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
Organization (S): EDF/MTI/MN
Handbook of Référence
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 traditional 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.
Handbook of Référence
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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
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 - Partie 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 - Partie 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 constraints .......................................................................................... 13
5 Conclusion ........................................................................................................................................... 13
Appendix 1 the transformation of the field ................................................................................................ 14
Appendix 2 Form ............................................................................................................................... 14
Appendix 3 Commutativité of derivations Lagrangian and temporal .................................................... 18
Appendix 4 Mise in numerical work ...................................................................................................... 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
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 starting of the rupture is ensured by coupling with
software PROBAN, with methods FORM/SORM. This evolution supposes to know them
variations of the constraints 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 constraints and
temperatures being calculated by Code_Aster, the traditional 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
constraints 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 principal 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.
Handbook of Référence
R4.01 booklet: Analyze sensitivity
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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
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, one studies a transformed field, 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: imposed temperature
2: imposed normal flow
3: convectif exchange
what gives the following variational equation:


T
C
T * +
T.T *

+ H T T * =
S T * + Q T * + H T T *
p




ext.

T







3

2
3
with:
T
Temperature
C
voluminal heat with constant pressure
p

thermal conductivity
H
coefficient of convectif exchange
S
thermal voluminal source
Q

normal heat flow entering imposed on edge 2
T *

function of test in H1 (), 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

F
will use the formula of Reynolds, after having defined the vector from for
transformation F (cf Annexe 2). represent the direction of variation of the field.
We choose functions of test T * which are independent of the parameter. In addition, one
is here in a case where Lagrangienne derivations and temporal commutate (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
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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:
5/32
2.2.1 Integral of the temporal variation

T

I =
C

*

T
1
p

T


·

dI
T
T
1
*


=

*
C

T + C

T
div


D
p


T
p

T







We suppose that voluminal heat CP 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:
·
·



T
·

T


T
C


p
= CP
C

T




+
T
p T





!
= (
T


T
CP).
+ C
T
p T

dI1
T! *
T
*

=

T
C
T

*
p
+ C
div T
p
+

(CP). T
D
T
T
T
2.2.2 Integral of thermal conductivity
I =
.

*

T T
2

dI

·

2

= T.T
* + T.T
* div
D

[
]





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 ®
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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:
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From where the result:
D I 2 =
T
T * +

!.
(
.) T
. T
*
D

- (T.). T
* - T
.(T
*.)
+ div T
. T
*
2.2.3 Integral of the convectif exchange - Partie 1
I =
hTT
3

*

3

We use this time the proposal 4 (cf Annexe 2) which establishes derivation for an integral
surface.
dI
·
3 =
hTT *
hTT * div
D

+
S

3



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

) TT * + hTdiv T
!
.
*


D
S

3

2.2.4 Integral of the internal thermal sources
I =
St *
4

dI
·
4

=
St * + St * div


D

[]





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


D

2.2.5 Integral of the boundary conditions with imposed flow
I =
Q T *
5
2
dI
·
5

=
Q
T * + Q T * div
D
S

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):
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Key:
R4.03.01-A
Page:
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We suppose that the external heat flow is independent of the parameter. Thus!Q = Q.
D I 5 = (Q
. ) T * + Q T *

div
D
S

2

2.2.6 Integral of the convectif exchange - Partie 2
I =
H T
T *
6
ext.
3

dI

·

6

= H T T * + H T T * div
D
ext.
ext.
S

3



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

.
*
*
ext.
(T
ext.) T + HT T div
D
ext.
S

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.
!

T
C
T *
*
*

p
+ T
!.T + hTT
!


=

T

3
-

-

(
T *


T
C
*
p).
T
C div
T
T
p

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

-




[. T *] div T.T *

+
(H
.)

(T - T *
*
*
ext.
) T + H
(T - T
ext.
) div T + H
S
(T
ext.) T
3
3
3
+ (S.) T * +

S div T *

+ (Q.) T * +

Q div T *
S
2
2
Border 1 of 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
T
interest, it remains to achieve the last operation:
T
!
= T - T
.
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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):
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Key:
R4.03.01-A
Page:
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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
) and!T
=
+
=!T (T)
!T
!T + -!T -
The derivative in time approaches thus by:
~
-
T
T

At the current moment we will use the approximation:
!T ~ -!T + + (1 -)!T
I
I
We will apply this technique for the principal variables of the problem: !
T, T, H, Q, S. Tous them
fields at the moment T being known, the equation discretized in time can be written:
CP T+!T * +


+
*
+ + *
I
T
! . T

+
H T! T

=

T
I


3
CP T!T

* - (1
-
*
- - *
I) T! . T

- (1-i) H T! T

T


3
+
-
+
-
-
*
-
-
T
T
T
T
C div
T

*
p
-
.

(CP)
T
T

T

+

+
-

+ 1

. *

([T
I
(I) T)] T
+
(T+ (
-
*
I) T
).[T
.]
I
+ 1 -
- (.)


1

(+
-
*
I T
+ (- I) T
). T

-
div

1

[+
-
*
I T

+ (- I) T]. T

+
1
(
+
-
*
I S
+ (- I) S
). T
+ div

1

(+
-
*
I S
+ (- I) S) T
+
1
(
+
-
*
I Q

+ (- I) Q). T
2
+
1
(
+
-
*
I Q
+ (- I) Q) divs T
2
+
With div

*
S T
3
+ (h+ + (
-
+
+
-
-
1 -
*
I
) H).(I (T - T
ext.
) + (1 - I) (T - T
ext.
)
I

T
3
+ (h+
-
+
-
*
I
+ (1 - I) H) (T
I
ext. + (1 - I
) T
ext.) .T
3
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
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Author (S):
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Key:
R4.03.01-A
Page:
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The term of flow of convectif exchange, A, takes two expressions distinct according to the dependence from H
and Text with respect to time.
If H and Text are independent of time, only the temperature is with impliciter. From where:
H

(T - T+ - (1 -) T



) div T *
ext.
I
I
S
3
If not, the whole of flow undergoes it - diagram:

*
3
[+
+
+
-
-
-
I H (ext.
T
- T) + (1-i) H (ext.
T
- T)]divs T
Note:
One will have noted the nuisance to have to treat with “Dupond and Dupont of numerical”, with
to know it - diagram and the method… Pour 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” which interests us it
more in this work, of the “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 constraints and the tensor of
deformations is then of the type:
= + K (T - Tref) Id
The field of calculation is noted, where is the real parameter of control of the variations of the field.
Handbook of Référence
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Code_Aster ®
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:
10/32
For this application, we retain only three types of behavior at the edge of:
1: border with imposed displacement
2: border with “uniform connection”
3: border with imposed external pressure
what gives the following variational equation:

(U): (v) = v
p. N

3
with:
U
field of displacement
(U)
tensor of the constraints related to displacement U
(v)
tensor of the deformations associated with displacement v
p

pressure distributed on 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 Annexe 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) =: S (v)
where S (v) is the symmetrized gradient of v.
The integral and its derivative are thus written:
I
=
U:
() S (v)

dI

·


= (U) S
: (v)
S
+ U: v div
D

[() ()]





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



= -

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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:
11/32
We have:

·

(U): S (v) =!(U): S
S
v + (U): !
(v)




=!(U): S
S
S
v + (U): [(v!) - (
T v,)]
=!(U): S
S
v - (U): F (
T (v),)
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 of
temperature:


!(U) =
!E +
!T
E
T
In the particular case of isotropic linear elasticity:
(E, T) = E + K (T - Tref) Id


=
E

= K Id
T
from where: !(U) =!E + K!T Id
It remains to express the derivative of the deformation E, starting from its expression according to
displacement U.
1
E =
T
2 [U
+ U
]
1
· T


E! =! U
+
U



2







1
1
E! = [U
! - (
FT U,)] + [U
! - (
FT U
,)]T
2
2
1
1
E! =
T
T
!
!
FT
,
FT
,
2 [U
+ U
] - 2 [(U) + (U
)]
The final expression of the derivation of the integral is then:
dI
T
S
1
T
S
=
!U +!U: v - FT U, + FT U,
:
1
v
D
2 [
] ()

2 [[
(
)
(
)]
()
+ K T
S
! Id: (v) - (U): F (S
T (v),
)
+ [(U) S
: (v)] div
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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
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Key:
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Page:
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3.2.2 Integral
surface
I =
p
3
v
3

D I 3



=
statement
statement div
·

D
S

3


+
By choice of functions v we have!v = 0. We suppose that the external pressure is
independent of the parameter. As follows:
·

statement
[p.] v

=
D I 3 = [p
.]v + p div v
D
S

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 are the symmetrized gradient of v.
1
!U +!C: v =

[
]
2
1
FT U, + FT U, T: v
2 [[
(
)
(
)]
- K!T Id: v

+ (U)

:F (
T v,)

- [(U): v] div

+ [.p] v
3

+ p divs
3

On border 1, displacement is imposed. Whatever the position of this border,
boundary condition follows the matter, which involves!U = 0.
On border 2, with uniform connection, the degrees of freedom are identical, but free: uy =
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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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
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Author (S):
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Key:
R4.03.01-A
Page:
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If we set out again of the initial variational equation:

(U): (v) =
v
p. N

3
we can transform it by using the expressions of and of: .

(U): v
=
statement

3



(U)
[
+ K (T - Tref) I] D: v = statement
3
[(U)]: v
= -

K
(T - Tref) Id: v
+
statement
3
1
Like (U) = [U
+ U
T], we find well the same expression with the first member as
2
for the transformed equation.
4
Determination of the gradient of the constraints
The following stage aims at determining the gradient of the constraints. 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]):
1
1
!(U) =
T
T
K T
2 [!U +!U] - 2 [

(
FT U
,) + F (
T U,)] +! Id
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 constraints 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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Titrate:
Sensitivity of the mechanical thermo fields to a variation of the field
Date:
19/01/00
Author (S):
G. NICOLAS
Key:
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Page:
14/32
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 F0
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

F
The field of vectors is noted.
We will use the following deduced sizes:
· field
scalar
div, voluminal divergence, and divs, surface divergence,
· tensor
.
Appendix 2 Form
We will recall here, the principal formulas useful for calculations of derivation. One will refer to [bib1] for
their demonstrations.
That is to say (, M) an unspecified field. We note:
(M) = (
,
, F (M))

Lagrangian derivative: !
=
()
(

) - 1
F
F
(


det F)
Proposal 1: (I)
= (II)
= - (III)
= div





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
X
K
K
· if
is a tensor, the same formula applied to each component of the tensor gives:
I, J
I,
!

J
I, J =
+



K
X
K
K
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 modification 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 (,)
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 +


y
X
y
!


2



=
+

X +


y
X

X

X X
y
!


2

2

2
X


=
+
X +
+
y +
y
X
X
X
X X
X y
y X



X
=


y
X



X +

X

X +

y

X y +
+
X X
y X
By applying the formula (2.i) to the first three terms of this sum, we have:
·


! X y
=
+
+
X X X X there X


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The same technique used while deriving (2.i) compared to makes it possible it to establish:
·


!

X
y


+




X
X
X X
y X


= -
!

X
y


y y
+


X y
y y



·


!
(
FT,)



= -

· if
is a scalar field, in axisymmetric 2D, its gradient in axisymmetric 2D is worth the vector:


=
er +

E
R
Z Z
The starting point is still the expression (2.i):


!
=
+
R +

R
Z 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 and
.
R
Z
From where the expression:
·


! R
Z



+

R
R
R R
Z R
= -
!


R
Z





+
Z
Z R Z
Z Z


Summary for a scalar:
·



!
K

=
-
X
I
xi
X
K
K xi


front
I
K
EC. and {X,}
y
{R
or,}
Z
· when
is a vector or a tensor, we apply the same reasoning to each one of its
components in Cartesian co-ordinates:
·


J! J
J K

=
-
X

I
X

I
X
X
K
K I


·


J L! J L,
J L, K

=
-
X

I
X

I
X
X
K
K I


with I, J, K, L {X,}
y
· 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

0

R
Z

=
R
R er + ze Z =

0
0
R

Z
Z

0
R
R
J
Derivations of the terms in are obtained as considering previously. It is necessary from now on
I
X

to apply the expression (2.i) at the end exchange R:
R
R
R
R
·






R

R

R

R
=
+


R +

R

R

Z
Z

·
1


R 1

R

R
1 R
R
1

=
+
R

R
+
R -

2 R +

R R

R R
R
R
Z
Z

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

R R
=!
1
-


2
rr
R
R
R

From where the expression
·


R
R R!
R!
0

0


R
Z
R
Z




R
R

!
0
0
= 0
0


R

R

Z
Z Z!
!Z
0

0


R
Z R
Z


R R R Z
R R R Z

+
0
+

R R
Z R
R Z
Z Z
R
-
0
-
R


0
R 2

Z R Z Z
Z R Z

+
0
+
Z
R R
Z R
Z Z
Z Z
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Proposal 3 (theorem of Reynolds)
With I =
D


,


field of R
3:


(D I
I)
= (! + div) D
D


(D I

II)
=
D +
.
Nd S


D


Proposal 4:
With J =
D S, S

surface of R3 and by noting N the normal external with S:
S

(D J
I)
= [! + divs] ds
D
S
Appendix
3
Lagrangian commutation of derivations and
temporal
Proposal:
·


!
=

T

T



Proof:
For the transformation F, we pose classically:
(M T) = (
,
,
, F (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 commutate.
!


=

=

(
, F (M), T)
T
T

T


=
(

, F (M), T)
T


·





=



T = T




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Appendix 4 Mise in numerical work
A4.1 Calcul of the variation in temperature
A4.1.1 Principe general
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 Algorithme total
More precisely, calculations of T and!
T are imbricated in the following way:
· Initialization of the field of temperatures T, and 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 and T
N
n+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)
Rock of the values of T
and T
n+1
N
! +1 in T and T
N
N
!
A4.1.3 Conditions in extreme cases 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 are
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 border 1.
A4.1.4 Détail 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 node I to the point of Gauss pg the element running in the calculation of the integrals by
formulate of Gauss, knowing that all its contributions are to be cumulated.
Term due to derivation
It is necessary to calculate the contribution of:
I = I1 + I2 + I3 + I4 + I5 + I6 + I7 + I8, with
C

I
p
*
1 =
T -
! T
T
I
*
2 = - (1 -
-
I
) T
! T

T + - T -
I
*
3 =
- div C

T

p

T

T + - T -
I
*
4 =
-



(CP).
T
T

I
*
5 =


([T+
-
I
+ (1 - I) T
)]. T

I
*
6 =


(T+
-
I
+ (1 - I) T) [T]
I
(.)

(I +T + (1-i) - T) *
7 =
-
.T
I = div

[I +T + (1
8
- I) -
T] *
T
· Calculation
of
I1
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. CP is supposed to be constant by
element. From where the contribution:
C

I
p
=
T -
1,
! (pg) ui (pg
I pg
)
T
pg

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· Calculation
of
I2
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:
ui
I
-
2 I pg = (1
,
- I) (T! )

pg, J
xj
pg

J
· Calculation
of
I3
In stationary regime, this term does not appear. CP 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:
C

I
p div
+
-
3,
= -
(pg) (T (pg) - T (pg) ui (pg
I pg
)
T
pg

· Calculation
of
I4
CP is supposed to be constant by element. Its gradient is thus null by element. From where:
I4 I pg 0
,
=
· Calculation
of
I5
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 T

+ (1-i) T
. The result is a vector of which them
components at the point of Gauss are:
ui
With (pg) =
[T+ (I) + (1 -) T


(I)] (pg
J
I
I
)
xj
I
The tensorial product contracted A
is written: (A) =

K
Aj jk
J

y

Example: (A) =
X +
Z
With
Ay
+
X
X
Z
With
X
X
X
etc, from where the formula:
ui
I5, = (
With) (pg)
(pg
I pg

K
).
xk
pg
K
In axisymmetric 2D, product A
is written:
(



With) =
R
With
+
Z

Z
With
+ A
R
R R
X

X
(



With) =
R
With
+
Z

Z
With
+ A
Z
R Z
Z
Z
(

With) =
R
With

R
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Here, component A is always null. We thus find the same expression as in
Cartesian co-ordinates 2D.
· Calculation
of
I6
is supposed to be constant in the element. +, -
T
T and div will have been calculated at the points of
Gauss. The vector
+
-
I T

+ (1-i) T
A will be noted, like previously. The product
tensorial contracted T * has on the knot slip I for component:
(
U
T *) = I (pg)
I, K
X
J K
J
J
Ex: (
U

U

y
U


T *) = I (pg) X + I (pg)
+ I (pg) Z
I, X
X
X

y

X

Z

X

from where the formula:
I
= A
*
6,
(pg) (T
I pg
K
).
I, K
pg
K
As for the preceding integral in axisymmetric 2D, component A is null.
The expression is thus the same one in Cartesian 2D or axisymmetric.
· Calculation
of
I7
is supposed to be constant in the element. Its gradient is thus null there. From where:
I7 I pg 0
,
=
· Calculation
of
I8
is supposed to be constant in the element. +, -
T
T and div will have been calculated at the points of
Gauss. The vector
+
-
I T

+ (1-i) T
A is noted, like previously. We have then:
U
I
div
8,
=
(pg) A (pg) I (pg
I pg
K
)
X
pg
K
K
Source term of energy
We calculate the two integrals:
I =
div
+
-
*
1
(if + (1 - I) S) T
I
+
-
*
2 = (S
I
+ (1 - I) S
). T
s+
S
and
are known at the points of Gauss. div was calculated at the point of Gauss. From where
contribution:
I
div
+
-
1
(pg) (S (pg) (1
,
=
+ -) S (pg) U (pg
I pg
I
I
I
) pg
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The source S being constant by element, its gradient is null. From where:
I2 I pg 0
,
=
Term of the boundary conditions of imposed flow
I
+
-
*
1 = (Q + (1 -) Q) div T
I
I
S
q+
Q
and
are known at the points of Gauss. div () is calculated at the point of Gauss.
I
+
-
1
[Q (pg) (1
,
=
+ -) Q (pg)] × div (pg) U (pg
I pg
I
I
S
I
) pg
Note:
The calculation of the J/xk 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/xk 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
R
of
,
, is multiplied by the component orthoradiale of normal N. However this
R
component is null.
I
+
-
*
2 = (Q

+ (1 -) Q
) .T
I
I
2

Q is supposed to be constant by element. Its gradient is thus null there. From where:
I2 I pg 0
,
=
Term of the boundary conditions of convectif exchange
If H and Text are independent of time, we calculate the following expression:
I = -
(1 -) H T!T * + H


(T
- T + - (1 -) T -


) div T *
I
ext.
I
I
S
3
3
What gives:
I
-
I, pg =
([1 - I) H T! (pg)] +
(HT - T+
-
ext.
I
(pg) - (1 - I) T (pg) × divs (pg) ui (pg) pg
If H or Text is independent of time, it is then necessary to calculate:
I = - (1
- - *
I
) H T! T +
3

+

*

3
[h+
+
+
-
-
-
I
(T - T
ext.
) + (1 - I) H (T - T
ext.
)]div T
S
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What gives:
I
-
-
I, pg =
([1 - I) H T (pg)]
+ [h+ +
+
-
-
-
I
(T - T
ext.
(pg) + (1 - I) H (T - T
ext.
(pg)] ×
div (
pg) ui (pg) pg
Same remarks that to the preceding paragraph apply to the calculation of the quantities J/xk.
The two integrals utilizing H and Text are null, insofar as H and Text are
presumedly constant by element.
A4.2 Calcul of the gradient of displacements and the constraints
A4.2.1 Principe general
As for thermics, the calculation of the gradient of displacement is inserted in the calculation of
displacement, i.e. operator MECA_STATIQUE.

Then the calculation of! and will be done in postprocessing, in commands CALC_ELEM and
CALC_NO.
A4.2.2 Conditions in extreme cases 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 Détail 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 node I and the point of Gauss pg for the current element.
Term due to derivation
It is necessary to calculate the contribution of:
1
I =
FT U,
FT U,
:
2 [[
() + () T] S
v

- K T
S
! Id:

v

(U)

: F (S
T v,)

-
[(U)
S
: v] div
where sv is the symmetrized gradient of v is 1 + T
2 (v
v).
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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One will remember (cf [§3.2.1]) that this integral results from derivation from:

(U): sv

and that we broke up it into:
·
[


(U): sv] + (U):

sv +

!(U): S
div

v





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:
v
vy
v
v

: sv =
X
X
y
xx
+ yy
+ xy
+

X

y

y

X
The divergence of the field is a data, calculated at the points of Gauss.
The tensor of the constraints (U) is known at the points of Gauss. Tensor v is known.
From where contributions:

v
v

I
I
I
1 I, pg, X = - xx (pg)
+ xy (pg)
div
() (pg).
X

y
pg



v
v

I
I
I
1 I, pg, y = - xy (pg)
+ yy (pg)
div
() (pg).
X

y
pg


The term of the integral second breaks up into:
·
·
·






v
v

v
X
y
v


X
y
xx
+



yy
+
+
X

xy
y
y
X










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

X
y
xx
+


X X
y X
vy
v
+
X
y
y
yy
+

X y
y y
v
v
v
X X
v

X
y
y


+
X
y
y
xy
+
+
+

X y
y y
X X
y X
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While distributing on the functions of form, we have:


v


y
y v
I
X
X
I
I
2 I, pg, X = - xx
+




X
xy
y
+
+

X
xx
X
xy

X

y
pg







v

y
y v
I
X
X
I
I
2 I, pg, y = - yy
+




y
xy

X
+
+
X
yy

y
xy

X

y
pg





The third integral is worth:
v
v

v
X
y
v

!
X
y
xx
+!yy
+!
+


X

xy
y
y
X
In the isotropic elastic case, we have:
U
U

X
y
xx = 1
+ 2
+ K (T - ref.
T)
X
y
U
U

X
y
yy = 2
+ 1
+ K (T - ref.
T)
X
y
U
U

X
y
zz = 2
+ 2
+ K (T - ref.
T)
X
y
1
U
U



X
y
xy =
3
+

2
y
X
E (1 -)
with
1 = (1+) (1 - 2)
E
2 = (1+) (1 - 2)
E
3 = 1+
Let us detail the calculation of!
xx:
·
·
uy
uy
!

= 1
+ 2
+ K!
xx

T
y
y








!uy
!uy
=

1
+

2
+ K!

T
X
y
U

U
U

X X
U

X
y
y


X
y
y
- 1
+


- 2
+

X X
y X
X y
y y




Handbook of Référence
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A

Code_Aster ®
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:
27/32
How it was seen with [§3.2], the terms in!U and!U
X
y are to the first member. The same one
technique of derivation applied to yy, zz and xy encourages to pose notation LAGUGT for
1
T
the expression
FT U, + FT U,
2 [
(
)
(
)]
ux X U
U
X
y
y
U
LAGUG (
T)
1 =

X
y
y
1
+
+ 2
+

X X
y X
X y
y y
ux X U
U
X
y
y
U
LAGUG (
T 2) =

X
y
y
2
+
+ 1
+

X X
y X
X y
y y
U


U

X X
U
U
LAGUG (
T)
3 =

X
y
y
X
y
y
2
+
+ 2
+

X X
y X
X
y
y y


1





LAGUG (
T 4)
U

U
U

X
X
ux
y
y
X
y
y
=

+
+
+

2
3
X
y
y

y

X
X
y

X



what gives:
!U
U
X
!
!

y
= 1
+ 2
- LAGU
xx
(
WP)
1 +!
kT
X
y
!U
U
X
!
!

y
= 2
+ 1
- LAGU
yy
(
WP 2) +!
kT
X
y
!U
U
X
!
!

y
= 2
+ 2
- LAGU
zz
(
WP)
3 +!
kT
X
y
1
!U
U
X
!
!

y
= 3
+
- LAGU
xy
(
WP 4)
2
y
X


The contribution to the second member is thus:

v
v
I
I
I
3 I, pg, X = [LAGUG

(
T)
1 - kT!]
+ LAGUG (
T 4)

X

y pg



v
v
I
I
I
3 I, pg, y = LAGUG

(
T 4)
+ [LAGUG (
T 2) - kT!]

X
y pg


· In axisymmetric 2D
The starting expression is:
v

v
v
v
v


S
R
Z
R
R
Z
: v = rr
+ zz
+
+ rz
+
R
Z

R

Z
R

Handbook of Référence
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A

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:
28/32
We thus find the same formal writing as in plane 2D, increased by a term
complementary:

v
v
v
I
I
I
I
1 I, pg R
, = - rr (pg)
+ rz (pg)
+ (pg)
div () (pg).


R
Z

R
pg



v

v
I
I
I
1 I, pg, Z = - rz (pg)
+ zz (pg)
div () (pg).


R

Z
pg

The second integral breaks up into:
R
v!
v!Z
R
v!
vr! v

!Z
rr
zz




R +
rz
Z +

R +
+

Z 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:


v


v
I
R
R
I
Z
Z
I
2 I, pg R
, = - rr
+



R
rz
Z

+
+
R
rr


R
rz
Z
Z

R

+


v

Weight
R 2 R


v


v
I
R
R
I
Z
Z
I
2 I, pg, Z = - zz
+




Z
rz

Z

+
+
R
zz


Z
rz

R Z
pg

The third integral is worth:
R
v
vz
vr
vr v


!

Z
rr
+!zz
+!
+!
+

R

rz
Z
R

Z R
In the isotropic elastic case, we have:
ur
U
U

Z
R
rr = 1
+ 2
+ 2
+ K T - T
R
Z
R
(ref.)


ur
U
U

Z
R
zz = 2
+ 1
+ 2
+ K T - T
R
Z
R
(ref.)


ur
U
U

Z
R
= 2
+ 2
+ 1
+ K T - T
R
Z
R
(ref.)


1
R
U
U

Z
rz =
3
+
2

Z R
where 1, 2, 3 is worth like previously.
Handbook of Référence
R4.01 booklet: Analyze sensitivity
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Code_Aster ®
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:
29/32
The detail of the calculation of!
rr gives:
·
·
·
ur
U
U
!

R
R
= 1
+ 2
+ 2
+!
rr

kT
R

Z

R





!ur
!uz
!ur
= 1
2

!

kT
R +

Z + 2 R +
ur R R
U Z
uz R uz
U
Z
R
-
R
1
2
+
+

+
R R
Z R -

R Z Z Z
R 2
By taking again the axisymmetric equivalent of LAGUGT:
ur R ur Z
uz R uz
U
Z
rr
LAGUG (
T)
1 = 1
+
2
2

R R
Z R +
+
+

R Z Z Z
R


ur R ur
U
Z
rr
uz R uz Z
LAGUG (
T 2) = 2
+
+

2
1

R R
Z R
R

+
+

R Z
Z Z
ur R ur
U
U
U
LAGUG (
T)
3 =
Z
Z
R
Z
Z
R R
2
+
+
+


R R
Z R
R

Z

Z
Z
+
1 r2
LAGUG (
1
T 4)
U
R R

U
R Z
U
Z R

U
Z Z
=
3
+
+
+

2

R Z
Z
Z
R
R

Z

R

we have the same expression symbolic system:
!ur
!uz
!U
!

R
= 1
+ 2
+ 2
- LAGUG
rr
(
T)
1 +!
kT
R
Z
R
!ur
!uz
!U
!

R
= 2
+ 1
+ 2
- LAGUG
zz
(
T 2) +!
kT
R
Z
R
!ur
!uz
!U
!

R
= 2
+ 2
+ 1
- LAGUG

(
T)
3 +!
kT
R
Z
R
1
!ur!U
!

Z
= 3
+
LAGUG
rz
(
T 4)
2

Z
R -
The contribution to the second member is thus:

v
v
I
I
I
3 I, pg R
, = [LAGUG (
T)
1 - kT!]
+ [LAGUG (
T)
3 - kT!]


R
R
v
+

LAGUGT (4) I
Z
pg


v

v
I
I
I
3 I, pg, Z = LAGUG (
T 4)
+ [LAGUG (
T 2) - kT!]



R

Z
pg

Term of the loading in pressure
I1 = [p.] v + pdivs v
3

Handbook of Référence
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A

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:
30/32
The loading in pressure is supposed to be known, therefore the tensor which expresses its gradient is calculable
easily:
p
p
X
X


X
y
p =

p
p
y
y

X y
p
p
X
X


X
+
y

X
y
p. =

p
p
y
y


X
+
y

X
y


[
p
p


X
X

p
p
y
y

p
. ] v =
X
+
y
v X +
X
+
y
v
X
y

X
y
y
The calculation of the divs term is done as in the case of thermics. From where contributions:
p
p

I
X
X
1 I, pg, X =
(pg) X
(pg) +
(pg) y
(pg) + px (pg) divs (
pg)]ui
X

y
pg


p

p
I
y
y
1 I, pg, y =
(pg) X
(pg) +
(pg) y
(pg) + py (pg) divs (
pg) ui
X
y

] pg

In axisymmetric 2D, the gradient of P comprises a complementary term out of Pr/R. Cette
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 with the gradient of the constraints
Knowing the Lagrangian derivative of the field of displacement U and temperature, us
let us calculate the Lagrangian derivative of the tensor of the constraints by (cf [§4]).
1
1
! = U
C
U
U
T
kT Id
2 [! +! ] -
2 (
(
FT,) + F (
T,)) +!
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 Calcul of derived the eulérienne from the constraints
The last stage of the processing is conversion Lagrangian/eulérien for the derivative of the tensor of
constraints. 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
X
y

=
-
+
X
y
I, J
Handbook of Référence
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A

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
6 Bibliography
[1]
P. MIALON: “Calculation of derived from a size compared to a bottom of fissure by
method théta ", EDF - Bulletin of Direction of Etudes and Recherches, Série 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 ", Rapport EDF HI-74/98/001/February 0, 26 1998
[3]
V. VENTURINI: “Probabilistic Study of the tank by a coupling mechanic-reliability engineer”, Fiche
P1-97-04 project
Handbook of Référence
R4.01 booklet: Analyze sensitivity
HI-72/99/011 - Ind A

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
Intentionally white left page.
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R4.01 booklet: Analyze sensitivity
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