Code_Aster
®
Version
7.4
Titrate:
Procedure
IMPR_STURM
Date:
21/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.01-H
Page
:
1/6
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
Organization (S):
EDF-R & D/AMA,
SINETICS
Instruction manual
U4.5- booklet: Methods of resolution
Document: U4.52.01
Procedure
IMPR_STURM
1 Goal
To calculate the number of eigenvalues included/understood in an interval and to print out it. This procedure is
advised like checking a priori of the model and to define intervals of search containing
a reasonable number of eigenvalues in order to optimize the calculating time of the operators
MODE_ITER_SIMULT
or
MODE_ITER_INV
.
The execution of this procedure requires two factorizations LDL
T
.
Code_Aster
®
Version
7.4
Titrate:
Procedure
IMPR_STURM
Date:
21/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.01-H
Page
:
2/6
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
2 Syntax
IMPR_STURM
(
MATR_A
=
With
/
[matr_asse_DEPL_R]
/
[matr_asse_TEMP_R]
/
[matr_asse_PRES_R]
MATR_B
=
B
/
[matr_asse_DEPL_R]
/
[matr_asse_TEMP_R]
/
[matr_asse_PRES_R]
TYPE_RESU=/
“DYNAMIC”
[DEFECT]
/“MODE_FLAMB”
# If
TYPE_RESU = “DYNAMIC”
FREQ_MIN=
/
f_min
[R]
/
0.
[DEFECT]
FREQ_MAX=
f_max
[R]
# If
TYPE_RESU = “MODE_FLAMB”
CHAR_CRIT_MIN=
_min
[R]
CHAR_CRIT_MAX=
_max
[R]
SEUIL_FREQ=
/
f_seuil [R]
/
0.01 [DEFECT]
PREC_SHIFT=
/
p_shift [R]
/
0.01 [DEFECT]
NMAX_ITER_SHIFT=/
n_shift [I]
/
5
[DEFECT]
NPREC_SOLVEUR =/
ndeci
[I]
/
8
[DEFECT]
UNIT
=/
nunite
[I]
/
8
[DEFECT]
);
Code_Aster
®
Version
7.4
Titrate:
Procedure
IMPR_STURM
Date:
21/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.01-H
Page
:
3/6
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3 Operands
3.1 Operands
MATR_A
and
MATR_B
MATR_A: A
MATR_B: B
With
and
B
being names of the assembled matrices, the problem generalized with the eigenvalues
studied is:
(
)
With
B v
-
=
0
In the conventional case of dynamics,
With
is the matrix of rigidity and
B
the matrix of mass.
eigenvalue
is then connected to the Eigen frequency
F
by the formula:
(
)
= 2
2
F
.
In the case of the theory of linear buckling,
With
is the matrix of rigidity and
B
the matrix of
geometrical rigidity. The eigenvalue
critical load is called.
This procedure allows, before carrying out the search for eigenvalues, to know it
numbers in a tape stipulated by the user.
Method of calculation:
One applies the property of the continuations of STURM and the theorem of SYLVESTER (cf [R5.01.01
§2.5 and §2.6]). If
µ
is a given spectral shift, the appearing number of negative pivots
during symmetrical factorization (by LDL
T
) of
(
)
With
B
-
µ
is equal to the number of values
clean real lower than
µ
.
3.2 Operand
TYPE_RESU
TYPE_RESU=/
“DYNAMIC”
[DEFECT]
/“MODE_FLAMB”
This key word makes it possible to define the nature of the modal problem to treat: search for frequencies of
vibration (conventional case of dynamics) or search for critical loads (case of the theory of
linear buckling).
3.3 Operands
FREQ_MIN
and
FREQ_MAX
FREQ_MIN =
F
min
FREQ_MAX =
F
max
These key words must be used if
TYPE_RESU = “DYNAMIC”
. They define the terminals
lower and higher in Hertz of the frequency band in which one seeks it
a number of Eigen frequencies. These two terminals are positive realities. One seeks then
the number of eigenvalues in the tape
[
]
min
max
,
with:
(
)
*
*
= 2
2
F
Action by defect:
If
FREQ_MIN
then one misses calculates the number of Eigen frequencies included/understood enters
0. and
F
max
.
Code_Aster
®
Version
7.4
Titrate:
Procedure
IMPR_STURM
Date:
21/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.01-H
Page
:
4/6
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3.4 Operands
CHAR_CRIT_MIN
and
CHAR_CRIT_MAX
CHAR_CRIT_MIN =
min
CHAR_CRIT_MAX =
max
These key words must be used if
TYPE_RESU = “MODE_FLAMB”
. They define the terminals
lower and higher of the tape of critical loads in which one seeks the number
clean critical loads. These two terminals are realities positive or negative. One seeks
then the number of eigenvalues in the tape
[
]
min
max
,
.
3.5 Operands
PREC_SHIFT
and
NMAX_ITER_SHIFT
PREC_SHIFT
=
p
shift
NMAX_ITER_SHIFT
=
N
shift
If
F
min
(
min
) or
F
max
(
max
) are detected as being eigenvalues or being
located near eigenvalues (loss of more than eight decimals (
ndeci
) at the time of
factorization of the shiftée matrix
(
)
With
B
-
), they are then modified:
·
(
)
F
F
p
shift
min
min
-
=
× -
1
(
(
)
min
min
-
=
× -
1
p
shift
)
·
(
)
F
F
p
shift
max
max
+
=
× +
1
(
(
)
max
max
+
=
× +
1
p
shift
)
One then seeks the number of eigenvalues in the new interval
[
]
F
F
min
max
,
-
+
(
[
]
min
max
,
-
+
)
One is not authorized more
N
shift amendments of the terminals of the interval.
3.6 Operand
SEUIL_FREQ
SEUIL_FREQ =
F
threshold
If
F
F
threshold
min
and if
F
min
is detected like eigenvalue, then one seeks it
a number of Eigen frequencies in the interval
[
]
- F
F
threshold
,
max
.
It is considered whereas
F
min
is associated a mode of rigid body. Amendment of
limit lower interval allows a priori to enter all the modes of body
rigid.
In the case of the theory of linear buckling, one replaces
F
by
and one defines:
(
)
threshold
threshold
F
= 2
2
Code_Aster
®
Version
7.4
Titrate:
Procedure
IMPR_STURM
Date:
21/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.01-H
Page
:
5/6
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3.7 Operand
NPREC_SOLVEUR
NPREC_SOLVEUR = ndeci
ndeci
represent the number of decimals which one is authorized to lose during factorization
shiftée matrix
(
)
With
B
-
.
3.8 Operand
UNIT
UNIT
Number of the logical unit corresponding to the file of writing. By defect, it is worth 8,
i.e. file RESU.
Code_Aster
®
Version
7.4
Titrate:
Procedure
IMPR_STURM
Date:
21/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.01-H
Page
:
6/6
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
Intentionally white left page.