Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
1/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R4.03 booklet: Analyze sensitivity
Document: R4.03.05
Parametric probabilistic models and not
parametric in dynamics
Summary:
This document describes two probabilistic approaches, one parametric and the other nonparametric in order to
to take into account uncertainties of model and modeling for the dynamic systems in mechanics
structures. The not-parametric approach is specific to the resolution on modal basis of the systems
linear (in transient or harmonic) and with the resolution about modal base of the systems with
localized non-linearities (in transient). The parametric approach is a priori universal, but it is more
particularly presented if it is combined with the nonparametric approach.
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
2/24
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
Count
matters
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
3/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
4/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
1 General information
Uncertainties, even when they are reduced, can change the prediction significantly of
vibratory behavior of the structure ([bib13], [bib16]). It is thus necessary to take them in
hope in a quantifiable and explicit way to increase the realism and the robustness of
forecasts. In this context, a probabilistic model of uncertainties contributes to the realism of
step.
The probabilistic approach conventional, known as parametric, makes it possible to incorporate in the mechanical analysis them
uncertainties on the data, i.e parametric uncertainties on the geometry, conditions with
limits or properties of materials. In this approach, each parameter identified like
source of random uncertainties is modelized by a random variable. Parameters of input of
model being thus characterized, the probabilistic numerical methods seek to characterize
probabilistic way the results quantities of the model. For complex structures, for
which the vibratory behavior depends on a great number of parameters, this type of analysis
probabilist is limited by the great quantity of information necessary to characterize them
parameters of input and difficulties of implementation of the propagation of variability.
A new approach, known as nonparametric probabilistic approach of random uncertainties in
dynamics of the structures was recently proposed by C. Soize ([bib20] with [bib24]). This approach
allows to take into account uncertainties of model (uncertainties on the geometry for example) and
uncertainties of modeling (uncertainties on the kinematics of beam or plate for example).
It is based on the construction of random matrices of the linear dynamic systems, afterwards
projection on modal basis.
These two probabilistic approaches, one parametric and the other nonparametric, are
complementary. Thus a mixed, parametric and nonparametric approach, can be is
developed (original method having given place to publications ([bib6] and [bib11]). In particular,
this mixed method is well adapted to the taking into account of uncertainties in the analysis of a system
nonlinear dynamics composed of a structure linaire reduced on modal basis and of non-linearities
located. Indeed, uncertainties on the level of the linear structure can be treated
naturally by the nonparametric approach and uncertainties on non-linearities can be
treated naturally by the parametric approach.
The basic digital model of the nonlinear dynamic system is a model finite elements which
will be called “model fine elements means”. That it is about the parametric approach or not
parametric, the laws of probability must be defined in an adequate way and most objectively
possible starting from this average model. A Gaussian model of the matrices random is not adapted to
dynamics in low frequency (negative Eigen frequencies). In order to build the law of
corresponding probability, one uses the principle of the maximum of entropy of Jayne ([bib14], [bib15],
[bib17]) as well as information available (model fine elements means, algebraic properties of
matrices, etc)
In this document, we present the nonparametric approach for transient resolutions or
harmonic of the dynamic system. The parametric approach is more particularly presented
if it combined with the non-parametric method.
The readers seeking the fundamental results of stochastic dynamics will be able to refer to
[18] and readers seeking the theoretical details of the probabilistic approach presented in it
document will be able to refer to [bib19]. Examples of uses of the approach are given in
[bib5], and [bib7]. In [bib10], experimental tests made it possible to show the predictive character of
approach.
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
5/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
2
Modelings of the dynamic system
2.1
Average model finite elements
2.1.1 Transitory resolution in absolute co-ordinates
In the absolute reference mark, the mechanical system is modelized by the method with the finite elements. It
basic model (in general that which would have been used in the deterministic study) is indicated under
name of “average model finite elements”. All the sizes relating to the average models are
underlined.
That is to say
)
(T
T
y
has
the transitory response in the absolute reference mark of the “model finite elements average” definite
on the interval of study
[0,]
T
and with value in
K
R
where
K
is the number of D.D.L. Matrices of mass,
of damping and rigidity are respectively noted
[]
M
,
[]
D
and
[]
K
.
The transitory answer
)
(T
y
“average model finite elements” checks the differential equation not
linear discretized following:
[] () [] () [] ()
(, (), (); )
()
C
T
T
T
T
T
T
T
+
+
+
=
y
y
y
F
y
y
W
F
&&
&
&
M
D
K
, T
[0,]
T
, éq
2.1.1-1
with the initial conditions,
(0)
(0) 0
=
=
y
y&
, éq
2.1.1-2
1)
()
T
F
m
R
represent the discretization by finite elements of the external forces.
2)
)
),
(
),
(
,
(
W
y
y
F
T
T
T
C
&
m
R
corresponds to nonthe localized linearities (for example due to
butted elastic of shock). Elements
1
,
W
W
L
vector
W
R
represent a play
parameters defining these nonlinearities (for example play, stiffness of shock,
damping of shock, etc).
2.1.2 Transitory resolution in relative co-ordinates (seism)
As in the transitory case in absolute co-ordinates, the mechanical system is modelized by one
basic model, the “average model finite elements”.
One notes
()
T
T
Z has
the transitory response in absolute co-ordinates of this model on the interval
of study
[0,]
T
with value in
K
R
(attention: notice the change of notation compared to
preceding paragraph.).
The transitory answer
() ((), ())
S
T
T
T
=
Z
Z
Z
“average model finite elements” checks the equation
following discretized nonlinear differential:
[]
[
]
()
[]
[
]
()
[]
[
]
()
[
]
[
]
()
[
]
[]
()
[
]
[
]
()
ls
ls
ls
T
T
T
ls
S
S
ls
S
S
ls
S
S
T
T
T
T
T
T
+
+
Z
Z
Z
Z
Z
Z
&&
&
&&
&
M
M
D
D
K
K
M
M
D
D
K
K
()
(, (), (); )
()
0
C
D
S
T
T
T
T
T
+
=
G
Z
Z
W
G
&
F
, T
[0,]
T
, éq
2.1.2-1
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
6/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
with the initial conditions,
(0)
(0)
=
Z
z&
,
(0)
(0)
S
S
=
Z
z&
,
éq 2.1.2-2
1)
()
T
G
m
R
represent the discretization by finite elements of the external forces and
()
S
T
G
D
R
corresponds to the discretization of the forces of reaction due to D conditions of
Dirichlet
()
S
T
Z
.
2)
(, (), (); )
C
T
T
T
Z
Z
W
&
F
m
R
corresponds to nonthe linearities located with like
previously
W
R
representing a set of parameters defining these nonlinearities.
After static raising, matric equations [éq 2.1.2-1] and [éq 2.1.2-2] in the absolute reference mark
are rewritten in “relative” co-ordinates:
[] () [] () [] ()
(, (), (); )
()
C
T
T
T
T
T
T
T
+
+
+
=
y
y
y
F
y
y
W
F
&&
&
&
M
D
K
, T
[0,]
T
, éq
2.1.2-3
(0)
(0) 0
=
=
y
y&
,
éq
2.1.2-4
1)
()
T
y
m
R
is the vector of D.D.L. free in the “relative” frame of reference such as
()
() [] ()
S
T
T
R
T
=
+
Z
y
Z
with
1
[]
[] [
]
ls
R
-
= -
K
K
.
2)
function
()
T
T
F
has
defined on
[0,]
T
and with value in
m
R
and the nonlinear application
()
(
)
W
y
X
F
y
X
;
,
,
,
T
C
has
of
m
R
×
m
R
in
m
R
are such as
()
() ([] [] [
]) () ([] [] [
]) ()
ls
ls
T
T
R
T
R
T
=
-
+
-
+
F
G
Z
Z
&&
&
M
M
D
D
, éq
2.1.2-5
(
)
[]
()
[]
()
(
)
.
;
,
,
;
,
,
W
Z
y
Z
X
W
y
X
F
T
R
T
R
T
T
S
S
C
C
&
+
+
=
F
éq
2.1.2-6
Note:
1) In the continuation, according to whether one carried out or not a static raising,
()
T
y
corresponds
maybe with the transitory response in absolute co-ordinates defined by [the §
2.1.1], is
transitory response in “relative” co-ordinates defined by [the §
2.1.2].
2) It is supposed that if them
D
conditions of Dirichlet were homogeneous no movement
rigid body could not occur. Consequently,
[]
K
is symmetrical defined
positive and its reverse
[]
K
- 1
is defined, which makes it possible to introduce the real matrix
1
[]
[] [
]
ls
R
-
= -
K
K
of dimension
(
)
m D
×
.
3) In Code_Aster the term of damping in [éq 2.1.2-5] is neglected.
2.1.3 Resolution
harmonic
As in the transitory case, the mechanical system is modelized by a basic model, it
“average model finite elements”. On a frequential tape
[
]
2
1
,
, the harmonic answer
)
(
Q
linear “model finite elements average” checks the following equation:
(-
2
[]
M
+ I
[]
D
+
[]
K
)
)
(
Q
=
F
(
),
[
]
2
1
,
éq
2.1.3-1
with
()
F
representing the discretization by finite elements of the external forces.
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
7/24
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
2.2
Average reduced matric model
It is supposed that the energy of vibration of the dynamic response is mainly localized in
field of the low frequencies. One can thus build the average matric model reduced in
projecting
()
T
y
or
)
(
y
on under clean space generated by N first modes of the system
dynamics linear (infinite plays) conservative homogeneous (locked supports) associated which is written,
[]
K
=
[]
M
.
éq
2.2-1
Matrices
[]
M
and
[]
K
being definite positive (for
[]
K
cf notices 2 [§ 2.1.2]), the values
clean
1
,
N
L
are real and positive,
1
2
0
N
<
L
. éq
2.2-2
Clean modes of vibration associated {
1
2
,
,
L
} checks the properties of orthogonality,
<
[]
M
,
> =
µ
, éq
2.2-3
<
[]
K
,
> =
2
µ
,
éq
2.2-4
with
=
.
éq 2.2-5
One respectively notes the matrix of generalized mass, the matrix of generalized stiffness and
stamp damping generalized by:
]
[
N
M
=
[
]
T
N
[]
M
[
]
N
, éq
2.2-6
]
[
N
K
=
[
]
T
N
[]
K
[
]
N
, éq
2.2-7
]
[
N
D
=
[
]
T
N
[]
D
[
]
N
,
éq
2.2-8
2.2.1 Resolution in transient
Projection
()
N
T
y
of
()
T
y
on under space generated by N first modes of the system
homogeneous dynamics linear conservative associated is written:
1
() [
] ()
()
N
N
N
N
N
T
T
Q T
=
=
=
y
Q
,
éq
2.2.1-1
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
Generalized displacements
()
N
T
Q
are solutions of the average reduced matric model (system
dynamics not linaire),
]
[
N
M
()
N
T
Q
&&
+
]
[
N
D
()
N
T
q&
+
]
[
N
K
()
N
T
Q
+
N
C
F
(T,
()
N
T
Q
,
()
N
T
q&
;
W
) =
()
T
N
F
, éq
2.2.1-2
(0)
(0) 0
N
N
=
=
Q
Q
&
,
éq
2.2.1-3
with
()
T
N
F
=
[
]
T
N
()
T
F
,
éq
2.2.1-4
(
)
W
p
Q
F
;
,
,
T
N
C
=
[
]
T
N
C
F
(
,
T [
]
N
,
Q
[
]
N
W
p;
). éq
2.2.1-5
2.2.2 Resolution in harmonic
Projection
)
(
N
y
of
)
(
y
on under space generated by N first modes of the system
homogeneous dynamics linear conservative associated is written
)
(
N
Q
=
[
]
T
N
()
,
F
éq
2.2.2-1
Generalized displacements
)
(
N
Q
are solutions of the model matric average tiny room
(-
2
]
[
N
M
+ I
]
[
N
D
+
]
[
N
K
)
)
(
N
Q
=
()
N
F
éq
2.2.2-2
with
()
N
F
=
[
]
T
N
()
F
, éq
2.2.2-3
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
9/24
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
3 Model
probabilist
3.1
Introduction of the probabilistic model into the dynamic problem
In order to take into account uncertainties of modeling and uncertainties on the data, one
mixed probabilistic formulation nonparametric parametric is used. For that, the vector of N
d.d.l. generalized
()
N
T
Q
(resp.
)
(
N
Q
) is replaced by a random variable
)
(T
N
Q
(resp.
)
(
N
Q
).
In transient, the stochastic process
()
N
T
T
Q
has
indexed by
[0,]
T
and with value in
N
R
is
solution of the nonlinear dynamic system,
[]
()
[]
()
[]
()
() ()
(
)
()
,
;
,
,
T
T
T
T
T
T
T
N
N
N
N
C
N
N
N
N
N
N
F
W
Q
Q
F
Q
K
Q
D
Q
M
=
+
+
+
&
&
&&
éq
3.1-1
(0)
(0) 0
N
N
=
=
Q
Q&
,
éq 3.1-2
and in harmonic, the stochastic process
)
(
T
N
Q
has
indexed on
[
]
2
1
,
and with value in
N
R
is solution of the system:
(-
2
[]
N
M
+ I
[]
N
D
+
[]
N
K
)
)
(
N
Q
=
N
F
(
),
éq
3.1-3
where, in the two transitory and harmonic cases,
]
[
N
M
,
]
[
N
D
and
]
[
N
K
are real matrices
symmetrical definite positive full random and where
W
is a random variable with value in
R
.
The introduction of random matrices into the equations [éq 3.1-1] and [éq 3.1-3] makes it possible to modelize
the random uncertainties associated the linear part of the dynamic system. The random variable
W
vectorial value introduced into the equation [éq 3.1-1] allows to modelize uncertainties
random concerning the parameters of nonthe linearities of shock.
The parametric probabilistic approach and the nonparametric probabilistic approach introduce
random matrices (
]
[
N
M
,
]
[
N
D
and
]
[
N
K
) and a random variable
W
of which laws of probability
are a priori nonknown. The choice of a probabilistic model rather than another must rest
only on information available (algebraic properties of the generalized matrices, values
averages of the parameters and the generalized matrices, etc). In order to build the laws objectively
of probability of the probabilistic model of uncertainties, ([bib20] with [bib24]), the principle of the maximum
of entropy ([bib14], [bib15], [bib17]) is used with a system of stresses defined by this
information available. Information available and the probabilistic model which results from this are presented
in the next paragraph.
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
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Key
:
R4.03.05-A
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:
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
3.2 Probabilistic model for the matrices of the dynamic system
(nonparametric uncertainties)
3.2.1 Information available on the matrices of the dynamic system
The nonparametric probabilistic model is built in substituent the matrices
]
[
N
M
,
]
[
N
K
and
]
[
N
D
by respectively noted random matrices
]
[
N
M
,
]
[
N
K
and
]
[
N
D
. So that the system
probabilistic dynamics thus built either mechanically and statistically correct, construction
random matrices
]
[
N
M
,
]
[
N
K
and
]
[
N
D
must be such as:
1)
]
[
N
M
,
]
[
N
K
and
]
[
N
D
are of the random variables of the second command with values in
the whole of the positive definite real matrices symmetrical and dimension
)
(
N
N
×
.
]
[
N
M
,
]
[
N
K
and
]
[
N
D
+
N
M
P.S. (almost surely),
éq
3.2.1-1
where
+
N
M
is the whole of the positive definite symmetrical matrices real of dimension
)
(
N
N
×
.
This algebraic property is absolutely required to have a random model of equation which
corresponds to that of a dynamic system of the second deadened command.
2) Average values of the random matrices
]
[
N
M
,
]
[
N
K
and
]
[
N
D
are respectively
]
[
N
M
,
]
[
N
K
and
]
[
N
D
:
[]
{
}
[]
N
N
M
=
M
E
,
[]
{}
[]
N
N
K
=
K
E
and
[]
{}
[]
N
N
D
=
D
E
,
éq 3.2.1-2
where
E
indicate the expectation.
3) So that the solution of the probabilistic dynamic system is also a variable of the second
command, one imposes on the moments of the second command standards of Frobenius of the matrices
opposite
1
]
[
-
N
M
,
1
]
[
-
N
K
and
1
]
[
-
N
D
to be finished:
E
{||
]
[
N
M
- 1
||
2
F
} <
+
,
E
{||
]
[
N
K
- 1
||
2
F
} <
+
,
E
{||
]
[
N
D
- 1
||
2
F
} <
+
, éq 3.2.1-3
with
[]
[] []
{
}
(
)
2
/
1
tr
T
F
With
With
With
=
.
Note:
The only property of positivity of the matrices is not enough and only their opposite should be secured
are of the second command, from where (3 (a random variable of the second command almost surely
invertible in the general case a random variable does not have reverses of the second command). For
details to see more [bib19].
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
11/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
3.2.2 Construction of the probabilistic model by the principle of the maximum of entropy
The entropy “measures” the level of uncertainty of a law of probability. Thus, if
]
[
With
p
is related to
density of probability corresponding to a random matrix
]
[A
(representing the matrices
]
[
N
M
,
]
[
N
K
or
]
[
N
D
) of law given, then entropy (or probabilistic uncertainty)
)
(
]
[
With
p
S
of
]
[
With
p
is defined
by:
With
D
With
p
With
p
p
S
N
M
~
]))
([
ln (
])
([
)
(
]
[
]
[
]
[
+
-
=
With
With
With
,
éq
3.2.2-1
The principle of the maximum of entropy of Jayne consists in building the function of density of probability
who maximizes the probabilistic entropy
)
(
]
[A
p
S
while checking a system of stresses. In the case
present, the system of stresses is defined by information available corresponding to the equations
[éq 3.2.1-1] with [éq 3.2.1-3]. For the random matrix
]
[A
, this system of stresses is written
+
N
M
]
[A
P.S.,
{}
]
[
]
[
With
E
=
With
,
E
{|| [
With
]
- 1
||
2
F
} <
+
.
éq 3.2.2-2
It is shown whereas the random matrix [
With
] is such as (see [bib20] with [bib24])
]
] [
[
]
[
]
[
With
With
T
With
L
L
G
With
=
,
éq
3.2.2-3
where
]
[
With
L
is the lower triangular matrix resulting from the factorization of Cholesky of the matrix
average
]
[A
and where the function of density of probability of the random matrix
]
[
With
G
is defined on
the unit
+
N
M
compared to the measurement
G
d~
such as:
ij
N
J
I
N
N
dG
G
D
-
=
1
4
/
)
1
(
2
~
,
éq
3.2.2-4
]
tr [
)
2
) (
1
(N
)
1
(
)
2
) (
1
(
]
[
1
2
With
1
2
2
E
])
(det [
])
([
])
([
G
N
G
M
With
With
With
N
With
G
C
G
G
p
-
-
+
+
-
+
-
×
×
×
=
1
G
, éq
3.2.2-5
with
)
(
2
1
2
1
2
1
2
1
)
2
) (
1
(
2
4
/
)
1
(
2
1
)
2
(
J
N
With
With
With
N
J
N
N
With
N
N
G
N
C
-
+
+
+
=
=
+
-
-
-
,
éq
3.2.2-6
where
+
-
-
=
0
T
1
E
)
(
dt
T
Z
Z
,
éq
3.2.2-7
and where
+
N
M
1
is the indicating function of
+
N
M
, and where the parameter
With
controlling the dispersion of
random matrix
]
[A
is defined by:
[] []
{
}
[]
2
/
1
2
2
E
-
=
F
With
F
With
With
With
G
G
G
, éq
3.2.2-8
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The theoretical construction of the model provides an acceptable terminal for the level of uncertainty introduced.
With
must be selected so that
0
0
1
0
5
With
N
N
+
<
<
+
,
éq
3.2.2-9
where
0
N
NR
is a constant of the probabilistic model selected so that
0
N
N
<
.
One shows moreover than, under the only stresses of the equations [éq 3.2.1-1] with [éq 3.2.1-3], it
principle of the maximum of entropy leads so that the random matrices
]
[
N
M
,
]
[
N
K
or
]
[
N
D
are
statistically independent as a whole.
This probabilistic model for the positive definite symmetrical matrices random real differs from
more conventional models of the random matrices based on the Gaussian Assemblies and them
circular assemblies (references in [bib19]). The orthogonal Gaussian unit used in addition
for fields high frequencies would lead in the field low frequencies (in which one
places itself) at negative eigenvalues, which one cannot admit for the systems
considered. Moreover, one matrix of the orthogonal Gaussian unit does not have in the case
General an opposite matrix of the second command, which would lead to a solution of the dynamic system
of infinite variance, which one cannot admit either.
3.3 Probabilistic model for the real variables (uncertainties
parametric)
3.3.1 Information
available on the real variables
In the mixed probabilistic approach, parametric probabilistic modeling consists in substituting it
parameter
W
non-linearities in the nonlinear dynamic systems given by [éq 2.1.1-1]
or [éq 2.1.2-3] by a noted random variable
(
)
W
,
,
W K
1
=
W
. In an approach purely
parametric (i.e without making random the matrices of the system dynamic), modeling
parametric probabilist consists in substituting certain parameters
W
matrices
[
]
)
(W
N
M
,
[
]
)
(W
N
K
and
[
]
)
(W
N
D
average dynamic system reduced by a random variable
W
. These
parameters can be for example parameters of material.
It is supposed that the components of
W
are independent real random variables enters
they and independent of the random matrices of the dynamic system. In the continuation, to reduce
the writing, one notes
W
an unspecified co-ordinate
J
W
. The construction of the probabilistic model
require to define the information available, which constitutes a system of stress under which
entropy of the density of probability of the random variable
W
is maximized.
Information available is as follows:
1) The support of the random variable
W
is an interval
D
of
R
.
.
.
,
S
p
D
W
éq
3.3.1-1
2) The average value of the random variable
W
is
W
:
{}
W
W
E
=
.
éq
3.3.1-2
3) Possibly, according to information indeed available, the moment of the second command of
the random variable
1
-
W
is finished:
{}
.
2
+
<
-
W
E
éq
3.3.1-3
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3.3.2 Construction of the probabilistic model by the principle of the maximum of entropy
If
W
p
is related to density of probability corresponding to the random variable
W
then entropy
(probabilistic uncertainty)
)
(
W
p
S
of
W
p
is defined by:
dw
W
p
W
p
p
S
W
W
W
))
(
(
ln
)
(
)
(
+
-
-
=
,
éq
3.3.2-1
By using the principle of the maximum of entropy, one obtains three densities of probability according to nature
support
D
and according to whether the stress corresponding to the equation [éq 3.2.2-6] is considered or not.
3.3.3 Closed support limited without information on the reverse
If there are two realities
has
and
B
such as
[]
B
has
D
,
=
and if information available is given by
equations [éq 3.3.1-1] and [éq 3.2.2-5], then the random variable
W
follows a truncated exponential law
whose function of density of probability is:
)
exp (-
)
(
)
(
)
(
]
,
[
kw
K
K
W
W
p
has
W
+
=
1
éq 3.3.3-1
where
]
,
[B
has
1
is the indicating function of
]
,
[B
has
and where
)
(K
and
K
are such as
0
)
(
)
(
)
1
(
=
-
-
K
K
K
K
W
, éq
3.3.3-2
with
K
ak
E
E
)
(
B
K
-
-
-
=
,
éq
3.3.3-3
and
K
ak
E
E
)
(
B
B
has
K
-
-
-
=
.
éq
3.3.3-4
3.3.4 Closed semi support not limited without information on the reverse
If there is a reality
has
such as
[
[
+
=
,
has
D
and if information available is given by the equations
[éq 3.3.1-1] and [éq 3.3.1-2] then the random variable
W
follows an exponential law of which the function of
density of probability is:
)
exp (-
1
)
(
)
(
]
,
[
has
W
has
W
has
W
W
W
p
has
W
-
-
-
=
+
1
, éq
3.3.4-1
where
[
,
0
[
+
1
is the indicating function of
[
[
+
,
has
.
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3.3.5 Closed semi support not limited with information on the reverse
If there is a reality
has
such as
[
[
+
=
,
has
D
and if information available is given by the equations.
[éq 3.3.1-1], [éq 3.3.1-2] and [éq 3.3.1-3], then the random variable
W
a law gamma follows of which
function of density of probability is,
-
-
-
-
=
-
+
2
)/
-
(1
2
/
1
2
2
]
,
[
)
(
-
exp
)
(
)
/
1
(
)
(
)
(
)
(
2
2
2
has
W
has
W
has
W
has
W
W
W
p
has
W
1
,
éq 3.3.5-1
where
is a parameter controlling the level of uncertainty of the random variable which is written
W
(of
way similar to the nonparametric case [éq 3.2.2-8]):
(
)
{
}
2
/
1
2
2
E
-
=
W
W
W
,
éq
3.3.5-2
3.4 Construction of the stochastic answer and the statistics
associated
3.4.1 Case
transient
3.4.1.1 Stochastic transitory answer
The excitations of the dynamic system are supposed to be deterministic, but in the paragraph [§3.1], of
matrices and of the random parameters were introduced into the reduced matric model. Therefore,
transitory answer
()
N
T
T
Q
has
is a nonstationary stochastic process indexed by
[0,]
T
with
value in
N
R
(by using some additional assumptions of existence, unicity and of
regularity of the deterministic solution, cf [bib19]).
Consequently, with the vector of
m
D.D.L. free
()
N
T
y
corresponds the stochastic process
)
(T
N
Y
indexed by
[0,]
T
and with value in
m
R
such as
()
[]
()
,
T
T
N
N
N
Q
Y
=
éq
3.4.1.1-1
In the case of the passage in relative co-ordinates, with the stochastic process
()
N
T
T
Y
has
indexed by
[0,]
T
and with value in
m
R
defined by the equation [éq 3.4.1.1-1] corresponds the stochastic process
()
N
T
T
Z
has
D.D.L. free of the structure in absolute co-ordinates indexed by
[0,]
T
and with value
in
m
R
such as
()
() [] ()
N
N
S
T
T
R
T
=
+
Z
Y
Z
.
éq
3.4.1.1-2
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3.4.1.2 Elastic spectrum of answer
One notes
()
N
J
Z T
the J
ème
component of the vector
()
N
T
Z
agent with a random realization of
stochastic response of the J
ème
D.D.L. free of the structure.
()
N
J
Z T
perhaps characterized by its spectrum
of elastic answer (also called spectrum of oscillator in Doc. of Code_Aster) that one
note
(,)
J
S
where
and
are respectively the associated rate of depreciation and the pulsation.
With reasonable assumptions, in particular on the regularity of the nonlinear application
(
)
(
)
W
p
Q
F
W
p
Q
;
,
,
;
,
,
T
T
N
C
has
, one can show that
()
N
T
Z
is a process of the second command of which
the trajectories are almost surely continuous. Consequently, for all
fixed in an interval
J
given,
(,)
J
S
has
is a stochastic process indexed on the tape of analysis
J
with value
in
+
R
. It is admitted that this process is of the second command, i.e.:
2
{(,)}
,
J
E S
J
< +
. éq
3.4.1.2-1
3.4.2 Case
harmonic
In the paragraph [§3.1], random matrices and parameters were introduced into the model
matric tiny room. The harmonic answer
)
(
T
N
Q
has
is thus a stochastic process indexed on
[
]
2
1
,
with value in
N
R
.
Consequently, with the vector of
m
D.D.L. free
)
(
N
y
the process corresponds
stochastic
)
(
N
Y
indexed on
[
]
2
1
,
and with value in
m
R
such as
()
[]
()
N
N
N
Q
Y
=
éq
3.4.2-1
)
(
N
J
Y
the J
ème
component of the vector
)
(
N
Y
is a random variable which one will admit of the second
command.
3.4.3 Construction of the stochastic response by the Monte Carlo method
3.4.3.1 Choice and implementation of the Monte Carlo method
The answers and the spectra of answer correspond to strongly nonlinear transformations
random matrices and random parameters which result from probabilistic modeling from
uncertainties. Moreover, one can of course build only numerical approximations as of these
answers and of these spectra of answer. The statistics (first moments statistical, probability
of going beyond of a threshold,…) are written formally like multiple integrals of very large
dimension because the number of random variables of the probabilistic model is by construction high.
Lastly, the number of sizes observed is very large (several ddl for several frequencies).
For all these reasons, method the most adapted to calculate the probabilistic solution (answer
stochastic and associated statistics) is the method of digital simulation of Monte Carlo.
The method of simulation of Monte Carlo has the advantage of giving results which one can
to control the precision (checking of convergence, cf [§ 4.1]), contrary to the majority of
methods based on approximations. It can be expensive in calculating times, but
the use of the techniques of reduction of the variance can make it possible to reduce the number of
simulations necessary (cf [bib8] or [bib9]).
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The implementation of the Monte Carlo method consists for the problem which concerns us with
to generate
S
N
achievements of the random matrices
]
[
N
M
,
]
[
N
K
and
]
[
N
D
dynamic system and/or
S
N
achievements of the vectorial random variable
W
. Resolutions of the dynamic system
determinist for each one of
S
N
achievements of (
]
[
N
M
,
]
[
N
K
,
]
[
N
D
,
W
) produce
S
N
achievements
stochastic process solution
()
N
T
T
Q
has
(resp.
)
(
T
N
Q
has
) and in consequence of
()
N
T
T
Y
has
, of
()
N
T
T
Z
has
and of
(,)
J
S
has
(resp.
)
(
N
J
Y
has
). The generation of the random matrices is
treated in the following paragraph; the generation of the random variable
W
is more conventional and is not
not recalled.
3.4.3.2 Generation of the pseudo-random matrices
In order to generate the achievements of the random matrix
[]
With
G
, the algebraic representation is used
following of the random matrix
[]
With
G
the law of probability is defined by the equations
[éq 3.2.1-2], [éq 3.2.2-1]:
[] [] []
L
L
G
T
With
=
,
éq 3.4.3.2-1
the triangular random matrix
[]
L
being such as:
1) Random variables {
[]
L
ij
, I
J} are independent.
2) For
I < J, real random variables
[]
L
ij
are written
[]
L
ij
=
N
U
ij
where
N
=
With
(n+1)
- 1/2
and
where U
ij
is a Gaussian real random variable of average 0 and variance 1.
3) For
I = J, real random variables
[]
L
jj
are written
[]
L
jj
=
N
(2 V
J
)
1/2
where
N
is defined
previously and where V
J
is variable real positive random of law gamma of which the function of
density of probability
)
(v
p
J
v
compared to measurement FD is written:
v
J
N
With
v
With
J
v
J
N
v
v
p
-
-
+
+
+
-
+
+
=
E
)
2
/
)
1
(
)
2
/(
)
1
((
)
(
)
(
2
/
)
1
(
)
2
/(
)
1
(
2
[
,
0
[
2
1
,
éq 3.4.3.2-2
where
[
,
0
[
+
1
is the indicating function of
[
,
0
[
+
.
3.4.4 Statistics on the spectra
In this chapter, one presents the definition of the statistics of the spectra of elastic answer
)
,
(
J
S
, in the case of a transitory resolution. In the harmonic case, the statistics on
random variables
)
(
N
J
Y
are defined in the same way and are thus not presented.
3.4.4.1 Estimate of the quantiles
For all
J
J
×
)
,
(
,
)
,
(
J
S
is a random variable with value in
+
R
. One seeks with
to consider the quantile associated with the probability
noted
)
,
(
,
J
S
and defined by:
)
1
(
)
,
(
1
,
,
-
=
-
F
S
J
éq
3.4.4.1-1
where
,
F
is related to unknown distribution of
)
,
(
J
S
.
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That is to say (
)
;
,
(
1
J
S
,…,
)
;
,
(
R
J
S
,…,
)
;
,
(
S
N
J
S
) the sample made up of
S
N
achievements of
)
,
(
J
S
and (
)
;
,
(
)
1
(
J
S
,…,
)
;
,
(
)
(R
J
S
,…,
)
;
,
(
)
(
S
N
J
S
) the associated ordered sample.
A natural estimator of the quantile
)
,
(
,
J
S
for
S
N
R
=
,
S
N
R
1
is:
)
;
,
(
)
,
(
)
(
,
R
J
N
R
J
S
S
S
=
éq
3.4.4.1-2
To obtain a more robust estimator of the quantile, one can “realize” the estimator on several
series of
S
N
achievements. If the desired probability is such as
S
N
/
1
<
, or if one wishes to reduce
the number of simulations, it is possible to use more sophisticated estimators, for example in
supposing that the function of distribution
,
F
belongs to a field of attraction given (theory of
extreme values) or for example by using a method of regularization bayésienne (cf [bib12]).
3.4.4.2 Extreme values of sample
For a sample of
S
N
achievements of
)
,
(
J
S
noted
)
;
,
(
1
J
S
,…,
)
;
,
(
S
N
J
S
, one defines
extreme values of sample by:
=
=
)
;
,
(
min
log
)
;
,
(
,…,
1
10
min
,
R
J
N
R
S
J
S
N
dB
S
has
éq
3.4.4.2-1
=
=
)
;
,
(
max
log
)
;
,
(
,…,
1
10
max
,
R
J
N
R
S
J
S
N
dB
S
has
éq
3.4.4.2-2
3.4.4.3 “Field of confidence” established starting from the inequality of Tchebychev
For a sample of
S
N
achievements of the process
(,)
J
S
has
noted
)
;
,
(
1
J
S
has
,…,
)
;
,
(
S
N
J
S
has
, one can build the “field of confidence” of the random variable
10
(,) log ((,))
J
J
dB
S
=
for all
(,) J
J
×
, by using the inequality of Tchebychev
associated a level of probability
C
P
:
{
}
C
J
J
J
P
dB
dB
dB
<
<
+
-
)
,
(
)
,
(
)
,
(
Proba
,
éq
3.4.4.3-1
where the lower envelope
(,)
J
dB
-
has
and the higher envelope
(,)
J
dB
+
has
are defined
by:
-
+
=
+
C
J
J
J
P
m
dB
1
)
,
(
)
,
(
log
)
,
(
1
10
,
éq 3.4.4.3-2
(
)
)
,
(
)
,
(
log
2
)
,
(
1
10
+
-
-
=
J
J
J
dB
m
dB
. éq
3.4.4.3-3
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
18/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
with
1
(,)
J
m
the average and
(,)
J
the standard deviation of
)
,
(
J
dB
:
1
(,)
{(,)}
J
J
m
E S
=
, éq
3.4.4.3-4
()
()
()
(
)
{
}
2
/
1
2
1
,
,
E
,
J
J
J
m
S
-
=
.
éq
3.4.4.3-5
The “field of confidence” thus built proved to be a good approximation of the values
extremes of sample for the case treated in [bib21]. However, this “field of confidence” does not utilize
that the first two moments whose consistent estimators more quickly with respect to the number
S
N
simulations that extreme values of sample. It can thus be interesting to use this
construction of the “field of confidence” rather than a construction based on the estimate of the quantiles
more expensive in a number of simulations.
Note:
The term “field of confidence”, can be considered by certain an abuse
language. One should rather use the less intuitive terminology “inter-quantiles field”. In
effect, in the statistical literature, a confidence interval is theoretically the interval
in which the true value of a parameter of a random variable is (for example its
average) with a given probability. This terminology is employed within the framework very
precis of the theory of the ensemblist estimate. The confidence interval is not one
characterization of the variability of a random variable, contrary to a standard deviation or to
quantiles. One nevertheless uses " field of confidence " with parsimony in the continuation,
because it is certainly a little more speaking for the non-specialists of the statistics.
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
19/24
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
4
Implementation in Code_Aster
4.1
Study of the stochastic convergence of the digital model
4.1.1 Case
transient
The convergence of the stochastic solution must be studied compared to number N of modes and with
N numbers
S
simulations of Monte Carlo. As the stochastic solution
()
N
T
Z
is a process of
second command (by assumption, cf [§3.4.1.2]), its convergence can be analyzed by studying them
applications
|||
|||
N
J
N
Z
&&
has
such as:
{
}
dt
T
Z
Z
T
N
J
N
J
=
0
2
2
)
(
E
&&
&&
,
éq
4.1.1-1
where
()
N
J
T
Z T
&&
has
is a stochastic process of the second command indexed by
[0,]
T
and with value in
R
representing the acceleration of the J
ème
D.D.L. structure.
In the framework of simulations of Monte Carlo, this standard
|||
|||
N
J
Z
&&
is estimated for N fixed to leave
of a whole of
S
N
random achievements
1
(; ),
(;
)
S
N
N
J
J
N
Z T
Z T
&&
&&
L
by the approximation
|||
||| conv (,)
N
J
J
S
Z
N N
&&
with
dt
T
Z
N
N
N
T
N
I
I
N
J
S
S
J
S
2
0
1
2
)
;
(
1
)
,
(
conv
=
=
&&
éq
4.1.1-2
The stochastic convergence of the model is thus analyzed according to the dimension of the small-scale model
(i.e. the number of mode
N
under clean space of the average model finite elements on which
the stochastic nonlinear dynamic system was projected in the paragraph [§ 2.2]) and numbers it
S
N
simulations of Monte Carlo by studying the function
conv (,)
S
J
S
N
N N
has
.
4.1.2 Case
harmonic
Convergence in the case of a transitory resolution can be transposed directly in the case
of a harmonic resolution, with the standard:
{
}
D
Z
Z
N
J
N
J
=
2
1
2
2
)
(
E
,
éq
4.1.2-1
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
20/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
4.2
Choice of the parameters of dispersion
To use the method, the parameters of dispersion
must be fixed. Two approaches can
to be used a priori to fix the value of these parameters.
The first approach consists in identifying the value of the parameters
for a given structure or
for a class of structure using suitable methods. For that, one can use results
experimental of the dynamic responses of the structure. One can also use simulations
numerical built by using a parametric approach of uncertainties. In this last case,
it should be noted that only the errors on the data of the model are taken into account, since them
errors of modeling cannot be taken into account by the parametric approach.
The second approach consists in not fixing a priori a fixed value of the parameters
but with
to vary in a range given (only 3 scalars to vary for the matrices of mass,
of stiffness and damping on the not-parametric part in comparison with the very large one
a number of parameters to vary simultaneously in a conventional parametric study). This
approach makes it possible to carry out a total analysis of sensitivity to uncertainties. In the case of absence
of objective information on the parameters dispersion to be chosen, it is preferable to use such
approach. The non-parametric method suggested seems a robust approach then and
simple of analysis of sensitivity to uncertainties.
4.3 Main
stages
The implementation in Code_Aster is made up of three main stages: the construction of
average reduced matric model, the generation of the achievements of the answer seen like a process
stochastic, and finally the statistical postprocessing of these achievements. Two last stages
constitute in fact the method of digital simulation of direct Monte Carlo.
Stage 1: construction of the average reduced matric model
The average reduced matric model is built using a conventional sequence operators
depending on the precise analysis carried out whose main ones can be:
ASSE_MATRICE
,
MODE_ITER_SIMULT
,
MODE_STATIQUE
,
CALC_CHAR_SEISME
,
MACRO_PROJ_BASE
…
Stage 2: generation of the achievements of the transitory answer
S
N
achievements of the stochastic transitory answer are calculated in a loop in language
Python made up of:
1) Generation
p
ième
achievements of the random generalized matrices of mass, stiffness
and of damping by
GENE_MATR_ALEA
(Doc. [U4.36.06]). These matrices are not
diagonals and thus require a full storage.
2) Generation
p
ième
achievements of the random variables of the parameters of non-linearities
by
GENE_VARI_ALEA
(Doc. [U4.36.07]).
3) Calculation of p
ième
realization
)
;
(p
T
N
Q
or
)
;
(
p
N
Q
solution of the matric system
stochastic S. This realization is the solution of the conventional matric system of which them
matrices and the second members are the achievements previously generated. Calculation is
thus carried out by
DYNA_TRAN_MODAL
or
DYNA_LINE_HARM
(with
matr_asse_GENE_R
and
vect_asse_GENE
in input).
4) 1 -
Extraction of the temporal observations of D.D.L. preset physiques (by
example
& & (; )
Z T p
I
N
or
)
;
(
p
Y
N
J
, but also possibly fields of displacement,
speed, stresses, etc) via
RECU_FONCTION
(after one
REST_BASE_PHYS
for
)
;
(
p
Y
N
J
).
2 -
Calculation of the spectra corresponding (by
CALC_FONCTION (SPEC_OSCI)
for
S has (; )
J
p and
CALC_FONCTION (MODULE)
for
)
;
(
p
Y
N
J
).
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
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:
21/24
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R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
5) Evaluation, via
CALC_FONCTION
key words
COMB
or
POWER
or
WRAP
, of
contributions to the estimators of the averages, the moments of command two, the values
max. extremes and min. of sample for the standardized spectra:
$m (; ) S (; ) $m (;
)
1
1
1
J
J
J
p
p
p
=
+
-
, $m (; ) S (; )
$m (;
)
2
2
2
1
J
J
J
p
p
p
=
+
-
,
{
}
$S
(; )
Max S (; ), $S
(;
)
, max
, max
J
J
J
p
p
p
=
-
1,
{
}
$S
(; )
Min S (; ), $S
(;
)
, min
, min
J
J
J
p
p
p
=
-
1.
Stage 3: statistical postprocessings
Averages, standard deviations, max. extreme values and min. of sample for the spectra
standardized can be evaluated via
CALC_FONCTION (COMB)
:
m (,)
$m (; )
1
1
1
J
S
J
S
N
N
=
,
m (,)
$m (; )
2
2
1
J
S
J
S
N
N
=
.
The confidence intervals can then be traced starting from the extreme values of sample or
terminals obtained by Tchebychev cf [§ 3.4.4].
In the transitory case, an example is given by a case test of a flexbeam with not
linearities of shock, cf Doc. [V5.06.001] [bib1]. Other details are given in Doc. [U2.08.05]
[bib2].
4.4
Numerical effectiveness of the nonparametric approach
The nonparametric approach is more economic in calculating times than an approach purely
parametric in which the parameters of geometry, materials, etc are random variables.
In the purely parametric approach, the model finite elements depends on the dubious parameters.
For each simulation of Monte Carlo, the model finite elements is different. It is thus necessary, for
each simulation, to calculate the elementary matrices, to carry out the assemblies, to pass in
relative co-ordinates, to solve the problem with the eigenvalues, to project on modal basis, to solve
the reduced system and to return in physical base then in relative co-ordinates.
In the nonparametric approach, only the reduced system is different with each simulation. It is thus
simply necessary, with each simulation, to solve the reduced system and to return in base
physics then in relative co-ordinates. In particular, the resolution of the problem to the eigenvalues
model average finite elements is carried out once and for all, before simulations of
Monte Carlo.
The saving of time of calculation which results from it is variable, but it can be important. In first
approximation, this saving of time calculation depends on the ratio between time CPU necessary to the resolution
at the eigenvalues and time CPU necessary to the resolution of the reduced system. More this ratio is
large, more the nonparametric approach is advantageous compared to the approach purely
parametric. In particular, the saving of time of calculation can be very important for structures
with a very great number of degrees of freedom and a low-size modal base.
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
22/24
Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
5 Bibliography
[1]
CAMBIER S., DESCELIERS C.: Nonparametric probabilistic model parametric of one
flexbeam with nonlocalized linearities of shock, Code_Aster documentation
[V5.06.001].
[2]
CAMBIER S., DESCELIERS C.: Digital simulation of Monte Carlo, documentation
Code_Aster [U2.08.05].
[3]
CAMBIER S., DESCELIERS C.: Operator
GENE_VARI_ALEA
, Code_Aster documentation
[U4.36.06].
[4]
CAMBIER S., DESCELIERS C.: Operator
GENE_MATR_ALEA
, Code_Aster documentation
[U4.36.07].
[5]
CAMBIER S., DESCELIERS C., SOIZE S.: Taking into probabilistic account of uncertainties
in the estimate of the seismic behavior of a primary education circuit, Notes EDF-R & D
HT-62/03/005/A, 2003
[6]
CAMBIER S., DESCELIERS C., SOIZE S.: Taking into probabilistic account of uncertainties
in the estimate of the seismic behavior of a primary education circuit. 7th National conference
AFPS, 2003.
[7]
CAMBIER S.: Application of the probabilistic approach to the fuel assembly in
tally from “3% AMA” in 2002, CR-AMA-03-004, December 2002
[8]
CAMBIER S.: Probabilistic approach for the taking into account of the dispersion of parameters
mechanics - Application to vibratory fatigue, Notes EDF-R & D HT-64/00/040/A, December
2000
[9]
CAMBIER S., GUIHOT P., COFFIGNAL G.: Computational methods for accounting off
structural uncertainties, applications to dynamic behavior prediction off piping systems.
Structural Safety, 24:29 - 50, 2002.
[10]
CHEBLI H.: Modeling of nonhomogeneous Random Uncertainties in Dynamics of
Structures for the Field of the Low frequencies Thesis CNAM-ONERA, 2002
[11]
DESCELIERS C., SOIZE C., CAMBIER S.: Nonparametricparametric model for random
uncertainties in nonlinear structural dynamics Application to earthquake engineering, subjected
in Earthquaque Engineering and Structural Dynamics in 2003
[12]
DURBEC V., TROTTIER C (INRIA), DIEBOLT J. (CNRS): Regularization of distributions
for a better extreme adequacy: review article of CRECO EDF/INRIA EP 902,
Note EDF-R & D HP-26/00/008/A
[13]
IBRAHIM R.A.: Structural dynamics with parameter uncertainties, Applied Mechanics
Reviews, vol. 40, No 3, pp.309-328, 1987
[14]
JAYNES E.T., Information theory and statistical mechanics, Physical Review, 106 (4),
620-630 and 108 (2), 171-190, 1957
[15]
KAPUR J. NR, KEYSAVAN H.K.: Entropy Optimization Principles with Applications, Academic
Close, San Diego, 1992
[16]
KATAFYGIOTIS L.S., PAPADIMITRIOU C.: Dynamic Response Variability off Structures with
Uncertain Properties, Earthquake Engineering and Structural Dynamics, vol. 25, pp.775-793,
1996
[17]
SHANNON E.C.: With mathematical theory off communication, Beautiful System Tech. J., 27,
379-423 and 623-659, 1948
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
Page
:
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Manual of Reference
R4.03 booklet: Analyze sensitivity
HT-66/03/005/A
[18]
SOIZE C.: Methods of studies of the dynamic conventional problems stochastic ones,
Techniques of the Engineer, treaty fundamental Sciences, A 1.346, 61p, 1988
[19]
SOIZE C.: Parametric Mixed Probabilistic model Non Parametric of Uncertainties in
Nonlinear dynamics of the Structures, scientific Report/ratio CRECO T62/E28858, 2001
[20]
SOIZE C.: Modeling of random uncertainties in elastodynamic transient, Accounts
Returned Academy of Science - Series IIB Mechanics, 329 (3), 225-230, 2001.
[21]
SOIZE C.: Nonlinear dynamical System with nonparametric model off random uncertainties,
The Uncertainties in Engineering Mechanics Newspaper, E-newspaper form Resonance Publication,
HTTP://www.resonance-pub.com/eprints.htm, 2001
[22]
SOIZE C.: Transient responses off dynamical systems with random uncertainties, Probabilistic
Mechanics engineering, 16 (4), 363-372, 2001
[23]
SOIZE C.: Maximum entropy approach for modeling random uncertainties in transient
elastodynamics, J. Acoust. Plowshare amndt, 109 (5), 1979-1996, 2001.
[24]
SOIZE C.: Nonlinear dynamical System with nonparametric model off random uncertainties,
The Uncertainties in Engineering Mechanics Newspaper, E-newspaper form Resonance Publication,
HTTP://www.resonance-pub.com/eprints.htm, 2001
Code_Aster
®
Version
6.4
Titrate:
Parametric and not-parametric probabilistic models in dynamics
Date
:
06/05/03
Author (S):
S. CAMBIER, C. DESCELIERS
Key
:
R4.03.05-A
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:
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