Code_Aster
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
B. QUINNEZ
Key:
R4.07.05-A
Page:
1/20
Manual of Reference
R4.07 booklet: Coupling fluid-structure
HI-75/97/035/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R4.07 booklet: Coupling fluid-structure
R4.07.05 document
Homogenization of a bathing network of beams
in a fluid
Summary:
This note describes a model obtained by a method of homogenization to characterize the behavior
vibratory of a periodic network of tubes bathed by an incompressible fluid. Then the development of one
finite element associated this homogenized model is presented.
The tubes are modelized by beams of Euler and the fluid by a model with potential.
This modeling is accessible in the control
AFFE_MODELE
by choosing modeling
3d_FAISCEAU
.
Code_Aster
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Key:
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Contents
1 Introduction ............................................................................................................................................ 3
2 initial physical Problem ....................................................................................................................... 3
2.1 Description of the problem ................................................................................................................. 3
2.2 Assumptions of modeling ........................................................................................................... 4
3 Problem homogenized ......................................................................................................................... 5
3.1 Homogenized problem obtained ...................................................................................................... 5
3.2 Matric problem ........................................................................................................................... 7
4 Resolution of the cellular problem .......................................................................................................... 8
4.1 Problem to be solved ....................................................................................................................... 8
4.2 Problem are equivalent to define
.............................................................................................. 9
4.3 Practical application in Code_Aster ........................................................................................... 10
5 Choices of the finite element for the problem homogenized ....................................................................... 10
5.1 Choice of the finite elements ............................................................................................................... 10
5.2 Finite elements of reference ........................................................................................................... 11
5.2.1 Net HEXA 8 ...................................................................................................................... 11
5.2.2 Net HEXA 20 .................................................................................................................... 12
5.3 Choice of the points of Gauss ........................................................................................................... 14
5.4 Addition of the problems of traction and torsion ............................................................................... 14
5.4.1 Problem of traction ............................................................................................................ 15
5.4.2 Problem of torsion ............................................................................................................. 15
5.5 Integration in Code_Aster of this finite element ....................................................................... 15
6 Use in Code_Aster ............................................................................................................. 16
6.1 Data necessary .............................................................................................................. 16
6.2 Orientation of the axes of the beams .................................................................................................. 16
6.3 Modal calculation .................................................................................................................................. 16
7 Characterization of the spectrum of the model homogenized .......................................................................... 17
7.1 Heterogeneous model ........................................................................................................................ 17
7.2 Homogeneous model ......................................................................................................................... 18
7.2.1 Continuous problem ................................................................................................................. 18
7.2.2 Discretized problem .............................................................................................................. 18
8 Conclusion ........................................................................................................................................... 19
9 Bibliography ........................................................................................................................................ 20
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1 Introduction
In nuclear industry, certain structures make up of networks quasi-periodicals of tubes
bathed by fluids: “combustible” assemblies, steam generators,… For
to determine the vibratory behavior such structures, the conventional approach (each tube is
modelized, the volume occupied by the fluid is with a grid) is expensive and tiresome even impracticable (in
private individual, development of an intricate mesh containing a great number of nodes). Structures
studied presenting a character quasi-periodical, it seems interesting to use methods
of homogenization.
Techniques of homogenization applied to a network of tubes bathed by a fluid were with
various already elaborate recoveries [bib1], [bib5], [bib4]. The models obtained differ by the assumptions
carried out on the fluid (compressibility, initial speed of the flow, viscosity). According to
allowed assumptions, the action of the fluid on the network of tubes corresponds to an added mass (drops
frequencies of vibration compared to those given in absence of fluid), with one
damping even added to an added rigidity [bib5].
At the beginning, finite elements associated two-dimensional models (network of deadheads
bathed by a fluid) were elaborate [bib2]. To study the three-dimensional problems (network of
tubes), a solution to consist in projecting the movement on the first mode of bending of the beams
[bib4]. Later on, of the three-dimensional finite elements were developed [bib3], [bib8].
2
Initial physical problem
2.1
Description of the problem
One considers a whole of identical beams, axis Z, laid out periodically (either
the period
of space). These beams are located inside a chamber filled with fluid (see [2.1-a]). One
wish to characterize the vibratory behavior of such a medium, while considering for the moment only
the effect of added mass of the fluid which is dominating [bib6].
L
F
X
y
Z
External edge
Side surface
L
beam N
°
L
Appear 2.1-Error! Argument of unknown switch.
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2.2
Assumptions of modeling
It is considered that the fluid is a true fluid initially at rest, incompressible. Like
the assumption of small displacements around the position of balance was carried out (fluid initially
at rest), the field of displacement of the fluid particles is irrotational so that there is one
potential of displacement of the noted fluid
. There is no flow of fluid through external surface
.
It is considered that the beams are homogeneous and with constant section according to
Z
. To modelize them
beams, the model of Euler is used and the movements of bending are only taken into account.
section of beam is rigid and the displacement of any point of the section is noted:
S
L
the bending of the beam n°
L
()
() ()
(
)
(
)
S Z
S Z S Z
L
X
L
y
L
=
,
.
The beams are embedded at their two ends.
The variational form of the problem fluid-structure (conservation of the mass, dynamic equation of
each tube) is written:
(
)
=
·
F
L
S N
V
L
L
éq 2.2-1
S
L
L
L
L
L
L
F
L
L
L
S
S
S
T S
I.E.(internal excitation)
S
Z
S
Z
T N S
S
V
L
2
2
0
2
2
2
2
0
2
2
0
+
= -
éq 2.2-2
with:
] [
(
)
(
)
()
V
H
L
V
S
F
=
=
02
2
1
0,
H
and
where:
·
N
is the normal entering to the beam n° L,
·
F
is the constant density of the fluid in all the field,
·
S
is the density of material constituting the beam,
·
S
is the section of the beam,
·
E
is the Young modulus,
·
I
is the tensor of inertia of the section of the beam.
The second member of the equation [éq 2.2-2] represents the efforts exerted on the beam by the fluid.
pressure
p
fluid is related to the potential of displacement by:
p
T
F
= -
2
2
. In the same way, it
second member of the equation [éq 2.2-1] represents the flow of fluid induced by the movements of
beams. At the border of each beam
L
one a:
S N
N
L
· = ·
.
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3 Problem
homogenized
3.1
Homogenized problem obtained
To take account of the periodic character of the studied medium, one uses a method of homogenization
based in this precise case on an asymptotic development of the variables intervening in
physical starting problem. With regard to the operational step, one returns the reader to
following references [bib2], [bib4], [bib5], [bib6]. One will be satisfied here to state the results obtained.
In the homogenized medium
0
(see [3.1-a]), the two following homogenized variables are
considered:
S
0
(displacement of the beams) and
0
(potential speeds of the fluid). In form
variational, these variables are connected by the equations:
With
Ds
V
M
S
T S
K
S
Z
S
Z
D
T
S
S
V
F
S
= -
+
=
0
0
2
0
2
2
0
2
2
2
2
0
2
0
0
0
0
0
hom
hom
éq 3.1-1
where:
()
()
] [
()
()
(
)
{
] [
()
}
V
L
H
S
L
H
v
X y
S
Z
v X y Z
H
L
shom
,
;
,
,
,
=
= ×
=
2
0
2
02
2
0
02
0
02
0
0
where
!
.
L
0
X
y
Z
Lower edge
0, low
Higher edge
0, high
External edge
side
0, lat
S
Appear 3.1-Error! Argument of unknown switch.
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Y
Y
F
S
1
y1
y
2
N
Appear 3.1-Error! Argument of unknown switch.
The various tensors which intervene in [éq 3.1-1] are defined using two functions
(
)
=
1 2
,
in the following way:
()
()
B
B
Y
y
y
y
y
With
has
Y
Y
B
ij
Y
Y
Y
Y
ij
F
ij
ij
F
F
F
F
=
=
=
=
-
1
0
0
0
0
0
1
1
1
2
2
1
2
2
éq 3.1-2
()
()
()
D
D
B
Y
Y
Y
M
m
B
Y
S
S
K
K
E
Y
I
I
I
I
ij
S
S
ij
F
S
S
ij
X
xy
xy
y
=
= +
=
=
+
=
=
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
µ
µ
éq 3.1-3
where
Y
Y
F
S
and
the surfaces of the fluid fields and structure of the airframe represent respectively
elementary of reference (cf [3.1-b]).
Y
represent the sum of the two preceding surfaces.
basic cell of reference is homothetic of report/ratio
µ
with the real airframe of interval of
heterogeneous medium.
Two functions
(
)
=
1 2
,
are solutions of a two-dimensional problem, called problem
cellular. On the basic cell of reference, the functions
(
)
=
1 2
,
are defined by:
(
)
=
=
v
N v
v V
Y
Y
F
F
0
to have a single solution
éq 3.1-4
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where:
()
()
{
}
V
v
Y
v y
y
F
=
H
,
1
periodical in
from period
1
Note:
It is shown that the two-dimensional part of B is symmetrical and definite positive [bib5].
Note:
In the matrix M, the term
F
B
play the part of a matrix of mass added suitable for
each beam in its airframe.
Note:
For the various tensors, one can put in factor the multiplicative term
1
Y
. It was added
in order to obtain the “good mass” of tubes in absence of fluid. One has then
M FD
0
=
mass tubes component
0
.
3.2 Problem
matric
By discretizing the problem [éq 3.1-1] by finite elements, one leads (with obvious notations) to
following problem:
“
“
“
“
“
With
Ds
M
S
T
Ks
D
T
F
T
0
0
2
0
2
0
2
0
2
= -
+
=
éq 3.2-1
what can be put in the form (one pre - multiplies the first equation by
F
):
~
~
“
“
“
“
“
M
S
T
T
K S
M
D
D
S
T
T
K
With
S
F
T
F
F
2
0
2
2
0
2
0
0
2
0
2
2
0
2
0
0
0
0
0
0
+
=
-
-
+
-
=
éq 3.2-2
Note:
The problem obtained is symmetrical. If instead of choosing the potential of displacement for
to represent the fluid, one had chosen the potential speed, one would have obtained a matric problem
nonsymmetrical also revealing a matrix of damping.
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4
Resolution of the cellular problem
4.1
Problem to be solved
On the two-dimensional basic cell (see [4.1-a]), one seeks to calculate the functions
(
)
=
1 2
,
checking:
(
)
=
=
v
N v
v V
Y
Y
*
*
0
to have a single solution
éq 4.1-1
where:
()
()
{
}
V
v H Y
v y
y
=
1
*
,
periodical in
from period
1
y 1
y2
N
Y *
Y
Appear 4.1-Error! Argument of unknown switch.
After having determined the functions
(
)
=
1 2
,
, the homogenized coefficients are calculated
(
)
B
=
=
1 2
1 2
;
,
defined by the formula:
B
y
Y
=
éq 4.1-2
By using the formula of Green and the periodic character of
, it is shown that the coefficients
B
can be written differently:
B
N
=
éq 4.1-3
To estimate this quantity, it is necessary at the time of a discretization by finite elements, to determine for each
element the outgoing normal, which can be tiresome. Another way then is operated; while taking
in the equation [éq 4.1-1]
v
=
, one obtains:
B
Y
=
éq 4.1-4
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From the function potential energy defined by the conventional formula:
()
W v
v v
Y
= -
1
2
éq 4.1-5
one can rewrite the coefficients homogenized in the form:
(
)
()
()
(
)
B
W
W
W
= -
+
-
-
éq 4.1-6
In the two-dimensional general case, one must calculate three coefficients of the homogenized problem
B
B
B
B
11
12
21
22
,
,
=
(it is known that the matrix
()
B
B
=
is symmetrical). One must solve both
following problems:
To calculate
To calculate
To calculate
1
1
1
2
2
2
1
2
=
=
=
+
V
v
N v
V
v
N v
V
Y
Y
Y
Y
/
/
*
/*
*
*
*
*
éq 4.1-7
One has then:
()
()
()
() ()
(
)
B
W
B
W
B
B
W
W
W
11
1
22
2
12
21
1
2
2
2
= -
= -
=
= -
-
-
*
éq 4.1-8
Note:
If the basic cell has symmetries, that makes it possible to solve the problem on one
part of the airframe with boundary conditions adapted well and to only calculate
certain coefficients of the homogenized problem. For example for the airframe of the figure
n°4.1 - has one a:
B
B
B
B
11
22
12
21
0
=
=
=
.
4.2
Problem are equivalent to define
In the equation [éq 4.1-1], the calculation of the second member requires the determination of the normal with
edge. To avoid a determination of the normal, one can write an equivalent problem, checked by
functions
.
Are the vectors
G
G
G
1
2
1
0
0
1
1
1
=
=
=
,
and
, the functions are sought
1
2
,
,
V
such as:
=
=
=
1
1
2
2
v
G
v
v V
v
G
v
v V
v
G
v
v V
Y
Y
Y
Y
Y
Y
éq 4.2-1
By using the formula of Green and the anti-interval of the normal
N
, it is shown that the problems
[éq 4.1-1] and [éq 4.2-1] are equivalent.
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4.3
Practical application in Code_Aster
In Code_Aster, to solve the problem [éq 4.2-1], the thermal analogy by defining one
material having a coefficient
C
p
equal to zero and one coefficient
equal to is used. To impose it
calculation of the second member utilizing the term in
G
, the key word
GRAD_INIT
in the control
AFFE_CHAR_THER
is selected. The thermal problem is solved by using the control
THER_LINEAIRE
. The calculation of the potential energy
W
is provided by the control
POST_ELEM
with
the option
ENER_POT
. In the general case, three calculations are carried out to determine the values
() ()
()
W
W
W
1
2
,
,
and then, the values of the coefficients of the homogenized problem are
deduced manually. To impose the periodic character of the space in which the solution is
sought, the key word
LIAISON_GROUP
in the control
AFFE_CHAR_THER
is used.
5
Choice of the finite element for the homogenized problem
5.1
Choice of the finite elements
In the model presented previously, the axis
Z
a dominating role as a principal axis has of
beams. The developed finite elements check this characteristic. The meshs are of the cylindrical type:
the quadrangular bases are contained in plans
Z
=
Cte
and cylinder centers it is parallel to
the axis
Z
(see [5.1-a]).
Center Z
L
Z
Appear 5.1-Error! Argument of unknown switch.
According to the equations [éq 3.1-1], derivative second following the co-ordinate
Z
intervene in
model, which requires finite elements
C
1
in the direction
Z
. Functions of form of the type
Hermit to represent the variations of
S
along the axis
Z
are thus used. At the points of
discretization, displacements
S
X
, S
y
but also the derivative
S
Z
S
Z
X
y
,
who are related to the degrees of
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freedom of rotation
X
y
,
by the formulas
X
y
y
X
S
Z
S
Z
=
= -
,
must be known. In what
relate to the variations according to
X y
,
, one limits oneself for the moment to functions of form
Q
1
.
For the degree of freedom of potential, functions of form
Q
1
or
Q
2
according to the three directions
X y Z
,
space are used.
The finite element thus has as unknown factors the following degrees of freedom:
S S
X
y
X
y
,
,
,
,
.
Note:
The command of the nodes of the meshs support is very important. Indeed, edges parallel with
the axis
Z
only the edges contained in the plans are represented in the same way
Z
=
Cte
. The nodes of the meshs are thus arranged in a quite precise order: list
nodes of the lower base, then list of with respect to the higher base (or vice versa).
With regard to the geometry, the functions of form allowing to pass from the element of
reference to the element running are
Q
1
. The finite element is thus under-parametric.
Two finite elements were developed:
·
a associate with a mesh HEXA 8. In each node of the mesh, the unknown factors are
S S
X
y
X
y
,
,
,
,
. Functions of form associated with the potential
are
Q
1
.
·
another associate with a mesh HEXA 20. In each node node of the mesh, unknown factors
are
S S
X
y
X
y
,
,
,
,
. In each node medium of the edges, the unknown factor is
. Functions
of form associated with the potential
are
Q
2
.
5.2
Finite elements of reference
5.2.1 Net HEXA 8
On the finite element of reference HEXA 8 (see [5.2-a]), the following functions of form are defined:
()
() () ()
NR
P
P
P
L
L
± ± ±
±
±
±
=
=
1 1 1
1
1
1
2
1
3
,
with L
or D or R
éq 5.2-1
Indices
±
1
represent the co-ordinates of the nodes of the mesh support of reference.
The functions which make it possible to define the functions of form write:
()
()
()
()
()
()
()
()
()
()
()
()
()
() (
)
()
() (
)
[]
P
P
P
P
P
P
P
P
D
D
R
R
-
-
-
-
= -
= +
= -
= +
=
-
+
=
+
-
=
- - +
=
- - + +
-
1
1
1
1
1
3
1
3
1
2
3
1
2
3
1
2
1
2
1
2
1
2
1
2 1
3
2
1
2
1
2 1
3
2
1
2
1
4 1
1
4
1
11
,
éq 5.2-2
Functions
P P
D
R
,
are related to Hermit.
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The unknown factors of the problem homogenized, on a mesh, break up in the following way:
()
()
()
()
()
()
()
()
(
)
S
DX NR
L
DRX NR
S
DY NR
L
DRY NR
NR
X
I
I
D
I
Z
I
I
R
I
y
I
I
D
I
Z
I
I
R
I
J
J
J
=
+
=
-
=
=
=
=
=
=
=
1
8
1
8
1
8
1
8
1
8
1
2
3
2
2
,
éq 5.2-3
where
DX DY DRX DRY
I
I
I
I
I
,
,
,
,
are the values of displacement according to
X
, of displacement according to
y
, of rotation around the axis
X
, of rotation around the axis
y
and of the potential of displacement with
node I of the mesh. In
Code_Aster
, for each node, the degrees of freedom are arranged
in the order quoted previously.
1
2
3
4
5
8
1
2
3
6
7
1
0
- 1
- 1
1
0
- 1
0
0
1
Appear 5.2-Error! Argument of unknown switch.
5.2.2 Net HEXA 20
On the finite element of reference HEXA 20 (see [5.2-b]), the following functions of form are
defined:
()
() () ()
NR
P
P
P
L
L
± ± ±
±
±
±
=
=
1 1 1
1
1
1
2
1
3
,
with L D or R
éq 5.2-4
()
()
NR
Q
J
J
J
=
=
3
1 20
,
éq 5.2-5
Indices
±
1
represent the co-ordinates of the nodes nodes of the mesh support of reference.
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Functions
P
P
L
±
±
1
1
,
were already defined in the paragraph [§5.2.1]. Functions
Q
I
are defined
by:
()
(
) (
) (
) (
)
()
()
(
)
(
) (
)
()
()
(
)
(
) (
)
()
()
(
)
(
)
Q
I
Q
I
Q
I
Q
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
=
+
+
+
+
+
-
=
=
-
+
+
=
=
-
+
+
=
=
-
+
+
1
8 1
1
1
2
1
8
1
4 1
1
1
9 11 17 19
1
4 1
1
1
10 12 18 20
1
4 1
1
1
1
1
2
2
3
3
1
1
2
2
3
3
1
1
2
2
2
3
3
2
2
2
1
1
3
3
3
3
2
1
1
,…,
,
,
,
,
,
(
)
2
2
13 14 15 16
I
I
=
,
,
éq 5.2-6
where
(
)
1
2
3
I
I
I
,
are the co-ordinates of the node
I
mesh.
The unknown factors of the problem homogenized, on a mesh, break up in the following way:
()
()
()
()
()
()
()
()
(
)
S
DX NR
L
DRX NR
S
DY NR
L
DRY NR
NR
X
I
I
D
I
Z
I
I
R
I
y
I
I
D
I
Z
I
I
R
I
J
J
J
=
+
=
-
=
=
=
=
=
=
=
1
8
1
8
1
8
1
8
1
20
1
2
3
2
2
,
éq 5.2-7
where
DX DY DRX DRY
I
I
I
I
I
,
,
,
,
are the values of fluid displacement according to X, displacement
according to y, rotation around axis X, of rotation around the axis y and the potential of displacement
at node I of the mesh
(
)
I
=
1 8
,
and
J
fluid potential of displacement to the node medium of the edges
(
)
J
=
9 20
,
.
1
2
3
4
5
8
2
1
3
6
7
1
0
- 1
- 1
1
0
- 1
9
10
11
12
13
14
1
15
16
17
18
19
20
0
Appear 5.2-Error! Argument of unknown switch.
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5.3
Choice of the points of Gauss
Each integral which intervenes in the forms of the elementary matrices, is transformed into
an integral on the element of reference (a change of variable is carried out) who is then
calculated by using a formula of quadrature of the GAUSS type.
The points of Gauss are selected in order to integrate exactly the integrals on the element of
reference. Families of different points of integration are used to calculate the matrices of
mass and the matrices of rigidity (the degrees of the polynomials to be integrated are different). But here, for
to calculate the various contributions of the matrix of mass, various families of points of Gauss
can still be used.
The element of reference being a HEXA 8 or one HEXA 20, the integral on volume can be separate in
a product of three integrals which correspond each one to a direction of the space of reference.
a number of points of integration necessary is determined by direction.
According to the mesh of reference, the number of points of integration by direction is as follows:
Net HEXA 8
Net HEXA 20
direction
X
or
y
direction
Z
direction
X
or
y
direction
Z
stamp
“K
2
2
2
2
stamp
“A
2
2
3
3
stamp
“D
2
3
2
3
stamp
“
M
2
4
2
4
Four families of points of Gauss were defined. Each family corresponds to one of the matrices of
problem to be solved.
On the segment [- 1,1], the co-ordinates of the points of integration and their weights are as follows [bib7]:
A number of points of integration
Co-ordinates
Weight
2
±
1
3
/
1
3
0
3 5
±
/
8 9
5 9
/
/
4
±
-
±
+
3 2 6 5
7
3 2 6 5
7
/
/
1
2
1
6.6 5
1
2
1
6.6 5
+
-
/
/
The weight of a point of Gauss in the three-dimensional element of reference is obtained by multiplying them
three weights corresponding to each co-ordinate of the point of Gauss.
5.4
Addition of the problems of traction and torsion
To supplement the problem of bending homogenized described previously, the problem of traction and it
problem of torsion are added in an uncoupled way.
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5.4.1 Problem of traction
The problem of traction homogenized is written in the following form:
E S
Y
U
Z
v
Z
S
Y
U
T v
v V
T
Z
S
T
Z
2
2
2
2
0
+
=
with
[]
()
V
H
L
=
1
0,
The finite element of reference is a HEXA 8 having for unknown factor displacement
DZ
in each node.
The associated functions of form are Q
1.
5.4.2 Problem of torsion
The problem of torsion homogenized is written in the following form:
(
)
E
Y J
U
Z
v
Z
Y
J
U
T v
v V
T
Z
Z
S
T
Z
Z
2
2
2
2
2 1
0
+
+
=
with
[]
()
V
H
L
=
1
0,
where
J
Z
is the constant of torsion.
The finite element of reference is a HEXA 8 having for unknown factor displacement
DRZ
in each
node. The associated functions of form are Q
1.
5.5
Integration in Code_Aster of this finite element
The finite element is developed in Code_Aster in 3D. A modeling was added in
catalog modelings:
·
“FAISCEAU_3D”
for the 3D.
In the catalog of the elements, the element can apply to the two following meshs:
Net
A number of nodes
in displacement and
rotation
A number of nodes
in fluid potential
Name of the element
in the catalog
HEXA 8
8
8
meca_poho_hexa8
HEXA 20
8
20
meca_poho_hexa20
In the routines of initialization of this element, one defines:
·
two families of functions of form respectively associated with displacements and rotation
beams (linear function of form in
X y
,
and cubic in
Z
) and under the terms of potential
fluid (function linear in
X y Z
,
),
·
four families of points of Gauss to calculate the matrix of rigidity and the various parts
matrix of mass.
During the calculation of the elementary terms, the derivative first or seconds of the functions of form
on the element running are calculated. In spite of the simplified geometry of the finite element (the axis of the mesh
cylindrical is parallel to the axis
Z
and the sections lower and higher are in plans
Z
=
Cte
),
a general subroutine to calculate the derivative second was written [bib7]. In addition, two
news subroutines was developed starting from the subroutines existing for the elements
isoparametric to take account of the under-parametric character of the element.
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6
Use in Code_Aster
6.1
Data necessary
Characteristics of the beams (section
S
, tensor of inertia
I
, constant of torsion
J
Z
) are
informed directly under the key word factor
BEAM
control
AFFE_CARA_ELEM
.
The characteristics of the homogenized coefficients and the airframe of reference are indicated under
the key word factor
POUTRE_FLUI
control
AFFE_CARA_ELEM
. For the single-ended spanner words,
correspondence is as follows:
B_T
:
B
11
B_N
:
B
22
B_TN
:
B
12
A_FLUI
:
Y
F
A_CELL
:
Y Y
Y
F
S
=
+
COEF_ECHELLE
:
µ
The characteristics of materials are indicated in the control
DEFI_MATERIAU
. For
tubes, the key word factor
ELAS
is used to indicate the Young modulus (
E
:
E
), the coefficient of
Poisson (
NAKED
:
) and density (
RHO
:
S
). For the fluid, the key word factor
FLUID
is used
to indicate the density of the fluid (
RHO
:
F
).
6.2
Orientation of the axes of the beams
The generators of the cylindrical meshs are obligatorily parallel to the axis of the beams and them
bases of the meshs perpendicular to this same axis. During the development of the mesh, it is necessary to be ensured
that the command of the nodes (local classification) of each cylindrical mesh is correct: nodes of
base lower then the nodes of the higher base (or vice versa). Direction of the axis of the beams
is well informed under the key word factor
ORIENTATION
control
AFFE_CARA_ELEM
.
The following assumption was carried out: the reference mark of reference is the same one as the main reference mark
of inertia of the characteristic tube representing the homogenized medium. That means that in
equations [éq 3.1-3], the term
I
xy
is null.
6.3 Calculation
modal
The developed finite element makes it possible to characterize the vibratory behavior of a network of beams
bathed by a fluid. It is interesting to determine the frequencies of vibration of such a network in
air and out of water.
To carry out a modal calculation in air (
F
=
0
), it is necessary to lock all the degrees of freedom corresponding
with the fluid potential of displacement
, if not rigidity stamps it (and even the matrix shiftée of
modal problem) is noninvertible [R5.01.01].
To carry out a modal water calculation (
F
0
), it is necessary to use in the control
MODE_ITER_SIMULT
, the option
CENTER
key word factor
CALC_FREQ
. The shiftée matrix
(
)
~
~
K
M
-
is then invertible if
is not eigenvalue or if
is different from zero.
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7
Characterization of the spectrum of the homogenized model
7.1 Model
heterogeneous
That is to say a network with square pitch of
N
beams fixed in their low ends and of which ends
higher move in the same way (uniform movement) (cf appears [7.1-a]). Only them
movements of bending are considered.
Embedding
Uniform movement
X
y
Z
H
H
L
Appear 7.1-Error! Argument of unknown switch.
The spectrum of vibration in air of this network to the following form. For each command of mode of
vibration of bending, the modal structure consists of a frequency doubles agent with one
mode in
X
and with a mode in
y
where all the higher part moves (all the beams have
even deformed) and of a frequency of multiplicity
(
)
2
2
N
-
agent with modes where all
higher part of the beams is motionless and where beams move in opposition of phase.
In the presence of fluid, the spectrum is modified. For each command of mode of vibration in bending, them
2 N
frequencies of vibration are lower than the frequencies of vibration obtained in air. The effect of
incompressible fluid is comparable with an added mass. There is always a double frequency
agent with a mode in
X
and with a mode in
y
where all the higher part moves (all them
beams have the same deformation). On the other hand, one obtains
(
)
N
-
1
couples different of double frequency
(one in
X
and one in
y
) agent with modes where all the higher part of the beams is
motionless and where beams move in opposition of phase.
frequency
of vibration
in air
out of water
Frequency
double
Frequency of
multiplicity 2n-2
Frequency
double
Spreading out of
spectrum
For a command of mode of bending
Appear 7.1-Error! Argument of unknown switch.
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7.2 Model
homogeneous
The heterogeneous medium was replaced by a homogeneous medium.
7.2.1 Problem
continuous
Recent work, concerning a problem of plane homogenization of a network of deadheads
reserves by springs and bathed by a fluid, show that the spectrum of the homogeneous model
continuous consists of a continuous part and two frequencies of infinite multiplicity [bib10].
spectrum of the Eigen frequencies of the water problem is also contained in an interval well
defined limited supérieurement by the fréqence of vibration in air of a deadhead [bib5].
These results are transposable for each command of bending of the network of tubes.
7.2.2 Problem
discretized
That is to say the homogeneous field with a grid by hexahedrons. That is to say
p
the number of generators parallel with
axis Z of the network of beams.
Embedding
Uniform movement
X
y
Z
H
H
L
Appear 7.2.2-a
One finds results similar to those obtained for the heterogeneous model. It is enough to replace
N
by
p
. For a command of bending of beam, the number of frequencies corresponding to modes where
beams do not vibrate all in the same direction, depends on the discretization used in
transverse directions with the axis of the beams.
According to the finite element used (mesh HEXA 8 or nets HEXA 20), the distribution of
(
)
2
2
p
-
last
frequencies is different. The first frequency doubles (that corresponding to the mode where the part
higher moves) is the same one for the two finite elements.
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Frequency
of vibration
Command of
frequencies
Frequency
double
Frequency of
multiplicity 2p-2
For a command of mode of bending
HEXA 8
HEXA 20
Appear 7.2.2-b
All in all, the homogeneous model makes it possible to obtain the frequencies of vibration easily
agent with modes where all the beams vibrate in the same direction. Other modes
obtained provide only one vision partial of the spectrum. In the discretized spectrum, one can turn over
one or two frequencies of infinite multiplicity present in the spectrum of the continuous model.
8 Conclusion
The use of the finite elements developed associated the homogenized model of a beam of tubes
periodical bathed by a fluid makes it possible to characterize the overall vibratory movements (all
the structure moves in the same direction) of such a structure.
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9 Bibliography
[1]
E. Sanchez-Palencia (1980), “Not homogeneous media and vibration theory”, Springer
Verlag.
[2]
“Asymptotic Study of the dynamic behavior of the fuel assemblies of one
nuclear jet engine " Siaka Berete, Thesis of Doctorate of the University Paris VI, constant it
April 19, 1991.
[3]
“Behavior under dynamic stresses of cores of pressurized water reactors”
E. Jacquelin, Thesis carried out at the Central School of Lyon with EDF-SEPTEN (Division: Ms,
Group: DS), December 1994.
[4]
“Study of the interaction fluid-structure in the beams of tubes by a method
of homogenization: application to the seismic analysis of cores RNR " L. Hammami, Thesis
university of Paris VI, 1990.
[5]
“Mathematical Problems in coupling fluid-structure, Applications to the beams
tubular " C. Conca, J. PLanchard, B. Thomas, Mr. Vanninathan, Collection of Management
studies and Search of Electricity of France, n°85, Eyrolles.
[6]
“Taken into account of an incompressible true fluid at rest like masses added on one
structure, bibliographical Synthesis " G. Rousseau, Internal report EDF - DER, HP-61/94/009.
[7]
“A presentation of the finite element method” G. Dhatt and G.Touzot, Maloine S.A.
Paris editor.
[8]
D. Brochard, F. Jedrzejewski and Al (1996), “3D Analysis off the fluid structure interaction in tub
bundles using homogenization methods ", PVP-Flight. 337, Fluid-Structure Interaction ASME
1996.
[9]
H. Haddar, B. Quinnez, “Modeling by homogenization of the grids of mixture of
assemblies fuel ", Internal report EDF-DER, HI-75/96/074/0.
[10]
G. Wing, C. Conca, J. Planchard, “Homogenization and Bloch wave method for fluid-tube
bundle interaction ", Article in preparation.