Code_Aster ®
Version
5.0
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
Organization (S): EDF/MTI/MN, SAMTECH
Handbook of Référence
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.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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
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
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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
1 Introduction
Our analysis leaves the weak formulation of balance under a loading of following the pressure type
activated by key word TYPE_CHARGE:“SUIV” in command STAT_NON_LINE [U4.32.01].
difference compared to a traditional 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 facets of the elements 3D [R3.03.04]. Indeed, this last method, based on
a Lagrangienne formulation brought up to date, 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 point out that the geometrical part of
stamp tangent is already nonsymmetrical [bib2].
2 Kinematics
For the elements of voluminal hull one defines a surface of reference, 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) of average surface and rise 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 (1,2) = X (1,2) + U
P
P
P (1,2).
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
4/12
N (1,2) = (1,2) N (1,2)
N (1,2)

uQ (1,2,3)
Q (3 0)
Q 3 0
·
·

(
)
P (3 = 0)

·
P (3 = 0)
·
H
U P (1,2, 3 = 0)
xQ (1,2, 3)
X P (1,2, 3 = 0)
X

Q (1, 2, 3)
X P (1, 2, 3 = 0)
E 2, y
e1, X
E 3, Z
Appear voluminal 2-a: Coque.
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:



X
X

dx
P
P
P (1,2) =
D +
D
1


2
1
1
dx


P (1,2) = D has
1
1 (1,2) + D
has
2
2 (1,2)
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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




where [has (,);has
1
1 2
2 (1
, 2
)] represent a nonorthogonal natural base (A. has
1
2)
0 and not


normalized (1 has; has
1
1)
1 tangent on the surface. The two basic vectors can be
dependant on displacements via the following formula:


X
X + U
has
P
p
p
1 (1
, 2
)
(
)
=
=
1

1



X
X + U
has
P
p
p
2 (1
, 2
)
(
)
=
=
2

2

what makes it possible to connect them to the vectors of the natural base related to initial surface by
relations:
U

has
p
1 (1
, 2
) = a1 (1, 2) + 1

U

has
p
2 (1
, 2
) = a2 (1, 2) + 2

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
):
a1 (1
, 2
) (1
, 2
) a1 (1
, 2
)
a2 (1
, 2
) (1
, 2
) a2 (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) d1d2
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
6/12
3 Formulation
variational
3.1 Work
virtual
N
(,
1)
2


p0

N
(,

N
1)
2
=
(,
1)
2

(,
1)
2


p = p0


Appear voluminal 3.1-a: Coque.
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 = -

up.p D

following
If one uses the element of isoparametric surface corresponding to our modeling of hull
voluminal, surface D is expressed directly according to the isoparametric co-ordinates
D D
1 2 and one obtains the following simple form of the equation above:

= -


U p. p (, 12)

a1 (,)

pressure
× has (,)
[-, 1+] 1 [× -, 1+]
D D
following
1
1 2
2
1 2
1
2
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
7/12
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 pennies the form:

(

i+)
1
(I)
(I)
L

pressure = pressure + pressure

following
following
following


(I)
where pressure is the increment between two iterations of the virtual work of the following pressure. If
following
pressure is given in the form:
p = p0
being the level of load which is fixed lasting the iterations (control = 0 charge some), one can
to write:





pressure = - [
U.
has
has
has
has
-, +] [
× -, +] P p (1 × 2 - 2 × 1) D D
1 2
1 1
1 1
following
Incremental variations of the vectors of the tangent local base to the transform of surface
average are given by:

a1 =
U P
1


a2 =
U P
2

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 = - [, 1] 1 [, 1] uP.p has
U
has
U
1
2

- + × - +1
[
×] P -
×
P D D
1 2
following

[]
2
2



where [× has] and [has
1
2 ×] are respectively the antisymmetric matrices of the tangent vectors
has and has
1
2 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.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
8/12
4 Discretization
At the points P of average surface, the interpolation of virtual displacement is written:
U
1
NB

U (
1
1, 2)
()
= NOR (1,2) v
I 1
=

W I
and the interpolation of incremental displacement between two iterations is written:
U

1
NB


U (
1
1, 2)
()
= NOR (1,2) v

I 1
=

W I
We rewrite the two preceding equations in the matric form:
U (
E
1, 2) = [NR] {}
U
U (
E

1,2) = [NR] {}
U
where [NR] is the matrix of the functions of form of translation on the average surface, of which the expression
is:

1 0

0 0 0

0
0 0



0

[]
() 1




NR =!NR I 0 1
0 0 0
0
!0 0
0





0 0 10 0
0

0 0 0

I 1, NB1
NB2
=
() 1
(2)
Functions of form NR
and NR
I
I
(used thereafter are given in appendix of [R3.07.04].
The nodes I = 1, NB1 are the nodes nodes and the mediums on the sides (for the quadrangle and it
triangle). Node NB2 is with the barycentre of the element.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
9/12
The vector {}
U.E. is the nodal vector of virtual displacements given by:

.




.


.

U



v


W


X

{
y





ue} =

Z I



.


.




.

I =,
1 NB


1




X





X




X NB2
The vector {}
U.E. is the nodal vector of displacements incremental between two iterations.

.




.


.

U



v


W


X


{
y


ue} =


Z I



.


.




.

I =,
1 NB

1




X





X



X NB2
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
10/12
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:



U (


E
1,2) =
NR {}

U
1

1



U (


E
1, 2) =
NR {}

U
2
2




where
and
are the matrices derived from the functions of forms of translation on the surface
NR
NR
1

2
average, whose expressions are:


() 1

0 0 0

0 0
0

0 0

1
NR I






=!


!
NR
0 1 0 0 0 0
0 0 0





1

1


0

0 1 0

0 0
0

0 0








I 1
=, NR 1
B
NB2


() 1 0 0 0

0 0
0 0

0

1
NR

I







0 0 0

NR =
0 1 0 0 0 0
!

!






2



2



0 0 1 0

0 0






0 0 0

I 1, NB1
NB2
=
Thus one can express the virtual work of the following pressure in the following matric form:



E
E

pressure = {U} .f pressure
following
following








with F E
pressure the nodal vector of the external forces which can be expressed in the following way:
following








F E

=
[NR] T

[

- 1, +]
1 [
× - 1, +]
1
(a1 ×a
pressure
1) D D
1
2
following




It is important to note that with our parameterization of the transform of average surface, it
jacobien det [(J (3 =)
0]) of this surface are 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:
(
NB2
p
2
1,2)
()
= NR (1,2) p
I
I
I 1
=
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
11/12
One can also express the increment between two iterations of the virtual work of the following pressure under
the matric form:





E
E
E
pressure = - {U} .K T pressure {
U}
following
following








where K E
T pressure is the contribution in the tangent matrix of rigidity of the external forces which can
following




to be expressed in the form:










K E

=
[NR] T
a1 ×
NR


1
2
-

1, 1
1, 1

[NR] T
p
D D

[1,] 1 [1,]
p
1
[a2 ×]
[
] [
]
[]
NR
T pressure
D D


- + × - +
- + × - +
1
2
following




2
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
facets of the finite elements of three-dimensional elasticity.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Code_Aster ®
Version
5.0
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:
12/12
5 Bibliography
[1]
Mr. Al MIKDAD: “Static and Dynamique of Poutres in Grandes Rotations and Résolution of
Problems of Instabilité Non Linéaire ", thesis of doctorate, Université de Technologie of
Compiegne, (1998)
[2]
E.G. CARNOY, NR. GUENNOUN & G. SANDER: “Static Buckling Analysis off Shells Submitted
to Follower Pressure by the Finite Element Method ", Computers and Structures, Vol. 19,
N° 1-2, 41-49, (1984)
[3]
PH. JETTEUR: “Kinematic Non Linéaire of Coques”, report/ratio SAMTECH, contract
PP/GC-134/96, (1998)
[4]
K. SCHWEIZERHOF & E. RAM: “Displacement Dependant Pressure Loads in Non Linear
Finite Element Analyzes ", Computers and Structures, Vol. 18, N° 6, 1099-1114, (1984)
[5]
J.C. SIMO, R.L. TAYLOR & P. WRIGGERS: “A Note one Finite-Element Implementation off
Press Boundary Loading ", Communications in Applied Numerical Methods, Vol. 7, 513-525
(1995)
[6]
I. VAUTIER: “Implemented of STAT_NON_LINE”, handbook of Descriptif Informatique
Code_Aster [D9.05.01]
[7]
P. MASSIN, Mr. Al MIKDAD: “Code_Aster: Voluminal elements of hulls in Non Linéaire
Geometrical ", handbook of Référence of Code_Aster [R3.07.05]
[8]
E. LORENTZ: “Efforts external of pressure in great displacements”, handbook of
Reference of Code_Aster [R3.03.04]
[9]
P. MASSIN and A. LAULUSA: “Modeling Numérique of Coques Volumiques”, handbook of
Reference of Code_Aster [R3.07.04]
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

Outline document