background image
Code_Aster
®
Version
8.2
Titrate:
Operator
GENE_MATR_ALEA
Date:
22/02/06
Author (S):
S. CAMBIER, C. DESCELIERS
Key
: U4.36.06-C1
Page:
1/6
Instruction manual
U4.3- booklet: Function
HT-62/06/004/A
Organization (S):
EDF-R & D/AMA














Instruction manual
U4.3- booklet: Function
Document: U4.36.06



Operator GENE_MATR_ALEA







1 Goal
To generate achievements of generalized matrices considered as random for structures
or of the substructures. The law of probability of the matrices is built according to the principle of the maximum
of entropy by considering information available (average and coefficient of variation) and theirs
algebraic properties (definite symmetry positivity) [R4.03.05].
Product a structure of data matr_asse_gene_R or macr_elem_dyna according to the type of
data input.
background image
Code_Aster
®
Version
8.2
Titrate:
Operator
GENE_MATR_ALEA
Date:
22/02/06
Author (S):
S. CAMBIER, C. DESCELIERS
Key
: U4.36.06-C1
Page:
2/6
Instruction manual
U4.3- booklet: Function
HT-62/06/004/A
2 Syntax
[
macr_elem_dyna
]
= GENE_MATR_ALEA

(
/
MATR_MOYEN
=
average [matr_asse_gene_R]

COEF_VAR
=/
[R]
/0.1
[DEFECT]

/
MATR_MOYEN
=
average [
macr_elem_dyna
]

COEF_VAR_RIGI =/
R
[R]
/0.1
[DEFECT]

COEF_VAR_MASS =/
M
[R]
/
0.
[DEFECT]

COEF_VAR_AMOR =/
C
[R]
/
0.
[DEFECT]

INIT_ALEA
=
nor
[I]
);

If
average = [matr_asse_gene_R]
then
Fr = [matr_asse_gene_R]
If
average = [
macr_elem_dyna
]
then
Fr = [
macr_elem_dyna
]
background image
Code_Aster
®
Version
8.2
Titrate:
Operator
GENE_MATR_ALEA
Date:
22/02/06
Author (S):
S. CAMBIER, C. DESCELIERS
Key
: U4.36.06-C1
Page:
3/6
Instruction manual
U4.3- booklet: Function
HT-62/06/004/A
3 Operands
With or without under-structuring, this operator consists in in fine generating achievements of one or
several noted random matrices in a generic way
]
[A
.
]
[A
is a random variable with value
in the whole of the positive definite real matrices of dimension
()
N
N,
whose law is parameterized
by its average value
]
[A
and its scatter coefficient [R4.03.05].
3.1 Word
key
MATR_MOYEN
MATR_MOYEN = average
average
indicate the average matrix
]
[A
random matrix
]
[A
.
If
average
is of type
[matr_asse_gene_R],
then
]
[A
is obtained by projection of one
stamp average assembly of the average model to the finite elements on a given number of modes
clean of the dynamic system (operator
MACRO_PROJ_BASE
for example).
]
[A
and them
achievements of
]
[A
generated by
GENE_MATR_ALEA
can thus be matrices of masses,
generalized stiffness or damping.
Caution:
The average matrix (
]
[A
) must be stored in mode of full storage (operator
NUME_DDL_GENE
, key word
STOCKAGE=' PLEIN'
or operator
MACRO_PROJ_BASE
, key word
PROFIL=' PLEIN'
.).
So average is of type [macr_elem_dyna] (under-structuring), then
]
[A
is a concept
containing the matrices of rigidity, mass and possibly of damping projected on
base modal substructure supplemented by the matrices of connection of the interfaces, the model
means.
3.2 Word
key
COEF_VAR
COEF_VAR =
/
/
0.1
[DEFECT]
This key word informs the parameter
of control of the dispersion of the generalized matrix
random
]
[A
who can be of mass, stiffness or dissipation. This coefficient of variation
is defined by:
[] []
{
}
[]
2
/
1
2
2
2
2
2
*
)
(
)
(
*
)
1
(
F
F
F
With
With
E
With
tr
With
tr
With
N
-
+
+
=
With
with:
·
[]
[] []
{
}
(
)
]
tr
2
/
1
T
F
With
With
With
=
·
[]
With
N
of
dimension
·
[] []
{
}
[]
1/2
2
2
F
F
With
With
E
-
With
the scatter coefficient of the matrix
]
[A
background image
Code_Aster
®
Version
8.2
Titrate:
Operator
GENE_MATR_ALEA
Date:
22/02/06
Author (S):
S. CAMBIER, C. DESCELIERS
Key
: U4.36.06-C1
Page:
4/6
Instruction manual
U4.3- booklet: Function
HT-62/06/004/A
can also be written:
{
}




-
=
2
2
[
[
]
[
E
F
With
F
With
With
G
G
G
with
[]
With
L
the lower triangular matrix resulting from the factorization of Cholesky
[]
[]
[]
With
With
T
With
L
G
L
With
=
]
[
average matrix
{}
]
[
]
[
E
With
=
With
.
One must have (cf [R4.03.05]):
5
1
0
0
0
+
+
<
<
N
N
With
,
where
0
N
NR
is a constant of the probabilistic model selected so that
N
N
<
0
.

COEFF
_VAR_RIGI
=/
R
[R]
/0.1
[DEFECT]
This key word informs the parameter
R
of control of the dispersion of the random matrix of
rigidity of a substructure. This coefficient of variation is defined in a way identical to
definition given for key word COEF_VAR.
COEFF
_VAR_MASS
=/
R
[R]
/
0.
[DEFECT]
This key word informs the parameter
R
of control of the dispersion of the random matrix of
mass of a substructure. This coefficient of variation is defined in a way identical to
definition given for key word COEF_VAR.
COEFF
_VAR_AMOR
=/
R
[R]
/
0.
[DEFECT]
This key word informs the parameter
R
of control of the dispersion of the random matrix of
dissipation of a substructure. This coefficient of variation is defined in a way identical to
definition given for key word COEF_VAR.
3.3 Operand
INIT_ALEA
INIT_ALEA
=
nor
[I]
Cause initialization with sound
nor
ième term of the continuation of pseudo-random numbers used
for the generation of the matrices.
If the key word
INIT_ALEA
misses, the terms used of the continuation are those immediately
consecutive with those already used. If no term were still used, the continuation is initialized with sound
first term.
Recommendation:
With less than one particular use, it is advised not to inform the key word
INIT_ALEA
in the operators according to: GENE_FONC_ALEA, GENE_VARI_ALEA and GENE_MATR_ALEA.
In this case, with the first call to the one of these operators, the continuation of pseudo numbers
random is initialized in its first term. The omission of the key word
INIT_ALEA
with each one
calls of these operators in the command file guarantees independence
statistics of the pseudo-random numbers used.
background image
Code_Aster
®
Version
8.2
Titrate:
Operator
GENE_MATR_ALEA
Date:
22/02/06
Author (S):
S. CAMBIER, C. DESCELIERS
Key
: U4.36.06-C1
Page:
5/6
Instruction manual
U4.3- booklet: Function
HT-62/06/004/A
Note:
The germ of the continuation remains identical of one execution to the other of Code_Aster; results
thus remain rigorously identical (one can thus test nonthe regression of results
statistics not converged). If one wishes to generate results statistically
independent from one execution to another, then the key word should be used
INIT_ALEA
with
values raising the number of terms used in the former executions.
Caution:
The generator of random variable used is that of the module “random” of Python. It
depends on the version of Python exploited by Code_Aster. Not converged results
statistically can thus vary from one version to another of Code_Aster or a punt
form with the other, if the version of Python is not the same one and that between the two versions it
modulate random evolved/moved (case between Python 2.1 and 2.3).
Note:
In version Python 2.3, the period of the generator is 2 ** 19937-1 [bib1].


4 Example
By call, the control generates only one realization of the random matrix to simulate. For
to generate several achievements of the same random matrix, it is necessary to repeat the control without
to change its parameters or to place the control in a loop of the process control language of
Code_Aster - the language python.
In the following example, one generates NS achievements of a random matrix of average value
MATR_MOYEN with one
= 0.1. These achievements are then used as values of matrix of
mass.
ns=100

for K in arranges (1, ns+1):

# Generation
MAT_ALEA=GENE_MATR_ALEA (
MATR_MOYEN=MAT_MOY,
COEF_VAR=0.1,
)

DYN=DYNA_TRAN_MODAL (
… MASS_GENE=
MAT_ALEA,
)
# Here for example, statistical processing of DYN

TO DESTROY (CONCEPT=_F (NOM= (DYN, MAT_ALEA)))

# End of the loop (indentation)

For more complete examples, to consult the cases test SDNS01 [V5.06.001], SDNL105d [V5.02.105]
and SHLS200a [V2.06.200], like [U2.08.05].
background image
Code_Aster
®
Version
8.2
Titrate:
Operator
GENE_MATR_ALEA
Date:
22/02/06
Author (S):
S. CAMBIER, C. DESCELIERS
Key
: U4.36.06-C1
Page:
6/6
Instruction manual
U4.3- booklet: Function
HT-62/06/004/A
5 Bibliography
[1]
Mr. Matsumoto and T. Nishimura, Mersenne Twister: With 623-dimensionally equidistributed
uniform pseudorandom number generator, ACM Transactions one Modeling and Computer
Simulation vol. 8, No 1, January pp.3-30 1998.