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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
1/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
Organization (S):
EDF/MTI/MN
Manual of Reference
R4.03 booklet: Analyze sensitivity
Document: R4.03.01
Sensitivity of the mechanical thermo fields
with a variation of the field
Summary
To know the influence of a variation of the field on the mechanical thermo fields, the conventional approach
consist in making several calculations and evaluating, by difference, the sensitivity. Method described in it
document makes it possible to obtain in only one calculation with Code_Aster the value of the fields of temperatures,
displacements and forced and their derivative compared to the variation of the field.
The method is initially exposed in its general information: thermics and linear static mechanics, 2D and 3D,
variation of an unspecified edge. Then various calculations in the case 2D and for some are detailed
loadings.
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
2/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
Contents
1 Introduction ............................................................................................................................................ 3
2 Determination of the gradient of the temperature ........................................................................................ 4
2.1 The problem ..................................................................................................................................... 4
2.2 Derivation of the variational equation ............................................................................................. 4
2.2.1 Integral of the temporal variation ........................................................................................ 5
2.2.2 Integral of thermal conductivity .................................................................................... 5
2.2.3 Integral of the convectif exchange - Part 1 ............................................................................. 6
2.2.4 Integral of the thermal sources intern ............................................................................ 6
2.2.5 Integral of the boundary conditions with flow imposed ................................................................. 6
2.2.6 Integral of the convectif exchange - Part 2 ............................................................................. 7
2.2.7 Assessment ....................................................................................................................................... 7
2.3 Comments on this equation ................................................................................................... 7
2.4 Discretization in time ................................................................................................................... 8
2.5 Space discretization ...................................................................................................................... 9
3 Calculation of the gradient of displacement ....................................................................................................... 9
3.1 The problem ..................................................................................................................................... 9
3.2 Derivation of the variational equation ........................................................................................... 10
3.2.1 Voluminal integral .............................................................................................................. 10
3.2.2 Surface integral .............................................................................................................. 12
3.2.3 Assessment ..................................................................................................................................... 12
3.3 Comments on the equation to solve ...................................................................................... 12
4 Determination of the gradient of the stresses .......................................................................................... 13
5 Conclusion ........................................................................................................................................... 13
Appendix 1 the transformation of the field ................................................................................................ 14
Appendix 2 Form ............................................................................................................................... 14
Appendix 3 Commutation of derivations Lagrangian and temporal .................................................... 18
Appendix 4 Implementation numerical ...................................................................................................... 19
6 Bibliography ........................................................................................................................................ 31
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
3/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
1 Introduction
The developments presented in this document aim at allowing probabilistic studies of
brutal rupture by a coupling mechanic-reliability engineer. The geometry of the field is treated like one
random field. The evaluation of the probability of priming of the rupture is ensured by coupling with
software PROBAN, with methods FORM/SORM. This evolution supposes to know them
variations of the stresses and the temperatures by report/ratio geometry. Thus, a first application
industrial seeks to determine the probability of rupture of the nuclear reactor vessel, of which
the thickness of the lining is regarded as a random variable. These stresses and
temperatures being calculated by Code_Aster, the conventional technique consists in carrying out series
of calculation for several values thickness of the coating. Then by difference, one deduces some
the influence thickness on these fields.
This technique has limits; in particular:
·
precision: how to choose the values of the parameter thickness so that the difference enters
does two calculations represent its influence well?
·
performance: for a value of the parameter, at least two calculations with Code_Aster are
necessary to calculate the influence.
The method developed in this work makes it possible to obtain in only one calculation with Code_Aster the value
stresses and temperatures and their derivative compared to the thickness of the coating.
The technique selected is based on a direct derivation of the equations expressed in form
variational. It takes again the method as under the name of “method
“already introduced for
calculation of the rate of refund of energy in Code_Aster. This fact a certain number of results of
base are not redémontrés but are the subject only of references in appendix.
The first part relates to the derivation of the temperature, in stationary regimes and transitory, in
linear thermics. The main loadings are studied: convectif exchange, temperature
imposed, internal source.
Then, we expose the derivation of the field of displacement in linear static mechanics.
loadings taken into account are limited to imposed displacements and pressures distributed.
derivation of the stress field is summarized then with a postprocessing of derived from the field from
displacement.
The method is presented in its general information: 2D or 3D, influence of an unspecified variation of one
edge of the field. In practice, the functionality is currently available only in 2D, for
loadings mentioned herebefore.
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
4/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
2
Determination of the gradient of the temperature
2.1
problem
A first stage of the calculation of the gradients of the rate of refund of energy G is the calculation of
gradient of the temperature compared to a real parameter. This parameter controls the variation of
field of calculation: starting from the area of reference
, a transformed field is studied
, where
is the parameter symbolizing the transformation. The required gradients are those which are expressed
at the place of the resolution of reference, i.e. for
=
0
. One will refer to appendix 1 for
notations employed.
We leave the variational equation governing it thermal of the problem on the field
transformed
. While following [R5.02.01], we define the borders by:


1
2
3
:
:
:
imposed temperature
imposed normal flow
convectif exchange
what gives the following variational equation:
C
T
T T
T T
H T T
S T
Q T
H T T
p
ext.
*
*
*
*
*
*
.
+
+
=
+
+
2
3
3
with:
T
Temperature
C
p
voluminal heat with constant pressure
thermal conductivity
H
coefficient of convectif exchange
S
thermal voluminal source
Q
normal heat transfer rate entering imposed on the edge
2
T
*
function of test in
()
H
1
, null on
1
Note:
We present here only the problem with boundary conditions of temperature
imposed, of imposed normal flow and convectif exchange. The taking into account of the conditions
of exchange between wall or radiation will be done later on.
2.2
Derivation of the variational equation
We will derive each integral successively forming the equation. Each time, us
will use the formula of Reynolds, after having defined the vector



from
F
for
transformation
F
(cf Appendix 2).



represent the direction of variation of the field.
We choose functions of test
T
*
who are independent of the parameter
. In addition, one
is here in a case where derivations Lagrangienne and temporal switch over (cf Annexe 3).
In all the formulas presented, we note with a point the Lagrangian derivative of
size:
!T
is the Lagrangian derivative of
T
.
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
5/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
2.2.1 Integral of the temporal variation
I
C
T
T T
dI
D
C
T
T T
C
T
T T
div
p
p
p
1
1
=
=




+


·
*
*
*



We suppose that voluminal heat
C
p
is independent of the parameter
i.e.
purely Lagrangian (attached to the material point).
By using proposal 2 of appendix 2,
!
.
=
+
, we have here
!
.
=
. From where
the expression:
()
()
C
T
T
C
T
T
C
T
T
C
T
T
C
T
T
dI
D
C
T
T T
C
T
T div T
C
T
T T
p
p
p
p
p
p
p
p
·
·
·




= +




=
+
=
+
+
.
!
!
.
*
*
*









1
2.2.2 Integral of thermal conductivity
[
]
I
T T
dI
D
T T
T T div
2
2
=
=




+
·
.
.
.
*
*
*



We suppose that thermal conductivity is also independent of the parameter
. Thus,
we have:




=
+




+






·
·
·
T T
T T
T
T
T
T
.
!
.
.
.
*
*
*
*
With the general result of appendix 2, we have:




= -






=
-
= -
=
·
·
T
T
T
T
T
T
T
!
.
!
.
.
!
.
*
*
*
*












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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
6/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
From where the result:
(
)
(
)
(
)
D I
D
T T
T T
T
T
T
T
div
T T
2
=
+
-
-
+
!.
.
.
.
.
.
.
.
*
*
*
*
*












2.2.3 Integral of the convectif exchange - Part 1
I
hTT
3
3
=
*
We use this time the proposal 4 (cf Annexe 2) which establishes derivation for an integral
surface.
=




+
·
dI
D
hTT
hTT div
S
3
3
*
*



We suppose that the coefficient of heat exchange by convection is him still independent of
parameter
. Thus,
!
.
H
H
=
.
(
)
D I
D
H TT
H
TT
hTdiv T
S
3
3
=
+
+
!
.
*
*
*






2.2.4 Integral of the internal thermal sources
[]
I
St
dI
D
St
St div
4
4
=
=




+
·
*
*
*



We suppose that the thermal voluminal source is independent of the parameter
. Thus,
!
.
S
S
=
.
()
(
)
D I
D
S
T
S div T
4
=
+
.
*
*






2.2.5 Integral of the boundary conditions with imposed flow
I
Q T
dI
D
Q T
Q T div
S
5
5
2
2
=
=




+
·
*
*
*



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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
7/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
We suppose that the external heat transfer rate is independent of the parameter
. Thus,
!
.
Q
Q
=
.
(
)
D I
D
Q
T
Q T div
S
5
2
=
+
.
*
*






2.2.6 Integral of the convectif exchange - Part 2
I
H T
T
dI
D
H T T
H T T div
ext.
ext.
ext.
S
6
6
3
3
=
=




+
·
*
*
*



We suppose that the coefficient of heat exchange by convection and the outside air temperature
are independent of the parameter
.
(
)
(
)
D I
D
H
T T
H
T
T
H T T div
ext.
ext.
ext.
S
6
3
=
+
+
.
.
*
*
*









2.2.7 Assessment
In all these expressions of derivations of integrals, only the Lagrangian derivative of
temperature is unknown. We can thus form a new variational equation of which
!T
is
the unknown factor.
()
(
)
[
]
[
]
(
)
(
)
(
)
(
)
C
T
T T
T T
hTT
C
T
T T
C div
T
T T
T T
T
T
T
T
div
T T
H
T
T T
H T
T div
T
H
T
T
p
p
p
ext.
ext.
S
ext.
!
!.
!
.
.
.
.
.
.
.
.
*
*
*
*
*
*
*
*
*
*
*
*
+
+
=
-
-
-
+
+
-
+
-
+
-
+
+
3
3
3
3



























(
)
(
)
+
+
+
S
T
S div T
Q
T
Q div
T
S
.
.
*
*
*
*












2
2
The border
1
boundary conditions of Dirichlet for the calculation of
T
corresponds to the same type of
boundary conditions:
!T
is imposed on a zero value along this border.
2.3
Comments on this equation
One can notice that the first member of this equation is, formally, identical to that of
the variational equation which allows the calculation of the temperature. It is thus a question of solving the same one
equation, with a second modified member.
The solution of this equation provides the Lagrangian derivative of
T
. To have the derivative which us
interest,
T
, it remains to achieve the last operation:

T
T
T
= -
!
.



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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
8/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
2.4
Discretization in time
The temporal resolution will be done by using the method known as of
- diagram, as for the equation
who controls the change of the temperature. To know the size at the moment
T
T
+
, us
will pose:
(
)
()
!
!
!
!
T
T T
T
T
T T
+
-
=
+
=
and
The derivative in time approaches thus by:
! ~!
!
T
T
T
T
T
-
-
+
-
At the current moment we will use the approximation:
(
)
! ~
!
!
T
T
T
I
I
-
+ -
+
-
1
We will apply this technique for the main variables of the problem:
!,
T T H Q S
. All them
fields at the moment T being known, the equation discretized in time can be written:
(
)
(
)
()
(
)
(
)
[
]
C
T T T
T
T
H T T
C
T T T
T
T
H T T
C div T
T
T
T
C
T
T
T
T
T
T
T
T
p
I
I
p
I
I
p
p
I
I
I
+
+
+ +
-
-
- -
+
-
+
-
+
-
+
+
+
=
-
-
-
-
-
-
-
-
+
+ -
+
+ -
!
! .
!
!
! .
!
.
.
.
*
*
*
*
*
*
*
*
*
3
3
1
1
1
1









(
)
(
)
[
]
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
I
I
I
I
I
I
I
I
I
I
I
I
I
S
S
I
T
T
T
T
T
div
T
T
T
S
S
T
div
S
S T
Q
Q
T
Q
Q
div
T
With div
T
-
+ -
-
+ -
+
+ -
+
+ -
+
+ -
+
+ -
+
+
-
+
-
+
-
+
-
+
-
+
-
+
-







.
.
.
.
.
.
.
*
*
*
*
*
*
*
*
























1
1
1
1
1
1
2
2
3
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
H
H
T
T
T
T
T
H
H
T
T
T
I
I
ext.
I
ext.
I
I
I
ext.
I
ext.
+
-
+
+
-
-
+
-
+
-
+ -
-
+ -
-
+
+ -
+ -

1
1
1
1
3
3
.
.
*
*






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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
9/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
The term of flow of convectif exchange,
With
, takes two expressions distinct according to the dependence from
H
and
T
ext.
with respect to time.
If
H
and
T
ext.
are independent of time, only the temperature is with impliciter. From where:
(
)
(
)
H T
T
T
div
T
ext.
I
I
S
-
- -
+
-
1
3



*
If not, the whole of flow undergoes it
- diagram:
(
)
(
)
(
)
[
]
I
ext.
I
ext.
S
H T
T
H T
T
div
T
+
+
+
-
-
-
-
+ -
-
1
3



*
Note:
One will have noted the nuisance to have to treat with “Dupond and Dupont of numerical”, with
to know it
- diagram and method



… To keep coherence with the remainder of
documentation of Code_Aster, we chose to preserve the notations



for both
parameters of these methods. Insofar as it is the “method”



who interests us it
more in this work, it
“diagram” was affected of an index
I
, like “implicitation”.
Let us hope that that will have been clearly…
2.5 Discretization
space
The space discretization of this equation is copied exactly on that employed for
resolution of thermics. We return to [R5.02.01] for his description.
3
Calculation of the gradient of displacement
3.1
problem
A second stage necessary to the calculation of the gradients of the rate of refund of energy is calculation
gradient of the field of displacement compared to the variation of the field. We begin again
exactly same conventions as in the preceding chapter for the calculation of the gradient of
temperature.
Only certain loadings are taken into account. The extension to other types of change would be done
while following the principles which will be stated. We place ourselves in the case of linear elasticity
isotropic, in two dimensions. The relation between the tensor of the stresses and the tensor of
deformations is then of the type:
(
)



=
+
-
K T T
ref.
Id
The field of calculation is noted
, where
is the real parameter of piloting of the variations of the field.
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
10/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
For this application, we retain only three types of behavior at the edge of
:


1
2
3
:
:
:
border with imposed displacement
border with “uniform connection”
border with imposed external pressure
what gives the following variational equation:
() ()






U
v
v N
:
.
=
p
3
with:
U
field of displacement
()



U
tensor of the stresses related to displacement
U
()



v
tensor of the deformations associated with displacement
v
p
pressure distributed on the edge
3
3.2
Derivation of the variational equation
We will derive the two integrals which constitute the variational equation by applying it
theorem of Reynolds (cf Appendix 2 and [§2.2]).
3.2.1 Integral
voluminal
() ()
I
=






U
v
:
Because of isotropy of the problem, we have the equality of the scalar products:
()
()






:
:
v
v
=
S
where
()
S
v
is the gradient symmetrized of
v
.
The integral and its derivative are thus written:
()
()
()
()
()
()
[
]
I
dI
D
div
S
S
S
=
=




+

·












U
v
U
v
U
v
:
:
:
We will use the property of derivation of a gradient of vector, where
FT
is a tensorial function,
described in appendix 2:
(
)




= -
·
v
v
v
FT
,



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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
11/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
We have:
()
()
()
()
()
()
()
()
(
)
[
]
()
()
()
(
)



























U
v
U
U
v
U
U
v
U
U
v
v
v
v
v
:
!
:
: !
!
:
:
!
T
,
!
:
: FT
,




=
+
=
+
-
=
-
·
S
S
S
S
S
S
S
S
because functions
v
are supposed to be independent of the variations of the field, therefore
!v
=
0
. For
to calculate the derivative not



, we notice that



is a function of the deformation and
temperature:
()
!
!
!









U
E E
=
+

T T
In the particular case of isotropic linear elasticity:
()
(
)









E
E
Id
E
Id
, T
K T T
T
K
ref.
=
+
-
=
=
from where:
()
!
!
!



U
E
Id
=
+
K T
It remains to express the derivative of the deformation
E
, starting from its expression according to
displacement
U
.
[
]
(
)
[
]
(
)
[
]
[
]
(
)
(
)
[
]
E
U
U
E
U
U
E
U
U
U
U
E
U
U
U
U
=
+
=
+








=
-
+ -
=
+
-
+
·
1
2
1
2
1
2
1
2
1
2
1
2
T
T
T
T
T
!
!
!
! FT
,
! FT
,
!
!
!
FT
,
FT
,












The final expression of the derivation of the integral is then:
[
]
()
(
)
(
)
[
]
[
]
()
()
()
()
(
)
()
()
[
]
dI
D
K T
div
T
S
T
S
S
S
S
=
+
-
+
+
-
+
1
2
1
2












!
! :
FT
,
FT
,
:
!
:
: FT
,
:
U
U
v
U
U
v
Id
v
U
v
U
v
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3.2.2 Integral
surface
I
p
D I
D
p
p div
S
3
3
3
3
=
=




+
·
v
v
v



By choice of the functions
v
we have
!v
=
0
. We suppose that the external pressure is
independent of the parameter
. As follows:
[
]
[
]
p
p
D I
D
p
p div
S
v
v
v
v
·


=
=
+
.
.









3
3
3.2.3 Assessment
In all these expressions of derivations of integrals, only the Lagrangian derivative of displacement
is unknown. We can thus form a new variational equation of which
!U
is the unknown factor and
where
v
is the gradient symmetrized of
v
.
[
]
(
)
(
)
[
]
[
]
()
(
)
()
[
]
[
]
1
2
1
2
3
3




+
=
+
-
+
-
+
+
























!
! :
FT
,
FT
,
:
!
:
:FT
,
:
.
U
U
v
U
U
v
Id v
U
v
U
v
v
T
T
S
K T
div
p
p div
On the border
1
, displacement is imposed. Whatever the position of this border,
boundary condition follows the matter, which involves
!U
=
0
.
On the border
2
, with uniform connection, the degrees of freedom are identical, but free:
U
y
=
constant for example. It is the same for the derivative
!U
.
3.3
Comments on the equation to be solved
We will note that, as for the problem in temperature, the problem to be solved here is
formally the same one as that of the determination of displacement
U
. The matrix is the same one. Only
the second member evolves/moves.
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If we set out again of the initial variational equation:
() ()






U
v
v N
:
.
=
p
3
we can transform it by using the expressions of



and of



:
.
()
()
(
)
[
]
()
[
]
(
)



U
v
v
U
Id
v
v
U
v
Id
v
v
:
:
:
:
=
+
-
=
= -
-
+
p
K T T
p
K T T
p
ref.
ref.
3
3
3
Like
()
[
]



U
U
U
=
+
1
2
T
, we find well the same expression with the first member as
for the transformed equation.
4
Determination of the gradient of the stresses
The following stage aims at determining the gradient of the stresses. It will be calculated from
knowledge of the gradients of the temperature and displacement.
We saw that in the particular case of isotropic linear elasticity,
!



could express itself under
form (cf [§3.2.1]):
()
[
]
(
)
(
)
[
]
!
!
!
FT
,
FT
,
!









U
U
U
U
U
Id
=
+
-
+
+
1
2
1
2
T
T
K T
This stage, all the quantities on the right of the sign = are known; there is not any more but to do one
postprocessing to obtain
!



.
In the same way, knowing that:




= -
!
,
the derivative eulérienne of the stresses is expressed in postprocessing of the quantities previously
calculated.
Note:
This phase of derivation imposes that calculation took place with quadratic elements.
5 Conclusion
To continue this work, it remains to make the calculation of derived from the rate of refund of energy. That
is envisaged in a later version.
The use of this functionality, planned initially for probabilistic mechanics, can extend to
other fields: optimization of forms, identification.
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Appendix 1 the transformation of the field
The technique used to calculate the various gradients during a variation of the field is that of
method known as “method
”. This method was developed for calculation of the rate of refund of
energy G; it is described in [bib1] and [R7.02.03]. We give here the various expressions which are
used in this document.
The field of calculation of reference is noted
. It is transformed into a noted field
,
where
is one
real parameter. The whole of the transformations is represented by the functions
F
. We agree that
F
0
corresponds to the identity.
In a general way, the only sizes which interest us are the gradients expressed at the point of
resolution. They are thus the derivative compared to the parameter
expressed for
=
0
. This is why, for
to reduce the notations, in all the document we write:
instead of




=
0
The field of vectors
F
is noted



.
We will use the following deduced sizes:
·
field
scalar
div



, voluminal divergence, and
div
S



, surface divergence,
·
tensor
.
Appendix 2 Form
We will recall here, the main formulas useful for calculations of derivation. One will refer to [bib1] for
their demonstrations.
That is to say
(
)
,
M
an unspecified field. We note:
(
)
()
(
)
,
, F
M
M
=
Lagrangian derivative:
!
=
Proposal 1:
()
()
()
()
()
(
)
I
II
III
div
=
= -
=
-
F
F
det F









1
Proposal 2:
()
I!
=
+
!
is the Lagrangian derivative for the movement
F
is the derivative eulérienne
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·
if
is a scalar field,
is a scalar product, which gives
!
=
+
X
K K
K
·
if



is a field of vector,
is a tensor, which gives:
!
I
I
I
K K
K
X
=
+
·
if



is a tensor, the same formula applied to each component of the tensor gives:
!
,
,
,
I J
I J
I J
K
K
K
X
=
+
Note:
The analytical expression of this formula is the same one in plane 2D or axisymmetric 2D. Indeed, it
term complementary to
if



is a vector is
R R
/
. It would be to multiply by the component
orthoradiale of the vector



; that one being null, there is no amendment of the expression. One passes
thus of the formula plane 2D to the axisymmetric formula 2D by the formal analogy
()
()
X y
R Z
,
,
.
Proposal 2:
()
()
(
)
II
= -
!
! FT
,
The operator
FT
is the matric operator which is connected formally with the matric product.
·
if
is a scalar field, in plane 2D:
On the basis of the expression
()
2i
and by deriving it compared to
X
:
!
!
!
=
+
+
=
+
+




=
+
+
+
+
=




+




+




+
+
X X
y y
X
X
X
X X
y y
X
X
X X
X
X
X there y
y
X
X
X
X
y
X
X
X
y
X
y
X
y
X
2
2
2
2
y
X
By applying the formula
()
2.i
with the first three terms of this sum, we have:
!
X
X
X
X
y
X
X
y
=




+
+
·
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Same technique used while deriving
()
2.i
compared to
y
allows to establish:
(
)


X
y
X
y
X
X
y
X
X
y
y
y
X
y
X
y
·
·








=




-
+
+










= -
!
!
! FT
,



·
if
is a scalar field, in axisymmetric 2D, its gradient in axisymmetric 2D is worth the vector:
=
+
R
Z
R
Z
E
E
The starting point is still the expression
()
2.i
:
!
=
+
+
R
Z
R
Z
By deriving this formula compared to
R
or with
Z
, we find the same expression formally
that in plane 2D, for the terms in
R
Z
and
.
From where the expression:


R
Z
R
Z
R
R
Z
R
R
Z
Z
Z
R
Z
R
Z
·








=




-
+
+




!
!
Summary for a scalar:
{}
{}
X
X
X
X
I
K
X y
R Z
I
I
K
K
K
I
·




=
-
!
,
,
with and
or
·
when



is a vector or a tensor, we apply the same reasoning to each one of its
components in Cartesian co-ordinates:
{}

J
I
J
I
J
K
K
K
I
J L
I
J L
I
J L
K
K
K
I
X
X
X
X
X
X
X
X
I J K L
X y
·
·




=




-




=




-
!
!
,
,
,
,
,
with
·
for a vector or a tensor into axisymmetric, it is necessary to take account of the characteristics of the gradient.
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We have indeed:






=
+
=










R R
Z Z
R
R
R
Z
Z
R
Z
R
R
R
E
E
0
0
0
0
Derivations of the terms in
J
I
X
are obtained as considering previously. It is necessary from now on
to apply the expression
()
2.i
at the end exchange
R
R
:
R
R
R
R
R
Z
R
R
R
R R
R R
R
Z
R
R
R
R
R
Z
R
R
R
R
R
R
R
Z
·
·




=


+ +




=
+


+
-
+
1
1
1
1
2
The derivative eulérienne of
1
R
is null by construction. Terms 1, 3 and 5 of the sum are
the expression
()
2.i
applied to
R
. What gives:
R
R
R R
R
R
R
·




=
-
!
1
2
From where the expression
R
R
R
Z
Z
R
R
R
Z
Z
R
Z
R
R
Z
R
Z
R
R
Z
0
0
0
0
0
0
0
0
·












=










!
!
!
!
!
-
+
+
-
+
+










R
R
R
Z
R
R
R
Z
R R
Z
R
Z
Z
Z
R
Z
Z
R
R
Z
R
R
Z
Z
Z
R
R
R
Z
R
Z
Z
Z
Z
0
0
0
0
2
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Proposal 3 (theorem of Reynolds)
With
I
D
R
=
,
:

field of
3
()
(
)
()
I D I
D
div
D
II D I
D
D
N D S
=
+
=
+
!
.






Proposal 4:
With
J
D S S
S
=
,
surface
R
3
and while noting
N
the normal external with
S
:
()
[
]
I D J
D
div
ds
S
S
=
+
!



Appendix
3
Lagrangian commutation of derivations and
temporal
Proposal:
!
T
T
=




·
Proof:
For the transformation
F
, we pose classically:
(
)
()
(
)
,
,
, F
,
M T
M T
=
By definition, the Lagrangian derivative is worth:
(
)
!
,
M T
=
If the transformation applied is the same one at every moment, the moment,
T
, and the parameter of
transformation are independent one of the other. Derivations compared to
T
and
can thus switch over.
()
(
)
()
(
)
!
, F
,
, F
,
T
T
T
M T
T
M T
T
T
=




=




=




=




=




·
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Appendix 4 Implementation numerical
A4.1 Calculation of the variation in temperature
A4.1.1 general Principle
We saw with [§2.2] whom the equation to solve is the same one as that which controls calculation
thermics, except for the second member. That encourages us to insert the calculation of derived from the temperature
in the calculation of the temperature itself (operator
THER_LINEAIRE
). It will be thus possible with
each moment to re-use the assembled matrices and to treat all the loadings of the problem
thermics.
A4.1.2 total Algorithm
More precisely, calculations of
T
and
!T
imbricate themselves in the following way:
·
Initialization of the field of temperatures
T
, and of its gradient,
!T
, with two possibilities:
-
resetting
-
resumption of a field previously calculated
·
Loop in time:
1)
Calculation of the elementary matrices, then assembly
2)
Calculation of the second member of the equation of thermics
3)
Resolution
This stage, one knows
T
T
N
N
and
+
1
. One can connect on the calculation of
!T
.
4)
Calculation of the second member of the calculation of
!T
-
term due to the thermal source and the boundary conditions of flow
-
term due to the derivation of the equation
-
term due to the method of implicitation. One uses the same program as for
the calculation of
T
, while having replaced the field
T
by the field
!T
5)
Resolution of the system to know the new value of
!T
6)
Tilt of the values of
T
T
N
N
+
+
1
1
and
!
in
T
T
N
N
and
!
A4.1.3 Boundary conditions of Dirichlet
Everywhere where one has boundary conditions of Dirichlet on the thermal problem, one finds
boundary conditions of Dirichlet for the calculation of
!T
. In these points,
T
being imposed,
T
is
independent of the variations of the field:
T
=
0
As we have the relation:
!
.
T
T
T
=
+




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we deduce the values from them from the boundary conditions of Dirichlet:
!
.
T
T
di
=
This value is thus calculated in each node of the border
1
.
A4.1.4 Detail of the various terms of the second member
We will gather under the same integral the obligatory terms due to the derivation of
the equation, then to examine each possible change. The result will be written in the form of
contribution of the node
I
at the point of Gauss
pg
the element running in the calculation of the integrals by
formulate Gauss, knowing that all its contributions are to be cumulated.
Term due to derivation
It is necessary to calculate the contribution of:
(
)
()
(
)
(
)
[
]
(
)
(
)
[
]
I
I
I
I
I
I
I
I
I
I
C
T T T
I
T T
I
div
C T
T
T
T
I
C
T
T
T
T
I
T
T
T
I
T
T
T
I
p
I
p
p
I
I
I
I
=
+
+
+
+
+
+
+
=
= -
-
=
-
-
=
-
-
=
+ -
=
+ -
=
-
-
-
+
-
+
-
+
-
+
-
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
1
1
1
,
!
!
.
.
*
*
*
*
*
*
with












(
)
(
)
(
)
(
)
[
]
+ -
=
+ -
+
-
+
-
.
.
*
*






I
I
I
I
T
T
T
I
div
T
T
T
1
1
8
·
Calculation
of
I
1
In stationary regime, this term does not exist. In transient, one will have expressed
!T
-
, derived
Lagrangian of
T
at the previous moment at the points of Gauss.
C
p
is supposed to be constant by
element. From where the contribution:
() ()
I
C
T T
pg ui pg
I pg
p
pg
1,
!
=
-
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·
Calculation
of
I
2
In stationary regime, this term does not exist. In transient, one will have expressed the gradient of
!T
with
points of Gauss.
is supposed to be constant by element. From where the contribution:
(
)
()
I
T
ui
xj
I pg
I
pg J
J
pg
2
1
,
,
!
= -
-
·
Calculation
of
I
3
In stationary regime, this term does not appear.
C
p
is supposed to be constant by element.
div T
T



,
+
-
and
will have been calculated at the points of Gauss as quoted in appendix 5. From where
contribution:
() ()
()
(
)
()
I
C
T div pg T
pg
T
pg ui pg
I pg
p
pg
3,
= -
-
+
-



·
Calculation
of
I
4
C
p
is supposed to be constant by element. Its gradient is thus null by element. From where:
I
I pg
4
0
,
=
·
Calculation
of
I
5
is supposed to be constant by element.
+
-
T
T
,
and



will have been calculated at the points of Gauss.
One starts by calculating the quantity
()
I
I
T
T
+ -
+
-
1
. The result is a vector of which them
components at the point of Gauss are:
()
()
()
()
[
]
()
With pg
T I
T I
ui
xj pg
J
I
I
I
=
+ -
+
-

1
The contracted tensorial product
With
is written:
(
)
With
=



K
J
J
jk
With
Example:
(
)
With
=
+
+



X
X
X
y
y
Z
Z
With
X
With
X
With
X
etc, from where the formula:
(
)
() ()
I
pg
ui
xk pg
I pg
K
K
pg
5,
.
=
With



In axisymmetric 2D, the product
With
is written:
(
)
(
)
(
)
With
With
With
=
+
+
=
+
+
=









R
R
R
Z
Z
Z
R
R
Z
Z
R
With
R
With
X
With
X
With
Z
With
Z
With
Z
With R
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Here, the component
With
is always null. We thus find the same expression as in
Cartesian co-ordinates 2D.
·
Calculation
of
I
6
is supposed to be constant in the element.
+
-
T
T
div
,
and



will have been calculated at the points of
Gauss. The vector
()
I
I
T
T
+ -
+
-
1
will be noted
With
, like previously. The product
tensorial contracted
T
*



has on the current node
I
for component:
(
)
()
(
)
()
()
()
=
=
+
+
T
U
X
pg
T
U
X pg
X
U
y pg
X
U
Z pg
X
I K
I
J
J K
J
I X
I
X
I
y
I
Z
*
,
*
,






Ex:
from where the formula:
()
(
)
I
With pg
T
I pg
K
I K
pg
K
6,
*
,
.
=



As for the preceding integral in axisymmetric 2D, the component
With
is null.
The expression is thus the same one in Cartesian 2D or axisymmetric.
·
Calculation
of
I
7
is supposed to be constant in the element. Its gradient is thus null there. From where:
I
I pg
7
0
,
=
·
Calculation
of
I
8
is supposed to be constant in the element.
+
-
T
T
div
,
and



will have been calculated at the points of
Gauss. The vector
()
I
I
T
T
+ -
+
-
1
is noted
With
, like previously. We have then:
()
() ()
I
div pg
With pg
U
X
pg
I pg
K
I
K
pg
K
8,
=



Source term of energy
We calculate the two integrals:
(
)
(
)
(
)
(
)
I
div
S
S T
I
S
S
T
I
I
I
I
1
2
1
1
=
+ -
=
+ -
+
-
+
-






*
*
.
S
S
+
-
and
are known at the points of Gauss.
div



was calculated at the point of Gauss. From where
contribution:
()
()
(
)
()
(
)
()
I
div pg
S pg
S pg U pg
I pg
I
I
I
pg
1
1
,
=
+ -
+
-



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Code_Aster
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
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Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
23/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
The source
S
being constant by element, its gradient is null. From where:
I
I pg
2
0
,
=
Term of the boundary conditions of imposed flow
(
)
(
)
I
Q
Q
div
T
I
I
S
1
1
=
+ -
+
-



*
Q
Q
+
-
and
are known at the points of Gauss.
()
div



is calculated at the point of Gauss.
()
(
)
()
[
]
() ()
I
Q
pg
Q
pg
div
pg U pg
I pg
I
I
S
I
pg
1
1
,
=
+ -
×
+
-



Note:
The calculation of
J
K
X
/
is done at the points of Gauss of the element of edge, for example on one
segment for a calculation 2D. However on this segment, one knows only the derivative curvilinear of
functions of form, i.e. the derivative tangential. It is thus necessary to calculate them as a preliminary
quantities
J
K
X
/
while basing itself on the elements of volume and that with the nodes of
elements of edge. Then, one evaluates their values at the points of Gauss of the element of edge
with the functions of this element of edge.
The expression is the same one in axisymmetric 2D as into Cartesian because the complementary term
of
,
R
R
, is multiplied by the component orthoradiale of the normal
N
. However this
component is null.
(
)
(
)
I
Q
Q
T
I
I
2
1
2
=
+ -
+
-
.
*



Q
is supposed to be constant by element. Its gradient is thus null there. From where:
I
I pg
2
0
,
=
Term of the boundary conditions of convectif exchange
If
H
T
ext.
and
are independent of time, we calculate the following expression:
(
)
(
)
(
)
I
H T T
H T
T
T
div T
I
ext.
I
I
S
= -
-
+
-
- -
-
+
-
1
1
3
3
!
*
*



What gives:
(
)
()
[
]
()
(
)
()
(
)
() ()
I
H T
pg
H T
T
pg
T
pg
div
pg U pg
I pg
I
ext.
I
I
S
I
pg
,
!
=
-
+
-
- -
×
-
+
-
1
1



If
H
T
ext.
or
are independent of time, it is then necessary to calculate:
(
)
(
)
(
)
(
)
[
]
I
H T T
H T
T
H T
T
div T
I
I
ext.
I
ext.
S
= -
-
+
+
-
+ -
-
- -
+
+
+
-
-
-

1
1
3
3
!
*
*



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Code_Aster
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Version
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
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Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
24/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
What gives:
(
)
()
[
]
()
(
)
(
)
()
(
)
[
]
() ()
I
H T
pg
H T
T
pg
H T
T
pg
div pg U pg
I pg
I
I
ext.
I
ext.
I
pg
,
=
-
+
-
+ -
-
×
-
-
+
+
+
-
-
-
1
1



Same remarks that to the preceding paragraph apply to the calculation of the quantities
J
K
X
/
.
Two integrals utilizing
H
T
ext.
and
are null, insofar as
H
T
ext.
and
are
presumedly constant by element.
A4.2 Calculation of the gradient of displacements and the stresses
A4.2.1 general Principle
As for thermics, the calculation of the gradient of displacement is inserted in the calculation of
displacement, i.e. the operator
MECA_STATIQUE
.
Then the calculation of
!






and
will be done in postprocessing, in the controls
CALC_ELEM
and
CALC_NO
.
A4.2.2 Boundary conditions of Dirichlet and uniform connection
Nothing is to be made for these two types of boundary conditions. Their processing is ensured by
standard operation of the calculation of static mechanics linear.
A4.2.3 Detail of the various terms of the second member
We will gather under the same integral the obligatory terms which had with the derivation of
the equation, then to examine each possible loading. The result will be written in the form of
contribution of the node
I
and of the point of Gauss
pg
for the current element.
Term due to derivation
It is necessary to calculate the contribution of:
(
)
(
)
[
]
[
]
()
(
)
()
[
]
I
K T
div
T
S
S
S
S
=
+
-
-

1
2












FT
,
FT
,
:
!
:
: FT
,
:
U
U
Id
v
U
v
U
v
v
where
S
v
is the gradient symmetrized of
(
)
v
v
v
that is to say
1
2
+
T
.
To format this writing symbolic system, we will leave the broken up analytical form
and to write derivations on the scalar terms.
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Code_Aster
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
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Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
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Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
One will remember (cf [§3.2.1]) that this integral results from derivation from:
()



U
v
:
S
and that we broke up it into:
()
[
]
()
()









U
v
U
v
U
v
:
:
!
:
+




+
·
S
S
S
div
We start by clarifying the first integral because that will make it possible to set up them
various terms according to the mode: plan or axisymmetric.
·
In plane deformations:



:
=
+
+
+




S
xx
X
yy
y
xy
X
y
v
X
v
y
v
y
v
X
v
The divergence of the field



is a data, calculated at the points of Gauss.
The tensor of the stresses
()



U
is known at the points of Gauss. The tensor
v
is known.
From where contributions:
()
()
()
()
()
()
()
()
I
pg vx
pg vy div pg
I
pg vx
pg vy div pg
I pg X
xx
I
xy
I
pg
I pg y
xy
I
yy
I
pg
1
1
,
,
,
,
.
.
= -
+




= -
+










The term of the integral second breaks up into:
xx
X
yy
y
xy
X
y
v
X
v
y
v
y
v
X
·
·
·




+




+
+




It is then enough to use the formulas shown in appendix 2 for the function
FT
and to establish
the following expression, knowing that
!
!
v
v
X
y
=
=
0
:
-
+




+
+




+
+
+
+




xx
X
X
X
y
yy
y
X
y
y
xy
X
X
X
y
y
X
y
y
v
X
X
v
y
X
v
X
y
v
y
y
v
X
y
v
y
y
v
X
X
v
y
X
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Code_Aster
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Version
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
26/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
While distributing on the functions of form, we have:
I
X
y
v
X
X
X
v
y
I
y
X
v
X
y
X
v
y
I pg X
xx
X
xy
X
I
xx
y
xy
y
I
pg
I pg y
yy
X
xy
X
I
yy
y
xy
y
I
pg
2
2
,
,
,
,
= -
+




+
+








= -
+




+
+








The third integral is worth:
!
!
!
xx
X
yy
y
xy
X
y
v
X
v
y
v
y
v
X
+
+
+




In the isotropic elastic case, we have:
(
)
(
)
(
)
()
() (
)
() (
)
xx
X
y
ref.
yy
X
y
ref.
zz
X
y
ref.
xy
X
y
U
X
U
y
K T T
U
X
U
y
K T T
U
X
U
y
K T T
U
y
U
X
E
E
E
=
+
+
-
=
+
+
-
=
+
+
-
=
+




=
-
+
-
= + -
= +
1
2
2
1
2
2
3
1
2
3
1
2
1
1
1 2
1
1 2
1
with
Let us detail the calculation of
!
xx
:
!
!
!
!
!
xx
y
y
y
y
X
X
X
y
y
X
y
y
U
y
U
y
K T
U
X
U
y
K T
U
X
X
U
y
X
U
X
y
U
y
y
=




+




+
=




+




+
-
+




-
+




·
·
1
2
1
2
1
2
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Code_Aster
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
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Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
27/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
Like it was seen with [§3.2], the terms in
!
!
U
U
X
y
and
are to the first member. The same one
technique of derivation applied to
yy
zz
xy
,
and
incite to pose notation LAGUGT for
the expression
(
)
(
)
[
]
1
2






FT
,
FT
,
+
U
U
T
()
()
()
LAGUGT
LAGUGT
LAGUGT
1
2
3
1
2
2
1
2
2
=
+




+
+




=
+




+
+




=
+




+
U
X
X
U
y
X
U
X
y
U
y
y
U
X
X
U
y
X
U
X
y
U
y
y
U
X
X
U
y
X
U
X
X
X
y
y
X
y
y
X
X
X
y
y
X
y
y
X
X
X
y
()
y
X
y
y
X
X
X
y
y
X
y
y
X
y
U
y
y
U
X
y
U
y
y
U
X
X
U
y
X
+




=
+
+
+




LAGUGT 4
1
2
3
what gives:
()
()
()
()
!
!
!
LAGUGT
!
!
!
!
LAGUGT
!
!
!
!
LAGUGT
!
!
!
!
LAGUGT
xx
X
y
yy
X
y
zz
X
y
xy
X
y
U
X
U
y
kT
U
X
U
y
kT
U
X
U
y
kT
U
y
U
X
=
+
-
+
=
+
-
+
=
+
-
+
=
+




-
1
2
2
1
2
2
3
1
2
3
1
2
4
The contribution to the second member is thus:
()
[
]
()
()
()
[
]
I
kT
X
y
I
X
kT
y
I pg X
I
I
pg
I pg y
I
I
pg
3
3
1
4
4
2
,
,
,
,
LAGUGT
!
LAGUGT
LAGUGT
LAGUGT
!
=
-
+




=
+
-




v
v
v
v
·
In axisymmetric 2D
The starting expression is:



:
=
+
+
+
+




S
rr
R
zz
Z
R
rz
R
Z
R
Z
R
Z
R
v
v
v
v
v
v
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Code_Aster
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
28/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
We thus find the same formal writing as in plane 2D, increased by a term
complementary:
()
()
()
()
()
()
()
()
()
I
pg vr
pg vz
pg vr div pg
I
pg vr
pg vz div pg
I pg R
rr
I
rz
I
I
pg
I pg Z
rz
I
zz
I
pg
1
1
,
,
,
,
.
.
= -
+
+




= -
+










The second integral breaks up into:
rr
R
zz
Z
R
rz
R
Z
v
R
v
Z
v
R
v
Z
v
R
!
!
!
!
!




+




+


+
+




In appendix 2, we established the Lagrangian expressions of each one of the derivative. It is enough
to defer them here, by sorting them by type of component:
I
R
Z
v
R
R
Z
v
Z
R v
I
Z
Z
v
R
Z
R
v
Z
I pg R
rr
R
rz
R
I
rr
Z
rz
Z
I
R
R
I pg Z
zz
R
rz
R
I
zz
Z
rz
Z
I
pg
2
2
2
,
,
,
,
= -
+




+
+






+


= -
+




+
+








Weight
The third integral is worth:
!
!
!
!
rr
R
zz
Z
R
rz
R
Z
v
R
v
Z
v
R
v
Z
v
R
+
+
+
+




In the isotropic elastic case, we have:
(
)
(
)
(
)
rr
R
Z
R
ref.
zz
R
Z
R
ref.
R
Z
R
ref.
rz
R
Z
U
R
U
Z
U
R
K T T
U
R
U
Z
U
R
K T T
U
R
U
Z
U
R
K T T
U
Z
U
R
=
+
+
+
-
=
+
+
+
-
=
+
+
+
-
=
+




1
2
2
2
1
2
2
2
1
3
1
2
where
1
2
3
,
,
are worth like previously.
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Code_Aster
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
29/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
The detail of the calculation of
!
rr
give:
!
!
!
!
!
!
rr
R
R
R
R
Z
R
R
R
R
Z
Z
R
U
R
U
Z
U
R
kT
U
R
U
Z
U
R
kT
U
R
R
U
Z
R
U
R
Z
=




+




+




+
=




+




+


+
-
+




-
+
·
·
·
1
2
2
1
2
2
1
2
U
Z
Z
U
R
Z
Z
R R
+




2
By taking again the axisymmetric equivalent of LAGUGT:
()
()
()
LAGUGT
LAGUGT
LAGUGT
1
2
3
1
2
2
2
2
1
2
=
+




+
+
+




=
+
+




+
+




=
+
U
R
R
U
Z
R
U
R
Z
U
Z
Z
U
R
U
R
R
U
Z
R
U
R
U
R
Z
U
Z
Z
U
R
R
U
Z
R
R
R
Z
Z
R
Z
Z
R R
R
R
R
Z
R R
Z
R
Z
Z
R
R
R
()
Z
Z
R
Z
Z
R R
R
R
R
Z
Z
R
Z
Z
R
U
R
Z
U
Z
Z
U
R
U
R
Z
U
Z
Z
U
R
R
U
Z
R
+
+




+




=
+
+
+




1
2
3
4
1
2
LAGUGT
we have the same expression symbolic system:
()
()
()
()
!
!
!
!
LAGUGT
!
!
!
!
!
LAGUGT
!
!
!
!
!
LAGUGT
!
!
!
!
LAGUGT
rr
R
Z
R
zz
R
Z
R
R
Z
R
rz
R
Z
U
R
U
Z
U
R
kT
U
R
U
Z
U
R
kT
U
R
U
Z
U
R
kT
U
Z
U
R
=
+
+
-
+
=
+
+
-
+
=
+
+
-
+
=
+




-
1
2
2
2
1
2
2
2
1
3
1
2
3
1
2
4
The contribution to the second member is thus:
()
[
]
()
[
]
()
()
()
[
]
I
kT vr
kT vr
LAGUGT
v
Z
I
v
R
kT vz
I pg R
I
I
I
pg
I pg Z
I
I
pg
3
3
1
3
4
4
2
,
,
,
,
LAGUGT
!
LAGUGT
!
LAGUGT
LAGUGT
!
=
-
+
-


+


=
+
-




Term of the loading in pressure
[
]
I
p
p div
S
1
3
=
+
.






v
v
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Code_Aster
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
30/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
The loading in pressure is supposed to be known, therefore the tensor which expresses its gradient is calculable
easily:
[]
=






=
+
+






=
+




+
+




p
p
X
p
y
p
X
p
y
p
p
X X
p
y y
p
X X
p
y y
p
p
X X
p
y y
p
X X
p
y y
X
X
y
y
X
X
y
y
X
X
X
y
y
y
.
. v
v
v
The calculation of the term
div
S
is done as in the case of thermics. From where contributions:
() ()
() ()
()
()
]
() ()
() ()
()
()
]
I
p
X pg X pg
p
y pg y pg
p pg div
pg ui
I
p
X pg X pg
p
y pg y pg
p pg div
pg ui
I pg X
X
X
X
S
pg
I pg y
y
y
y
S
pg
1
1
,
,
,
,
=
+


+
=
+


+






In axisymmetric 2D, the gradient of
P
comprise a complementary term in
Pr
/
R
. This
component would be to multiply by the component orthoradiale field



. That one being null, there is not
no the particular contribution and we thus use formally the same expression as in 2D
Cartesian.
A4.2.4 Passage to the gradient of the stresses
Knowing the Lagrangian derivative of the field of displacement
U
and of the temperature, us
let us calculate the Lagrangian derivative of the tensor of the stresses by (cf [§4]).
[
]
(
)
(
)
(
)
!
!
!
FT
,
FT
,
!









=
+
-
+
+
1
2
1
2
U
U
U
U
Id
T
T
kT
Analytical expressions of the various components of the tensor
!
were seen in the paragraph
precedent. It is enough to apply them in postprocessing.
A4.2.5 Calculation of derived the eulérienne from the stresses
The last stage of the processing is conversion Lagrangian/eulérien for the derivative of the tensor of
stresses. It is enough to apply the formula:




= -
!
Like the vector



does not have component orthoradiale, the expression of the product
is the same one
in plane 2D or axisymmetric 2D. We have as follows:








=
-
+
I J
I J
I J
I J
X
X
y
y
,
,
,
,
!
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
31/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
6 Bibliography
[1]
P. MIALON: “Calculation of derived from a size compared to a bottom of fissure by
method théta ", EDF - Bulletin of the Management of the Studies and Search, Series C, n_3, 1998,
pp. 1-28.
[2]
I. EYMARD, A.M. DONORE: “Deterministic Study axisymmetric 2D of the tank for coupling
mechanic-reliability engineer in thermo elasticity ", Report/ratio EDF HI-74/98/001/February 0, 26 1998
[3]
V. VENTURINI: “Probabilistic Study of the tank by a coupling mechanic-reliability engineer”, Card
P1-97-04 project
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Code_Aster
®
Version
5.0
Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
R4.03.01-A
Page:
32/32
Manual of Reference
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A
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