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Organization (S): EDF-R & D/AMA
Handbook of Référence
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 a modification 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 U
/p
. After having given some indications
on this calculation of U and U
/p
(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.
Handbook of Référence
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Count
matters
1 Recall of the calculation of sensitivity for the basic variable (displacement) ............................................ 3
1.1 Direct problem ................................................................................................................................ 3
1.2 Derived problem ............................................................................................................................... 3
2 Calculation of derived from the strains and the stresses ..................................................................... 4
2.1 Derived from the deformations ................................................................................................................ 4
2.2 Derived from the constraints if p does not depend on the material .......................................... 4
2.3 Derived from the constraints if p depends on the material ...................................................... 4
3 Calculation of derived from G ....................................................................................................................... 5
3.1 Recall of the formulation of G ......................................................................................................... 5
3.2 Derived from G .................................................................................................................................. 6
3.2.1 Derived from G compared to the Young modulus ....................................................................... 6
3.2.1.1 Derived from the traditional term ....................................................................................... 6
3.2.1.2 Derived from the thermal term ...................................................................................... 7
3.2.1.3 Derived from the deformations term and initial constraints .............................................. 8
3.2.1.4 Derived from the term forces voluminal ............................................................................. 9
3.2.1.5 Derived from the term forces surface ............................................................................. 9
3.2.2 Derived from G compared to the loading ............................................................................. 9
3.2.2.1 Derived from the traditional term ..................................................................................... 10
3.2.2.2 Derived from the thermal term .................................................................................... 11
3.2.2.3 Derived from the deformations term and initial constraints ............................................ 12
3.2.2.4 Derived from the term forces voluminal ........................................................................... 12
3.2.2.5 Derived from the term surface ........................................................................... 13 force
4 Bibliography ........................................................................................................................................ 14
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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]):
R U
() = L
with:
[R U ()] =
K
With U (): (wk)
D
where:
A is the matrix of elasticity
K
W is related to form of the kth degree of freedom of the modelled structure
That is written in matric form:
KU = L
1.2 Problem
derived
The derivation of the matric writing above gives:
(K
/p
) U + K (U
/p
) = L
/p
From where:
K (U
/p
) = L
/p
- (K
/p
) U
If p is of loading type, one a: K/p = 0 and the second member is reduced to L
/p
.
If p is of material type, there is L/p = 0 and the second member is worth: - (K
/p
) U. One has
in particular in a more precise way:
K
With
-
U = -
U
(): (W) D
K
p
p
K
Term A
/p
is calculated in routine DMATMC.
Handbook of Référence
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2
Calculation of derived from the strains and the stresses
One considers in this paragraph derivation compared to a parameter p given, p which can be
a loading or a characteristic of material. It is also supposed that the field of
displacement U and its derivative U
/p
, 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. En effet, in
the assumption of the small disturbances, one a:
= (U + C)/2 = Drunk
From where while deriving compared to p:
1
U
U
T
U
=
+
(
)
= B
p
2
p
p
p
It is thus enough in the routine to operator CALC_ELEM to provide in input to the subroutine
calculating the deformations the field U
/p
in the place of the field U.
2.2
Derived from the constraints if p does not depend on material
By noting A the tensor of elasticity, one a:
= ABu
Knowing that A and B do not depend on p, one a:
/p
= AB (U
/p
)
As to the preceding paragraph, in operator CALC_ELEM, one provides in input to
subroutine calculating the constraints the field U
/p
in the place of the field U.
2.3
Derived from the constraints if p depends on material
B not depending on p, one a:
/p
= (A
/p
) Drunk + AB (U
/p
)
Term AB (U
/p
) is in the same way calculated that in the paragraph [§2.2].
For the other term, it is necessary to calculate A
/p
, i.e. to re-use the routine DMATMC which had been
developed for the calculation of U
/p
if p depends on the material (cf [§1.2]).
Handbook of Référence
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3
Calculation of derived from G
3.1
Recall of the formulation of G
One considers a fissured elastic solid occupying the field of space R2 or R3. 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 Sd part of
.
F
S
G
Sd
Appear elastic 3.1-a: fissured Solide
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 constraints,
·
0
the tensor of the initial constraints,
· (, 0
, 0
, T) density of free energy.
Then the rate of refund of energy associated with a virtual field of propagation with the fissure
is written:
G ()
=
1
1
(
U
)
[(
)
(
)
]
,
,
0
,
0,
0 0,
ij
I p
p J -
K K
D
-
T
D
K
K
+
ij -
ij
ij K
K -
ij -
HT
ij -
ij
ij K
K
D
T
2
2
+ (F U
)
[
]
,
,
,
,
,
I
I
K K + F
U
I K
K
I
D
+ G
U
I K
K
I + G U
I
I
K K -
nk
D
-
N U
ij
J
I K
K
D
N
S
K
Sd
The last term associated with a boundary condition with Dirichlet is not established in
Code_Aster. One will thus not take it into account.
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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: G () = CLA
T
+ THER
T
+ TEPSINI + TFVOL + TFSUR
5 terms being given in the order of [§3.1].
dG
dT
dT
dT
dT
dT
CLA
THER
EPSINI
FVOL
FSUR
=
+
+
+
+
of
of
of
of
of
of
3.2.1.1 Derived from the traditional term
dT
D
CLA
ij
I, p
D
TCLA= (U
ij I, p p, J - K, K)
D
= (
ui, p p, J + ij
p, J -
K, K)
D
of
of
of
of
with in 3D and axi:
((
9
U), T) =
2
2
kk + µijij - HT
with HT = 3K (T - D
T F) kk - K (T - D
T F)
2
2
E
E
E
where 3K
=
; =
; 2
=
1 -
2
(1+) (1 - 2)
µ
1+
it comes:
D
D
D
µ
(T - T)
2
kk
ij
ref.
D kk
3
=
+
+ + 2µ
-
(+ E
- (T - T))
of
2
kk
kk
ij ij
ij
E
of
E
of
1 - 2
kk
of
2
ref.
in plane deformations:
(,)
(1 -) E
2
2
T =
(
E
E
+ +
+
2 -
xx
yy)
(21+) (1 -
2)
(1+) (1 -
2) xx yy (1+) xy
HT
that is to say:
D
1
(-) E
D
D
E
D
xx
yy
1
D
=
(
+
+
(2
2
+)) +
(
yy
xx
xx yy
+
+
)
of
1
(+ 1
) (- 2) xx
yy
of
of
2
xx
yy
E
1
(+ 1
) (- 2) xx
yy
of
of
E
2nd
D xy
1
(T - T)
2
ref.
D kk
3
+
+
-
(+ E
- (T - T))
1
xy
+
of
1
xy
+
1 - 2
kk
of
2
ref.
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in plane constraints:
(,)
E
2
2
T =
2 (
E
E
+ +
+
2 -
xx
yy)
(21 -)
(1 -) 2 xx yy (1+) xy
HT
D
E
D
D
E
D
xx
yy
1
D
:
that is to say
=
(
+
+
(2
2
+)) +
(
yy
xx
xx yy
+
+
)
xx
yy
xx
yy
xx
yy
of
1
(-) 2
of
of
2nd
1
(-) 2
of
of
E
2nd
D xy
1
(T - T)
2
ref.
D kk
3
+
+
-
(+ E
- (T - T))
1
xy
+
of
1
xy
+
1 - 2
kk
of
2
ref.
3.2.1.2 Derived from the thermal term
TTHER =
T, kk
D
T
((
1
U), T)
dK (T)
=
(
D T
-
3 T - T
- 3K T
(
) +
T - T
-
3 T - T
kk
(ref.)
() (ref.) (kk (ref.)
T
2 dT
dT
dTTHER
= - D (
T
), kk
D
of
of
T
in 3D and axi:
D
1
(+ 2 2
)
2
D
D
+
kk
kk
of
E 1
(
2 2)
D
(
) =
+
(
+
)
T
1
(+) 2 1
(- 2) 2 dT 2
kk
of
1
(+ 1
) (- 2) dT
1
(+) 2 1
(- 2) 2 dT
D
1
dij 1
of
E D
-
+
(
-
)
ij ij dT 1
(
2 +) 2
ij of 1+ dT 1+ dT
1
D
D
(T - T)
kk
ref.
2
D
of
2nd D
D
-
(+
(T - T))(+ E
) -
(
+ (
+
)
kk)
1 - 2
dT
ref.
kk
of
1 - 2
1 - 2 dT kk
dT
1 - 2 dT of
3
D
D
+
(T - T) (+ (T - T)
+
(T - T)
)
1 - 2
ref.
ref.
dT
1 - 2
ref.
dT
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in plane deformations:
D
(2 -)
D
D
D
-
of
- E
D
2
2
1
2 (2
)
(
) =
(
+) + (
XX
YY
+
) (
+
)
T
1
(+) 2 1
(- 2) 2 dT
XX
YY
XX
of
YY
of
1
(+ 1
) (- 2) dT
1
(+) 2 1
(- 2) 2 dT
1 + 2 2
D
D
D
+
YY
XX
of
1
(
2 2) E
D
+ (
+
) (
+
)
1
(+) 2 1
(- 2) 2 dT XX YY
XX
of
YY
of
1
(+ 1
) (- 2) dT
1
(+) 2 1
(- 2) 2 dT
1
D
D
XY
of
E
D
2
1
-
+ 2
(
-
)
1
(+) 2 dT XY
XY
of
1
(+) dT
1
(+) 2 dT
1
D
D
(T - T)
kk
ref.
2
D
of
2nd D
D
-
(+
(T - T))(+ E
) -
(
+ (
+
)
kk)
1 - 2
dT
ref.
kk
of
1 - 2
1 - 2 dT kk
dT
1 - 2 dT of
3
D
D
+
(T - T) (+ (T - T)
+
(T - T)
)
1 - 2
ref.
ref.
dT
1 - 2
ref.
dT
in plane constraints:
D
D
D
D
XX
YY
of
E
D
2
2
1
2
(
) =
(
+) + (
+
) (
+
)
T
1
(
2
-) 2 dT XX
YY
XX
of
YY
of
1
(
2
-) dT
1
(
2
-) 2 dT
1
2
+
D
D
D
+
YY
XX
of
1
(
2) E D
+ (
+
) (
+
)
1
(
2
-) 2 dT XX YY
XX
of
YY
of
1
(
2
-) dT
1
(
2
-) 2 dT
1
D
D
XY
of
E
D
2
1
-
+ 2
(
-
)
1
(+) 2 dT XY
XY
of
1
(+) dT
1
(+) 2 dT
1
D
D
(T - T)
kk
ref.
2
D
of
2nd D
D
-
(+
(T - T))(+ E
) -
(
+ (
+
)
kk)
1 - 2
dT
ref.
kk
of
1 - 2
1 - 2 dT kk
dT
1 - 2 dT of
3
D
D
+
(T - T) (+ (T - T)
+
(T - T)
)
1 - 2
ref.
ref.
dT
1 - 2
ref.
dT
3.2.1.3 Derived from the deformations term and initial constraints
1 0 0
1
T
0
EPSINI = [(
0
ij - ij) ij, kk - (ij - HT
ij - ij) ij, kk)]
D
2
2
dT
D
D
EPSINI =
ij
(
0,
0,
)
ij
-
ij
K
ij K
K
D
of
of
of
Handbook of Référence
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3.2.1.4 Derived from the term forces voluminal
T
(
)
FVOL =
F U
I
I
K, K + F
U
I, K
K
I
D
dT
FVOL = (F
I
)
I
K, K + F
I
I, K
K
D
of
of
of
3.2.1.5 Derived from the term forces surface
T
[
]
FSUR =
G
U
I, K
K
I + G U
I
I
K, K -
nk
D
S
nk
dT
FSUR = [G
I
]
I, K
K
+ G
I
I
K, K -
nk
D
of
of
of
S
nk
3.2.2 Derived from G compared to the loading
The significant parameter can be one (or several) component of forces F voluminal, surface or
I
nodal, and (or) one (or several) pressure on an edge, which returns to a surface force.
In all the cases one can write:
G
ncha N
dim
G
F icha
I
=
additivity coming owing to the fact that contributions from the loadings from
PS
icha
=1 =1
icha
I
F
PS
I
G cumulate
F icha
with
I
= 1
if the significant parameter intervenes in component I of
PS
loading icha
F icha
I
= 0
if not
PS
Example:
The user defines a significant parameter being worth 1. This significant parameter is used to define a force
voluminal F, a surface force of components (F, F) and a pressure p.
Z
X
Y
G
G
G
G
G
then
=
+
+
+
PS
F
F
F
p
Z
X
Y
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 traditional 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.
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3.2.2.1 Derived from the traditional term
dT
D
CLA
ij
I, p
D
T
(
)
= (
U
)
I, p
p, J +
ij
p, J -
D
K, K
CLA =
U
ij
I, p
p, J -
K, K
D
df
df
df
df
with in 3D and axi:
((
U), T)
2
=
+ µ - 3K T - T + K T - T
kk
ij ij
(ref.)
9
2
kk
(ref.) 2
2
2
E
E
E
where 3K
=
; =
; 2µ =
1 -
2
(1+) (1 -
2)
1+
D
D
D
kk
ij
D
it comes:
=
+ 2µ
- 3K
kk
ij
(T - Tréf) kk
df
df
df
df
in plane deformations:
(,)
(1 -) E
2
2
T =
(
E
E
+ +
+
2 -
xx
yy)
(21+) (1 -
2)
(1+) (1 -
2) xx yy (1+) xy
HT
D
1
(-) E
D
D
xx
yy
E
D yy
D xx
that is to say
=
(
+
) +
(
+
)
df
xx
1
(+ 1
) (-
2)
df
yy
df
xx
1
(+ 1
) (-
2)
df
yy
df
2nd
D
T
(- T
xy
ref.)
D
+
-
E
kk
xy
1 +
df
1 -
2
df
in plane constraints:
(,)
E
2
2
T =
2 (
E
E
+ +
+
2 -
xx
yy)
(21 -)
(1 -) 2 xx yy (1+) xy
HT
D
E
D
D
xx
yy
E
D yy
D xx
that is to say
=
(
+
) +
(
+
)
df
xx
1
(-) 2
df
yy
df
xx
1
(-) 2
df
yy
df
2nd
D
T
(- T
xy
ref.)
D
+
-
E
kk
xy
1 +
df
1 -
2
df
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3.2.2.2 Derived from the thermal term
T
,
THER =
T K K
D
T
(
1 dK T
D
(
T
U), T)
()
=
kk -
3 T - T
- 3K
ref.
+
T - Tréf kk -
3 T - T
T
2 dT (
(
)
()
dT (
) (
(ref.)
dT
THER = - D (
T
), D
K
K
df
df T
in 3D and axi:
D
D
+
kk
of
E 1
(
2 2)
D
(
) =
(
+
)
df T
kk
df
1
(+ 1
) (- 2) dT
1
(+) 2 1
(- 2) 2 dT
dij 1
of
E D
+
(
-
)
ij df 1+ dT 1+ dT
1
D
D
(T - T)
kk
ref.
2
D
of
2nd D
D
-
(+
(T - T))(+ E
) -
(
+ (
+
)
kk)
1 - 2
dT
ref.
kk
df
1 - 2
1 - 2 dT kk
dT
1 - 2 dT df
3
D
D
+
(T - T) (+ (T - T)
+
(T - T)
)
1 - 2
ref.
ref.
dT
1 - 2
ref.
dT
in plane deformations:
D
D
D
1
of
2 (2 -) E
D
(
) = (
XX
YY
+
) (
+
)
df T
XX
df
YY
df
1
(+ 1
) (- 2) dT
1
(+) 2 1
(- 2) 2 dT
D
D
+
YY
XX
of
1
(
2 2) E
D
+ (
+
) (
+
)
XX
df
YY
df
1
(+ 1
) (- 2) dT
1
(+) 2 1
(- 2) 2 dT
D
XY
1
of
E
D
+ 2
(
-
)
XY
df
1
(+) dT
1
(+) 2 dT
1
D
D
(T - T)
kk
ref.
2
D
of
2nd D
D
-
(+
(T - T))(+ E
) -
(
+ (
+
)
kk)
1 - 2
dT
ref.
kk
df
1 - 2
1 - 2 dT kk
dT
1 - 2 dT df
3
D
D
+
(T - T) (+ (T - T)
+
(T - T)
)
1 - 2
ref.
ref.
dT
1 - 2
ref.
dT
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
Code_Aster ®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES Clé
:
R4.03.07-B Page
: 12/14
in plane constraints:
D
D
D
XX
YY
1
of
2nd
D
(
) = (
+
) (
+
)
df T
XX
df
YY
df
1
(
2
-) dT
1
(
2
-) 2 dT
D
D
+
YY
XX
of
1
(
2) E D
+ (
+
) (
+
)
XX
df
YY
df
1
(
2
-) dT
1
(
2
-) 2 dT
D
XY
1
of
E
D
+ 2
(
-
)
XY
df
1
(+) dT
1
(+) 2 dT
1
D
D
(T - T)
kk
ref.
2
D
of
2nd D
D
-
(+
(T - T))(+ E
) -
(
+ (
+
)
kk)
1 - 2
dT
ref.
kk
df
1 - 2
1 - 2 dT kk
dT
1 - 2 dT df
3
D
D
+
(T - T) (+ (T - T)
+
(T - T)
)
1 - 2
ref.
ref.
dT
1 - 2
ref.
dT
3.2.2.3 Derived from the deformations term and initial constraints
1 0 0
1
T
[(
)
(
0
) 0
)]
EPSINI =
ij -
ij
ij, K
K -
ij -
HT
ij -
ij
ij, K
K
D
2
2
dT
D
D
EPSINI =
ij
(
0,
0,
)
ij
-
ij
K
ij K
K
D
df
df
df
3.2.2.4 Derived from the term forces voluminal
T
(
)
FVOL =
F U
I
I
K, K + F
U
I, K
K
I
D
if F = fl
is a voluminal component of force:
dT
FVOL = ((F
I
)
)
I
+ ul K, K + F
I
I, K
K
D
df
df
df
L
L
L
if not:
dT
FVOL = (F
I
)
I
K, K + F
I
I, K
K
D
df
df
df
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
Code_Aster ®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES Clé
:
R4.03.07-B Page
: 13/14
3.2.2.5 Derived from the term forces surface
T
[
]
FSUR =
G
U
I, K
K
I + G U
I
I
K, K -
nk
D
S
nk
if F = gl
is a surface component of force:
dT
FSUR = [G
I
(
)
]
I, K
K
+ G
I
I
+ ul
K, K -
nk
D
dg
dg
dg
L
S
L
L
nk
if not:
dT
FSUR = [G
I
]
I, K
K
+ G
I
I
K, K -
nk
D
df
df
df
S
nk
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
Code_Aster ®
Version
7.2
Titrate:
Postprocessing of sensitivity
Date:
07/09/04
Author (S):
P. of BONNIERES, X.DESROCHES Clé
:
R4.03.07-B Page
: 14/14
4 Bibliography
[1]
TARDIEU NR.: Calculation of sensitivity in mechanics. Application to Code_Aster. note EDF-DER
HI-75/01/016-Index A, 2001
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/04/002/A
Outline document