Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
1/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R4.03 booklet: Analyze sensitivity
Document: R4.03.07
Postprocessing of sensitivity
Summary:
Classically, the digital simulations provide the response of a structure to a stress. One
seek here to determine in addition to this answer the tendency of the response to an amendment of parameters
of input of simulation (material, loading…). This tendency is obtained by calculating the derivative of
response compared to a parameter
p
given.
In this document, one places oneself in the case of linear elasticity and one supposes that the basic variable of
calculation (displacement
U
) was calculated thus that its derivative
p
U
/
. After having given some indications
on this calculation of
U
and of
p
U
/
(for more detail to see [R4.03.03]), one will be interested in derived from
variables which result from this (strains and stresses) like with derived from the rate of refund from energy G.
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
2/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
Count
matters
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
3/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
1
Recall of the calculation of sensitivity for the basic variable
(displacement)
One is interested in this paragraph in the sensitivity of displacement
U
compared to a parameter
p
given: loading (imposed displacement or forces imposed) or characteristic material (module
of Young, Poisson's ratio, or characteristics anisotropic).
1.1 Problem
direct
In the case of elasticity, the direct problem is written in a simplified way (cf [bib1]):
L
U
R
=
)
(
with:
[
]
=
D
W
U
With
U
R
K
K
)
(
:
)
(
)
(
where:
With
is the matrix of elasticity
K
W
is related to form of the kth degree of freedom of the modelized structure
That is written in matric form:
L
KU
=
1.2 Problem
derived
The derivation of the matric writing above gives:
(
)
(
)
p
L
p
U
K
U
p
K
=
+
/
/
/
From where:
(
)
(
)
U
p
K
p
L
p
U
K
-
=
/
/
/
If
p
is of type loading, one a:
0
/
=
p
K
and the second member is reduced to
p
L
/
.
If
p
is of type material, one has
0
/
=
p
L
and the second member is worth:
(
)
U
p
K
-
/
. One has
in particular in a more precise way:
-
-
=
D
W
U
p
With
U
p
K
K
K
)
(
:
)
(
The term
p
With
/
is calculated in routine DMATMC.
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
4/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
2
Calculation of derived from the strains and the stresses
One considers in this paragraph derivation compared to a parameter
p
given,
p
being able to be
a loading or a characteristic of material. It is also supposed that the field of
displacement
U
and its derivative
p
U
/
, all the two resulting one from a mechanical calculation (via
MECA_STATIQUE
), are known.
2.1
Derived from the deformations
In this case, the step is always the same one, whatever the nature of
p
. Indeed, in
the assumption of the small disturbances, one a:
(
)
Drunk
U
U
T
=
+
=
2
/
From where while deriving compared to
p
:
p
U
B
p
U
p
U
p
T
=
+
=
)
(
2
1
It is thus enough in the routine to the operator
CALC_ELEM
to provide in input to the subroutine
calculating the deformations the field
p
U
/
in the place of the field
U
.
2.2
Derived from the stresses if p does not depend on material
While noting
With
the tensor of elasticity, one a:
ABu
=
Knowing that
With
and
B
do not depend on
p
, one a:
(
)
p
U
AB
p
=
/
/
As in the preceding paragraph, in the operator
CALC_ELEM
, one provides in input to
subroutine calculating the stresses the field
p
U
/
in the place of the field
U
.
2.3
Derived from the stresses if p depends on material
B
not depending on
p
, one a:
(
)
(
)
p
U
AB
Drunk
p
With
p
+
=
/
/
/
The term
(
)
p
U
AB
/
is in the same way calculated that in the paragraph [§2.2].
For the other term, it is necessary to calculate
p
With
/
, i.e. to re-use the routine DMATMC which had been
developed for the calculation of
p
U
/
if
p
depends on the material (cf [§1.2]).
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
5/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
3
Calculation of derived from G
3.1
Recall of the formulation of G
One considers a fissured elastic solid occupying the field
space
R
2
or
R
3
. That is to say:
·
U
the field of displacement,
·
T
the field of temperature,
·
F
the field of voluminal forces applied to
,
·
G
the field of surface forces applied to a part
S
of
,
·
U
the field of displacements imposed on a part
D
S
of
.
S
S
D
F
G
Appear fissured elastic Solid 3.1-a:
To simplify, one places oneself in linear elasticity and small deformations, but this approach
generalize without sorrow with plasticity, the great deformations….
One indicates by:
·
the tensor of the deformations,
·
0
the tensor of the initial deformations,
·
HT
the tensor of the deformations of thermal origin,
·
the tensor of the stresses,
·
0
the tensor of the initial stresses,
·
(
)
T
,
,
,
0
0
density of free energy.
Then the rate of refund of energy associated with a virtual field of propagation with the fissure
is written:
()
-
-
+
+
+
+
-
-
-
-
+
-
-
=
D
U
N
D
N
N
U
G
U
G
D
U
F
U
F
D
D
T
T
D
U
K
K
I
J
S
ij
K
K
K
K
I
I
I
K
S
K
I
I
K
K
I
K
K
I
I
K
K
ij
ij
HT
ij
ij
K
K
ij
ij
ij
K
K
K
K
J
p
p
I
ij
D
,
,
,
,
,
0
,
0
0
,
0
,
,
,
,
]
[
)
(
]
)
2
1
(
)
2
1
[(
)
(
G
The last term associated with a boundary condition with Dirichlet is not established in
Code_Aster. One will thus not take it into account.
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
6/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
3.2
Derived from G
3.2.1 Derived from G compared to the Young modulus
One supposes known the derivative of displacements, the strains and the stresses by report/ratio
with E (see [§1] and [§2]).
G is in fact the sum of 5 terms:
FSUR
FVOL
EPSINI
THER
CLA
T
T
T
T
T
G
+
+
+
+
=
)
(
5 terms being given in the order of [§3.1].
of
dT
of
dT
of
dT
of
dT
of
dT
of
dG
FSUR
FVOL
EPSINI
THER
CLA
+
+
+
+
=
3.2.1.1 Derived from the conventional term
-
=
D
U
T
K
K
J
p
p
I
ij
CLA
)
(
,
,
,
-
+
=
D
of
D
of
U
of
D
of
dT
K
K
J
p
p
I
ij
J
p
p
I
ij
CLA
)
(
,
,
,
,
,
with in 3D and axi:
()
(
)
(
)
(
)
ref.
kk
ref.
HT
HT
ij
ij
kk
T
T
K
T
T
K
T
U
-
-
-
=
-
+
=
2
2
2
9
3
2
,
µ
with
where
(
) (
)
µ
+
=
-
+
=
-
=
1
2
;
2
1
1
;
2
1
3
E
E
E
K
it comes:
))
(
2
3
(
2
1
)
(
2
2
2
ref.
kk
kk
ref.
ij
ij
ij
ij
kk
kk
kk
T
T
of
D
E
T
T
of
D
E
of
D
E
of
D
-
-
+
-
-
-
+
+
+
=
µ
µ
in plane deformations:
()
(
)
(
) (
)
(
)
(
) (
)
(
)
HT
xy
yy
xx
yy
xx
E
E
E
T
-
+
+
-
+
+
+
-
+
-
=
2
2
2
1
2
1
1
2
1
1
2
1
,
that is to say:
))
(
2
3
(
2
1
)
(
1
1
1
2
)
(
)
2
1
) (
1
(
))
(
2
1
(
)
2
1
) (
1
(
)
1
(
2
2
2
ref.
kk
kk
ref.
xy
xy
xy
yy
xx
xx
yy
yy
xx
yy
xx
yy
yy
xx
xx
T
T
of
D
E
T
T
of
D
E
E
of
D
of
D
E
E
of
D
of
D
E
of
D
-
-
+
-
-
-
+
+
+
+
+
+
-
+
+
+
+
+
-
+
-
=
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
7/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
in plane stresses:
() ()
(
)
(
)
(
)
HT
xy
yy
xx
yy
xx
E
E
E
T
-
+
+
-
+
+
-
=
2
2
2
2
2
1
1
1
2
,
))
(
2
3
(
2
1
)
(
1
1
1
2
)
(
)
1
(
))
(
2
1
(
)
1
(
2
2
2
2
2
ref.
kk
kk
ref.
xy
xy
xy
yy
xx
xx
yy
yy
xx
yy
xx
yy
yy
xx
xx
T
T
of
D
E
T
T
of
D
E
E
of
D
of
D
E
E
of
D
of
D
E
of
D
-
-
+
-
-
-
+
+
+
+
+
+
-
+
+
+
+
-
=
:
that is to say
3.2.1.2 Derived from the thermal term
-
=
D
T
T
T
K
K
THER
,
()
(
)
()
(
)
(
)
()
(
)
(
)
(
)
ref.
kk
ref.
ref.
kk
T
T
T
T
dT
T
D
T
K
T
T
dT
T
dK
T
U
T
-
-
-
+
-
-
-
=
3
)
(
3
3
2
1
,
-
=
D
T
T
of
D
of
dT
K
K
THER
,
)
(
in 3D and axi:
)
)
2
1
(
)
1
(
)
2
1
(
)
2
1
) (
1
(
(
2
)
2
1
(
)
1
(
)
2
1
(
)
(
2
2
2
2
2
2
2
dT
D
E
dT
of
of
D
dT
D
T
of
D
kk
kk
kk
-
+
+
+
-
+
+
-
+
+
=
)
1
(
1
1
)
1
(
2
1
2
dT
D
E
dT
of
of
D
dT
D
ij
ij
ij
ij
+
-
+
+
+
-
)
)
2
1
2
(
2
1
2
(
2
1
)
(
)
))(
(
(
2
1
1
of
D
dT
D
E
dT
of
dT
D
T
T
of
D
E
T
T
dT
D
kk
kk
ref.
kk
kk
ref.
-
+
+
-
-
-
-
+
-
+
-
-
)
)
(
2
1
)
(
) (
(
2
1
3
dT
D
T
T
dT
D
T
T
T
T
ref.
ref.
ref.
-
-
+
-
+
-
-
+
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
8/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
in plane deformations:
)
)
2
1
(
)
1
(
)
2
(
2
)
2
1
) (
1
(
1
) (
(
)
(
)
2
1
(
)
1
(
)
2
(
)
(
2
2
2
2
2
2
dT
D
E
dT
of
of
D
of
D
dT
D
T
of
D
YY
YY
XX
XX
YY
XX
-
+
-
+
-
+
-
+
+
+
-
+
-
=
)
)
2
1
(
)
1
(
)
2
1
(
)
2
1
) (
1
(
) (
(
)
2
1
(
)
1
(
2
1
2
2
2
2
2
2
dT
D
E
dT
of
of
D
of
D
dT
D
XX
YY
YY
XX
YY
XX
-
+
+
+
-
+
+
+
-
+
+
)
)
1
(
)
1
(
1
(
2
)
1
(
1
2
2
2
dT
D
E
dT
of
of
D
dT
D
XY
XY
XY
+
-
+
+
+
-
)
)
2
1
2
(
2
1
2
(
2
1
)
(
)
))(
(
(
2
1
1
of
D
dT
D
E
dT
of
dT
D
T
T
of
D
E
T
T
dT
D
kk
kk
ref.
kk
kk
ref.
-
+
+
-
-
-
-
+
-
+
-
-
)
)
(
2
1
)
(
) (
(
2
1
3
dT
D
T
T
dT
D
T
T
T
T
ref.
ref.
ref.
-
-
+
-
+
-
-
+
in plane stresses:
)
)
1
(
2
)
1
(
1
) (
(
)
(
)
1
(
)
(
2
2
2
2
2
2
2
dT
D
E
dT
of
of
D
of
D
dT
D
T
of
D
YY
YY
XX
XX
YY
XX
-
+
-
+
+
+
-
=
)
)
1
(
)
1
(
)
1
(
) (
(
)
1
(
1
2
2
2
2
2
2
2
dT
D
E
dT
of
of
D
of
D
dT
D
XX
YY
YY
XX
YY
XX
-
+
+
-
+
+
-
+
)
)
1
(
)
1
(
1
(
2
)
1
(
1
2
2
2
dT
D
E
dT
of
of
D
dT
D
XY
XY
XY
+
-
+
+
+
-
)
)
2
1
2
(
2
1
2
(
2
1
)
(
)
))(
(
(
2
1
1
of
D
dT
D
E
dT
of
dT
D
T
T
of
D
E
T
T
dT
D
kk
kk
ref.
kk
kk
ref.
-
+
+
-
-
-
-
+
-
+
-
-
)
)
(
2
1
)
(
) (
(
2
1
3
dT
D
T
T
dT
D
T
T
T
T
ref.
ref.
ref.
-
-
+
-
+
-
-
+
3.2.1.3 Derived from the deformations term and initial stresses
-
-
-
-
=
D
T
K
K
ij
ij
HT
ij
ij
K
K
ij
ij
ij
EPSINI
)]
)
2
1
(
)
2
1
[(
0,
0
0,
0
-
=
D
of
D
of
D
of
dT
K
K
ij
ij
K
ij
ij
EPSINI
)
(
0
,
0
,
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
9/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
3.2.1.4 Derived from the term forces voluminal
+
=
D
U
F
U
F
T
I
K
K
I
K
K
I
I
FVOL
)
(
,
,
+
=
D
of
F
of
F
of
dT
I
K
K
I
K
K
I
I
FVOL
)
(
,
,
3.2.1.5 Derived from the term forces surface
-
+
=
D
N
N
U
G
U
G
T
K
K
K
K
I
I
I
K
S
K
I
FSUR
]
[
,
,
-
+
=
D
N
N
of
G
of
G
of
dT
K
K
K
K
I
I
I
K
S
K
I
FSUR
]
[
,
,
3.2.2 Derived from G compared to the loading
The significant parameter can be one (or several) component of forces
I
F
voluminal, surface or
nodal, and (or) one (or several) pressure on an edge, which returns to a surface force.
In all the cases one can write:
PS
F
F
G
PS
G
icha
I
ncha
icha
N
I
icha
I
=
=
=
1
dim
1
additivity coming owing to the fact that contributions from the loadings from
G cumulate
with
1
=
PS
F
icha
I
if the significant parameter intervenes in component I of
loading icha
0
=
PS
F
icha
I
if not
Example:
The user defines a significant parameter being worth 1. This significant parameter is used to define a force
voluminal
Z
F
, a surface force of components
)
,
(
Y
X
F
F
and a pressure
p
.
then
p
G
F
G
F
G
F
G
PS
G
Y
X
Z
+
+
+
=
In the same way that for the derivative of G compared to E, one derives term in the long term.
The derivative of the terms conventional and thermal includes/understands less terms because the coefficients of Lamé
and K do not depend on the loading (whereas they depend on E).
On the other hand the voluminal terms surface forces and forces comprise a term moreover if it
significant parameter intervenes in the corresponding loading.
To simplify, one will note
F
the component of the loading by report/ratio to which one derives.
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
10/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
3.2.2.1 Derived from the conventional term
-
=
D
U
T
K
K
J
p
p
I
ij
CLA
)
(
,
,
,
-
+
=
D
df
D
df
U
df
D
df
dT
K
K
J
p
p
I
ij
J
p
p
I
ij
CLA
)
(
,
,
,
,
,
with in 3D and axi:
()
(
)
(
)
(
)
2
2
2
2
9
3
2
,
ref.
kk
ref.
ij
ij
kk
T
T
K
T
T
K
T
U
-
+
-
-
+
=
µ
where
(
) (
)
µ
+
=
-
+
=
-
=
1
2
;
2
1
1
;
2
1
3
E
E
E
K
it comes:
(
)
df
D
T
T
K
df
D
df
D
df
D
kk
ref.
ij
ij
kk
kk
µ
-
-
+
=
3
2
in plane deformations:
()
(
)
(
) (
)
(
)
(
) (
)
(
)
HT
xy
yy
xx
yy
xx
E
E
E
T
-
+
+
-
+
+
+
-
+
-
=
2
2
2
1
2
1
1
2
1
1
2
1
,
df
D
E
T
T
df
D
E
df
D
df
D
E
df
D
df
D
E
df
D
kk
ref.
xy
xy
xx
yy
yy
xx
yy
yy
xx
xx
2
1
)
(
1
2
)
(
)
2
1
) (
1
(
)
(
)
2
1
) (
1
(
)
1
(
-
-
-
+
+
+
-
+
+
+
-
+
-
=
that is to say
in plane stresses:
() ()
(
)
(
)
(
)
HT
xy
yy
xx
yy
xx
E
E
E
T
-
+
+
-
+
+
-
=
2
2
2
2
2
1
1
1
2
,
df
D
E
T
T
df
D
E
df
D
df
D
E
df
D
df
D
E
df
D
kk
ref.
xy
xy
xx
yy
yy
xx
yy
yy
xx
xx
2
1
)
(
1
2
)
(
)
1
(
)
(
)
1
(
2
2
-
-
-
+
+
+
-
+
+
-
=
that is to say
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
11/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
3.2.2.2 Derived from the thermal term
-
=
D
T
T
T
K
K
THER
,
()
(
)
()
(
)
(
)
()
(
)
(
)
(
)
T
U T
dK T
dT
T T
K
D T
dT
T T
T T
kk
ref.
ref.
kk
ref.
,
=
-
-
-
+
-
-
-
1
2
3
3
3
-
=
D
T
T
df
D
df
dT
K
K
THER
,
)
(
in 3D and axi:
)
)
2
1
(
)
1
(
)
2
1
(
)
2
1
) (
1
(
(
)
(
2
2
2
dT
D
E
dT
of
df
D
T
df
D
kk
kk
-
+
+
+
-
+
=
)
1
(
1
1
dT
D
E
dT
of
df
D
ij
ij
+
-
+
+
)
)
2
1
2
(
2
1
2
(
2
1
)
(
)
))(
(
(
2
1
1
df
D
dT
D
E
dT
of
dT
D
T
T
df
D
E
T
T
dT
D
kk
kk
ref.
kk
kk
ref.
-
+
+
-
-
-
-
+
-
+
-
-
)
)
(
2
1
)
(
) (
(
2
1
3
dT
D
T
T
dT
D
T
T
T
T
ref.
ref.
ref.
-
-
+
-
+
-
-
+
in plane deformations:
)
)
2
1
(
)
1
(
)
2
(
2
)
2
1
) (
1
(
1
) (
(
)
(
2
2
dT
D
E
dT
of
df
D
df
D
T
df
D
YY
YY
XX
XX
-
+
-
+
-
+
-
+
=
)
)
2
1
(
)
1
(
)
2
1
(
)
2
1
) (
1
(
) (
(
2
2
2
dT
D
E
dT
of
df
D
df
D
XX
YY
YY
XX
-
+
+
+
-
+
+
+
)
)
1
(
)
1
(
1
(
2
2
dT
D
E
dT
of
df
D
XY
XY
+
-
+
+
)
)
2
1
2
(
2
1
2
(
2
1
)
(
)
))(
(
(
2
1
1
df
D
dT
D
E
dT
of
dT
D
T
T
df
D
E
T
T
dT
D
kk
kk
ref.
kk
kk
ref.
-
+
+
-
-
-
-
+
-
+
-
-
)
)
(
2
1
)
(
) (
(
2
1
3
dT
D
T
T
dT
D
T
T
T
T
ref.
ref.
ref.
-
-
+
-
+
-
-
+
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
12/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
in plane stresses:
)
)
1
(
2
)
1
(
1
) (
(
)
(
2
2
2
dT
D
E
dT
of
df
D
df
D
T
df
D
YY
YY
XX
XX
-
+
-
+
=
)
)
1
(
)
1
(
)
1
(
) (
(
2
2
2
2
dT
D
E
dT
of
df
D
df
D
XX
YY
YY
XX
-
+
+
-
+
+
)
)
1
(
)
1
(
1
(
2
2
dT
D
E
dT
of
df
D
XY
XY
+
-
+
+
)
)
2
1
2
(
2
1
2
(
2
1
)
(
)
))(
(
(
2
1
1
df
D
dT
D
E
dT
of
dT
D
T
T
df
D
E
T
T
dT
D
kk
kk
ref.
kk
kk
ref.
-
+
+
-
-
-
-
+
-
+
-
-
)
)
(
2
1
)
(
) (
(
2
1
3
dT
D
T
T
dT
D
T
T
T
T
ref.
ref.
ref.
-
-
+
-
+
-
-
+
3.2.2.3 Derived from the deformations term and initial stresses
-
-
-
-
=
D
T
K
K
ij
ij
HT
ij
ij
K
K
ij
ij
ij
EPSINI
)]
)
2
1
(
)
2
1
[(
0
,
0
0
,
0
-
=
D
df
D
df
D
df
dT
K
K
ij
ij
K
ij
ij
EPSINI
)
(
0
,
0
,
3.2.2.4 Derived from the term forces voluminal
+
=
D
U
F
U
F
T
I
K
K
I
K
K
I
I
FVOL
)
(
,
,
if
L
F
F
=
is a voluminal component of force:
+
+
=
D
df
F
U
df
F
df
dT
L
I
K
K
I
K
K
L
L
I
I
L
FVOL
)
)
((
,
,
if not:
+
=
D
df
F
df
F
df
dT
I
K
K
I
K
K
I
I
FVOL
)
(
,
,
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
13/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
3.2.2.5 Derived from the term forces surface
-
+
=
D
N
N
U
G
U
G
T
K
K
K
K
I
I
I
K
S
K
I
FSUR
]
[
,
,
if
L
G
F
=
is a surface component of force:
-
+
+
=
D
N
N
U
dg
G
dg
G
dg
dT
K
K
K
K
L
L
I
I
L
I
K
S
K
I
L
FSUR
]
)
(
[
,
,
if not:
-
+
=
D
N
N
df
G
df
G
df
dT
K
K
K
K
I
I
I
K
S
K
I
FSUR
]
[
,
,
Code_Aster
®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES
Key
:
R4.03.07-B
Page
:
14/14
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
4 Bibliography
[1]
TARDIEU NR.: Calculation of sensitivity in mechanics. Application to Code_Aster. note EDF-DER
HI-75/01/016-Index A, 2001