Code_Aster ®
Version
4.0
Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
B. QUINNEZ
Key:
R4.07.05-A
Page:
1/20
Organization (S): EDF/IMA/MN
Handbook of Référence
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 modelled by beams of Euler and the fluid by a model with potential.
This modeling is accessible in command AFFE_MODELE by choosing modeling
3d_FAISCEAU.
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
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Code_Aster ®
Version
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
B. QUINNEZ
Key:
R4.07.05-A
Page:
2/20
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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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
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Key:
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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,… Pour
to determine the vibratory behavior such structures, the traditional approach (each tube is
modelled, the volume occupied by the fluid is with a grid) is expensive and tiresome even impracticable (in
private individual, development of a complicated grid 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 runners
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 inflection 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 an enclosure 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].
Side surface L of beam N ° L
External edge
L
Z
y
X
F
Appear 2.1-Error! Argument of unknown switch.
Handbook of Référence
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
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Key:
R4.07.05-A
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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 model them
beams, the model of Euler is used and the movements of inflection are only taken into account.
section of beam is rigid and the displacement of any point of the section is noted:
S
(SSL Z (SSL Z SSL
=
,
Z
X
y
)
L the inflection of the beam n° 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:
=
(SSL ·N)
V
éq 2.2-1
L
F
L
L
2 L
L
2 L
2 L
L
S
S
S
2
L
L
L
S
S + I.E.(internal excitation)
= -
N S
S V
éq 2.2-2
S
2
2
2
T
Z
F
Z
2
S
T
0
0
0
L
with:
V = (H2
2
1
0]
(0, [L) and V
S
= H (F)
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.
2
pressure p of the fluid is related to the potential of displacement by: p = -
. In the same way, it
F
T
2
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: SSL · N = · N.
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
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Key:
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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 (see [3.1-a]), the two following homogenized variables are
0
considered: S (displacement of the beams) and (potential speeds of the fluid). In form
0
0
variational, these variables are connected by the equations:
With
hom
0 = -
Ds
0
V
0
0
2
éq 3.1-1
S
2
S 2
S
2
M
0 S
0
0
hom
2
+ K
2
2
=
D
S
2
S V
T
Z
Z
F
T
S
0
0
0
where:
2
V hom = L2
2
2
S
(0) H0 ()
where
0 = S ×], 0 [
L
H2
2
0 (0) = {v; (X, y) S
Z
v
! (X, y, Z) H0]
(, 0 [L)}
Higher edge
0, high
External edge
side 0, lat
L
Z
y
X
0
Lower edge
0, low
S
.
Appear 3.1-Error! Argument of unknown switch.
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
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Key:
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1
YF
y
2
YS
N
y1
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:
1
1
0
y
y
Y
1
Y
2
F
F
1
2
2
Y
B = (b) =
0
With
= (has
F
=
-
éq 3.1-2
ij)
B
ij
Y
y
y
Y
ij
ij
Y
1
Y
2
F
F
0
0
0
Y
0
0
S
0
0
S
2
S
1
µ
D = (D = +
0
0
=
=
+
0
0
ij)
B
Y
M
S
(mij) B
S
Y
F
Y
S
0 0 0
0
0
0
éq 3.1-3
I
I
0
E 2
X
xy
µ
K = (K =
0
ij)
I
I
Y
xy
y
0 0 0
where Y and Y respectively represent the surfaces of the fluid fields and structure of the cell
F
S
elementary of reference (cf [3.1-b]). Y represents the sum of the two preceding surfaces.
basic cell of reference is homothetic of µ report/ratio to the real cell of periodicity 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
F
V
Y
éq 3.1-4
=
0 (to have a solution uniqu)
E
YF
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Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
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Key:
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where:
V = {v H
periodical in period 1
1 (Y), v (y)
y
F
}
Note:
It is shown that the two-dimensional part of B is symmetrical and definite positive [bib5].
Note:
In the matrix M, the term B plays the part of a matrix of mass added suitable for
F
each beam in its cell.
Note:
1
For the various tensors, one can put in factor the multiplicative term
. It was added
Y
in order to obtain the “good mass” of tubes in absence of fluid. One has then
M FD
=
0
mass tubes composing 0.
3.2 Problem
matric
By discretizing the problem [éq 3.1-1] by finite elements, one leads (with obvious notations) to
following problem:
“A = - “Ds
0
0
2
2
éq 3.2-1
“
s0
M
+ “Ks = “DT
0
T 2
0
F
T 2
what can be put in the form (one pre - multiplies the first equation by):
F
2s
2
0
s0
~
T
t2
~ S
0
“
M
- “D
2
“K
0
S
M
0
2
+ K
F
T
=
+
=
éq 3.2-2
2
0
-
-
0
0
“D
0
0
0
“A
F
F
0
t2
t2
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.
Handbook of Référence
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
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Author (S):
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Key:
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Page:
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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
éq 4.1-1
0
to have a single solution
* =
(
)
Y
where:
V = {v H *,
periodical in period 1
1 (Y) v (y)
y
}
y2 Y *
Y
N
y 1
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
=
éq 4.1-2
Y y
By using the formula of Green and the periodic character of, one shows 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 =
éq 4.1-4
Y
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Homogenization of a network of beams bathing in a fluid
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From the function potential energy defined by the traditional formula:
1
W (v) = -
v
éq 4.1-5
2
v
Y
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 (one knows that the matrix B = (b) is symmetrical). One must solve both
11
12
21
22
following problems:
Ca
lculer V/* v = N v
1
1
Y
* 1
Y
Ca
lculer V/
éq 4.1-7
*
v =
N v
2
2
Y
* 2
Y
To calculate
* V/* = +
1
2
One has then:
B
= - 2W
11
(1)
B
= - 2 W
éq 4.1-8
22
(2)
B
= B = -
* -
-
12
21
(W () W (1) W (2)
Note:
If the basic cell has symmetries, that makes it possible to solve the problem on one
part of the cell with boundary conditions adapted well and to only calculate
certain coefficients of the homogenized problem. For example for the cell of the figure
n°4.1 - has one a: B = B
B = B = 0.
11
22
12
21
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.
1
0
1
G = G are the vectors =
G
,
and
=, the functions are sought,
, V
1
2
1
2
0
1
1
such as:
v
G
v
v V
1
Y
= 1
Y
v
G
v
v V
éq 4.2-1
2
Y
= 2
Y
v
Y
G v
v V
=
Y
By using the formula of Green and the anti-periodicity of normal N, one shows that the problems
[éq 4.1-1] and [éq 4.2-1] are equivalent.
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Homogenization of a network of beams bathing in a fluid
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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 equal to zero and one coefficient equal to is used. To impose it
p
calculation of the second member utilizing the term in G, key word GRAD_INIT in the command
AFFE_CHAR_THER is selected. The thermal problem is solved by using the command
THER_LINEAIRE. The calculation of the potential energy W is provided by command POST_ELEM with
option ENER_POT. In the general case, three calculations are carried out to determine the values
W (,
,
and then, the values of the coefficients of the homogenized problem are
1) W (2) W ()
deduced manually. To impose the periodic character of the space in which the solution is
sought, key word LIAISON_GROUP in command 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, axis Z has a dominating role as a principal axis 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 the axis of the cylinder is parallel to
axis Z (see [5.1-a]).
Lz
Center Z
Appear 5.1-Error! Argument of unknown switch.
According to the equations [éq 3.1-1], derivative second following co-ordinate Z intervene in
model, which requires finite elements C1 in direction Z. Functions of form of the type
Hermit to represent the variations of S following axis Z are thus used. At the points of
S S
discretization, displacements S
X
y
X, sy but also the derivative,
who are related to the degrees of
Z
Z
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Homogenization of a network of beams bathing in a fluid
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sy
S
freedom of rotation,
by the formulas
X
=
,
= -
must be known. In what
X
X
y
y
Z
Z
relate to the variations according to X, y, one limits itself for the moment to functions of form Q.
1
For the degree of freedom of potential, functions of form Q or Q following the three directions
1
2
X, y, Z of 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
axis Z are in the same way represented only the edges contained in the plans
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. The finite element is thus under-parametric.
1
Two finite elements were developed:
· a associate with a mesh HEXA 8. In each node of the mesh, the unknown factors are
S, S. The functions of form associated with the potential are Q.
X
y
X
y
1
· another associate with a mesh HEXA 20. In each node node of the mesh, unknown factors
are S, S. In each node medium of the edges, the unknown factor is. Functions
X
y
X
y
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 L
with L
or D or R
éq 5.2-1
1, 1, 1 () = P 1 (1) P 1 (2) P L
± ± ±
±
±
±1 (3)
=
The 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:
1 -
1+
P
1
=
1
=
- ()
P ()
()
()
2
2
1 -
1+
P
1
=
1
=
- ()
P ()
()
()
2
2
1
3
1
1
3
1
[
-,]
11 éq 5.2-2
3
PD
3
1
1
1
=
-
+
- ()
PD1 ()
()
()
2
2
2
=
+
-
2
2
2
1
1
PR
2
3
2
3
1
1
1
=
- - +
1
=
- - + +
- ()
(
) PR () (
)
()
()
4
4
The functions P D P R
,
are related to Hermite.
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Homogenization of a network of beams bathing in a fluid
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The unknown factors of the problem homogenized, on a mesh, break up in the following way:
8
L 8
S
D
Z
R
=
+
X ()
DX NR
I
I ()
DRX NR
I
I ()
2
i=1
I
=1
8
L 8
S
D
Z
R
=
-
=,
éq 5.2-3
1 2
y ()
DY NR
I
I ()
DRY NR
I
I ()
(
3)
2
i=1
I
=1
8
(
) = NR
J
J ()
J
=1
where DX, DY, DRX, DRY, are the values of displacement according to X, displacement according to
I
I
I
I
I
y, of rotation around axis X, rotation around the axis y and 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.
8
7
3
6
1
5
2
4
3
1
0
0
- 1
- 1
1
2
- 1
0
1
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 L
with L D or R
éq 5.2-4
1, 1, 1 () = P 1 (1) P 1 (2) P L
± ± ±
±
±
±1 (3)
=
NR () = Q (
1, 20
éq 5.2-5
3)
J =
J
J
The indices ± 1 represent the co-ordinates of the nodes nodes of the mesh support of reference.
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Date:
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The functions P, P L
±
were already defined in the paragraph [§5.2.1]. The functions Q are defined
1
±1
I
by:
1
Q
I
I
I
I
I
I
= 1+
,…,
1
1
1
+ 2 1
2
+ 3 3 1 1 + 2 2 + 3 3 - 2
= 1
8
I ()
(
) (
) (
) (
)
I
8
1
2
Q
I
I
I
= 1 -
,
,
1
1
1
+ 2 1
2
+ 3 3
= 9 11 17 19
I ()
4 (
()) (
) (
)
I
1
éq 5.2-6
2
Q
I
I
I
= 1 -
,
,
,
2
1
2
+ 1 1
1
+ 3 3
= 10 12 18 20
I ()
4 (
()) (
) (
)
I
1
2
Q
I
I
= 1 -
(
I
I = 13, 14, 15, 16
2
2)
3
1
3
+ 1 1
1
+
I ()
4 (
()) (
)
where (I
I
I
,
are the co-ordinates of node I of the mesh.
1
2 3)
The unknown factors of the problem homogenized, on a mesh, break up in the following way:
8
L 8
S
D
Z
R
=
+
X ()
DX NR
I
I ()
DRX NR
I
I ()
2
i=1
I
=1
8
L 8
S
D
Z
R
=
-
=,
éq 5.2-7
1 2
y ()
DY NR
I
I ()
DRY NR
I
I ()
(
3)
2
I =1
I
=1
20
(
) = NR
J
J ()
J
=1
where DX, DY, DRX, DRY, are the values of fluid displacement according to X, of displacement
I
I
I
I
I
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 fluid potential of displacement to the node medium of the edges
J
(J = 9,2) 0.
8
19
7
20
18
16
15
1
6
5
17
4
3
1
11
3
13
0
12
14 10 0 2
- 1
- 1
1
9
2
- 1
0
1
1
Appear 5.2-Error! Argument of unknown switch.
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HI-75/97/035/A
Code_Aster ®
Version
4.0
Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
B. QUINNEZ
Key:
R4.07.05-A
Page:
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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 type GAUSS.
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 “
With
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
0
8/9
3
± 3/5
5/9
3 - 2 6/5
1
1
±
+
4
7
2 6.6/5
1
1
3 + 2 6/5
-
±
2 6.6/5
7
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 inflection homogenized described previously, the problem of traction and it
problem of torsion are added in an uncoupled way.
Handbook of Référence
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Titrate:
Homogenization of a network of beams bathing in a fluid
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Author (S):
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5.4.1 Problem of traction
The problem of traction homogenized is written in the following form:
E S 2 U
v
2 S 2u
Z
S
Z
+
v = 0
v
V
with V = H1 [(0, L])
Y
Z
Z
Y
T
2
T
T
The finite element of reference is a HEXA 8 having for unknown factor displacement DZ in each node.
The associated functions of form are Q1.
5.4.2 Problem of torsion
The problem of torsion homogenized is written in the following form:
E 2
U v
2
2u
Z
S
Z
with V = H1 [(0, L])
(
+
=
+)
J
J
v 0
v V
Y
Z Z Z
Y
Z
t2
2 1
T
T
where J is the constant of torsion.
Z
The finite element of reference is a HEXA 8 having for unknown factor displacement DRZ in each
node. The associated functions of form are Q1.
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
A number of nodes
Name of the element
in displacement and
in fluid potential
in the catalog
rotation
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 in X, y and cubic function of form in Z) and under the terms of potential
fluid (linear function 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 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.
Handbook of Référence
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
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Key:
R4.07.05-A
Page:
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6
Use in Code_Aster
6.1
Data necessary
The characteristics of the beams (section S, tensor of inertia I, constant of torsion J) are
Z
informed directly under the key word factor POUTRE of command AFFE_CARA_ELEM.
The characteristics of the homogenized coefficients and the cell of reference are indicated under
the key word factor POUTRE_FLUI of command AFFE_CARA_ELEM. For the single-ended spanner words,
correspondence is as follows:
B_T: b11
B_N: b22
B_TN: b12
A_FLUI: YF
A_CELL: Y = Y + Y
F
S
COEF_ECHELLE: µ
The characteristics of materials are indicated in command 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: ). For the fluid, the key word factor
S
FLUIDE is used
to indicate the density of 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 grid, 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 of command AFFE_CARA_ELEM.
The following assumption was carried out: the reference mark of reference is the same one as the principal reference mark
of inertia of the characteristic tube representing the homogenized medium. That means that in
equations [éq 3.1-3], term I is null.
xy
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 (= 0), it is necessary to block all the degrees of freedom corresponding
F
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 (0), it is necessary to use in the command
F
~
~
MODE_ITER_SIMULT, option CENTER of the key word factor CALC_FREQ. The shiftée matrix (K - M)
is then invertible if is not eigenvalue or if is different from zero.
Handbook of Référence
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
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Key:
R4.07.05-A
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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 step 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 inflection are considered.
Uniform movement
Z
y
L
X
H
Embedding
H
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 inflection, the modal structure consists of a frequency doubles correspondent with one
mode in X and with a mode in where all the higher part moves (all the beams have there
even deformed) and of a frequency of multiplicity (2 N - 2) correspondent 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 inflection, 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
correspondent with a mode in X and a mode in there 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) correspondent with modes where all the higher part of the beams is
motionless and where beams move in opposition of phase.
For a command of mode of inflection
Frequency
Frequency of
double
multiplicity 2n-2
in air
frequency
of vibration
out of water
Frequency
Spreading out of
double
spectrum
Appear 7.1-Error! Argument of unknown switch.
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HI-75/97/035/A
Code_Aster ®
Version
4.0
Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
B. QUINNEZ
Key:
R4.07.05-A
Page:
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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 runners
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 runner [bib5].
These results are transposable for each command of inflection 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.
Uniform movement
Z
y
L
X
H
Embedding
H
Appear 7.2.2-a
One finds results similar to those obtained for the heterogeneous model. It is enough to replace N
by p. Pour a command of inflection 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 p - 2) 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.
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
Author (S):
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Key:
R4.07.05-A
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For a command of mode of inflection
Frequency
Frequency
Frequency of
double
of vibration
multiplicity 2p-2
HEXA 8
HEXA 20
Command of
frequencies
Appear 7.2.2-b
All in all, the homogeneous model makes it possible to obtain the frequencies of vibration easily
correspondent 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.
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Homogenization of a network of beams bathing in a fluid
Date:
06/01/98
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9 Bibliography
[1]
E. Sanchez-Palencia (1980), “Non homogeneous media and vibration theory”, Springer
Verlag.
[2]
“Asymptotic Study of the dynamic behavior of the fuel assemblies of one
nuclear engine " Siaka Berete, Thèse de Doctorat of Université Paris VI, constant it
April 19, 1991.
[3]
“Behavior under dynamic stresses of cores of pressurized water reactors”
E. Jacquelin, Thèse carried out in Ecole Centrale 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, Thèse
of Université of Paris VI, 1990.
[5]
“Mathematical Problems in coupling fluid-structure, Applications with the beams
tubular " C. Conca, J. PLanchard, B. Thomas, Mr. Vanninathan, Collection of Direction
of Etudes and Recherches d' Electricité de France, n°85, Eyrolles.
[6]
“Taken into account of an incompressible true fluid at rest like masses added on one
structure, bibliographical Synthèse " G. Rousseau, Rapport Intern EDF - DER, HP-61/94/009.
[7]
“A presentation of the method of Eléments finished” 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-Vol. 337, Fluid-Structure Interaction ASME
1996.
[9]
H. Haddar, B. Quinnez, “Modélisation by homogenization of the grids of mixture of
assemblies fuel ", internal Rapport EDF-DER, HI-75/96/074/0.
[10]
G. Al, C. Conca, J. Planchard, “Homogenization and Bloch wave method for fluid-tube
bundle interaction ", Article in preparation.
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
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