Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
1/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R4.03 booklet: Analyze sensitivity
Document: R4.03.03
Calculation of sensitivity in mechanics
Summary
Classically, the digital simulations provide the response of a system to a stress. It appears
currently an important evolution aiming at providing in addition to this answer the tendency of the answer to
an amendment of parameters of input of simulation (material, loading, geometry,…). These
tendencies are obtained by calculating the derivative of the response compared to parameters given.
The object of this note is thus the determination of the sensitivity of the results of a calculation of mechanics of
solids with various data input by the method of direct differentiation. These data input are them
data material and loadings. Moreover will be detailed the linear calculation cases (operator
MECA_STATIQUE) and nonlinear (operators STAT_NON_LINE and DYNA_NON_LINE).
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
2/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
3/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
1 Introduction
Classically, the digital simulations provide the response of a system to a stress. It
currently appears an important evolution aiming at providing in addition to this answer the tendency
response to an amendment of parameters of input of simulation (material, loading,
geometry,…). The possible applications are important:
·
probabilistic calculations related to the existence of an uncertainty on the value of a parameter,
·
problems opposite, retiming, optimization for which the knowledge of derived from one
field can be capital in term of effectiveness,
·
reliability of the studies (which credit to bring to a simulation where the response of the system can
to strongly vary following a small variation of a parameter?).
These tendencies are generally obtained by calculating the derivative of the answer, which can be done
various manners: finished differences, direct differentiation or method of the assistant state.
finished differences are to be excluded from their weak precision and their important numerical cost.
method of the associated state, though powerful and specifies, requires developments particular to
each study and it will not be retained. We will thus concentrate here on the method of
differentiation direct, powerful, specifies, general and very adapted to nonlinear calculations.
The object of this note is thus the determination of the sensitivity of the results of a calculation of mechanics
solids with various data input by the method of direct differentiation. These data
of input will be the data material and the loadings.
Moreover, the choice was made carry out our reasoning on the equations of the mechanical problem
discretized. This choice seems important to us insofar as it ensures the coherence of derived calculation
compared to direct calculation. We insist on the fact that this coherence is essential to
precision of the results obtained.
Lastly, we will treat two model cases within the framework of the calculation of sensitivity of linear problems
and nonlinear: linear elasticity and the plasticity of Von Mises with linear isotropic work hardening. The goal
of these examples is well to clarify the difference in nature of the problems derived in both
preceding cases. In the linear case, the derived problem is very similar to the direct problem in
the measurement where only the second member of the equations is modified. In the nonlinear case, the problem
derived is appreciably different from the direct problem: the second member but also the law of
behavior are modified. Nevertheless, in these two cases, the derived problem preserves one
extremely interesting property insofar as it consists of a succession of linear problems
whose matrices already were calculated and factorized (in the case of the direct use of solveurs).
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
4/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
2
Sensitivity to the data materials
2.1
The direct problem
We place ourselves in this part within the framework of the resolution of non-linear calculations. Us
let us use the notations of [bib1] and return to the reading of this document for any precision on
technique of resolution approached in the continuation.
In Code_Aster, any non-linear static calculation is solved incrémentalement. It thus requires with
each pitch of load
}
,
1
{I
I
the resolution of the non-linear system of equation:
=
=
+
di
I
I
I
T
I
I
T
U
Drunk
L
B
U
R
)
,
(
éq
2.1-1
with
=
D
T
K
I
K
I
I
)
(
:
)
(
))
,
(
(
W
U
U
R
éq
2.1-2
·
K
W
is related to form of K
ième
degree of freedom of the modelized structure,
·
))
,
(
(
I
I
T
U
R
is the vector of the nodal forces.
The resolution of this system is done by the method of Newton-Raphson:
=
+
-
=
+
+
+
+
0
)
,
(
1
1
1
N
I
N
I
T
I
nor
I
N
I
T
N
I
nor
T
B
B
U
R
L
B
U
K
éq
2.1-3
where
)
,
(
I
N
I
T
N
I
U
U
R
K
=
is the tangent matrix with the pitch of load
I
and with the iteration of Newton
N
.
The solution is thus given by:
+
=
+
=
=
-
=
-
NR
N
N
I
I
I
NR
N
N
I
I
I
0
1
0
1
U
U
U
with
NR
, the iteration count of Newton which was necessary to reach convergence.
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
5/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
2.2
Derived calculation
2.2.1 Preliminaries
Within the framework of the calculation of sensitivity, it is necessary to insist on the dependences of a size
compared to the others. We thus will clarify that the results of preceding calculation depend
of a parameter
given (modulus elastic Young, limit, density,…) and that of
following manner:
)
(
=
I
I
U
U
,
)
(
=
I
I
.
But that is not sufficient. Also we within the framework of an incremental calculation with law place ourselves
of elastoplastic nonlinear behavior with linear isotropic work hardening [bib2]. If one considers
the interdependences of the parameters on an algorithmic level, one can write [bib3]:
))
(
),
(
),
(
(
1
1
=
-
-
U
R
R
I
I
p
)
),
(
),
(
),
(
(
)
(
1
1
1
+
=
-
-
-
U
I
I
I
I
p
)
),
(
),
(
),
(
(
)
(
1
1
1
+
-
-
-
=
U
I
I
I
I
p
p
p
p
Where
U
is the increment of displacement to convergence with the pitch of load
I
.
Let us specify the direction of the notations which we will use for the derivative:
·
Y
X
indicate the partial derivative clarifies
X
compared to
Y
,
·
Y
X,
indicate the total variation of
X
compared to
Y
.
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
6/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
2.2.2 Derivation of balance
Taking into account the preceding remarks, let us express the total variation of [éq 2.1-1] compared to
:
=
=
+
+
+
+
-
-
-
-
0
,
0
,
,
,
,
1
1
1
1
U
B
B
R
R
U
U
R
R
I
T
I
I
I
I
p
p
éq
2.2.2-1
Let us notice that here
0
=
R
:
R
does not depend explicitly on
but implicitly like us
will see in detail in the continuation.
That is to say:
=
-
=
+
0
,
,
,
,
)
(
U
B
R
B
U
K
U
U
I
T
NR
I
éq
2.2.2-2
Where
·
NR
I
K
is the last tangent matrix used to reach convergence in
iterations of Newton,
·
)
(
,
U
U
R
is the total variation of
R
, without taking account of the dependence of
U
by
report/ratio with
.
The problem lies now in the calculation of
)
(
,
U
U
R
.
Note:
In [éq 2.2.2-2], one used the fact that
U
U
R
K
=
)
,
(
I
I
NR
I
T
whereas in [éq 2.1-3] one has it
defined by
NR
I
I
I
NR
I
T
U
U
R
K
=
)
,
(
. There is well equivalence of these two definitions in the measurement
where
U
U
U
+
=
- 1
I
I
and that
R
depends indeed on
U
(and as well sure of
1
-
I
and
1
-
I
p
).
Note:
If one derives compared to
directly [éq 2.1-3], one finds
1
/
/
1
,
,
,
+
+
-
-
=
+
=
N
N
T
N
N
U
K
R
B
U
K
U
U
. What is the same thing
with convergence and reveals that the error on
U
depends on
-
,
1
K
K
.
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
7/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
2.2.3 Calculation of derived from the law of behavior
In the continuation, by preoccupation with a clearness, we will give up the indices
1
=
I
.
According to [éq 2.1-2], one can rewrite
)
(
,
U
U
R
in the form:
(
)
+
=
D
K
)
(
:
,
,
,
)
(
)
(
W
R
U
U
U
U
éq
2.2.3-1
One must thus calculate
)
(
,
U
U
. With this intention, we will use the expressions which
intervene in the numerical integration of the law of behavior.
2.2.3.1 Case of linear elasticity
Within the framework of linear elasticity, the law of behavior is expressed by:
=
µ
=
))
(
(
.
3
)
(
)
(
~
.
2
~
U
U
Tr
K
Tr
or:
Id
U
U
)).
(
(
.
)
(
~
.
2
+
µ
=
Tr
K
éq
2.2.3.1-1
where
Id
is the tensor identity of command 2.
Then, by calculating the total variation of [éq 2.2.3.1-1] compared to
, one obtains:
Id
U
U
Id
U
U
)).
,
(
(
.
)
,
(
~
.
2
)).
(
(
.
,
)
(
~
.
,
2
,
+
µ
+
+
µ
=
Tr
K
Tr
K
éq
2.2.3.1-2
That is to say:
Id
U
U
U
U
)).
(
(
.
,
)
(
~
.
,
2
|
,
)
(
+
µ
=
Tr
K
2.2.3.2 Case of elastoplasticity with linear isotropic work hardening
The elastoplastic law of behavior to linear isotropic work hardening is written:
+
+
+
+
=
-
)
'.(
)
(
)
(
~
~
2
3
:
)
(
p
p
R
p
eq
eq
S
U
where
S
is the tensor of the elastic flexibilities and
'
R
is the slope of work hardening defined by:
T
T
E
E
E
E
R
-
=
.
'
where
T
E
E
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
8/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
In numerical terms, this law of behavior is integrated using an algorithm of return
radial: one makes an elastic prediction (noted
E
) that one corrects if the threshold is violated. With
notations of [bib4], one thus writes:
µ
+
-
-
=
=
µ).
+
-
-
µ
-
µ
=
3
'
'.
))
(
(
.
3
)
(
~
3
'
(
'.
3
)
(
~
.
2
~
R
p
R
p
Tr
K
Tr
R
p
R
y
E
eq
E
E
eq
y
E
eq
U
U
éq
2.2.3.2-1
We will distinguish two cases.
1
Er
case:
0
=
p
What comes down to saying that at the time of these pitch of load, the point of Gauss considered did not see
of increase in its plasticization. One finds oneself then in the case of linear elasticity:
Id
U
U
U
U
)).
(
(
.
,
)
(
~
.
,
2
,
)
(
+
µ
=
Tr
K
2
eme
case:
0
>
p
Taking into account the dependences between variables in [éq 2.2.3.2-1], one can write:
+
+
+
=
+
+
+
=
),
(
)
(
,
,
,
)
,
(
)
(
,
,
,
U
U
U
U
p
p
p
p
p
p
p
p
p
éq 2.2.3.2-2
Moreover, in agreement with the algorithmic integration of the law, we will separate parts deviatoric
and hydrostatic.
+
+
=
+
+
+
+
+
=
,
,
,
,
)
(
3
1
,
~
,
)
(
3
1
,
~
)
(
3
1
~
,
)
(
)
(
p
p
p
p
p
p
p
p
Tr
p
p
Tr
Tr
Id
Id
Id
U
U
U
U
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
9/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
And thus, one calculates:
µ).
+
-
-
µ
-
µ)
+
+
µ
+
µ).
+
-
-
µ
+
µ).
+
-
-
µ
-
µ).
+
-
-
µ
-
µ
=
E
E
eq
y
E
eq
E
E
eq
E
eq
E
eq
y
E
eq
E
E
eq
y
E
eq
E
E
eq
y
E
eq
R
p
R
R
R
R
p
R
R
p
R
R
p
R
U
~
3
'
(
'.
3
~
3
'
(
3
'
)
3
'
((
'.
3
~
3
'
(
'
3
~
3
'
(
'.
3
)
(
~
2
~
2
))
(
(
3
)
(
U
=
Tr
K
Tr
J
µ).
+
-
-
µ
-
-
-
-
µ).
+
µ
-
=
E
eq
y
E
eq
E
E
eq
E
eq
y
E
eq
E
eq
R
p
R
p
R
R
3
'
(
'.
3
~
'.
1
3
'
(
3
~
where
J
is the operator deviatoric defined by:
~
:
=
J
0
)
(
=
Tr
p
E
E
eq
R
R
p
~
3
'
(
'
3
~
µ).
+
µ.
=
0
)
(
=
p
Tr
,
p
The fact is used that:
y
eq
p
p
R
+
+
=
)
'.(
-
+
-
+
+
=
+
y
eq
p
p
R
R
p
p
)
(
'
)
~
~
(
:
)
,
~
,
~
(
2
3
'
1
,
,
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
10/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
Note:
In these calculations were or must be used the following results:
)
(
~
2
~
U
µ
=
E
E
eq
E
eq
µ.
+
µ
=
))
(
~
2
~
(
:
))
(
~
2
(
2
3
U
U
Tensor of command 2
Scalar
E
eq
E
E
eq
=
~
2
3
J
=
E
~
Tensor of command 2
Tensor of command 4
2.2.3.3 Calculation of derived from displacement
Once
)
(
,
U
U
calculated, one can constitute the second member
)
(
,
U
U
R
while using
[éq 2.2.3-1]. One solves then the system [éq 2.2.2-2] and one obtains the increment of derived displacement
compared to
.
2.2.3.4 Calculation of derived from the other sizes
Now that one has
,
U
, one must calculate the derivative of the other sizes. One separates
still two cases:
Linear elasticity
According to [éq 2.2.3.2-1], one as follows calculates the derivative of the increment of stress:
Id
U
U
U
U
)).
,
(
(
.
)
,
(
~
.
2
,
,
)
(
+
µ
+
=
Tr
K
The increment of cumulated plastic deformation, as for him, does not see evolution:
0
,
=
p
Elastoplasticity with linear isotropic work hardening
If
0
=
p
, the preceding case is found.
If not, one obtains according to [éq 2.2.3.2-2]:
)
,
(
:
)
(
,
,
)
(
+
=
U
U
U
U
And for the cumulated plastic deformation:
-
+
-
+
+
=
+
y
eq
p
p
R
R
p
p
)
(
'
)
~
~
(
:
)
,
~
,
~
(
2
3
'
1
,
,
Once all these calculations are finished, all the derived sizes are reactualized and one passes
with the pitch of load according to.
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
11/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
2.2.3.5 Synthesis
To summarize the preceding paragraphs, one represents the various stages of calculation by
following diagram:
2.3 Establishment
data processing
For each finite element of Code_Aster, it is necessary to create two new options of calculations
to allow calculations above:
·
for the calculation of
)
(
,
U
U
,
·
for the calculation of
,
and
,
p
.
Derived calculation is controlled by a routine which launches elementary calculations of
)
(
,
U
U
,
carry out the assembly of these terms to create the second member, solves the linear system
[éq 2.2.2-2], then lance the calculation of
,
and
,
p
. This routine is called by OP0070 after
convergence was detected.
Note:
For the user, the sequence of various calculations will be transparent. The definition of
significant variables will be done according to the diagram [bib5]:
…
v = DEFI_PARA_SENSI (VALE = < value of the parameter >)
subdue = DEFI_MATERIAU (VMIS_ISOT_LINE = _F (SY =
v),…)
chmat = AFFE_MATERIAU (AFFE = _F (GROUP_MA = < group (S)
>,
MATER
=
matt
))
…
resu = STAT_NON_LINE (CHAM_MATER = chmat,
SENSITIVITY
=
(
v),
…)
…
Convergence of the pitch of
charge
N
direct calculation
Calculation of the terms
)
(
,
U
U
Assembly of
)
(
,
U
U
R
Resolution of the system
[éq 2.2.2-2]
,
U
Calculation of
,
and
,
p
Passage to the pitch
of load
1
+
N
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
12/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
3
Sensitivity to the loading
3.1
The direct problem: expression of the loading
Until now we expressed the direct problem in the form:
=
=
+
di
I
I
I
T
I
I
T
U
Drunk
L
B
U
R
)
,
(
éq
3.1-1
The loadings are gathered with the second member and include/understand the imposed forces
I
L
and them
imposed displacements
di
U
.
Let us suppose that the loading in imposed force
I
L
depends on a scalar parameter
following manner:
)
(
)
(
2
1
I
I
I
L
L
L
+
=
éq
3.1-2
Where
·
1i
L
is a vector independent of
,
·
2i
L
depends linearly on
.
One wishes to calculate the sensitivity of the results of direct calculation to a variation of the parameter
.
3.2
The derived problem
3.2.1 Derivation of balance
As in the preceding chapter, by taking account of the dependences between the various fields,
one derives balance [éq 3.1-1] by report/ratio
:
-
=
=
+
+
+
+
-
-
-
-
-
,
1
2
1
1
1
1
,
)
1
(
,
,
,
,
I
I
I
T
I
I
I
I
p
p
Drunk
U
B
L
B
R
R
U
U
R
R
éq
3.2.1-1
One used the fact that
2i
L
depends linearly on
.
That is to say:
-
=
-
=
+
-
,
1
)
(
2
,
,
)
1
(
,
,
I
I
I
T
NR
I
Drunk
U
B
R
L
B
U
K
U
U
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
13/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
Where
·
NR
I
K
is the last tangent matrix used to reach convergence in
iterations of Newton,
·
)
(
,
U
U
R
is the total variation of
R
, without taking account of the dependence of
U
by
report/ratio with
.
The problem lies like previously in the calculation of
)
(
,
U
U
R
.
Note:
If one seeks to calculate the sensitivity compared to the loading of Dirichlet, the system with
to solve becomes:
-
=
-
=
+
-
,
1
2
)
(
)
1
(
,
,
,
,
I
I
I
T
NR
I
D
Drunk
U
U
B
R
B
U
K
U
U
Where
D
I
2
U
is the part of the loading of Dirichlet which depends linearly on
.
3.2.2 Calculation of derived from the law of behavior
According to [éq 2.1-2], one can rewrite
)
(
,
U
U
R
in the form:
(
)
+
=
D
K
)
(
:
,
,
,
)
(
)
(
W
R
U
U
U
U
éq
3.2.2-1
With this intention, we will use the expressions which intervene in the numerical integration of
law of behavior to calculate
)
(
,
U
U
.
3.2.2.1 Case of linear elasticity
Within the framework of linear elasticity, the law of behavior is expressed by:
Id
U
U
)).
(
(
.
)
(
~
.
2
+
µ
=
Tr
K
éq
3.2.2.1-1
where
Id
is the tensor identity of command 2.
Then, by calculating the total variation of [éq 3.2.2.1-1] compared to
, one obtains:
Id
U
U
Id
U
U
Id
U
U
)).
,
(
(
.
)
,
(
~
.
2
.
0
.
0
)).
,
(
(
.
)
,
(
~
.
2
)).
(
(
.
,
)
(
~
.
,
2
,
+
µ
+
+
=
+
µ
+
+
µ
=
Tr
K
Tr
K
Tr
K
éq 3.2.2.1-2
That is to say:
.
0
,
)
(
=
U
U
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
14/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
3.2.2.2 Case of elastoplasticity with linear isotropic work hardening
As previously, we will distinguish two cases.
1
Er
case:
0
=
p
What comes down to saying that at the time of these pitch of load, the point of Gauss considered did not see
of increase in its plasticization. One finds oneself then in the case of linear elasticity:
.
0
,
)
(
=
U
U
2
eme
case:
0
>
p
Taking into account the dependences between variables, one can write:
+
+
+
=
+
+
+
=
),
(
)
(
,
,
,
)
,
(
)
(
,
,
,
U
U
U
U
p
p
p
p
p
p
p
p
p
Moreover, in agreement with the algorithmic integration of the law, we will separate parts deviatoric
and hydrostatic.
+
+
=
+
+
+
+
+
=
,
,
,
,
)
(
3
1
,
~
,
)
(
3
1
,
~
)
(
3
1
~
,
)
(
)
(
p
p
p
p
p
p
p
p
Tr
p
p
Tr
Tr
Id
Id
Id
U
U
U
U
And thus, one calculates:
Insofar as one does not have dependence clarifies
compared to
, one obtains:
.
0
~ =
.
0
)
(
=
Tr
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
15/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
J
µ).
+
-
-
µ
-
-
-
-
µ).
+
µ
-
=
E
eq
y
E
eq
E
E
eq
E
eq
y
E
eq
E
eq
R
p
R
p
R
R
3
'
(
'.
3
~
'.
1
3
'
(
3
~
where
J
is the operator deviatoric.
0
)
(
=
Tr
p
E
E
eq
R
R
p
~
3
'
(
'
3
~
µ).
+
µ.
=
0
)
(
=
p
Tr
,
p
The fact is used that:
y
eq
p
p
R
+
+
=
)
'.(
eq
R
p
p
+
+
=
+
)
~
~
(
:
)
,
~
,
~
(
2
3
'
1
,
,
3.2.2.3 Calculation of derived from displacement
Once
)
(
,
U
U
calculated, one can constitute the second member
)
(
,
U
U
R
. One solves
then the system [éq 3.2.1-1] and one obtain the increment of derived displacement compared to
.
3.2.2.4 Calculation of derived from the other sizes
Now that one has
,
U
, one must calculate the derivative of the other sizes. One separates
still two cases:
Linear elasticity
According to [éq 3.2.2.1-1], one as follows calculates the derivative of the increment of stress:
Id
U
U
)).
,
(
(
.
)
,
(
~
.
2
.
0
,
+
µ
+
=
Tr
K
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
16/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
The increment of cumulated plastic deformation, as for him, does not see evolution:
0
,
=
p
Elastoplasticity with linear isotropic work hardening
If
0
=
p
, the preceding case is found.
If not, one obtains:
)
,
(
:
)
(
,
,
)
(
+
=
U
U
U
U
And for the cumulated plastic deformation:
eq
R
p
p
+
+
=
+
)
~
~
(
:
)
,
~
,
~
(
2
3
'
1
,
,
Once all these calculations are finished, all the derived sizes are reactualized and one passes
with the pitch of load according to.
3.2.2.5 Synthesis
To summarize the preceding paragraphs, one represents the various stages of calculation by
following diagram:
Convergence of the pitch of
charge
N
direct calculation
Calculation of the terms
)
(
,
U
U
Assembly of
)
(
,
U
U
R
Resolution of the system
[éq 3.2.1-1]
,
U
Calculation of
,
and
,
p
Passage to the pitch
of load
1
+
N
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
17/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
3.3 Establishment
data processing
The data-processing establishment is in any point similar to the preceding one insofar as one adds
two options:
·
for the calculation of
)
(
,
U
U
,
·
for the calculation of
,
and
,
p
.
Let us notice that the organization of the calculation of the loadings (where all the contributions are summoned
in only one vector) obliges with the revaluation of the force compared to which one calculates them
sensitivities.
Note:
The definition of the significant variables will be done like previously:
…
v = DEFI_PARA_SENSI (VALE = < value of the parameter >)
force = AFFE_CHAR_MECA (
PRES_REP=_F (GROUP_MA=< groups (S)
>,
PRES=
v),…)
…
resu = STAT_NON_LINE (EXCIT = force,
SENSITIVITY
=
(
v),
…)
…
Code_Aster
®
Version
7.4
Titrate:
Calculation of sensitivity in mechanics
Date:
19/04/05
Author (S):
NR. TARDIEU
Key
:
R4.03.03-B
Page
:
18/18
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A
4
Availabilities within Code_Aster
The following table recapitulates the derivative available in Code_Aster. It is updated at the fur and at
measure developments of the derivatives.
Linear mechanics (operator
MECA_STATIQUE)
Nonlinear mechanics
(operators STAT_NON_LINE
and DYNA_NON_LINE)
Sensitivity to the data
materials in elasticity
linear isotropic
All the elements continuous mediums
Not
Sensitivity to the data
materials in elasticity
linear orthotropic
All the elements continuous mediums
Not
Sensitivity to the data
materials in elasticity
linear isotropic transverse
All the elements continuous mediums
Not
Sensitivity to
loadings
·
nodal forces
·
force divided voluminal into 3D
·
force divided surface into 3D
·
force divided linear into 3D
·
force divided surface into 2D
·
force divided linear into 2D
·
force divided linear into 1D
·
force distributed for the hulls
“2D”
·
force distributed for the hulls
“3D”
·
pressure distributed
Without restriction of finite elements.
Not
5 References
[1]
Quasi-static nonlinear algorithm, Reference material of Code_Aster
[R5.03.01]
[2]
Tangent Operators and Design Sensitivity Formulations for Transient Nonlinear Coupled
Problems with Applications to Elastoplasticity, P. Michaleris and Al, Int. J. Num. Meth. Eng.
1994
[3]
Parameter Sensitivity in Nonlinear Mechanics, Mr. Kleiber and Al, Wiley 1997
[4]
Numerical integration of the Relations of Elastoplastic Behavior, Documentation of
Reference of Code_Aster [R5.03.02]
[5]
Impact of calculations of uncertainties on the architecture of Aster, G.Nicolas, J.Pellet, Note
HI-72/01/009/A