensional container with rigid walls.
The analytical solution of this case gives for the first four modes: 0.88 Hz, 1.25 Hz, 1.53 Hz and 1.77
Hz
Sloshing freq.
Mode (1,0)
Mode (1,1)
Mode (2,0)
Mode (1,2)
Analytical
0.88 Hz
1.05 Hz
1.25 Hz
1.32 Hz
Computed
0.88 Hz
1.04 Hz
1.22 Hz
1.28 Hz
Table III. Analytical/numerical comparison of the first four sloshing frequencies of a cubic container
with rigid walls.
natural frequencies.
Closing this section, the sloshing frequencies of a cylindrical rigid container of diameter
φ = 2m and water height H = 2m are computed. The finite element mesh is composed of 803
nodes and 640 eight-node hexahedral fluid elements, presenting 16 circumferential divisions
and 10 elements in the vertical direction with mesh refinement under the free surface.
Analytical solution for sloshing in a rigid cylinder for symmetric (S) and antisymmetric (AS)
modes can be found from the zeros of Bessel functions derivatives [15], resulting in the values
presented in Table IV for our particular case. Comparing with numerical results, it is clear
that the agreement is good for the first two AS modes and the first S mode.
The mode shapes and corresponding sloshing frequencies for this case are presented in
Fig. 14 with color fields representing elevation. These surface sloshing mode shapes are more
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Figure 13. Sloshing modes and natural frequencies in a cubic container with rigid walls. The analytical
solution of this case gives for the same four modes: 0.88 Hz, 1.05 Hz, 1.25 Hz and 1.32 Hz
Sloshing freq.
Mode (1,S)
Mode (2,S)
Mode (1,AS)
Mode (2,AS)
Analytical
0.98 Hz
1.32 Hz
0.67 Hz
0.87 Hz
Computed
0.99 Hz
1.41 Hz
0.67 Hz
0.88 Hz
Table IV. Analytical/numerical comparison of the first four sloshing frequencies for symmetric (S)
and antisymmetric (AS) modes of a cylindrical container with rigid walls.
Figure 14. Symmetric and antisymmetric sloshing modes and natural frequencies in a cylindrical
container with rigid walls.
complicated than in the case of the cube, requiring a more refined finite element mesh in the
circumferential direction.
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Figure 15. Time records of the free-surface elevation at the right wall of a rigid container (point A)
excited horizontally with three different frequencies: 0.5 Hz (top), 1 Hz (medium) and 1.4 Hz (bottom).
10.2.2. Sloshing in rigid containers: transient analysis We now investigate the water motion
inside a rigid tank during low severe sloshing phenomena with smooth free-surface. Transient
problem for the fluid is solved integrating in time system (75) by using the trapezoidal rule
with time step ∆t = 1 · 10−2 s
The geometry of the tank, subject to pure swaying motion of a single amplitude and period,
with the measurement point for surface elevation are indicated in Fig. 15 (left). The motions
imposed to the tank are sinusoidal: xs = a sin(ωt), where a is the amplitude of motion
(maximum displacement of the rigid walls from the reference location) and ω is the excitation
frequency. For the present case, the liquid fills 1m of the tank height, the maximum amplitude
used is a = 1cm and the excitation frequencies are 0.5Hz, 1Hz and 1.4Hz.
Solutions are given on the right plots of Fig. 15 in terms of wave-elevation time history at a
point near the wall. Snapshots of the fluid motion in time for the different swaying frequencies
are shown in Fig. 16. In the second case, with an excitation frequency of 1Hz, the contribution
of the first sloshing mode in the response is very clear, changing to the third sloshing mode
when the swaying frequency is increased to 1.4Hz.
Specially for the second excitation frequency, it is patent the effect of approaching to a
sloshing frequency, generating higher waves. If we come very close to the natural frequency,
the nonlinearities involved increase and a threshold condition is reached above which wave
breaking phenomena characterize the sloshing phenomena. Obtained results for the studied
regimes agree qualitatively well with experimental observations [9].
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Figure 16. Vertical displacement evolution of the free-surface for different excitation frequencies of the
container.
10.3. Sloshing in flexible containers
When the structure containing the fluid is flexible, the hydrodynamic pressure on the walls
arising due to the free surface oscillation causes the wall to deflect and move, which in turn
alters the free surface oscillation and the hydrodynamic forces on the wall. This coupled effect
is studied in the next two examples for the square and cubic tanks of the previous section.
In the 2D case, represented in Fig. 17, the elastic tank is modeled using a regular mesh with
30 beam elements with 3-dof per node, restricting displacements and rotation of a node located
at the center of the bottom plate. The material properties of the container are Es = 2.1·1011 P a,
νs = 0.3 with density ρs = 2500kg/m3 and wall thickness ts = 2mm. The mode shapes and
natural frequencies of the empty structure are also represented in Fig. 17 up to the fourth
mode.
The results of performing a vibration analysis of the coupled system are presented in Fig.
18 using for the fluid exactly the same mesh than in the rigid case. It can be observed the
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Figure 17. Dimensions of the flexible container, boundary conditions and first four mode shapes of the
structure vibrating in vacuo.
Figure 18. Coupled mode shapes of the 2D Results of the 3D sloshing problem with flexible walls.
First natural frequencies of the fluid container (dry structure) are: 1.62 Hz, 5.67 Hz, 16.1 Hz and 44.39
Hz
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Figure 19. Mode shapes and natural frequencies of the cubic container vibrating in vacuo.
well known added mass effect on the first two coupled modes, structure-driven coupled modes
presenting a lower frequency than their counterparts in the dry structure. Note that in the
coupled case the sloshing modes are appearing at higher frequencies: the third coupled mode
with frequency 1.25Hz corresponds to the first analytical sloshing mode that appears at 0.88Hz
in a rigid container and the sixth coupled mode with frequency 1.77Hz corresponds to the
second sloshing mode of 1.25Hz. More complicated patterns for the fluid are observed at higher
frequencies.
In the 3D case, coupled vibration analysis of the cubic tank is revisited introducing six
flexible walls for the container. The mesh used for the fluid is exactly the same than in the
rigid-wall case and the mesh for the structure is composed of 125 quadrilateral plate elements
with 3-dof per node; the normal displacement and two in-plane rotations. Boundary conditions
are somewhat different than in the previous example, each face of the tank is considered as
a simply supported plate. Properties used for the structure are Es = 2.1 · 109 P a, νs = 0.3,
density ρs = 2500kg/m3 and plate thickness ts = 2mm.
The natural frequencies and mode shapes of the dry structure can be found in Fig. 19 and
the coupled mode shapes with their corresponding frequencies in Fig. 20. The first four coupled
modes obtained are structure-driven modes with added mass effect, the third one is produced
by the vibration of the bottom plate. Sloshing appears in the form of coupled modes after the
fourth frequency.
10.4. Wave generator
This example is devised just to demonstrate that the proposed interfacing strategy can
handle scenarios with drastic mesh transitions from structure to fluid without any further
modification. A flexible structure modeled using only three quadrilateral elements, as
represented in Fig. 21, is used to generate waves on a rectangular basin by imposing a harmonic
horizontal motion xs (t) = a sin(ωt) at the left bottom node of the structure with frequency
ω = 6.28rad/s and amplitude a = 5cm. Fluid and structure present elastic properties,
boundary conditions and dimensions given in Fig. 21 with meshes composed of three planestrain quadrilateral elements for the structure and 800 quadrilateral fluid elements with 861
nodes for the fluid.
The dynamic calculation is performed integrating in time the equations of motion (75) using
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Figure 20. Coupled mode shapes for the fluid contained in a cubic tank with corresponding natural
frequencies.
Figure 21. Wave generator: geometry, boundary conditions and material properties.
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Figure 22. Wave generator: deformed configurations at different time steps.
a time-step size ∆t = 1 · 10−2 s. In Fig. 22 the waves generated on the fluid at different time
steps are depicted, showing that the dynamic interface-frame definition algorithm can handle
with no difficulty the contact of sharp edges. This underlines the powerful interface strategy
behind the combination of LLMs and ZMR techniques, validating the interface construction
algorithm presented in Section 8.
10.5. Cylindrical container
In our last example, a cylindrical container partially filled with water up to a height H = 2m
is subject to a forced motion in the horizontal plane. The container is modeled using a regular
mesh of 160 four node structural shell elements with 176 nodes and six degrees of freedom per
node. The structural properties are Es = 2.1 · 109 P a, νs = 0.3, density ρs = 2500kg/m3 and
shell thickness ts = 2mm. All the nodes of the shell structure laying on the floor are allowed
to move freely in the X-Y plane except point A, see Fig. 23, that is subject to a sinusoidal
forced motion in the X-direction of amplitude a = 2cm an a frequency of 1Hz.
The water inside the cylinder, with properties ρf = 2500kg/m3 and cf = 1500m/s, is
modeled with 640 hexahedral finite elements and 803 nodes with three degrees of freedom per
node, exactly the same mesh used for the rigid case. A detail of the spatial discretization is
shown in Fig. 23 emphasizing the presence of non-matching between the fluid and the structure.
Transient simulation is performed integrating in time system (75) using the trapezoidal rule
with time step ∆t = 1 · 10−2 s.
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Figure 23. Dimensions and meshes of the cylindrical container (a), internal fluid (b) and coupled
system presenting non-matching interfaces.
The pictures in Fig. 24 show the evolution in time of the free-surface due to the prescribed
swaying motion imposed to the tank. Solution combines stretching of the container walls in
the horizontal direction and large amplitude waves in the free-surface with a swirl effect. Note
that, when the fluid free-surface runs up and down the walls of the container, activation and
deactivation of structural interface nodes takes place so they are not affected by the fluid.
Time history of free-surface elevation is given in Fig. 25 for two points near the wall. Points
A and B are aligned with the direction of external excitation.
11. Conclusions
The main objective of the paper is to demonstrate applicability of the partitioned FSI
formulation proposed by Park et. al. [25] for the simulation of structures experiencing
significant FSI effects with coupled acoustic and gravity waves, using 2D and 3D models.
Theoretical and implementation issues behind the techniques proposed in [25] are reviewed,
extended and explained in detail as appropriate to understand its possibilities. Also additional
theory topics are included, as treatment of fluid quasi-incompressiblity, procedures for
interfacing non-matching meshes in the case of inviscid fluids and treatment of the interface
using LLMs.
A computational fully implicit strategy for fluid-structure interaction has been presented
that is able to represent correctly acoustic and gravitational effects on the fluid. Numerical
examples have been provided, which give evidence of its accuracy, robustness and efficiency.
The adopted computational framework allows the independent discretization of fluid and
structure, introducing an unique discretization for the interface that passes the patch test
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Figure 24. Cylindrical container: deformed configurations for different time steps. Half cylinder is
removed for a clear visualization of the free surface evolution.
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0.6
Point B
Point A
Surface elevation (m)
0.4
A
B
0.2
0
-0.2
-0.4
0
1
2
3
4
5
6
7
Time (s)
Figure 25. Cylindrical container: evolution of the free-surface elevation at opposite points near the
wall aligned with the direction of imposed motion.
following the fluid and structure interfaces during their relative motion. The strategy seems
very well suited for the modeling of internal fluids with a free-surface interacting with flexible
structures.
12. Acknowledgements
The research reported in this paper has been supported by the WCU (World Class University)
Program through the Korea Science and Engineering Foundation funded by the Ministry of
Education, Science and Technology, Republic of Korea (Grant Number R31-2008-000-100450). J. A. González has been partially supported by the project Proyectos Investigación de
Excelencia 2008, Consejeria de Innovación Ciencia y Empresa, Junta de Andalucı́a, Spain,
with contract number P08-TEP-03804.
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