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Titrate:
Following pressure for the voluminal elements of hulls
Date:
19/12/00
Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
R3.03.07-A
Page:
1/12
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R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A
Organization (S):
EDF/MTI/MN, SAMTECH
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
R3.03.07 document
Following pressure for the elements of hulls
voluminal
Summary:
We present in this document, the model used to calculate the loading of following the pressure type
acting on the average surface of the finite elements of voluminal hulls corresponding to modeling
COQUE_3D
. Discretization of the loading led to a nodal vector of the external forces and to one
nonsymmetrical contribution in the tangent matrix of rigidity. These finite elements objects are evaluated with
each iteration of the algorithm of Newton of
STAT_NON_LINE
.
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Code_Aster
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
19/12/00
Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
R3.03.07-A
Page:
2/12
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HI-75/00/006/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Kinematics ........................................................................................................................................... 3
2.1 Parameterization of the transform of average surface ............................................................. 4
3 variational Formulation ...................................................................................................................... 6
3.1 Virtual work ................................................................................................................................... 6
3.2 Tangent operator ........................................................................................................................... 7
4 Discretization .......................................................................................................................................... 8
5 Bibliography ........................................................................................................................................ 12
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Code_Aster
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
19/12/00
Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
R3.03.07-A
Page:
3/12
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R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A
1 Introduction
Our analysis leaves the weak formulation of balance under a loading of following the pressure type
activated by the key word
TYPE_CHARGE:“SUIV”
in the control
STAT_NON_LINE
[U4.32.01].
difference compared to a conventional geometrical linear analysis is that the pressure acts on
geometry deformed and either on the initial geometry. This new geometry is obtained to leave
transform of the initial average surface subjected to great displacements and the large ones
rotations [R3.07.05]. The notations are inspired by [R3.07.05].
This transform can be paramétrisée exactly as initial surface by using them
reduced co-ordinates of the associated isoparametric element: Co-variable or counter-variable reference marks
build themselves in each point of deformed surface. The writing of the virtual work of the pressure with
this parameterization is done in the configuration deformed by using the isoparametric elements
associated. It results an independence from it from the field of integration with displacements that one
use to express the variation of the virtual work of the efforts external of pressure compared to the known as ones
displacements. That has an important advantage compared to the method applied for
pressure which follows the breakages of the elements 3D [R3.03.04]. Indeed, this last method, based on
a brought up to date Lagrangian formulation, led to nonlinear terms difficult to linearize,
coming from the transformation jacobienne compared to the configuration of reference.
The finite elements objects obtained by linearization compared to incrémentaux displacements of
virtual work of the efforts external of pressure are to be reactualized with each iteration of the algorithm of
Newton of
STAT_NON_LINE
. We underline the fact that the contribution of the following pressure to
stamp tangent rigidity is nonsymmetrical, and we remind the meeting that the geometrical part of
stamp tangent is already nonsymmetrical [bib2].
2 Kinematics
For the elements of voluminal hull
a surface of reference is defined
, or surfaces
average, left (of curvilinear co-ordinates
1 2
for example) and a thickness
()
H
,
1 2
measured according to the normal on the average surface. The position of the points of the hull is given by
curvilinear co-ordinates
()
1 2
,
average surface
and front elevation
3
compared to this
surface.
One points out the great transformation undergone by the hull:
(together of the points
P
with
3
0
=
) is the transform of initial average surface
(together of the points
P
with
3
0
=
).
The position of the point
P
on the deformed configuration can be established according to the position of
not initial
P
as follows:
()
()
()
X
X
U
P
P
P
1 2
1 2
1 2
,
,
,
=
+
.
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Code_Aster
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
19/12/00
Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
R3.03.07-A
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·
H
·
·
·
()
N
1 2
,
(
)
U
Q
1 2
3
,
(
) (
) (
)
N
N
1
2
1
2
1
2
,
,
,
=
(
)
Q
3
0
(
)
P
3
0
=
(
)
Q
3
0
(
)
P
3
0
=
(
)
U
P
1
2
3
0
,
=
(
)
X
Q
1
2
3
,
(
)
X
P
1
2
3
0
,
,
=
E
2
, y
(
)
X
Q
1
2
3
,
,
E
3
, Z
E
1
, X
(
)
X
P
1
2
3
0
,
,
=
Appear voluminal 2-a: Hull.
Great transformations of an initially normal fiber on the average surface
2.1
Parameterization of the transform of average surface
The transform
can be paramétrisée in a way similar to parameterization of surface
initial. Thus one can define the infinitesimal element of tangent vector in
:
(
)
(
)
(
)
(
)
dx
X
X
dx
has
has
P
P
P
P
D
D
D
D
1 2
1
1
1
2
1 2
1
1
1 2
2
2
1 2
,
,
,
,
=
+
=
+
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Code_Aster
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
19/12/00
Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
R3.03.07-A
Page:
5/12
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HI-75/00/006/A
where
(
) (
)
[
]
has
has
1
1 2
2
1 2
,
;
,
represent a nonorthogonal natural base
(
)
has has
1
2
0
.
and not
normalized
(
)
has
has
1
1
1
1
;
tangent on the surface
. The two basic vectors can be
dependant on displacements via the following formula:
()
(
)
()
(
)
has
X
X
U
has
X
X
U
1
1 2
1
1
2 1 2
2
2
,
,
=
=
+
=
=
+
P
p
p
P
p
p
what makes it possible to connect them to the vectors of the natural base related to initial surface
by
relations:
() ()
() ()
has
has
U
has
has
U
1
1 2
1 1 2
1
2 1 2
2 1 2
2
,
,
,
,
=
+
=
+
p
p
It is important to note that these vectors are distinct from the vectors obtained by great rotation
vectors
(
) (
)
has
has
1 1 2
2
1 2
,
;
,
:
(
)
(
) (
)
(
)
(
) (
)
has
has
has
has
1
1 2
1 2
1 1 2
2
1 2
1 2
2
1 2
,
,
,
,
,
,


Indeed, because of deformation due to transverse shearing, the turned vectors are not any more
tangent with
. The illustration of that is given by [Figure 3.1-a].
With this parameterization, the infinitesimal vector element of surface which is perpendicular to
can be written:
() () ()
D
has
has
1 2
1
1 2
2
1 2
1
2
,
,
,
=
×
D D
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Code_Aster
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
19/12/00
Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
R3.03.07-A
Page:
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3 Formulation
variational
3.1 Work
virtual
)
,
(
2
1
N
=
)
,
(
)
,
(
)
,
(
2
1
2
1
2
1
N
N
0
p
0
p
p
=
Appear voluminal 3.1-a: Hull.
Following pressure on initial average surface and its transform
The virtual work of a following pressure
p
(i.e. acting on transformed average surface
and moving with) can be expressed in the form:
pressure
following
p
p D
= -
U.
If one uses the element of isoparametric surface corresponding to our modeling of hull
voluminal, surface
D
express yourself directly according to the isoparametric co-ordinates
D D
1
2
and one obtains the following simple form of the equation above:
()
[
] [
]
()
()
pressure
following
p
p
D D
= -
×
- + × - +
U
has
has
.
,
,
,
,
,
1 2
1
1 1
1 1
1 2
2
1 2
1
2
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
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Author (S):
P. MASSIN, Mr. Al MIKDAD
Key:
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3.2 Operator
tangent
As the virtual work of the following pressure depends on the current configuration, its variation
linear
is not null and must be taken into account. The tangent operator associated this virtual work
is written with the iteration
(
)
I
+
1
in the form:
()
()
()
L
pressure
following
I
pressure
following
I
pressure
following
I
+


=
+
1
where
()
pressure
following
I
is the increment between two iterations of the virtual work of the following pressure. If
pressure is given in the form:
p
p
=
0
being the level of load which is fixed lasting the iterations (piloting in load
=
0
), one can
to write:
(
)
[
] [
]
pressure
following
P
p
D D
= -
×
-
×
- + × - +
U
has
has
has
has
.
,
,
1
2
2
1
1
2
1 1
1 1
Incremental variations of the vectors of the tangent local base to the transform of surface
average are given by:
has
U
has
U
1
1
2
2
=
=
P
P
since initial surface average “does not move” not during the iterations what involves
X
P
=
0
.
These calculations finally make it possible to establish the expression of the increment of the virtual work of pressure
following in the form:
[]
[]
[
] [
]
pressure
following
P
P
P
p
D D
= -
×
-
×




- + × - +
U
has
U
has
U
.
,
,
1
2
2
2
1
2
1 1
1 1
where
[] []
has
has
1
2
×
×
and
are respectively the antisymmetric matrices of the tangent vectors
has
has
1
2
and
respectively.
Note:
In the reference [bib2], an integration by part is undertaken on the expression above. It
is shown that the tangent matrix can be broken up into a symmetrical part resulting
of an integration on the field and an antisymmetric part resulting from integration on
contour. II is as shown as the assembly of the antisymmetric parts of the matrices
elementary tangents leads to a null matrix when the pressure is continuous of one
finite element with another, because of existence of a potential associated with work with the pressure
in this case there.
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Following pressure for the voluminal elements of hulls
Date:
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P. MASSIN, Mr. Al MIKDAD
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4 Discretization
At the points
P
average surface, the interpolation of virtual displacement is written:
(
)
()
(
)


U
1
2
1
1
1
1 2
,
,
=




=
NR
U
v
W
I
I
NB
I
and the interpolation of incremental displacement between two iterations is written:
(
)
()
(
)
U
1
2
1
1
1
1 2
,
,
=




=
NR
U
v
W
I
I
NB
I


We rewrite the two preceding equations in the matric form:
(
)
[]
{}
(
)
[]
{}
U
NR
U
U
NR
U
1
2
1 2
,
,
=
=
E
E
where
[]
NR
is the matrix of the functions of form of translation on the average surface, of which the expression
is:
[]
()
NR
=






















=
!
!
NR
I
I
NB
NB
1
1
1
2
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
,
Functions of form
()
()
NR
NR
I
I
1
2
and
(used thereafter are given in appendix of [R3.07.04].
Nodes
I
NB
=
1
1
,
are the nodes nodes and the mediums on the sides (for the quadrangle and it
triangle). The node
NB2
is with the barycentre of the element.
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Following pressure for the voluminal elements of hulls
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The vector
{}
U
E
is the nodal vector of virtual displacements given by:
{}







U
E
X
y
Z I
X
X
X NB
U
v
W
I
NB
=














=


















































.
.
.
.
.
., 1 1
2
The vector
{}
U
E
is the nodal vector of displacements incremental between two iterations.
{}







U
E
X
y
Z
I
X
X
X NB
U
v
W
I
NB
=














=
















































.
.
.
.
.
.,




1
1
2
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Titrate:
Following pressure for the voluminal elements of hulls
Date:
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P. MASSIN, Mr. Al MIKDAD
Key:
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Page:
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HI-75/00/006/A
This discretization also enables us to establish the expression of derived from incremental displacement
average surface compared to the surface isoparametric co-ordinates in the form:
(
)
{}
(
)
{}
1
1 2
1
2
1
2
2
U
NR
U
U
NR
U
,
,
=


=


E
E
where
1
2
NR
NR








and
are the matrices derived from the functions of forms of translation on the surface
average, whose expressions are:
()
()
1
1
1
1
1
2
2
1
2
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
NR
NR




=


























=




=
!
!
!
NR
NR
I
I
NB
NB
I
,


















=
I
NB
NB
1
1
2
0 0 0
0 0 0
0 0 0
,
!
Thus one can express the virtual work of the following pressure in the following matric form:
{}
pressure
following
E
pressure
following
E
=








U
F
.
with
F
pressure
following
E








the nodal vector of the external forces which can be expressed in the following way:
[]
(
)
[
] [
]
F
NR
has
has
pressure
following
E
T
D D








=
×
- + × - +
1
1
1
2
1 1
1 1
,
,
It is important to note that with our parameterization of the transform of average surface, it
jacobien
(
)
[
]
(
)
det J
3
0
=
of this surface is not implied in the calculation of the finite elements objects.
It will be also noted that the pressure is discretized with an isoparametric interpolation of the values with
NB2 nodes:
(
)
()
(
)
p
NR
p
I
I
NB
I
1 2
2
1
2
1 2
,
,
=
=
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Titrate:
Following pressure for the voluminal elements of hulls
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Key:
R3.03.07-A
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One can also express the increment between two iterations of the virtual work of the following pressure under
the matric form:
{}
{}
pressure
following
E
T pressure
following
E
E
= -








U
K
U
.
where
K
T pressure
following
E








is the contribution in the tangent matrix of rigidity of the external forces which can
to be expressed in the form:
[]
[]
[
] [
]
[]
[]
[
] [
]
K
NR
has
NR
NR
has
NR
T pressure
following
E
T
T
p
D D
p
D D








=
×


-
×


- + × - +
- + × - +
1
2
1
2
1 1
1 1
2
1
1
2
1 1
1 1
,
,
,
,
Note:
It is noted that the finite elements formulations resulting from this approach do not make
to intervene degrees of freedom of rotations. The processing is thus also valid for
breakages of the finite elements of three-dimensional elasticity.
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Titrate:
Following pressure for the voluminal elements of hulls
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5 Bibliography
[1]
Mr. Al MIKDAD: “Static and Dynamics of the Beams in Great Rotations and Resolution of
Problems of Nonlinear Instability ", thesis of doctorate, University of Technology of
Compiegne, (1998)
[2]
E.G. CARNOY, NR. GUENNOUN & G. SANDER: “Static Buckling Analysis off Shells Submitted
to Follower Presses by the Finite Element Method ", Computers and Structures, vol. 19,
N° 1-2, 41-49, (1984)
[3]
PH. JETTEUR: “Kinematic Nonlinear of the Hulls”, report/ratio SAMTECH, contract
PP/GC-134/96, (1998)
[4]
K. SCHWEIZERHOF & E. RAM: “Displacement Dependant Presses Loads in Not Linear
Finite Element Analyzes ", Computers and Structures, vol. 18, N° 6, 1099-1114, (1984)
[5]
J.C. SIMO, R.L. TAYLOR & P. WRIGGERS: “A Notes one off Finite-Element Implementation
Press Boundary Loading ", Communications in Applied Numerical Methods, vol. 7, 513-525
(1995)
[6]
I. VAUTIER: “Implemented of
STAT_NON_LINE
“, Data-processing manual of Description
Code_Aster [D9.05.01]
[7]
P. MASSIN, Mr. Al MIKDAD: “Code_Aster: Voluminal elements of hulls into Nonlinear
Geometrical ", manual of Reference of Code_Aster [R3.07.05]
[8]
E. LORENTZ: “Efforts external of pressure in great displacements”, manual of
Reference of Code_Aster [R3.03.04]
[9]
P. MASSIN and A. LAULUSA: “Numerical Modeling of the Voluminal Hulls”, manual of
Reference of Code_Aster [R3.07.04]