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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 1/56
Organization (S): EDF-R & D/AMA, SAMTECH
Handbook of Référence
R3.07 booklet: Machine elements on average surface
R3.07.05 document
Voluminal elements of hulls into nonlinear
geometrical
Summary:
We present in this document the theoretical formulation and the numerical establishment of a finite element of
voluminal hull for analyzes into nonlinear geometrical. This approach must make it possible to take in
count great displacements and great rotations of mean structures, of which the thickness report/ratio on
characteristic length is lower than 1/10. One will take care that these rotations remain lower than 2.
This formulation is based on an approach of continuous medium 3D, degenerated by the introduction of
kinematics of hull in plane constraints in the weak form of balance. The measurement of the deformations
that we retain is that of Green-Lagrange, combined énergétiquement with the constraints of Piola-Kirchhoff
of second species. The formulation of balance is thus Lagrangian total.
The geometrical entirely nonlinear problem is examined in first. The case of linear buckling is
treaty like a borderline case of the first approach.
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Voluminal elements of hulls into nonlinear geometrical
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:
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Count
matters
1 Introduction ............................................................................................................................................ 4
2 Formulation ............................................................................................................................................ 5
2.1 Geometry of the elements of voluminal hull ................................................................................ 5
2.2 Kinematics of the voluminal hulls .............................................................................................. 6
2.3 Law of behavior ....................................................................................................................... 9
2.3.1 Taking into account of transverse shearing ......................................................................... 10
3 Principle of virtual work ...................................................................................................................... 10
3.1 Virtual work interns ..................................................................................................................... 10
3.1.1 Incremental form of virtual work interns ....................................................................... 11
3.1.2 Passage of the total reference mark to the local reference mark ............................................................................ 11
3.1.3 Relation deformation-displacement ...................................................................................... 13
3.1.4 Calculation of the constraints of Cauchy ........................................................................................ 16
3.1.4.1 General case .............................................................................................................. 16
3.1.4.2 Approximation in small deformations .................................................................... 16
4 numerical Discretization of the variational formulation resulting from the principle of virtual work ............. 19
4.1 Finite elements ................................................................................................................................. 19
4.2 Discretization of the field of displacement ...................................................................................... 20
4.3 Discretization of the gradient of displacement ................................................................................... 22
4.3.1 Gradient of total displacement ............................................................................................. 22
4.3.2 Gradient of virtual displacement .......................................................................................... 24
4.3.3 Gradient of iterative displacement .......................................................................................... 25
4.3.4 Gradient of the iterative variation of virtual displacement ..................................................... 26
4.4 Discretization of the variational formulation resulting from the principle of virtual work ......................... 27
4.4.1 Vector of the forces intern .................................................................................................. 27
4.4.2 Stamp tangent rigidity .................................................................................................. 28
4.4.3 Diagrams of integration .......................................................................................................... 30
4.4.3.1 Operators of deformations of substitution ............................................................ 30
4.4.3.2 Substitution of the geometrical part of the tangent matrix of rigidity ................. 32
5 Rigidity around the transform of the normal .................................................................................. 34
5.1 Singularity of the tangent matrix of rigidity ................................................................................ 34
5.2 Principle of virtual work for the terms associated with rotation around the normal ................ 34
5.3 Notice ...................................................................................................................................... 37
5.4 Borderline case analysis geometrically linear ......................................................................... 37
5.5 Determination of the coefficient K ....................................................................................................... 38
6 linear Buckling ............................................................................................................................ 39
7 Establishment of the elements of hull in Code_Aster .................................................................. 42
7.1 Description ..................................................................................................................................... 42
7.2 Use ....................................................................................................................................... 42
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7.3 Calculation in geometrical nonlinear “elasticity” ........................................................................... 42
7.4 Establishment ................................................................................................................................... 43
7.4.1 Modification of the TE0414 ........................................................................................................ 43
7.4.2 Addition of a routine VDGNLR .................................................................................................. 43
7.5 Calculation in linear buckling ....................................................................................................... 43
8 Conclusion ........................................................................................................................................... 44
9 Bibliography ........................................................................................................................................ 45
Appendix 1
: Flow chart of calculation in linear buckling .................................................. 46
Appendix 2
: Flow chart of geometrical nonlinear calculation ............................................... 50
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Voluminal elements of hulls into nonlinear geometrical
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1 Introduction
The great transformations of hull are characterized by great displacements of surface
average and of great rotations of initially normal fibers on this surface. The transformation
thus is represented exactly, at least in the continuous problem. The derivation of the objects
finite elements associated the linearized system of equations resulting from the principle of virtual work is carried out
without any simplifying assumption on displacements or rotations. Moreover, one new
diagram of selective numerical integration is presented in order to solve the problem of blocking in
membrane and in transverse shearing.
The degrees of freedom of rotation retained are the components of the vector of space iterative rotation.
Between two iterations, it is the vector of the infinitesimal rotation superimposed on the configuration
deformation. This choice led to a tangent matrix of rigidity which is not symmetrical. This is due to
nonvectorial character of great rotations which actually belong to the differential variety
SO (3). Rotations must remain lower than 2 because of the choice of update of large
rotations established in Code_Aster, for which there is not bijection between the vector of full slewing
and the orthogonal matrix of rotation.
An important difference compared to the linear analysis is to be announced. The finite elements objects are
directly built in the total reference mark; displacements and rotations nodal are measured
in the total reference mark.
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Voluminal elements of hulls into nonlinear geometrical
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:
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2 Formulation
In this chapter, we present the various equations controlling the problem of deformation
hull within the framework of a theory of great transformations.
2.1
Geometry of the elements of voluminal hull
The voluminal hull is represented by volume (together points (
Q 3)
0) built
around the average surface (together of the points (
P 3 =)
0). In any point Q of, one
built a local orthonormé reference mark [T (,): T (,): N
1 1 2
3
2
1 2
3
(1, 2)]. The vector (n1,2)
represent the normal on the surface.
N (,),
1
2
3
T (,)
0
2
1
2
3
T (, = 0)
2
1
2
3
2
Q (0) ·
3
P (= 0) ·
3
T (, 0)
1
1
2
3
T (, = 0)
1
1
2
3
H
1
E, y
2
E, X
1
E, Z
3
Appear voluminal 2.1-a: Coque. Local reference marks on the configuration of reference
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In the initial configuration, the position of an unspecified point Q normal on the average surface can
to be expressed, according to the position of the revolved center P of normal fiber, the manner
following:
H
X (1,2, 3) = X (1,2) + 3
(N
Q
P
1,2)
2
2.2
Kinematics of the voluminal hulls
N
(,
N
1)
2
=
(,
1)
2
(,
1)
2
N
(,
1)
2
U
Q (,)
1
2
3
Q ()
0
3
·
Q ()
0
3
·
P (=)
0
3
·
P (=)
0 ·
3
H
uP (, =)
0
1
2
3
xQ (,)
1
2
3
xP (, =)
0
1
2
3
X Q (,)
1
2
3
E, y
2
xP (, = 0)
1
2
3
E, X
1
E, Z
3
Appear voluminal 2.2-a: Coque.
Great transformations of an initially normal fiber on the average surface
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In the deformed configuration, the position of the point Q can also be expressed according to
position of the point P:
H
X (1,2, 3) = X (1,2) + N
Q
P
3
(1,2)
2
where N is the unit vector obtained by great rotation of normal N.
Vector N is not necessarily normal on the deformed average surface, because of
transverse shearing strain. It is connected to the initial normal vector by the relation:
N = (1,2) N
is the orthogonal operator of the great rotation around the vector, angle, undergone by fiber
who was initially normal on the average surface whose expression is given by:
sin
1 - cos
= exp [×] = cos [I] +
[×] +
[]
2
where [×] is the antisymmetric operator of the vector of full slewing of which the matric expression
is:
0
- Z
y
[×] =
0
Z
- X
- y
0
X
and [] is the symmetrical operator given by [] = T.
More details on great rotations and their digital processing can be found in [bib1]
or [R5.03.40]. One can also write:
T1 = (1
, 2
) T1
t2 = (1
, 2
) t2
One can express the virtual variation of the operator of great rotation in the form:
= [W ×]
where [W ×] is the antisymmetric operator of the vector of space virtual rotation W which is also
rotation part of the functions tests:
[W ×] B = W B B R3
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Its matric expression is:
0
- wz
wy
[
W ×] =
W
0
Z
- wx
- wy
W
0
X
One can also express the iterative variation of the operator of great rotation in the form:
= [W ×]
where W is the vector of space iterative rotation, which is also the rotation part of the solution of
system of linearized equations.
This vector can be connected to the vector of total iterative rotation. There are thus the relations:
W = T () and W = T (
)
where T () is the differential operator of rotation, of which the expression according to the vector of rotation
total is given by:
sin
1 - cos
- sin
T () =
[I] -
[×] +
[]
2
3
This matrix has the same values and clean vectors that the matrix and checks the relation:
T () = () TT ()
In addition, the iterative variation of the matrix of virtual rotation can be put in the form:
= [W ×] [W ×]
The total displacement of the point Q on fiber can be connected to the displacement of the center of gravity P:
H
U (,) = U (,) + (N
1 2
3
1 2
3
(1,2) - (N
Q
P
1,2)
2
In order to lead to a system of linearized equations, obtained starting from the weak form of balance,
we need to calculate various differential variations of this total displacement.
virtual displacement has as an expression:
(,) = (,) + H
1 2
3
1 2
3
(1,2)
U
U
W
N (1,2); N
Q
P
= 0
2
Iterative displacement has as an expression:
H
U (,) = U (,) + W (,) N
1 2
3
1 2
3
1 2
(1,2); N
Q
P
= 0
2
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The iterative variation of virtual displacement has as an expression:
H
(1,2,3) = 3 (1,2) ((1,2)
U
W
W
N
Q
(1,2)
2
Note: The formulation suggested remains limited to rotations lower than 2. This limit is
had with the particular choice of update of the great rotations established in Code_Aster. This is
had with nonthe bijection enters the vector of full slewing and the orthogonal matrix of rotation.
2.3
Law of behavior
We consider a linear law of behavior hyper elastic: local constraints of
Piola-Kirchhoff of second species are proportional to the local deformations of
Green-Lagrange:
~
~
S = OF
Hereafter, the symbol ~ indicates the quantities expressed in the orthonormé reference mark
[T (,): T (,): N
1 1 2
3
2
1 2
3
(1, 2)].
The matrix of elastic behavior linear in plane constraints is written as follows:
E
E
0
0
0
1 - 2
1 - 2
E
0
0
0
1 - 2
E
D =
(
0
0
2 1 +)
Ek
sym
(
0
2 1 +)
Ek
(
2 1 +)
E being the Young modulus, the Poisson's ratio and K the coefficient of correction of
transverse shearing.
~
In the local reference mark, the state of Piola-Kirchhoff stress of second species is plane (Snn =)
0 and
can be characterized by a vector with 5 components:
~
S
T T
~ 1 1
St T
~2 2
~
S
S = T t12
~
St N
1
~
St N
2
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The vector of the deformations of Green-Lagrange is also expressed him in the local reference mark by one
vector with 5 components:
~
E
T T
~ 1 1
And T
~ 2 2
~
E = T t12
~
T N
1
~
T N
2
~
Here, we were unaware of the Enn term which is normal on the average surface and which is not inevitably
no one. This is a consequence of the assumption of the plane constraints.
2.3.1 Taking into account of transverse shearing
The correction of the transverse shear stress is carried out by extension of equivalences
energy given in the case of small deformations and of small displacements [R3.07.03].
3
Principle of virtual work
The principle of virtual work is the weak formulation of the static balance of the internal forces and
external forces:
int - ext. = 0
The non-linearity of the equilibrium equations leads us to solve the system above way
iterative by a method of Newton. We carry out thus the exact linearization of the principle of
virtual work with each iteration, which leads to the equality:
- ext. = ext. -
int
int
3.1
Internal virtual work
The virtual work of the internal forces can be written on the initial configuration in the form:
~ ~
int = (.
E S) D
~
~
where E and S are the vectors of deformation of Green-Lagrange and Piola-Kirchhoff constraint of
second species respectively, expressed in the local reference mark. Indeed, like the state of stress
is plane for Piola-Kirchhoff of second species, we use the formulation of the principle of work
virtual in the local reference mark. However, to limit the passages of the local reference mark to the total reference mark and
vice versa, the vectors of strains and local stresses are not calculated explicitly
in the local reference mark but they are obtained by the rotation of their representation in the total reference mark.
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3.1.1 Form
incremental
internal virtual work
The iterative variation of the work of virtual work interns is written:
~ ~
~ ~
int = (.
E S +.
E S) D
In this equality, iterative variation of the vector of local constraints of Piola-Kirchhoff of
second species is calculated by the iterative discrete form of the relation of behavior:
~ = ~
S
D E
3.1.2 Passage of the total reference mark to the local reference mark
In tensorial form one passes from the tensor of the total constraints to the tensor of the constraints
local 3 × 3 (see [bib4] p. 111 for the constraints of Cauchy, the same relations applying to
constraints of Piola-Kirchhoff of second species) while using:
~
[S]
[
P S] PT
=
and of the tensor of the local constraints to the tensor of the total constraints by the inversion of the relation
the preceding one:
T ~
[S] = P [S] P
In the two preceding expressions, the matrix of passage of the local reference mark to the total reference mark is
an orthogonal matrix P 1
- = Pt, and its expression clarifies according to the unit vectors of
locate orthonormé local is:
tT
1 (1,2, 3)
P (
T
1,2, 3) = t2 (1,2, 3)
NT
(1,2)
Within the framework of the conventional notation, one will be able to note:
T1 (1
, 2
, 3
) = E
0 1
t2 (1
, 2
, 3
) = E
0 2
T3 (1
, 2
, 3
) = (
N 1
, 2
) = E
0 3
with the orthogonal matrix of passage (initial rotation):
0 (1
, 2
, 3
) = [T1
(1
, 2
, 3
): t2 (1, 2
, 3
): T3 (1, 2
, 3
)]
It will be noticed that:
=
0
Pt
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The two relations of rotation of the constraints are also valid for the tensors of the deformations
of Green-Lagrange. Nevertheless, a writing which connects the vectors of local and total deformation is
necessary. This relation makes it possible to pass from vector 6 × 1 of the total deformations to the vector
6 × 1 of the local deformations:
~
6 1
×
6× 6.6 1
×
E = H E
with the form of the matrix of transformation of vectors 6 × 1 of deformation (see [bib2]
p. 258):
2
2
2
1
L
1
m
1
N
1
L 1
m
1
m 1
N
1
N 1l
2
2
2
L
2
2
m
N2
l2 2
m
2
m N2
N2 l2
6×6
2
2
2
L
m
N
L m
m N
N L
H
3
3
3
3 3
3 3
3 3
=
2 1ll2 2 1
m2
m
2 1
N N2
1
L 2
m + l2 1
m
1
m N2 + 2
m 1
N
1
N l2 + N2 1l
2l
2l3
2 2
m
3
m
2n2 n3 l2 3
m + l3 2
m
2
m n3 + 3
m N2 N2 l3 + n3l2
2l L
2m m
2n N
L m
+ L m
m N + m N
N L + N L
3 1
3 1
3 1
3 1
1 3
3 1
1 3
3 1
1 3
and components of the unit vectors of the local reference mark:
L = T .e
m = T .e
N
1
1
1
1
1
2
1 = t1.e3
L = T .e
m = T .e
N
2
2
1
2
2
2
2 = T 2 .e3
L = T .e
m = T .e
N
3
3
1
3
3
2
3 = t3.e3
These expressions are general for the curvilinear reference marks. In the Cartesian total reference mark
[E: E: E
1
2
3], these components are:
L = T
1
1 ()
1
m = T
1
1 (2)
N = T
1
1 ()
3
L = T
2
2 ()
1
m = T
2
2 (2)
N = T
2
2 ()
3
L = T
3
3 ()
1
m = T
3
3 (2)
N = T
3
3 ()
3
We have, actually, need for a writing which connects the vector of local deformation 5 × 1 and it
vector of total deformation 6 × 1:
~
5 1
×
5× 6.6 1
×
E = H E
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6×6
For that, one forgets the third line of the expression of H (line associated with Snn):
(
2
2
2
T1 ()
1)
(T1 (2))
(T1 () 3)
2
2
2
5×6
(t2 () 1)
(t2 (2))
(t2 () 3)
H = 2
T1 ()
1 t2 ()
1
2t1 (2) t2 (2) 2t1 ()
3 t2 ()
3
2t2 () 1t3 () 1 2t2 (2) T3 (2) 2t2 () 3t3 () 3
2
T3 ()
1 T1 ()
1
2t3 (2) T1 (2) 2t3 ()
3 T1 ()
3
T1 ()
1 T1 (2)
T1 (2) T1 ()
3
T1 ()
3 T1 ()
1
t2 ()
1 t2 (2)
t2 (2) t2 ()
3
t2 ()
3 t2 ()
1
T
1 ()
1 t2 (2) + t2 ()
1 T1 (2) T
t2 ()
3 + t2 (2) T1 ()
3
T1 ()
3 t2 ()
1 + t2 ()
3 T1 ()
1 (2)
1
t2 ()
1 T3 (2) + T3 ()
1 t2 (2) t2 (2) T3 ()
3 + T3 (2) t2 ()
3
t2 ()
3 T3 ()
1 + T3 ()
3 t2 ()
1
T
3 ()
1 T1 (2) + T1 ()
1 T3 (2) T3 (2) T1 ()
3 + T1 (2) T3 ()
3
T3 ()
3 T1 ()
1 + T1 ()
3 T3 ()
1
The same preceding relations can be applied for the passage of the vectors of
total deformation with the local deformation.
3.1.3 Relation
deformation-displacement
Tensor 3 × 3 of the total deformations of Green-Lagrange is defined by (see for example
[bib2]):
1
[E] = (U
+ C + U
You)
2
with the tensor of the gradient of displacements:
U v W
X
X X X
U v W
U = U v W
y <
> = there there y
U v W
Z
Z Z Z
The tensor of deformation of Green-Lagrange can be also written:
[
1
E] = (FTF - I)
2
with F the tensor gradient of deformations 3 × 3 which is not symmetrical:
F = X = I + U
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and I the tensor identity:
1 0
0
I = 0 1
0
0 0
1
Vector 6 × 1 of the total deformations of Green-Lagrange is ordered as follows (see [bib4]
p 117):
1
2
2
2
E
, X
, X
, X
xx
U, X
(U +v +w)
2
1
E yy
v, y
2
2
2
2 (U, y + v, y + W, y)
E W
zz
, Z
E =
=
+ 1
2
2
2
xy U, y + v, X
(U, Z +v, Z +w, Z)
2
U
xz
, Z +
W
U U
, X, y + v v
, X, y + W W
, X
, X, y
U U + v v + W W
yz
v, Z + W, y
, X, Z
, X, Z
, X, Z
U
, y U, Z + v, y v, Z + W, y W, Z
It as follows is calculated:
1
U U
E = Q + A (
)
2
X
X
with:
1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1
Q =
0 1 0 1 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 1 0 1 0
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and the vector of the gradient of displacements:
U, X
U, y
U
, Z
v, X
U
v
=
X
, y
v, Z
W, X
W, y
W, Z
and tensor A depend on the gradient of displacements:
U
0
0
v
0
0
W
, X
, X
, X
0
0
0
U
0
0
v
0
0
W
, y
, y
, y
0
U
0
0
U
0
0
v
0
0
W
, Z
, Z
, Z
With
X = U
U
0
v
v
0
W
W
, y
, X
, y
, X
, y
, X
0
U
0
U
v
0
v
W
0
W
, Z
, X
, Z
, X
, Z
, X
0
U
U
0
v
v
0
W
W
, Z
, y
, Z
, y
, Z
, y
The virtual variation, noted, from the deformations of Green-Lagrange is obtained by a calculation
differential:
U
U
E = Q
+
With
X X
In this expression and that which follows, we took account of the following property (see
[bib4] p 141):
1 U
U 1 U U
With
= A
2 X X
2 X X
The iterative variation is it also obtained by a differential calculus:
U
U
E = Q
+ A
X
X
The iterative variation of the virtual deformation of Green-Lagrange is put thus in the form:
U U
U
U
E = A
+ Q + A
X X
X X
traditional term
nontraditional term
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Whereas the first term of this expression is traditional for the continuous mediums 3D, it
second, which translates the taking into account of great rotations, is less.
3.1.4 Calculation of the constraints of Cauchy
3.1.4.1 Case
General
Tensor 3 × 3 of the total constraints of Piola-Kirchhoff of second species is connected to the tensor
3 × 3 of the total constraints of Cauchy by the relation:
[S] = (F) F [] F
det
1
T
Thus, knowing the state of the constraints of Piola-Kirchhoff of second species, one can calculate the state
constraints of Cauchy by the relation:
[]
1
=
F S FT
det (F) []
It should be noted that the state of stresses of Cauchy is not plane, in general, contrary to the state of
constraints of Piola-Kirchhoff of second species. In addition, the choice of a local reference mark in which
to represent this tensor is not at all obvious. It will be however shown, in the following paragraph, that
within the framework of the small deformations, there is a local reference mark, easily identifiable, in which
the state of stresses of Cauchy is him-also plane.
In the case of completely general laws, a detailed attention will have to relate to the diagrams
of numerical integration allowing to calculate the values of substitution of the gradient F at the points
of normal numerical integration.
3.1.4.2 Approximation in small deformations
It is pointed out [bib4] that the gradient F can be written thanks to the polar decomposition under two
forms:
F = RU = VR
where R = R - T is an orthogonal tensor, and where U and V are symmetrical matrices of elongation
defined positive.
Into the geometrical nonlinear field, we can introduce an important simplification
in the polar decomposition of the gradient of the deformations if the deformations remain small. This
simplification is not introduced into nonlinear calculation but in postprocessing of the constraints.
Elongation at the point Q being minor in front of the great rotation of the section:
U V I
One can then write:
F R =
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where is the tensor of great rotation which transforms normal N into N:
N = N
Simplification translates the fact that on a section, the transformation is reduced to a great rotation.
With this approximation of the gradient of the deformations, one can write:
F R =
and thus, by exploiting the orthogonality of one obtains:
F-1 T
and:
det (F) 1.
These simplifications lead to the final relation:
[] [
S] T
This relation translates the fact that the constraints of Cauchy are quite simply obtained by
great rotation of the constraints of Piola-Kirchhoff of second species.
One can now rewrite the property of plane constraints of the tensor of Piola-Kirchhoff of
second species N.[S] N = 0 pennies the new form:
n.T [] N
= 0
who leads in addition to the property:
N.[] N
= 0
That is to say still:
~
= 0
N N
Constraints of Cauchy [(1,2, 3)] are also plane in the local reference mark
[T (
1
, 2
, 3
): T (1, 2
, 3
): N (1, 2
) obtained by great rotation of the local reference mark on
1
2
]
initial configuration:
[T T N
:
:
] = [T: T:N
1
2
1
2
]
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In this reference mark, we can write all the components of the tensor as follows []:
~
~
~
T.
1 [] T
T.
1
1 [] T
T.
2
1 [] N
T T
T T
T N
~1 1
~1 2
~1
= T.
2 [] T
T.
1
2 [] T
T.
2
2 [] N
T T
T T
T N
~2 1 ~2 1
2
0
N
T
N
T
N
N
.
[]
.
1
[]
.
2
[]
T N
T N
1
1
By taking again the relation [] [
S] T, one obtains:
T
1.[] T
T
1
1.[] T
T
2
1.[] N
T
1.[S] T
T
1
1.[S] T
T
2
1.[S] N
T
2.[] T
T
1
2.[] T
T
2
2.[] N
= T
2.[S] T
T
1
2.[S] T
T
2
2.[S] N
N.[] T N
1
.[] T
N
2
.[] N
N.
[S] T
N
1
.[S] T
N
2
.[S] N
from where the final result:
~
~
~
~
~
~
S
S
S
T T
T T
T N
T T
T T
T N
~1 1
~1 2
~1
~1 1 ~1 2 ~1
= S
S
S
T T
T T
T N
T T
T T
T N
2 2
2 2
2
~2 1 ~2 1
2
~
~
0
S
S
0
T N
T N
T N
T N
1
2
1
1
In so far as the deformation remains small, components of the tensor of the constraints of
Cauchy in the local reference mark attached to the deformed configuration are identical to the components
tensor of the constraints of Piola-Kirchhoff of second species in the local reference mark attached to
initial configuration.
We take the party in the continuation, to consider only the constraints of Piola-Kirchhoff of
second species. We must note that within the framework of a more general constitutive law, one
will be able to pass from a stress measurement to another as indicated in the preceding paragraph.
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4
Numerical discretization of the variational formulation
exit of the principle of virtual work
4.1 Elements
finished
The three figures below summarize the finite elements choices concerning the voluminal hulls
[R3.07.04]. The selected finite elements are isoparametric quadrangles or triangles.
quadrangle is represented below. One chooses among the elements with functions of interpolation
quadratic, the element hétérosis whose displacements are approached by the functions
of interpolation of the Sérendip element and rotations by the functions of the element of Lagrange.
All the justifications as for these choices are given in [R3.07.04].
NB1 = 8
NB2 = 9
Sérendip element
Element of Lagrange
Hétérosis element
~
the U.K.,
~
Appear 4.1-a: Familles of finite elements for the isoparametric quadrangle
2
3 3 =1
7
3
4
= -
1
1
1 8
6
1
5
2
2 = 1
-
Appear voluminal 4.1-b: Elément of reference
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2
4
7
3
3
8
P
1
6
5
1
2
Appear 41-c: Real voluminal element
4.2
Discretization of the field of displacement
With an aim of avoiding the explicit calculation of the curvatures, which becomes extremely heavy in the case of them
great rotations, we choose to interpolate the normal on the initial average surface with the place
to interpolate rotations:
NB2
N (
2
1,2)
()
= NOR (1,2) nor
I 1
=
(2)
where NR I (1
, 2
) the function of interpolation to node I indicates among the NB2 nodes of Lagrange.
The same interpolations are adopted for the transform of the initial normal:
NB2
N (
2
1, 2)
()
= N1 (1, 2) N
I
I 1
=
The interpolation of the initial position of a point on the average surface of the hull (not P) is given
by:
X
(
1
NB
X
1
1,2)
()
=
NR I (1,2) y
I 1
=
Z I
() 1
where NR I (1
, 2
) the function of interpolation to node I among NB1 indicates = NB2 - 1 nodes of
Serendip.
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The interpolation of the initial position of an unspecified point of the hull (not Q) can then be written
in the form:
X
N
NB
X
(
1
NB
2
H
X
1
2
NR
y
I
1,2, 3)
()
=
(1,2)
()
+ 3
NR I (1,2) ny
2
I 1
=
I 1
Z
=
nz
I
The same interpolations are adopted for the deformed position of an unspecified point of fiber:
X
N
1
NB
NB2
H
X
X (
1
2
NR
y
I
1,2, 3)
()
=
(1,2)
()
+ 3
NR I (1,2)
ny
2
I 1
=
I 1
=
Z
nz
I
The interpolations for the positions initial and deformation being the same ones, we can adopt them
for the real displacement of an unspecified point of the hull:
U
N
N
1
NB
NB2
H
X
X
U (
1
2
NR
v
NR
N
N
I
1,2, 3)
()
=
(1,2)
()
+ 3
I
(1,2)
y - y
2
I 1
=
I 1
=
W
I
nz
nz
I
I
Thus, the interpolation of virtual displacement becomes:
U
0
- N
N
NB
Z
y
W
1
NB
2
H
X
U (
1
2
NR
v
I
1,2, 3)
()
=
(1,2)
()
- 3
NR I (1,2)
N
0
Z
- nx wy
2
I 1
=
I 1
=
W
-
N
N
0
y
X
I
wz
I
I
In the same way, the interpolation of iterative displacement becomes:
U
0
- N
N
NB
Z
y
W
1
NB
2
H
X
U (
1
2
NR
v
I
1,2, 3)
()
=
(1,2)
()
- 3
NR I (1,2)
N
0
Z
- nx wy
2
I 1
=
I 1
=
W
-
N
N
0
y
X
I
wz
I
I
Moreover, the interpolation of the iterative variation of virtual displacement is:
NB2
H
(,)
(2)
=
NOR (1,2)
2
((
U
W
W N
1 2
3
3
)
I 1
=
I
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4.3
Discretization of the gradient of displacement
4.3.1 Gradient of total displacement
The vector of the gradient of real displacement can be connected to the isoparametric gradient of
real displacement by the following relation:
U ~-1 U
= J
X
The isoparametric gradient of displacement is organized as follows:
U,
1
U, 2
U,
3
v, 1
U
v,
=
2
v,
3
W,
1
W,
2
W, 3
~
The matrix jacobienne generalized 9 × 9 J - 1 can be expressed according to the matrix jacobienne
isoparametric transformation 3 × 3 as follows:
J-1
0
0
~
J - 1 =
0
J -
1
0
0
0
J - 1
The isoparametric gradient of real displacement can be calculated as follows:
U NR E
=
p
1
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with the first matrix of derived from the functions of form:
2
()
()
1
NR
NR
3
I
0
0
0
0
,
I
1
,
1
()
(2)
1
0
0
NR
NR
3
I
0
0
,
I
2
,
2
(2)
0
0
0
NR
I
0
0
()
1
(2)
0
NR
0
0
NR
0
I
3
,
I
,
1
1
()
H
1
(2)
L
0
NR
0
0
NR
0
I
L = 1, NR 1
B
I
3
,
I
,
2
2
2
(2)
0
0
0
0
NR
0
()
I
0
0
NR 1
(2)
I
,
0
0
NR
1
3
I
,
()
1
0
0
NR 1
(2)
I
,
0
0
NR
2
3
I, 2
0
0
0
(
2)
NR
0
0
NR
I
=
(2)
1
NR
0
0
3
NB2,
1
(2)
NR
0
0
3
NB2,
2
(2)
NR
0
0
NB2
(
2)
0
NR
0
3
NB2,
1
H
(2)
0
NR
0
2
3
NB2,
2
0
(2)
NR
NB2
0
(
2)
0
0
NR
3
NB2, 1
(
2)
0
0
NR
3
NB2, 2
(
2)
0
0
NR
NB2
and the vector of “generalized nodal real displacement”:
M
U
v
W
nx - nx
N
y - ny
EP
N
=
Z - nz
I
M
I = 1, NB1
M
N
X - nx
N
y - N y
N
Z - nz NB2
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Finally, one will be able to write the gradient of real displacement in the form:
U ~
- 1 NR
E
= J
p
X
1
4.3.2 Gradient of virtual displacement
While proceeding similarly to the gradient of real displacement, one can connect the two gradients of
virtual displacement:
U ~-1 U
= J
X
The isoparametric gradient of virtual displacement can be calculated as follows:
U NR
=
E
U
2
with the second matrix of derived from the functions of form:
(2)
(2)
() 1
0
NR N
3
,
Z
-
I
NR
3
I
1
N
NR
,
y
1
I
0
0
, 1
(2)
(2)
()
1
0
NR
N
3
,
Z
-
0
0
I
NR
3
I
2
N
NR
,
y
2
I
, 2
0
NR (2)
(2)
0
0
0
1 N
- NR
Z
1 ny
()
(2)
(2)
0
NR 1
- NR
N
0
NR N
I
0
,
3
I
,
Z
3
I
,
X
1
1
1
()
H
L
(2)
(2)
0
NR 1
0
- NR
N
0
NR
N LI = 1, NB1
I
,
3
I
,
Z
3
I
,
X
2
2
2
2
0
0
0
(2)
(2)
- NR N
0
NR
N
()
1
Z
1
X
0
0
NR 1
(2)
(2)
I
,
NR N
- NR N
0
1
3
I
,
y
3
I
,
X
()
1
1
0
0
NR 1
(2)
(2)
I
,
NR
N
- NR
N
0
2
3
I
,
y
3
2
I, 2
X
0
0
0
(2)
(2)
NR
NR
1 N
- NR
y
1 N
0
X
=
(2)
(2)
2
0
NR
N
3
- NR
N
NB2, Z
3
NB2, y
1
1
(2)
(2)
0
NR
N
3
- NR
N
NB2,
Z
3
NB2,
y
2
2
(2)
(2)
0
NR
N
NB2 Z
- NR
N
NB2 y
(
2)
(2)
- NR
N
0
NR
N
3
NB2, Z
3
NB2, X
H
1
1
(2)
(2)
-
3 NR
N
0
,
Z
NR
N
2
NB2
3
NB2
2
,
X
2
(2)
(2)
- NR
NB2n
0
NR
Z
NB2nx
(2)
(2)
3 NR NB2 N
,
y
- 3N NB2 N
0
1
, 1 X
(2)
(2)
NR
N
NR
N
0
3
NB2
,
y
- 3 NB2
2
, 2 X
(
2)
(2)
NR
NB2ny
-
NR NB2n
0
X
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and the vector of the virtual nodal variables:
M
U
v
W
wx
W
y
ue = wz
I
M
I = 1, NB1
M
W
X
W
y
wz NB2
Finally, one will be able to write the gradient of virtual displacement in the form:
U ~
- 1 NR
J
E
=
U
X
2
4.3.3 Gradient of iterative displacement
The step here is similar to virtual calculation. It is enough to replace by:
U ~
- 1 NR
E
= J
U
X
2
with the vector of the iterative nodal variables:
M
U
v
W
wx
W
y
ue = wz
I
M
I = 1, NB1
M
W
X
W
y
W
Z NB2
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4.3.4 Gradient of the iterative variation of virtual displacement
U ~
- 1 NR
E
= J
Q
X
3
with:
(2)
NR
3
I
0
0
, 1
(2)
NR
0
0
3
I
, 2
(2)
NR
0
0
I
(2)
0
NR
3
0
,
I
NR
H
1
(2)
= L
0
0
,
,
L = 1
2
NR
I
NB
3
I
2
2
3
(2)
0
NR
I
0
(2)
0
0
NR
3
I
, 1
(2)
0
0
NR
3
I
, 2
(
2)
0
0
NR
I
and the vector of the iterative variation “nodal virtual displacement” generalized:
.
.
.
(W
(W N
E
)
Q
I
=
.
.
.
I =,
1 NB2
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4.4 Discretization of the variational formulation resulting from the principle of
virtual work
We take again the iterative variation (between two iterations) of internal virtual work:
~
~
~ ~
int = (.
E S +.
E S) D
and iterative variation of the vector of local constraints of Piola-Kirchhoff of second:
~ = ~
S
D E
Then, the linearized form of the principle of virtual work of the §3 can be written for the finite element described
above in the following matric form:
ue K E
E
E
E
E
.
T U
= U.(F - R)
where ue is the nodal vector of the functions tests. One deduces the system from it from equations:
K E
E
E
E
T U
= F - R
where:
KeT
is the tangent matrix of rigidity
ue
is the elementary vector of the solution of the linearized system of equations (nodal vector
between two iterations)
is the external level of load
F E
is the nodal vector of the external forces (associate with = 1)
Re
is the nodal vector of the internal forces
4.4.1 Vector of the internal forces
It is a vector (6 ×
1
Nb +)
3 × 1 entirely expressed in the total reference mark and which must be
evaluated with each iteration by the relation:
Re =
BT S
Jd D D
~ ~ det
2
1
2
3
with the vector of the local constraints Piola-Kirchhoff of second species:
~
~
S = OF
It is pointed out that the symbol ~ indicates an object expressed in the local reference mark.
The local deformations of Green-Lagrange are updated to each iteration:
~
~
E = B EP
1
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where the operator of the total deflections (first operator of the deformations) is written:
~
1 U ~ NR
B = H Q
+ A
J 1
-
1
2 X
1
with the gradient of real displacement:
U ~
- 1 NR
E
= J
p
X
1
The operator of the virtual deformations (second operator of the deformations):
~
U ~ NR
B = H Q + A
J 1
-
2
X
2
is highlighted by the relations:
~ ~
E = B
E
2u
~
~
E = B ue
2
4.4.2 Stamp tangent rigidity
The tangent matrix of rigidity which is serious (6 ×
1
NB +)
3 × (6 ×
1
NB +)
3 is expressed too
entirely in the total reference mark. One must be able to evaluate it with each iteration if it is wanted that
convergence of the method of Newton is quadratic. In a traditional way into nonlinear
geometrical, it takes the form:
K E = K E + K E
T
m
G
where the first part represents the material part:
K E =
BT DB
J
m
D D D
~ ~ det
2
2
1
2
3
and the second part represents the geometrical part, it even made up of two parts:
K E = K E
+ K E
G
G
G
traditional
nontraditional
with the traditional part of the geometrical part (see [bib4] p. 141):
T
NR
NR
K E
=
J - 1
SJ -
~
~ 1 det J
G
D D D
traditional
1
2
3
2
2
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where S the generalized tensor of the constraints expressed in the total reference mark is written:
3×3
[S]
0
0
9×9
S = 0
[S] 0
0
0
[S]
The nontraditional part of the geometrical part not represents terms uncoupled from rotation
symmetrical which has as a form:
3×3
K E
(I, I) = [Z ×] [N
G
I
I ×]
nontraditional
where N I is the transform of the initial normal to node I and zI a vector 3 × 1 with the node
I = 1, NB2 of the nodal vector 3 × (NB2 ×)
1 Z I
.
.
.
Z
Z
I
I =
.
.
.
I =,
1 NB
2
Nodal vector Z I is similar to a vector of internal force and its expression is:
Z =
BT S
J
I
D D D
~ ~ det
3
1
2
3
with the operator of the iterative variation of the virtual deformations (third operator of
deformations):
~
U ~ NR
B = H Q
+ A
J 1
-
3
X
3
who is highlighted by the relation:
~ T ~
~
~
B S det Jd
D D =
E
.S
3
1
2
3
D
nontraditional
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4.4.3 Diagrams
of integration
The integration of the terms of rigidity in the thickness of the hull is identical to the method used in
analyze linear geometrical [R3.07.04] for nonlinear behaviors. The initial thickness is
divided into NR identical layers thicknesses. There are three points of integration per layer. Points
of integration are located in higher skin of layer, in the middle of the layer and in lower skin
of layer. A layer in the thickness of the hull appears sufficient in the majority of the cases.
In order to be able bearing with the problem of blocking out of membrane of the curved hulls and to solve it
problem of blocking in transverse shearing, it is necessary to modify the diagram of integration
on average surface. If the technique is completely known in linear analysis, it is it less
in geometrical nonlinear analysis.
The procedure is presented in the form of a generalization of the separation of the effects of membrane, of
inflection and of transverse shearing if one uses the deformations of Green-Lagrange:
~
~
E
E = m
~
ES
~
E
T T
~
~
~ 11
~
T N
where E
1
m = And T represent the deformation of membrane-inflection and ES =
~
deformation of
~ 2 2
T N
2
T t12
transverse shearing.
During the numerical evaluation of the deformations at the points of normal numerical integration of Gauss
~
~
(9 points for the quadrilateral and 7 points for the triangle), one uses the relation E = B EP
1
.
modification is introduced on the level of the first operator of the deformations:
~
substitution
~
B
MF
B
1
1 =
~
B substitution
S
1
~
~
B
substitution
substitution
MF
and B
are the first operators of the deformations of substitution of
1
s1
membrane-inflection and of transverse shearing, respectively.
During the calculation of the nodal vector of the internal forces and material part of the tangent matrix of
rigidity, the modification is introduced in a way similar to the level of the second operator of
deformations:
~
known
~
B
MF
B
2
2 =
~
B known
S
2
4.4.3.1 Operators of deformations of substitution
In what follows the points of normal and reduced numerical integration of Gauss, on surface
average, are NPGSN = 9 and NPGSR = 4, respectively, for the element
quadrilateral, and NPGSN = 7 and NPGSR = 3, respectively, for the triangular element.
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Membrane-inflection part
At point INTSN among the NPGSN points of normal numerical integration of Gauss of surface
average, one will calculate:
normal
normal
~
B substitution
complete
incomplete
MF
(INTSN) ~
= Bmf
(INTSN) ~
- Bmf
(INTSN) +
1
1
1
reduced
NR
incomplete
I (INTSN) ~
Bmf
(INTSR)
1
INTSR=,
1 NPGSR
where INTSR is a point among the NPGSR points of reduced numerical integration of Gauss of
surface average.
normal
~
complete
~
In the expression above, B MF
represent the first three lines of B
1
1 calculated with
NR
points of normal numerical integration by considering the complete matrix
. The operator
1
normal
~
~
B
incomplete
MF
represent the first three lines of B
1
1 calculated at the points of numerical integration
NR
normal by considering a matrix
incomplete where the columns of rotation are cancelled:
1
()
NR 1
I
0
0
0 0 0
,
0 0 0
1
()
NR 1
0
0
0 0
0 0 0
I
0
, 2
0
0
0
0 0
0
0 0 0
()
1
Inc
0
NR
0
0 0 0
0 0 0
I
NR
,
1
()
= L
1
0
0
0 0
0 L = 1,
1 0 0 0
NR
I
NB
I
,
1
2
0
0
0
0 0
0
0 0 0
()
0
0
NR 1
0 0 0
I
0 0 0
, 1
()
0 0 0
0
0
NR 1
0 0
I
0
, 2
0 0 0
0
0
0
0 0
0
reduced
~
incomplete
~
B MF
(INTSR) the first three lines of B represent
1
1 calculated at the points
NR Inc
of reduced numerical integration with the matrix
incomplete above definite. They are thus
1
stored to be extrapolated at each point of normal numerical integration.
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Transverse shearing part
For the transverse shearing part, one will calculate:
reduced
~
B substitution (INTSN) =
NR (INTSN) ~B complete
S
1
I
S
(INTSR)
1
INTSR=,
1 NPGSR
reduced
~ complete
~
where B S
(INTSR) represents the two last lines of B
1
1 calculated at the points of integration
NR
numerical reduced with a matrix
complete. They are also stored to be extrapolated
1
at each point of normal numerical integration.
4.4.3.2 Substitution of the geometrical part of the tangent matrix of rigidity
The nontraditional part of the tangent matrix of rigidity K eg
is numerically
nontraditional
integrated into the points of normal integration of Gauss. No operation of substitution is necessary.
For the traditional part of the tangent matrix of rigidity, we use substitution:
normal
normal
reduced
reduced
complete
incomplete
incomplete
complete
membrane
membrane
membrane
shearing
substitution
K E
= K E inflection
- K E inflection
+ K E inflection
+ K E transverse
G
G
G
G
G
traditional
traditional
traditional
traditional
traditional
where:
normal
complete
membrane
K E inflection
G
is numerically integrated on the points of normal integration with a matrix
traditional
NR
supplements, and the local constraints of membrane inflection only;
2
normal
incomplete
membrane
K E inflection
G
is numerically integrated on the points of normal integration with a matrix
traditional
NR
incomplete, and local constraints of membrane inflection only;
2
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reduced
incomplete
membrane
K E inflection
G
is numerically summoned on the points of integration reduced with a matrix
traditional
NR
incomplete, and integrated local constraints of membrane inflection only;
2
reduced
complete
shearing
K E transverse
G
is numerically summoned on the points of integration reduced with a matrix
traditional
NR
supplements, and the integrated local constraints of transverse shearing only;
2
To be able to calculate the two last tangent matrices in the preceding equation, us
let us carry out the numerical integration of the local constraints on the NPGSN points of integration
normal:
~
S (INTSR) =
NR (INTSR) ~S (INTSN) det J D D
1
D
I
2
3
INTSN =,
1 NPGSN
This equation contains the terms of weight of the points of Gauss.
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5
Rigidity around the transform of the normal
5.1
Singularity of the tangent matrix of rigidity
Although the finite elements objects of the hull are expressed directly in the total reference mark
[E: E: E
1
2
3] (the degrees of freedom are displacements and rotations in the total reference mark),
the tangent matrix of rigidity presents a singularity compared to the component of rotation
around the transform of the normal in each node:
K E N
T
= 0
W
W
I=1, NB2
The contributions (W N)
are null.
I =1, NB2
In the preceding equation, this matrix represents the rigidity of rotation in the total reference mark. Its
structure is full:
K
K
K
[
11
12
13
K and] = K
K
K
I
12
22
23
W
W
K
K
K
31
32
33 I
it is a nonsymmetrical matrix.
This singularity is a direct consequence of the kinematics of hull. It is due to the product
vectorial appearing in linearized displacements (virtual and incremental). Thus displacement
between two iterations is given by:
H
U (1,2, 3) = U (1,2) + 3 W (1,2) N
Q
p
(1,2)
2
H
It is noticed that the contribution
W (,)
3
1 2 N (1,2) is perpendicular to N. One
2
interpret this singularity in the following way: the rotation of an initially normal fiber on the surface
average does not lead to an elongation of this one, and consequently does not induce deformation.
5.2 Principle of virtual work for the terms associated with rotation
around the normal
We propose to define the full slewing around the transform of the normal in the hull like
the projection of the vector of full slewing on the transform of the normal:
= .n
N
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It is pointed out that the vector of rotation is an invariant of the matrix of rotation = exp [×]
=
The vector of rotation is a clean vector of the matrix of rotation associated with the eigenvalue
identity. So the first relation is rewritten:
= ().(N)
N
=.
N
= N
This relation translates an important result:
The projection of the vector of full slewing on the transform of the normal is equal to
projection of the vector of full slewing on the initial normal
In discrete form, one defines a deformation energy associated with this rotation:
1
2
=
K
2
(
N
N) I
I =1, NB2
where K is a rigidity of torsion of which the determination of the value will be discussed further. One supposes
that this rigidity remains constant and undergoes neither virtual variation nor incremental variation.
The existence of the potential is supposed:
1
=
K ((.
N) (.
N))
N
2
I
I =1, NB2
that one can rewrite in a more elegant form:
1
=
K ([
N N])
N
2
I
I =1, NB2
By exploiting the property of orthogonality - 1 = T of the matrix of rotation:
T
N
N
N N
=
=
N
N T
= NT
(
) (
)
= N N
This property will be exploited in the double linearization of the potential energy.
One rewrites the potential in the form:
1
=
K ([
N N])
N
2
I
I =1, NB2
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The first linearization of, makes it possible to obtain the virtual variation:
N
1
=
K
(
[
N N] + [
N N
])
N
2
I
I =1, NB2
= K
(
[
N N]) I
I =1, NB2
It is necessary to express this form according to the function tests of rotations retained in
variational form.
= -
T1 (
) W
with the form of the matrix reverses of the differential operator of rotation:
1
1
T-1 () = 2 [I] - [×] +
1 - 2 []
2
2
tan
tan
2
2
From where the final form of the virtual work which makes it possible to deduce the vector from the interior forces:
- T
= K
(wT () [N N])
N
I
I =,
1 NB2
One carries out the second linearization of:
N
- T
- T
= K
W. T () [N N] + T () [N N]
N
((
)
=
I
I
,
1 NB2
with the particular choice of the ddls of rotation W = 0, and owing to the fact that the initial normal “does not move
not “during the iterations N = 0.
The expression of the tangent operator who gives rise to the terms corresponding to the ddls of
rotation around the transform of the normal of tangent matrix is as follows:
- T
- 1
- T
= K
W. T () [N N T
]
() W
+ K
W. T
() [N N]
N
((
)) I
((
)
I =,
1 NB2
=
I
I
,
1 NB2
In this relation, the last term is a differential term due to the nonlinear relation between
parameters of rotation. Its linearization is heavy to carry out and its contribution will be neglected in
the expression of the tangent operator.
With the property: N
N
= N N, we give the final expression:
- T
- 1
K
W. T () [N N T
]
() W
N
((
))
=
I
I
,
1 NB2
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The contribution of operator T-1 is noted () [I] in the tangent matrix of rigidity.
5.3 Notice
The potential energy brought by the terms of rotation around the transform of the normal is
nonnull even for a rigid rotational movement. This energy does not correspond to one
deformation. For this reason it must be nonsignificant. The default value of
COEF_RIGI_DRZ must guarantee that.
5.4
Borderline case analysis geometrically linear
In the case of the analysis geometrically linear, initial configuration and configuration
deformation are confused what leads us to confuse initial normal N with its transform
N:
N N
Rotations become small in this case and the operator of great rotation becomes:
= exp [×] [I] + [×]
The differential operator of rotations becomes:
T () [I]
and the parameters of rotations become confused:
W and W
All these approximations introduced into virtual work lead to its simplification:
K
(
[
N N])
N
I
K =,
1 NB2
The same approximations introduced into the differential part of virtual work also lead to
its simplification:
K (
[N N
])
N
I
K =,
1 NB2
The two last equations are those of the analysis geometrically linear. They show that
contributions in the vector of the interior forces and the tangent matrix of rigidity
cover the borderline case well with [R3.07.04].
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5.5
Determination of the coefficient K
The coefficient K is calculated with each iteration of each step of time. With each iteration of Newton
of each step of time, one builds with the NB2 nodes the matrix of passage
I = [T
: T
: N
1
2
]; I =1, NB2
I
who allows to pass from the vector W I, vector of iterative rotation to the node expressed in the reference mark
total [E: E: E
1
2
3] with the vector ~
W I expressed in local reference mark [T: T: N
1
2
]: I
~w I = Iw; =,
I
I
1 NB2
One can build in each node, the matrix [KeT] of size 3× 3
I
~
W W
[K E]
T
= [K E]
~ ~T I
I
T I
I
W W
W
W
This matrix represents the rigidity of rotation in the local reference mark. Its structure, nonsymmetrical, is:
K
K
0
T T
tt
1 1
1 2
[K E] = K
K
0
~ ~T I
T T
T T
2 1
2 2
W W
K K
0
N T
NT
1
2
I
The coefficient K is then calculated according to the relation:
K = COEF _ RIGI _ DRZ × KMIN
where COEF_RIGI_DRZ is a coefficient introduced like a mechanical characteristic of hull by
the user. In traditional linear analysis of the hulls or plates, this coefficient is selected enters
0.001 and 0.000001. By defect it is worth 0.00001. In the case of great rotations calculated with
great steps of load, one advise to use value 0.001.
~
KMIN is the minimum of the nonnull terms of rotation on the diagonal of K and.
KMIN =
MIN
K, K
I
, NB
T T
T T
=1
2
1 1
2 2
I
Note:
It would be undoubtedly more rigorous to calculate K with the first iteration of the first step of time
and to store this value like invariant information during the iterations and the steps of
following times. The experiment shows that this way of proceeding is often not optimal in
measurement where the values of the coefficients of the matrices of rigidity can evolve/move in way
important during a calculation in great displacements. The value of K determined initially
can then become too small and the matrix rigidity to end up being singular.
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6 Buckling
linear
Linear buckling is presented in the form of a particular case of the geometrical nonlinear problem. It
is based on the assumption of a linear dependence of the fields of displacements, deformations and
constraints compared to the level of load, and this, before the critical load is not reached.
In a total Lagrangian formulation, one recalls that linearized balance can be written under
variational form:
int - ext. = ext. - int
maybe in matric form after discretization:
the U.K.U
T = U (F - R)
where the dependence of the tangent matrix of rigidity K T is nonlinear compared to the vector of
nodal displacement generalized p =
U EP.
e=, Nel
1
If we suppose the linear dependence of displacement compared to the level of load, one can
to write:
U = u0
where u0 is the solution obtained following a linear analysis for = 1 by:
K U
F
0 0 =
where K 0 is the tangent matrix of initial rigidity. One can then develop the tangent matrix of
rigidity in a linear way compared to the level of load:
U
E
E
E
E
K T = K 0 + (Ku + K) +….
e=1, Nel
where K E
E
U is the matrix of initial displacements depending on p0, traditionally neglected in
Code_Aster, and K E the matrix of the initial constraints depending on the total tensor of the constraints
~
of Piola-Kirchhoff of second species [S0] and local vector S0. These constraints are
voluntarily confused with the constraints of Cauchy. They are obtained by a postprocessing
linear analysis.
For the rotation part of EP, the assumption of linearity of the deformations according to the level of
load results in the equality of:
N = N
I
I
who leads us to confuse the initial normals nor with their transforms N I.
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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:
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The matrix of the initial constraints K E represents the constant part in geometrical part
tangent matrix of rigidity. It is evaluated at the point pe0 and = 1 with a transform of
normal replaced by the initial normal:
K E = K E
+ K E
traditional
nontraditional
with the traditional part of the geometrical part (see [bib4] volume 1 p. 141):
T
E
- 1 NR
- 1 NR
K
=
J
~
~
SJ
det J
D 1
D 2
D
traditional
3
2
2
where the second matrix of derived from the functions of form becomes:
(2)
(2)
() 1
0
NR N
3
,
Z
-
I
NR
3
I
1
N
NR
,
y
1
I
0
0
, 1
(2)
(2)
()
1
0
NR
N
3
,
Z
-
0
0
I
NR
3
I
2
N
NR
,
y
2
I
, 2
(2)
0
NR
N
(2)
0
0
0
NR
N
I
Z
-
I
y
()
(2)
(2)
0
NR 1
- NR
N
0
NR N
3
,
3
I
0
,
I
Z
I
,
X
1
1
1
()
H
L
(2)
(2)
0
NR 1
0
- NR
N
0
NR
N
LI = 1, NB1
3
I
,
Z
3
I
,
X
I
, 2
2
2
2
2
2
0
0
0
()
()
- NR N
0
NR
N
I
Z
I
X
()
0
0
NR 1
(2)
(2)
I
,
NR N
3
- NR N
0
1
I
,
y
3
I
,
X
()
1
1
0
0
NR 1
(2)
(2)
NR N
NR
N
0
I
,
3
-
2
I
,
y
3
I
,
X
2
2
0
0
0
(2)
(2)
NR N
NR
N
0
NR
I
y
-
I
X
=
(2)
(2)
2
0
NR
N
3
- NR
N
NB2, Z
3
NB2, y
1
1
(2)
(2)
0
NR
N
3
- NR
N
NB2,
Z
3
NB2,
y
2
2
(2)
(2)
0
NR
N
NB2 Z
- NR N
1
y
(
2)
(2)
- NR
N
0
NR
N
3
NB2, Z
3
NB2, X
H
1
1
(2)
(2)
- NR
N
0
NR
N
3
NB2,
Z
3
NB2,
X
2
2
2
- NR (2)
(2)
NB N
2 Z
0
NR NB N
2 X
(2)
(2)
NR
3
-
NB
N
2
,
y
NR
3
NB
N
2
,
X
0
1
1
(2)
(2)
NR
3
-
NB
N
2
,
y
NR
3
NB
N
2
,
X
0
2
2
(
2)
(2)
NR
NB N
2 y
- NR NB N
2 X
0
Handbook of Référence
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Code_Aster ®
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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:
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and the generalized tensor of the total constraints:
3×3
[S]
0
0
9×9
S = 0
[S] 0
0
0
[S]
The nontraditional part of the geometrical part represents the terms uncoupled from rotation, not
symmetrical. Since the current algorithm of resolution of the problem to the eigenvalues [R5.06.01]
[K + (Ku +K
0
)] = 0 (being the level of critical load) consider only matrices
symmetrical, one makes symmetrical, while dividing by two the sum with its transposed, the matrix:
1
K E
(I, I) =
×
×
nontraditional
[(zI] [N
I
])
2
where N I is the normal with node I and zI a vector 3× 3 with the node I = 1, NB2 of the nodal vector
3 × (3 × NB2) Z I:
.
.
.
Z
Z
I
I =
.
.
.
I =,
1 NB
2
Nodal vector Z I is similar to a vector of internal force and its expression is:
Z =
BT S
J
I
D D D
~ ~ det
3
1
2
3
with the operator of the iterative variation of the virtual deformations (third operator of
deformations):
~
~ NR
B = HQJ 1
-
3
3
who is highlighted by the relation:
~ T ~
~
~
B S det Jd
D D = Enon traditional S
3
1
2
3
D
Note:
For numerical integration in the thickness of the various terms of rigidity, we retain one
diagram of Gauss at two points just like in elasticity for the geometrical linear hulls
[R3.07.04].
Handbook of Référence
R3.07 booklet: Machine elements on average surface
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Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
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:
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7
Establishment of the elements of hull in Code_Aster
7.1 Description
These elements (of names MEC3QU9H and MEC3TR7H) are pressed on meshs QUAD9 and TRIA7 which are
of geometry curves [R3.07.04].
7.2 Use
These elements are used in the following way:
AFFE_MODELE (MODELING: “COQUE_3D”…) for the triangle and the quadrangle.
One calls upon routine INI080 for standard calculations of numerical integration.
AFFE_CARA_ELEM (HULL:(EPAISSEUR:“EP”
ANGL_REP
:
(
'' '')
COEF_RIGI_DRZ
:
“CTOR”)
to introduce the characteristics of hull.
ELAS: (E:NAKED Young: ALPHA:. RHO:. )
For an elastic thermo behavior isotropic homogeneous in the thickness one uses the key word
ELAS in DEFI_MATERIAU where the coefficients E are defined, Young modulus, coefficient of
Poisson, thermal dilation coefficient and density.
AFFE_CHAR_MECA (DDL_IMPO: (
DX:. DY:. DZ:. DRX:. DRY:. DRZ:. DDL of plate in the total reference mark.
FORCE_COQUE: (FX:. FY:. FZ:. MX:. MY:. MZ:. )
They are the surface efforts on elements of plate. These efforts can be given in
total reference mark or in the reference mark user defined by ANGL_REP.
FORCE_NODALE: (FX:. FY:. FZ:. MX:. MY:. MZ:. )
They are the efforts of hull in the total reference mark.
7.3
Calculation in geometrical nonlinear “elasticity”
Calculation imposes the following instructions user:
COMP_ELAS: (RELATION: “elas”
COQUE_NCOU: 1 (or more)
DEFORMATION: “green_gr”)
Numerical integration in the thickness is based on an approach multi-layer with 3 points
of integration by layer. It is about the approach currently used in nonlinear hardware
[R3.07.04]. Options of postprocessing SIEF_ELNO_ELGA of the constraints and VARI_ELNO_ELGA of
variables intern (here null) by defect are activated with the convergence of each filed step.
Handbook of Référence
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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:
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7.4 Establishment
Options FULL_MECA, RIGI_MECA_TANG, and RAPH_MECA are already active in the catalogs
elementary mec3qu9h. cata and mec3tr7h. cata for material non-linearity. They direct it
calculation towards/extremely/te0414.f, then towards/extremely/vdxnlr.f to calculate and store, inter alia,
stamp tangent symmetrical rigidity in the address corresponding to mode MMATUUR PMATUUR.
For the geometrical nonlinear analysis, the calculation of the tangent matrix of rigidity is directed towards
new routine VDGNLR. This matrix is not symmetrical and must be stored in the address
corresponding to mode MMATUNS PMATUNS.
One defines the two local modes at the same time symmetrical and nonsymmetrical, at output of the catalogs
elementary. The tangent matrix of nonsymmetrical rigidity into nonlinear geometrical is stored
with the address reserved for a nonsymmetrical matrix. On the other hand, if it is about nonmaterial linearity in
small deformations, all the tangent matrix of rigidity is stored with the address corresponding to
nonsymmetrical mode. The lower triangular part is duplicated starting from the triangular part
higher. Material nonlinear calculation in small deformations thus proceeds also in not
symmetrical.
7.4.1 Modification
TE0414
Calculation is directed towards/extremely/vdgnlr.f when the type of behavior COMP_ELAS is checked,
i.e. when the problem is nonlinear geometrical.
7.4.2 Addition of a routine VDGNLR
According to the option, the routine/extremely/vdgnlr.f has as a role of:
Options: FULL_MECA and RAPH_MECA:
To calculate the 6 components of the state of the local constraints of Cauchy (confused with the state of
constraints of Piola-Kirchhoff of second species) at the points of normal numerical integration and it
nodal vector of the internal forces. They are stored in local modes ECONTPG PCONTPR and
MVECTUR PVECTUR respectively.
Options: FULL_MECA and RIGI_MECA_TANG:
To calculate and store the tangent matrix of nonsymmetrical rigidity in mode MMATUNS PMATUNS.
7.5
Calculation in linear buckling
Option RIGI_MECA_GE, inactive until now, is activated in the elementary catalogs
mec3qu9h. cata and mec3tr7h. cata.
The new TE0402 is dedicated to the calculation of the matrix of geometrical rigidity due to the constraints
initial for the buckling of Euler. One recovers the plane states of the local constraints of Cauchy
(null Snn component) at the points of normal numerical integration of Gauss. These states of
constraints must be obtained by postprocessing with the option of calculation SIEF_ELGA_DEPL following
a linear analysis (mode ECONTPG PCONTRR).
In analysis of buckling of Euler, the constraints [] of Cauchy can be confused with
constraints [S] of Piola-Kirchhoff of second species. Therefore we will keep
notation [S].
Handbook of Référence
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
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:
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The matrix of rigidity of the initial constraints can be broken up into a traditional part
symmetrical and a nonsymmetrical nontraditional part. First is calculated according to
total tensor of constraints 3 × 3, contrary to the second which, it, is calculated according to
vector of the local constraints 5 × 1.
Since the current algorithm of resolution of the problem to the eigenvalues [R5.06.01] does not consider
that symmetrical matrices, we force the symmetry of the nontraditional part of the matrix
geometrical before storing the higher triangular part of all the matrix in the mode
MMATUUR PMATUUR.
8 Conclusion
The formulation that we have just described applies to the curved mean structural analyzes in
great displacements, whose thickness report/ratio over characteristic length is lower than 1/10. It
comes in direct object from the finite elements described in the preceding reference material
[R3.07.04] and used within the framework of small displacements and deformations. They rest on
same meshs quadrangle and triangle.
Their formulation rests on an approach of continuous medium 3D into which one introduces one
kinematics of hull of the Hencky-Mindlin-Naghdi type, in plane constraints, in the formulation
weak of balance. The measurement of the deformations retained is that of Green-Lagrange,
combined énergétiquement with the constraints of Piola-Kirchhoff of second species. The formulation
balance is thus Lagrangian total. The transverse distortion is treated same manner
that in [R3.07.04]. Rotations must remain lower than 2 because of the choice of update of
great rotations established in Code_Aster for which there is not bijection between the vector of
full slewing and the orthogonal matrix of rotation.
Because of singularity of the tangent matrix of rigidity compared to the component of rotation
around the transform of the normal, one defines a fictitious deformation energy associated this
rotation. With this rotation, one associates a rigidity of constant torsion. Interior efforts associated
this potential energy are taken into account. This potential energy, nonnull, does not correspond
with a physical deformation. One thus needs that it remains negligible, which the user can control in
imposing a value of the being worth COEF_RIGI_DRZ of 103 on 105.
For the postprocessing of the constraints, one limits oneself to the framework of the small deformations. One then could
to prove the identity enters the tensor of the constraints of Piola-Kirchhoff observed the initial geometry
and the tensor of the constraints of Cauchy in the deformed geometry. Moreover, the state of the constraints
being plane for the tensor of Piola-Kirchhoff, one finds this property for the state of stresses of
Cauchy. It should be noted that in more general contexts, this property is not preserved.
Linear buckling is treated like a particular case of the geometrical nonlinear problem. It
rest on the assumption of a linear dependence of the fields of displacements, deformations and
constraints compared to the level of load, before the critical load. It results from it that one can
to linearly develop the tangent matrix of rigidity compared to the level of load. One finds
then the geometrical part of the matrices of nonlinear the geometrical General obtained while identifying
deformed normal on the average surface and the initial normal, which is coherent with
linearity of the deformations according to the level of load.
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
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:
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9 Bibliography
[1]
Al Mikdad Mr., “Statique 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]
Bathe K.J., “Finite Element Proceedings in Engineering Analysis”, Prentice Hall, New Jersey,
1982.
[3]
Cardonna A., “Year Integrated Approach to Mechanism Analysis”, thesis of doctorate,
University of Liege, 1989.
[4]
Crisfield Mr. A., “Non-linear Finite Element Analysis off Solids and Structures”, Volume 1:
Essentials, John Wiley, Chichester, 1994.
[5]
Crisfield Mr. A., “Non-linear Finite Element Analysis off Solids and Structures”, Volume 2:
Advanced topics, John Wiley, Chichester, 1994.
[6]
Jettor pH., “
Non Linéaire kinematics of Coques
“, report/ratio SAMTECH, contract
PP/GC-134/96, 1998.
[7]
Simo J.C., “
The (symmetric) Hessian
for Geometrically Nonlinear Models in Solid
Mechanics: Intrinsic Definition and Geometric Interpretation ", Comp. Methods Appl. Mech. 96
: 189-200, 1992.
[8]
Vautier I., “Mise in work of STAT_NON_LINE”, handbook of Code_Aster development
[D9.05.01].
[9]
Massin P., “Eléments of plate DKT, DST, DKQ, DSQ and Q4g”, Manuel de Référence of
Code_Aster [R3.07.03].
[10]
Massin P., Laulusa A., Al Mikdad Mr., Bui D., Voldoire F., “Modélisation Numérique of
Hulls Volumiques “, Manuel de Référence of Code_Aster [R3.07.04].
[11]
Lorentz E., “Relation de Comportement Elastique Non Linéaire”, Manuel de Référence of
Code_Aster [R5.03.20].
[12]
Jacquart G., “Méthode de Ritz in linear and nonlinear dynamics”, Manuel de Référence
of Code_Aster [R5.06.01].
[13]
Aufaure Mr., “Modélisation Statique and Dynamique of Poutres in Grandes Rotations”,
Handbook of Référence of Code_Aster [R5.03.40].
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 46/56
Appendix 1: Flow chart of calculation in linear buckling
Local reference marks with the NB2 nodes [T: T: N
1
2
] I
Loop on the points of normal numerical integration of Gauss
~
~
S S
T T
T T
~ 1 1
~1 1
St T St T
2 2
~2 2
~
~
2S
·
T T
T T
recovery of the vector of the local constraints S = 1 2 =
1 2
~
~
T N
2
1
St n1
~
~
T N
2
2
St N
2
~
S
T T
~1 1
St T
2 2
0
starting from the 6 components tensors stored in mode PCONTRR ~
St T
~1 2
S
T N
~1
S
T N
2
~
· formation of the symmetrical tensor 3 × 3 of the local constraints [S]
tT
1 (1,2, 3)
· construction of the matrix of transformation P (
T
1,2, 3) = t2 (1,2, 3) where T3 (
1 23) = (
N
1 2)
tT
3 (1,2, 3)
T ~
· calculation of the symmetrical tensor 3 × 3 of the total constraints [S] = P [S] P
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
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:
R3.07.05-B Page
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×
3 9
[HSFM]
·
for the nontraditional term, calculation of HQ = 2×9
[HSS]
HQ =
(
2
2
2
T
1 ()
1)
(T1 (2))
(T1 () 3)
T1 ()
1 T1 (2)
T1 (2) T1 ()
3
T1 ()
3 T1 ()
1
(
2
2
2
T
2 ()
1)
(t2 (2))
(t2 () 3)
t2 ()
1 t2 (2)
t2 (2) t2 ()
3
t2 ()
3 t2 ()
1
(
2
2
2
T3 ()
1)
(T3 (2))
(T3 () 3)
T3 ()
1 T3 (2)
T3 (2) T3 ()
3
T3 ()
3 T3 ()
1
2t2 ()
1 T3 ()
1
2t2 (2) T3 (2) 2t2 ()
3 T3 ()
3
t2 ()
1 T3 (2) + T3 ()
1 t2 (2) t2 (2) T3 ()
3 + T3 (2) t2 ()
3
t2 ()
3 T3 ()
1 + T3 ()
3 T3 ()
1
2t
() 1t () 1 2t (2) T (2) 2t ()
3 T ()
3
T ()
1 T (2) + T ()
1 T (2)
T (2) T ()
3 + T (2) T ()
3
T ()
3 T ()
1 + T ()
3 T
3
1
3
1
3
1
3
1
1
3
3
1
1
3
3
1
1
3 ()
1
1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1
0 1 0 1 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 1 0 1 0
· calculation of the matrix jacobienne opposite J - 1 and of the determinant det J
~
NR
· calculation of J -
1 with:
3
(2)
3
NR I,
0
0
1
(2)
NR
0
0
3
I,
2
(2)
NR
0
0
I
(2)
J - 1
0
0
0
3
NR I,
0
1
~-
1
-
NR
1
H
(2)
J
= 0
J
0
;
=
L
0
= 1,
2
3
NR
I,
0
LI
NB
2
2
1
0
0
J -
3
(2)
0
NR
0
I
(
2)
0
0
3
NR
I,
1
(
2)
0
0
3
NR I,
2
(2)
0
0
NR I
~
~ NR
· calculation of the third operator of the deformations B = HQJ 1
-
3
3
· calculation and numerical integration Z =
BT S
J
I
D D D
~ ~ det
3
1
2
3
· calculation of the generalized tensor of the total constraints
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
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:
R3.07.05-B Page
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3×3
[S]
0
0
9×9
S = 0
[S] 0
0
0
[S]
~
NR
· calculation of J -
1 with:
2
(2)
(2)
() 1
0
NR N
3
,
Z
-
I
NR
3
I
1
N
NR
,
y
1
I
0
0
, 1
(2)
(2)
()
1
0
NR
N
3
,
Z
-
0
0
I
NR
3
I
2
N
NR
,
y
2
I
, 2
(2)
0
NR
N
(2)
0
0
0
NR
N
1
Z
-
1
y
()
(2)
(2)
0
NR 1
- NR
N
0
NR N
3
,
3
I
0
,
I
Z
I
,
X
1
1
1
()
H
L
(2)
(2)
0
NR 1
- NR
N
0
NR
N
LI = 1, NB1
3
I
,
Z
3
I
,
X
I
0
, 2
2
2
2
2
2
0
0
0
()
()
- NR N
0
NR
N
1
Z
1
X
()
0
0
NR 1
(2)
(2)
I
,
NR N
3
- NR N
0
1
I
,
y
3
I
,
X
()
1
1
0
0
NR 1
(2)
(2)
NR N
NR
N
0
I
,
3
-
2
I
,
y
3
I
,
X
2
2
0
0
0
(2)
(2)
NR N
NR
N
0
NR
1
y
-
1
X
=
(2)
(2)
2
0
NR
N
3
- NR
N
NB2, Z
3
NB2, y
1
1
(2)
(2)
0
NR
N
3
- NR
N
NB2,
Z
3
NB2,
y
2
2
(2)
(2)
0
NR
N
NB2 Z
- NR N
1
y
(
2)
(2)
- NR
N
0
NR
N
3
NB2, Z
3
NB2, X
H
1
1
(2)
(2)
- NR
N
0
NR
N
3
NB2,
Z
3
NB2,
X
2
2
2
- NR (2)
(2)
NB N
2 Z
0
NR NB N
2 X
(2)
(2)
NR
3
-
NB
N
2
,
y
NR
3
NB
N
2
,
X
0
1
1
(2)
(2)
NR
3
-
NB
N
2
,
y
NR
3
NB
N
2
,
X
0
2
2
(
2)
(2)
NR
NB N
2 y
- NR NB N
2 X
0
· calculation and numerical integration of the matrix of geometrical rigidity traditional
T
E
- 1 NR
- 1 NR
K
=
J
~
~
SJ
det J
D D
1 D
2
traditional
3
2
2
Fine loops on the points of integration
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 49/56
Loop on all the nodes of Lagrange with distinction of the super node
.
.
.
zI
· calculation of [zI ×] knowing that ZI =
.
.
.
I
=,
1 NB2
· calculation of the vector normal N I and its antisymmetric matrix [nor ×]
· calculation of the nontraditional matrix of geometrical rigidity
3×3
K E
(I, I) =
,
nontraditional
[zI ×] [N
I ×]
I = 1 NB2
3×3
· addition of K E
(I, I) with distinction of the super node
nontraditional
Fine loops on the nodes
Storage of the higher triangular part of K E
FIN
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
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Appendix 2: Flow chart of geometrical nonlinear calculation
Local reference marks with the NB2 nodes [T: T: N
1
2
] I
Beginning Boucle JN on the NB2 nodes
IF JN NB1
· recovery of the vector of total displacement already updated by MAJOUR:
U (II) ZR (IDEPLP IDEPLM 1 6 * (JN
) 1 II); II 13; (JN 1, NB
I
=
+
- +
- +
=
=
) 1
· recovery of the vector of full slewing already updated by MAJOUR
I (II) = ZR (IDEPLP - 1+ 6 * (JN -) 1 + II +)
3
; II = 1 3
,
; (JN = 1, NB)
1
ELSE JN
· recovery of the vector of full slewing
I (II) = ZR (IDEPLP - 1+ 6 * NB1+ II); II = 1 3; (JN = NB2)
END IF JN
· calculation of the matrix of rotation I =
[
exp I ×]
· transform of the normal N = N
I
I
I
End Boucle on the NB2 nodes
M
U
v
W
nx - nx
N
y - ny
Calculation of EP
N
=
Z - nz I
M
I = 1, NB
1
M
N
X - nx
ny - ny
nz - nz
NB2
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 51/56
Beginning Boucle INTSR on the points of normal reduced integration of Gauss
~
~
· construction of part of the operators B, B
1
2
with the J = 1, INTSR points of integrations to be able to extrapolate them
End Boucle INTSR on the points of normal reduced integration of Gauss
Beginning Boucle INTSN on the points of normal numerical integration of Gauss
· construction of the matrix of transformation:
< T
T
1 (1,2, 3) >
T1 (1,2, 3)
P (
T
1,2, 3) = < t2 (1,2, 3) > = t2 (1,2, 3)
< T
T
3 (1,2, 3) >
T
3 (1,2, 3)
where T (,) = (
N
3 1 2
3
1
, 2
)
· calculation of the matrix jacobienne opposite J - 1 and of the determinant det J
J-1
0
0
~
· calculation of J - 1 =
0
J -
1
0
0
0
J - 1
NR
· calculation of the second matrix of derived from the functions of form
1
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 52/56
2
()
()
1
NR
NR
3
I
0
0
0
0
,
I
1
,
1
()
(2)
1
0
0
NR
NR
3
I
0
0
,
I
2
,
2
(2)
0
0
0
NR
I
0
0
()
1
(2)
0
NR
0
0
NR
0
I
3
,
I
,
1
1
()
H
1
(2)
L
0
NR
0
0
NR
0
I
L = 1, NR 1
B
I
3
,
I
,
2
2
2
(2)
0
0
0
0
NR
0
()
I
0
0
NR 1
(2)
I
,
0
0
NR
1
3
I
,
()
1
0
0
NR 1
(2)
I
,
0
0
NR
2
3
I, 2
0
0
0
(
2)
NR
0
0
NR
I
=
(2)
1
NR
0
0
3
NB2,
1
(2)
NR
0
0
3
NB2,
2
(2)
NR
0
0
NB2
(
2)
0
NR
0
3
NB2,
1
H
(2)
0
NR
0
2
3
NB2,
2
0
(2)
NR
NB2
0
(
2)
0
0
NR
3
NB2, 1
(
2)
0
0
NR
3
NB2, 2
(
2)
0
0
NR
NB2
U ~
- 1 NR
· calculation of
E
= J
p
X
1
U
0
0
v
0
0
W
, X
, X
, X
0
0
0 U
0
0
v
0
0
W
, y
, y
, y
0
U
0
0
U
0
0
v
0
0
W
, Z
, Z
, Z
· calculation of A
X = U
U
0
v
v
0
W
W
, y
, X
, y
, X
, y
, X
0
U
0
U
v
0
v
W
0
W
, Z
, X
, Z
, X
, Z
, X
0
U
U
0
v
v
0
W
W
, Z
, y
, Z
, y
, Z
, y
1 U
·
calculation of H Q
+ A
2 X
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 53/56
H =
(
2
2
2
T
1 ()
1)
(T1 (2))
(T1 () 3)
T1 ()
1 T1 (2)
T1 (2) T1 ()
3
T1 ()
3 T1 ()
1
(
2
2
2
T
2 ()
1)
(t2 (2))
(t2 () 3)
t2 ()
1 t2 (2)
t2 (2) t2 ()
3
t2 ()
3 t2 ()
1
(
2
2
2
T3 ()
1)
(T3 (2))
(T3 () 3)
T3 ()
1 T3 (2)
T3 (2) T3 ()
3
T3 ()
3 T3 ()
1
2t2 ()
1 T3 ()
1
2t2 (2) T3 (2) 2t2 ()
3 T3 ()
3
t2 ()
1 T3 (2) + T3 ()
1 t2 (2) t2 (2) T3 ()
3 + T3 (2) t2 ()
3
t2 ()
3 T3 ()
1 + T3 ()
3 T3 ()
1
2t
() 1t () 1 2t (2) T (2) 2t ()
3 T ()
3
T ()
1 T (2) + T ()
1 T (2)
T (2) T ()
3 + T (2) T ()
3
T ()
3 T ()
1 + T ()
3 T
3
1
3
1
3
1
3
1
1
3
3
1
1
3
3
1
1
3 ()
1
· calculation of the first operator of the deformations
~
1 U ~ NR
B = H Q
+ A
J 1
-
1
2 X
1
~
~
· calculation of the vector of the local deformations E = B EP
1
NR
· calculation of the second matrix of derived from the functions of form
2
(2)
(2)
() 1
0
NR N
3
,
Z
-
I
NR
3
I
1
N
NR
,
y
1
I
0
0
, 1
(2)
(2)
()
1
0
NR
N
3
,
Z
-
0
0
I
NR
3
I
2
N
NR
,
y
2
I
, 2
0
NR (2)
(2)
0
0
0
1 N
- NR
Z
1 ny
()
(2)
(2)
0
NR 1
- NR
N
0
NR N
I
0
,
3
I
,
Z
3
I
,
X
1
1
1
()
H
L
(2)
(2)
0
NR 1
- NR
N
0
NR
N LI = 1, NB1
I
0
,
3
I
,
Z
3
I
,
X
2
2
2
2
0
0
0
(2)
(2)
- NR N
0
NR
N
()
1
Z
1
X
0
0
NR 1
(2)
(2)
I
,
NR N
- NR N
0
1
3
I
,
y
3
I
,
X
()
1
1
0
0
NR 1
(2)
(2)
I
,
NR
N
- NR
N
0
2
3
I
,
y
3
2
I, 2
X
0
0
0
(2)
(2)
NR
NR
1 N
- NR
y
1 N
0
X
=
(2)
(2)
2
0
NR
N
3
- NR
N
NB2, Z
3
NB2, y
1
1
(2)
(2)
0
NR
N
3
- NR
N
NB2,
Z
3
NB2,
y
2
2
(2)
(2)
0
NR
N
NB2 Z
- NR
N
NB2 y
(
2)
(2)
- NR
N
0
NR
N
3
NB2, Z
3
NB2, X
H
1
1
(2)
(2)
-
3 NR
N
0
,
Z
NR
N
2
NB2
3
NB2
2
,
X
2
(2)
(2)
- NR
NB2n
0
NR
Z
NB2nx
(2)
(2)
3 NR NB2 N
,
y
- 3N NB2 N
0
1
, 1 X
(2)
(2)
NR
N
NR
N
0
3
NB2
,
y
- 3 NB2
2
, 2 X
(
2)
(2)
NR
NB2ny
-
NR NB2n
0
X
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 54/56
· calculation of the second operator of the deformations
~
U ~ NR
B = H Q + A
J 1
-
2
X
2
· calculation and numerical integration Re =
BT S
Jd D D
~ ~ det
2
1
2
3
· calculation of the matrix of behavior D
· calculation and numerical integration K E =
BT DB
J
m
D D D
~ ~ det
2
2
1
2
3
~
· construction of the symmetrical tensor 3 × 3 of the local constraints [S]
T ~
· calculation of the symmetrical tensor 3 × 3 of the total constraints [S] = P [S] P
~
T
~
NR
· calculation of B = [
H S] J 1
-
3
3
calculation and numerical integration Z =
BT S
J
I
D D D
~ ~ det
3
1
2
3
3×3
[S]
0
0
9×9
· calculation of the tensor generalized of the total constraints S = 0
[S] 0
0
0
[S]
· calculation and numerical integration of traditional rigidity
T
NR
NR
K E
=
J - 1
SJ -
~
~ 1 det J
G
D D D
traditional
1
2
3
2
2
Fine INTSN loops on the points of integration
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 55/56
Beginning Boucle JN on the NB2 nodes
.
.
.
zI
· calculation of [zI ×] knowing that ZI =
.
.
.
I =,
1 NB
2
· calculation of [nor ×]
3×3
· calculation of the matrix nonsymmetrical K E
(I, I) = [zI ×] [N
G
I ×
nontraditional
]
IF JN NB1
3×3
· addition of K eg
(I, I) with distinction of the extra-node
nontraditional
ELSE JN
3×3
· assignment of K eg
(I, I) with distinction of the extra-node
nontraditional
END IF JN
Storage of all the matrix nonsymmetrical K and
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Key Mr. Al MIKDAD
:
R3.07.05-B Page
: 56/56
Intentionally white left page.
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Outline document