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Abstract
A hierarchical finite element is developed for the free vibration
analysis of a liquid in a rigid cylindrical tank with or without a
free surface. It is a hierarchical quadrilateral element and has the
advantage that the hierarchical mode number is allowed to vary
independently of direction. Liquid behavior in tanks with large
aspect ratios can therefore be solved very accurately by using a
higher hierarchical mode number in the longer direction than in
the shorter one. Furthermore, it is possible to idealize the liquid
by using only one element. The solution can therefore be obtained
to any desired degree of accuracy simply by increasing the hierar-
chical mode number. In this method, the liquid behavior is de-
scribed by the displacements alone. The pressure and velocity
potential are not considered as unknowns. The results are com-
pared with other methods and show good agreement.
Keywords
Hierarchical finite element method; liquid; cylindrical rigid tank;
free vibration; free surface; sloshing; displacement based.
Free Vibration Analysis of a Liquid in a Circular Cylindrical
Rigid Tank Using the Hierarchical Finite Element Method
1 INTRODUCTION
There are many cases of fluid structure interaction, such as sloshing in liquid storage structures and
dams, and vibration of components of a nuclear reactor in a fluid. In most cases, it is difficult or
impossible to obtain analytical solutions for coupled systems. Thus we must approach the solution
by numerical methods based specifically on the finite element method.
In the added mass approach, a part of the mass of the fluid is added to the structural model
along the interface between the two fields. This approach neglects the compressibility of the fluid.
Several methods based on the added mass approach for the analysis of the liquid storage structures
were proposed (Housner, 1957, ASCE, 1986).
In the Eulerian approach (Haroun, 1983, Haroun, 1984, Veletsos and al, 1990, Balendra 1982,
Hughes et al, 1981, Soria and Casadei, 1997, Legay and al, 2006, Dunne, 2007, Cottet and al, 2008,
Sidi Mohammed Hamza Cherif a
Mohammed Nabil Ouissi a *
a Computational Mechanics Laboratory
Faculty of Technology
University of Tlemcen - Algeria
B.P. 230 Chetouane - Tlemcen – Algeria
* Author email:
n_ouissi@mail.univ-tlemcen.dz
http://dx.doi.org/10.1590/1679-78251774
Received 13.12.2014
In revised form 29.01.2016
Accepted 29.02.2016
Available online 23.03.2016
1266 S.M.H. Cherif and M.N. Ouissi / Free Vibration Analysis of a Liquid in a Circular Cylindrical Rigid Tank Using the Hierarchical
Latin American Journal of Solids and Structures 13 (2016) 1265-1280
Rannacher and Richter, 2010, Sanches and coda, 2010, He and Qiao, 2011, Wick, 2013), the velocity
potential, the pressure, or the velocity describes the behavior of the fluid, while the displacements
represent the movements of the structure. The solution of the coupled system can then be obtained
by solving the two systems separately, but by considering the effects of the interaction in an itera-
tive way. The coupled system can also be solved as being only one system, but this leads to non-
symmetrical matrix equations which require special solution techniques. Mixed Lagrangian-Eulerian
methods were also developed to solve this type of coupled systems using a velocity based formula-
tion (Hughes et al, 1981, Nomura and Hughes, 1992).
In the Lagrangian approach, the fluid behavior is in general described by a displacement field
(Shugar and Katona, 1975, Zienkiewicz and Bettess, 1978, Hamdi and al, 1978, Akkas and al, 1979,
Wilson and Khalvati, 1983, Chen and, Taylor, 1990, Idelson, 2003, Ryzhakov and al, 2010, and
Fuyin and al 2014). The motion of the fluid and structure being both described by only one dis-
placement field, this approach has the advantage of very easily satisfying the compatibility and
equilibrium conditions along the interface. So, a fluid structure system in interaction with a
geometry can be analyzed efficiently by this method.
Many displacement based fluid elements using the Lagrangian approach were proposed for vis-
cous and non-viscous liquids, like those available in the computer codes ADINA and ANSYS.
Shugar and Katona, (1975), developed a displacement based fluid element with four nodes using a
solid element with a shear modulus equal to zero and a specific bulk modulus.
When a non-fine grid is used to model a fluid in a rigid cavity, parasitic modes at non zero fre-
quencies appear as well as multiple modes with the same frequency. To solve this problem, Chen
and Taylor, (1990), used a projection of the mass matrix combined with a reduction of the stiffness
matrix.
The liquid in this analysis is regarded as non-viscous, irrotational and incompressible. Such
simplification allows displacements, pressures, or velocity potentials to be the field variables of the
liquid. In this study, only the displacements will be taken as variables. This allows the fluid element
to be easily incorporated into the available structural analysis programs.
The purpose of this work is to present a new displacement based bi-hierarchical finite element
and to apply it to the free vibration analysis of a non-viscous, irrotational and incompressible liquid
in a cylindrical rigid cavity with a free surface. This element coupled with other structural hierar-
chical finite elements such as that developed by Ouissi and Houmat, (2009) can easily idealize liquid
structure interaction. Because of its hierarchical nature, only one element can idealize the whole
liquid.
The hierarchical finite element method, also known as the p-version of the finite element meth-
od is more accurate and converges faster than the h-version. Indeed, when the exact solution is ana-
lytic everywhere the rate of convergence of the p-version is exponential, whereas that of the h-
version is only algebraic. The quality of the solution is not very sensitive to the distortions of the
elements, which allows the use of flattened elements or great ratio on sides without penalizing the
accuracy too much. In addition, as a hierarchical formulation is adopted for the representation of
displacements, the stiffness matrix relative to a given degree imbricates those of lower degrees. This
makes it possible to obtain in an economical way a sequence of solutions instead of only one solu-
tion as it is the case of the h-version (Babuska and al., 1982), (Szabo, 1990).
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The h-p version of the finite element method possesses the features of both the p-version and
the h-version. In this method, the accuracy is improved by increasing both the degree of the inter-
polating polynomial, and the number of elements (Szabo and al., 1991).
The hierarchical shape functions are generally selected in the Serendipity space. In this paper,
the shape functions are constructed from the shifted Legendre orthogonal polynomials. These shape
functions were introduced by (Houmat, 2004). They are different from those introduced by (Peano,
1975). Indeed, these polynomials are defined in the interval [0,1], whereas those of Peano are de-
fined in the interval [-1,1].
2 GOVERNING EQUATION
The liquid motion is governed by Euler’s equation:
vp l
(1)
Where p is the pressure,
is a gradient operator l
is the liquid density, and ν is the velocity
vector.
In this formulation, velocity and pressure fields describe the motion of the liquid. Using the re-
lationship between pressure and volume, one can write:
ukp T
l (2)
Where u is the displacement vector and l
kis the bulk modulus. Introducing the small amplitude
motion assumption, Euler’s equation is reduced to the acoustic wave equation:
p
c
p
2
21
(3)
Where c is the acoustic speed given by:
l
l
k
c
(4)
In Equation (3), only the pressure field describes the liquid motion. The same wave equation
can be derived from the Navier’s equation for an isotropic, homogeneous, and elastic medium as:
uuGkuG l
T
l
2 (5)
Where, G is the shear modulus. For an inviscid liquid, the shear modulus equals zero. Equation (5)
becomes:
uuk l
T
l
(6)
A weak form of the above expression for a liquid region Ω subjected to natural boundary condi-
tions and critical respectively on the boundaries Γp and Γf can be expressed as:
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Free surface
z
r
Liquid
ξ
2
1
3
4
η
lp
T
TTT TT T
ll l g
uu k u u d g un nud unpd
(7)
Where pis the prescribed pressure on p
, g
nis the gravity direction vector, nis the direction vector
normal to corresponding boundary, g is the gravitational acceleration, l
is the free surface, and u
~is
a virtual displacement field. The second term in the left part of the above equation results from the
free surface boundary condition relative to the sloshing movement.
3 HIERARCHICAL FINITE ELEMENT FORMULATION
The liquid is discretized by four nodes hierarchical axisymmetric quadrilateral isoparametric finite
elements (see Figure 1). All The liquid (inside and at the free surface) can be discretized into only
one element if the geometry is varying linearly or if it is constant.
The radial, circumferential, and axial displacement components u, v, and w along the respective
directions R, Z and θ may be written in the form
nzruzru cos,,,
(8-a)
nzrvzrv cos,,,
(8-b)
nzrwzrw sin,,,
(8-c)
In the above, the dependence on time is removed by assuming that the displacements vary si-
nusoidally in phase at the same frequency. The displacements
zru ,,
zrv ,, and
zrw , may be
expressed in terms of the hierarchical finite element generalized displacements using appropriate
shape functions.
Figure 1: Hierarchical axisymmetric liquid finite element
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3.1 Shape Functions Selection
They can be classified in three categories: nodal shape functions, side shape functions, and internal
shape functions. The hierarchical shape functions for a one-dimensional element are:
,...3,2,1d
1
0
2
2
1
iPxf
xxf
xxf
x
ii
(9)
Where
i
P are the shifted Legendre orthogonal polynomials defined as:
,...3,2,12412
1
1
12
1
11
1
0
iiPPii
i
P
P
P
iii
(10)
For C° continuous problems, the first two linear shape functions of the standard finite element
method are retained. Higher order C° shape functions vanish at each end of the element. These are
used to describe the displacements in the interior of the element.
The displacements
zru ,,
zrv , and
zrw , for a quadrilateral hierarchical finite element can
be expressed as
n
ii uzrNzru
1
,, (11-a)
n
ii vzrNzrv
1
,, (11-b)
n
ii wzrNzrw
1
,, (11-c)
Where
1,...1,1,...1,
qlpkzfrfzrN lki (12)
In the above, p and q denote the hierarchical mode numbers along ξ and η, respectively. The
matrix form of equations (11) is
qN
(13)
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Where
zrw
zrv
zru
,
,
,
(14)
)1()1(,...,1,...,,,...,,, 111 qpiwvuwvuq T
iii (15)
)1)(1(21 ,...,,...,,
qpi NNNNN (16)
zgrf
zgrf
zgrf
N
lk
lk
lk
i
.00
0.0
00.
(17)
Using the same shape functions matrix for the displacement fields uand u
~, one can write
qNu
(18)
qNu ~~
(19)
Where
qis the nodal displacement vector and
q
~ is an arbitrary constant vector.
Using the variational form of (7) and replacing u and u
~by their expressions in (18) and (19),
one can obtain the following matrix equation
fqSKqM ll
. (20)
Where,
l
K is the stiffness matrix associated with the volumetric deformation,
s
Kis the stiffness
matrix associated to the sloshing, and
f is the load vector. They are expressed as
dNNM T
ll
(21)
dBBkK l
T
lld (22)
l
dNnnNgK T
g
T
ls
(23)
p
dnNpf T (24)
Where kl is the liquid bulk modulus, Ω is the volume of the liquid,
N is the shape function matrix,
and
l
B is a matrix defined as
NdB ll
(25)
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3.2 Liquid Stiffness Matrix
The stiffness matrix is the superposition of the volumetric deformation stiffness matrix [Kd] and the
sloshing one [Ks].
sdl KKK
(26)
3.2.1 Volumetric Deformation Stiffness Matrix [Kd]
The liquid deformation is given by
w
r
r
u
z
v
r
u1
. (27)
Where
is a differential operator defined as
rzrr
11 (28)
Generalized displacements expressed in terms of the deformation are written as
ll d (29)
Where
f
d is a differential operator defined as
n
r
n
n
z
n
rr
dlcoscoscos
1 (30)
Substituting exp. (25) into (22), the element stiffness matrix can be written as
1
0
1
0
1
1
1
1
p
i
q
jl
T
lld ddJrBBkkK ji
(31)
Where k = 2 for n = 0 and k = 1 for n = 1,2,… (n: circumferential wave number)
3.2.2 Sloshing Stiffness Matrix
The vectors {n} and {ng}are orthogonal then, the sloshing stiffness matrix (23) can be written
l
rdrNNgK T
ls
2 (32)
To evaluate the sloshing stiffness matrix
s
K, the global cylindrical coordinates (r, z) must be
expressed in terms of dimensionless local coordinates (ξ,η). A characteristic of the liquid element is
that the surface local coordinate ξ varies from 0 to 1, while the local coordinated η remains constant
and is equal to 1 because of its position at the free surface.
The final expression of the sloshing stiffness is
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1
0
1
1
1
1
4343 12
p
i
q
jj
T
ils drrNNrrgK
(33)
As a result, the stiffness matrix, superposition of the deformation stiffness matrix and the slosh-
ing one, is
1
0
1
0
1
1
1
1
1
0
1
1
1
1
4343 1)(2
p
i
q
j
p
i
q
jj
T
ifl
T
lll drrNNrrgddJrBBkkK ji
(34)
3.3 Liquid Mass Matrix
By replacing the matrix
N by the expression (16), the liquid mass matrix can be written
1
0
1
0
1
1
1
1
drdJNNkM p
i
q
jj
T
i (35)
4 RESULTS AND DISCUSSIONS
The convergence and comparison studies must be carried out to ensure the reliability of the results.
The results are given by the frequency parameter Ω which is expressed in terms of the vibration
frequency ω by
2
(36)
4.1. Convergence and Validation
The effectiveness of the proposed liquid element was examined for the vibration of a liquid in a
rigid circular cylindrical cavity and in a rigid circular cylindrical tank.
4.1.1 Rigid Circular Cylindrical Cavity
It is assumed that the cavity has a radius equal to 1 and a height equal to 1 as shown in Figure 2.
The bulk modulus and the density are equal to 1. To validate the results, a standard FEM program
has been elaborated, where 2100 axisymmetric liquid elements with four nodes were used. Because
spurious modes cannot be eliminated, the real patterns will be recognized by plotting each mode.
Table 1 shows the convergence of the first six modes in terms of both of hierarchical mode
numbers p and q following respectively the radial and axial directions for one and two elements.
Considering the square shape of the cavity, the two hierarchical mode numbers will be equal.
Table 1 shows that for the two idealization one and two elements, an accuracy of two digits af-
ter the comma is reached for the first two modes for p=2 and q=2.
For modes 3, 4, 5; the convergence is reached for p=6 and q=6 for a one element model, and for
p=4 and q=4 for a two elements model. The sixth mode reaches the convergence for p=8 and q=8
for one element, and p=6 and q=6 for two elements.
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Figure 2: Liquid in a cylindrical rigid cavity
Element number p q Modes
1 2 3 4 5 6 7 8
1Elt 2 2 0.501 0.609 0.789 0.991 1.106 1.152 1.221 1.864
4 4 0.498 0.605 0.773 0.983 1.091 1.092 1.186 1.483
6 6 0.496 0.603 0.766 0.970 1.049 1.084 1.157 1.366
8 8 0.496 0.603 0.766 0.969 1.071 1.079 1.132 1.306
FEM 0.496 0.602 0.757 0.965 1.068 1.077 1.130 1.287
2Elt 2 2 0.499 0.608 0.785 0.984 1.101 1.136 1.216 1.787
4 4 0.497 0.604 0.770 0.979 1.082 1.087 1.183 1.411
6 6 0.496 0.603 0.766 0.969 1.073 1.082 1.145 1.364
8 8 0.496 0.603 0.766 0.969 1.071 1.079 1.132 1.296
FEM 0.496 0.602 0.757 0.965 1.068 1.077 1.130 1.287
Table 1: Convergence and frequency parameter comparison of a liquid in a circular cylindrical cavity (R =1, H=1).
It is apparent that by increasing the number of elements, the hierarchical mode numbers p and
q necessary to reach convergence are smaller. But the degrees of freedom number become more
important. The matrices size is smaller if one increases p and q rather than the element number.
Figure 3 shows the eight independent modes of a four node liquid element. The first six modes
are constant-strain modes and the other two are bending modes. One can find these two kind of
modes for the bi-hierarchical liquid element.
q
p
q
p
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The first eight acoustic modes of the considered example are shown in figure 4. The first five
modes are constant-strain modes and the other three are bending modes .These modes show the
shift of a liquid relative to the walls while being in contact, which reflects the modeling of the inter-
action that is, imposing the same normal displacement to the sides. In this case, the normal dis-
placement is equal to zero because the liquid is in a rigid cavity.
Figure 3: Eight independent displacement modes of a four node liquid element
Figure 4: First Eight acoustic modes of a liquid in a cylindrical rigid cavity
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4.1.2 Rigid Circular Cylindrical Tank
In order to verify its effectiveness, the hierarchical liquid element is used for the free vibration of a
liquid with respectively a volumetric mass and a bulk modulus equals to 103 Kg/m3 and 2.068 109
KPa contained firstly in a rigid circular cylindrical tank of a radius equal to 6.48m for a height
equal to 6.24m and secondly in a rigid circular cylindrical tank of a radius equal to 1.88m for a
height equal to 4.75m (see figure 5).
(a) (b)
Figure 5: Liquid in a rigid tank
The results are compared with those obtained by a standard FEM program where the axisym-
metric liquid element with four nodes is used and with the theoretical frequencies of a liquid in a
circular cylindrical storage tanks given by (Blevins, 1980)
R
H
R
g
fnnn
tanh
2
1 (37)
Where g is the gravity acceleration, R is the radius of the cylinder, H is the liquid height and εn is
the nth root of J(ε) Bessel function.
For the first example (Fig. 5.a), the comparison is carried out for one and two hierarchical ele-
ments with radial and axial hierarchical mode numbers varying from 2 to 8. The number of the
axisymmetric liquid finite elements used in the standard FEM program is equal to 4212. The rigid
tank of the second example (Fig. 5.b) being long, the comparison is carried out for only one hierar-
chical element with radial hierarchical mode number varying from 2 to 4 and axial hierarchical
mode number varying from 2 to 8. The number of the axisymmetric liquid finite elements used in
the standard FEM is equal to 2408.
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Element number p q Modes
1 2 3 4 5 6
1Elt 2 2 0.275 0.376 0.470 0.536 0.583 0.632
4 4 0.269 0.364 0.461 0.503 0.571 0.610
6 6 0.265 0.352 0.421 0.486 0.552 0.586
8 8 0.262 0.346 0.406 0.457 0.504 0.546
FEM 0.259 0.341 0.396 0.446 0.497 0.540
Theory 0.261 0.345 0.405 0.456 0.501 0.542
2Elt 2 2 0.271 0.371 0.465 0.528 0.574 0.617
4 4 0.266 0.359 0.458 0.491 0.563 0.596
6 6 0.263 0.348 0.418 0.473 0.545 0.569
8 8 0.262 0.346 0.406 0.458 0.503 0.545
FEM 0.259 0.341 0.396 0.446 0.497 0.540
Theory 0.261 0.345 0.405 0.456 0.501 0.542
Table 2: Convergence and frequency parameter comparison of a liquid in a rigid circular
cylindricaltank (R =6.48, H=6.24).
p q Modes
1 2 3 4 5 6
2 2 0.551 0.642 0.751 0.792 0.875 0.882
4 4 0.547 0.638 0.740 0.756 0.831 0.850
4 6 0.541 0.625 0.696 0.729 0.792 0.813
4 8 0.538 0.620 0.685 0.717 0.784 0.799
FEM 0.535 0.616 0.678 0.706 0.771 0.792
Theory 0.537 0.619 0.683 0.715 0.782 0.796
Table 3: Convergence and frequency parameter comparison of a liquid in a rigid circular
cylindrical tank (R =1.88, H=4.75).
Table 2 shows the convergence study for the first example (Fig. 5.a) of the first six modes with
an increasing of the two hierarchical mode numbers p and q following respectively the radius and
the axis directions for one and two elements. For the two idealization (Table 2), one and two ele-
ments, an accuracy of two digits after the comma is reached for the first mode with p=q=4. For the
second mode, an accuracy of two digits is reached for p=q=8 with one element and for p=q=6 with
two elements. For the modes 3, 4, 5; the convergence is reached for p=q=8 for the two idealization,
but the matrices size is smaller if one increases p and q rather than the element number. For the
second example, (Fig. 5.b) where the rigid tank is long, the convergence of the first six modes is
given in table 3. The liquid is idealized by only one element. The height of the liquid being greater
than the tank radius, the increasing of the two hierarchical mode numbers p and q isn’t the same.
q
p
q
p
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More the size is greater; more the increase of the hierarchical mode number is significant. As shown
in table 3, the hierarchical mode number q increase until 8, while p increase until 4. So, with this
example, one can see that for different dimensions, it is not necessary to use the same hierarchical
mode number for the radial and axial directions. This gives the advantage of having good results
with smaller matrices and reduced computation time.
4.2. Applications
The vibration analysis of a liquid in a rigid circular cylindrical tank is carried out (see figure 6).
The radius and the liquid height are equal to 1m. The liquid volumetric mass and the bulk modulus
are respectively equal to 103 Kg/m3 and 2.068 109 KPa.
The liquid will be idealized by only one hierarchical finite element; indeed, its hierarchical na-
ture provides this advantage. The height and the radius being equal, the radial and axial hierar-
chical mode numbers are equal to 8 (p=q=8).
Figure 6: Liquid in a rigid tank (H=1m, R=1m)
Figures 7 and 8, shows respectively the first eight hierarchical sloshing modes and the first eight
hierarchical acoustic modes. The liquid motion and then the deformed shapes are not the same by
comparing the sloshing modes and the acoustic ones.
Indeed, for the first eight modes, the liquid moves and then deform only at the free surface, the
liquid inside is almost steady state. This happens because at smaller frequencies, the liquid moves
only at the free surface resulting in sloshing modes.
The first eight hierarchical sloshing modes vary from 0.664 to 1.549. The deformed shapes are
the same as a beam vibration resting on a series of springs that is why there are used to model the
1278 S.M.H. Cherif and M.N. Ouissi / Free Vibration Analysis of a Liquid in a Circular Cylindrical Rigid Tank Using the Hierarchical
Latin American Journal of Solids and Structures 13 (2016) 1265-1280
free surface in some studies. The shape of the deformed is sinusoidal and the number of periods
increases with the modes.
In the eight modes shown in figure 8, only the internal liquid deforms with high frequencies
varying from 35.86 to 197.1 in the case studied. One can note the large gap with the sloshing fre-
quencies. For these modes the liquid behaves as if it were in a rigid cavity.
The deformations are similar to those of acoustic modes of the example in Figure 2 shown in
figure 4. Then, these modes are called acoustic modes.
Figure 7: First Eight sloshing modes of a liquid in a cylindrical rigid tank
Figure 8: First Eight acoustic modes of a liquid in a cylindrical rigid tank
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5 CONCLUSION
The hierarchical liquid finite element presented in this study is able to give accurate frequencies for
liquids in axisymmetric cavities with or without free surface. The results show clearly that this ele-
ment can be easily used for the analysis of the free vibration of a liquid in closed cavity or with a
free surface. With this element, one is not constrained any more to have the same number of hierar-
chical modes in the two main directions (radial and axial) to idealize a liquid in a cavity or a tank
which can be slender or lowered. Also, only one element can idealize a liquid in a rigid cavity (with-
out a free surface) or a liquid in a rigid tank (with a free surface). Finally, this hierarchical finite
element allows a triple increase in the accuracy, finite element number, and radial and axial hierar-
chical modes numbers.
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