Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
1/16
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: U4.52.03
Operator
MODE_ITER_SIMULT
1 Goal
To calculate clean values and vectors by methods of the subspace type. For the problem
conventional of dynamics (without damping) or the problem of buckling of Euler, three
algorithms are available: Sorensen, Lanczos, Bathe and Wilson. For the problem of dynamics
with damping, only the methods of Sorensen and Lanczos are usable. Product one
concept
mode_meca_ *
(dynamic case) or
mode_flamb
(case buckling of Euler).
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
2/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
2 Syntax
mode_ [*] = MODE_ITER_SIMULT
# MODAL FACT OF THE CASE
(
MATR_A
=
With
/
[matr_asse_DEPL_R]
/
[matr_asse_DEPL_C]
/
[matr_asse_PRES_R]
/
[matr_asse_GENE_R]
MATR_B
=
B
/
[matr_asse_DEPL_R]
/
[matr_asse_PRES_R]
/
[matr_asse_GENE_R]
MATR_C
=
C
/
[matr_asse_DEPL_R]
# STANDARD OF PROBLEM
TYPE_RESU
=
/
“DYNAMIC”
[DEFECT]
/
“MODE_FLAMB”
# CHOICE OF THE METHOD
METHOD
=
/
“SORENSEN” [DEFECT]
/
“TRI_DIAG”
/“JACOBI”
# If METHOD = “TRI_DIAG”
OPTION
=
/
“WITHOUT”
[DEFECT]
/
“MODE_RIGIDE”
# STANDARD OF MODAL CALCULATION
CALC_FREQ =_F (
OPTION
=/“CENTER”
/
“TAPE”
/
“PLUS_PETITE”
[DEFECT]
# CHARACTERISTIC OF CALCULATION
#
If TYPE_RESU = “DYNAMIC”
APPROACH
=/
“REAL”
[DEFECT]
/
“IMAG”
/
“COMPLEX”
#
If
OPTION
=
“PLUS_PETITE”
NMAX_FREQ
=
/
10
[DEFECT]
/
nf
[I]
#
If
OPTION
=
“CENTER”
FREQ
=
l_f
[l_R]
AMOR_REDUIT
=
l_a
[l_R]
NMAX_FREQ
=
/
10
[DEFECT]
/
nf
[I]
#
If
OPTION
=
“TAPE”
FREQ
=
l_f
[l_R]
#
If TYPE_RESU = “MODE_FLAMB”
APPROACH
=/
“REAL”
[DEFECT]
/
“IMAG”
#
If
OPTION
=
“PLUS_PETITE”
NMAX_FREQ
=
/
10
[DEFECT]
/
nf
[I]
#
If
OPTION
=
“CENTER”
CHAR_CRIT
=
l_c
[l_R]
NMAX_FREQ
=
/
10
[DEFECT]
/
nf
[I]
#
If
OPTION
=
“TAPE”
CHAR_CRIT
=
l_c
[l_R]
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
3/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
# CHARACTERISTIC OF THE SPACE OF PROJECTION
DIM_SOUS_ESPACE
= of
[I]
COEF_DIM_ESPACE
= mse
[I]
EXCLUDE (“DIM_SOUS_ESPACE”,
“COEF_DIM_ESPACE”)
# FOR PRE AND POSTPROCESSINGS
PREC_SHIFT
=
/
0.05
[DEFECT]
/
PS [R]
NMAX_ITER_SHIFT =
/5 [DEFECT]
/
NS [I]
NPREC_SOLVEUR =/8 [DEFECT]
/
ndeci [R]
SEUIL_FREQ
=
/
1.E-2 [DEFECT]
/
sf [R]
# PARAMETER SETTING INTERNS METHODS
#
If METHOD = “SORENSEN”
PREC_SOREN =
/0 [DEFECT]
/
pso
[R]
NMAX_ITER_SOREN =
/20
[DEFECT]
/
nso
[I]
PARA_ORTHO_SOREN =/0.717
[DEFECT]
/
porso
[I]
#
If METHOD = “TRI_DIAG”
PREC_ORTHO
=
/
1.E-12
[DEFECT]
/Po
[R]
NMAX_ITER_ORTHO =
/5 [DEFECT]
/
nio
[I]
PREC_LANCZOS =
/1.E-8
[DEFECT]
/pl
[R]
NMAX_ITER_QR
=
/
30 [DEFECT]
/
nim
[I]
#
If METHOD = “JACOBI”
PREC_BATHE
=
/
1.E-10
[DEFECT]
/
pbat
[R]
NMAX_ITER_BATHE
=/
40
[DEFECT]
/
nbat
[I]
PREC_JACOBI
=
/
1.E-2 [DEFECT]
/
pjaco [R]
NMAX_ITER_JACOBI =/12
[DEFECT]
/
njaco [I]
)
# FOR FINAL CHECKS
VERI_MODE
= _F (
STOP_ERREUR
=
/
“YES”
[DEFECT]
/
“NOT”
THRESHOLD
=/
1.E-6
[DEFECT]
/R
[R]
PREC_SHIFT
=
/
0.05 [DEFECT]
/
prs
[R]
STURM
=/
“YES”
[DEFECT]
/
“NOT”
)
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
4/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
# SENSITIVITY
SENSITIVITY = (
… to see [U4.50.02]….
)
# OTHERS
STOP_FREQ_VIDE
=
/
“YES”
[DEFECT]
/
“NOT”
INFORMATION
=
/
1
[DEFECT]
/
2
[I]
TITRATE = Ti
);
# GIVEN RESULT
If
MATR_C
= [matr_asse_DEPL_R]
then
[*]
- > meca_c
If
TYPE_RESU
=
“MODE_FLAMB”
then
[*]
- > mode_flamb
If
MATR_A
= [matr_asse_DEPL_C]
then
[*]
- > meca_c
If
MATR_A
= [matr_asse_DEPL_R]
then
[*]
- > meca
If
MATR_A
= [matr_asse_PRES_R]
then
[*]
- > acou
If
MATR_A
= [matr_asse_GENE_R]
then
[*]
- > embarrassment
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
5/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3 Operands
3.1 Principles
This operator solves the problem generalized with the eigenvalues according to [R5.01.01]:
To find (
,
X
) such as
X
B
X
With
=
,
X
0, where
With
and
B
are symmetrical matrices with coefficients
realities. To modelize a damping hysteretic in the study of the free vibrations of a structure,
the matrix
With
can be complex [U2.06.03] [R5.05.04]. This type of problem corresponds, in
mechanics, in particular with:
·
The study of the free vibrations of a not deadened and nonrevolving structure. For this
structure, one seeks the smallest eigenvalues or those which are in one
interval given to know if an exiting force can create a resonance. In this case,
the matrix
With
is the matrix of material rigidity, noted
K
(real or complex),
_éventuellement increased geometrical matrix of noted rigidity
K
G
, if the structure
is précontrainte_, and
B
is the matrix of mass or noted inertia
M
. Eigenvalues
obtained are the squares of the pulsations associated with the sought frequencies.
The system to be solved can be written:
(
)
{X
B
M
X
With
K
K
=
+ 43
42
1
G
where
(
)
2
2 F
=
is the square of
pulsation
,
F
the Eigen frequency and
X
the vector of associated clean displacement.
If
K
is complex,
and
F
are too.
·
The search for linear mode of buckling. Within the framework of the linearized theory, in
supposing a priori that the phenomena of stability are suitably described by
system of equations obtained by supposing the linear dependence of displacement by
report/ratio at the level of critical load, the search of the mode of buckling
X
associated it
level of critical load
µ
= -
, brings back itself to a problem generalized to the eigenvalues
form:
(
)
{
{X
B
K
X
With
K
0
X
K
K
G
G
µ
=
=
+
with
K
stamp material rigidity and
K
G
stamp geometrical rigidity.
Caution:
In the code, one treats only the eigenvalues of the generalized problem, them
. For
to obtain the true critical loads, them
µ
, they should be multiplied by 1.
This operator allows also the study of the dynamic stability of an involved structure
depreciation viscous (and/or quadratic) and gyroscopic effects. That led to
resolution of a modal problem of a nature higher, known as quadratic [R5.01.02]. One seeks then
complex values and clean vectors by the method of Lanczos after having carried out a reduction
linear of the problem.
·
The problem consists in finding
()
(
)
,
X
C C
NR
such as
(
)
2
0
B
C
With X
+
+
=
where
typically, in linear mechanics,
K
With
=
will be the matrix of rigidity,
M
B
=
the matrix of
mass and
C
the matrix of damping. Matrices
K
,
M
and
C
are matrices with
real coefficients. The complex eigenvalue
is connected to the Eigen frequency
F
and with
reduced damping
by:
(
) (
)
=
±
-
2
2
1
2
F
I
F
.
K
can be also complex to moreover simulate, one damping hysteretic
[U2.06.03] [R5.05.04].
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
6/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
To solve these generalized or quadratic modal problems, Code_Aster proposes
various approaches. Beyond their numerical specificities and functional calculuses which are taken again
in the document [R5.01.01], one can synthesize them in the shape of table below (the values
by defect are materialized in fat).
Operator/
Perimeter
of application
Algorithm Key word Advantages
Disadvantages
MODE_ITER_INV
1
era
phase
(heuristics)
Calculation of some
modes
Bisection
“SEPARATE”
Calculation of some
modes
Bisection +
Secant (gene.)
Muller (quad.)
“ADJUSTS”
Better precision
Cost calculation
Improvement of
some estimates
Initialization by
the user
“NEAR”
Resumption of values
clean estimated
by another
process.
Cost calculation of this
phase quasi-no one
No the capture
of multiplicity
2
ième
phase
(method of
powers properly
said)
Basic method
Powers
opposite
“DIRECT”
Very good
construction of
clean vectors
Not very robust
Option of acceleration
Quotient of
Rayleigh
“RAYLEIGH”
Improve
convergence
Cost calculation
Not carried in
quadratic
MODE_ITER_SIMULT
Calculation of part of
spectrum
Bathe & Wilson
“JACOBI”
Little
robust
Not carried in
quadratic
Lanczos
(Newman- Pipano)
“TRI_DIAG”
Little
robust
IRAM
(Sorensen)
“SORENSEN” increased Robustness.
Better
calculation complexities
and memory.
Control
quality of the modes.
Table 3.1-1: Summary of the modal methods of Code_Aster
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
7/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
When it is a question of determining some simple eigenvalues discriminated well or to refine
some estimates, operator MODE_ITER_INV, is often clearly shown. On the other hand, for
to capture a part significant of the spectrum, one A resorts to MODE_ITER_SIMULT, via the methods
known as “of subspace”.
It is this class of method which will interest us here.
It consists in projecting advisedly the operator of work in order to obtain a modal problem
standard of more reduced size and comprising a canonical matrix of form (tridiagonale or of
Higher Hessenberg). It is on the latter that total modal solveurs will be able
then to operate (algorithm QR, QL or Jacobi). They are in general very robust, but they provide
all the spectrum of the treated operator and they are very expensive. From where the idea to fix quotas for their efforts
on only one “projected” spectrum.
It is completely recommended besides to benefit from the strong points of the two classes from
method by refining the clean vectors obtained by MODE_ITER_SIMULT, via
MODE_ITER_INV (OPTION=' PROCHE'). That will make it possible to reduce the standard of the final residue
(cf [§3.7.2]).
Note:
One strongly advises a preliminary reading of the reference materials [R5.01.01],
[R5.01.02]. It gives to the user the properties and the limitations, theoretical and practical,
modal methods approached while connecting these considerations, which can sometimes
to appear a little éthérées, with a precise parameter setting of the options.
3.2 Operands
MATR_A
,
_B
,
_C
MATR_A
= A
Stamp assembly of concept
[matr_asse_ * _R/C]
system to be solved.
MATR_B
= B
Stamp assembly of concept
[matr_asse_ * _R]
system to be solved.
MATR_C
= C
Stamp assembly of concept
[matr_asse_ * _R]
quadratic system to solve.
3.3 Word
key
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 with or without damping) or search for loads
critical (case of the theory of linear buckling). According to this class of membership, them
results are displayed and stored differently in the structure of data:
·
In dynamics, the frequencies are ordered by order ascending of the module of their
variation with the shift (cf [§2.9], [§4.4] [R5.01.01]). It is the value of the variable of access
NUM_ORDRE
structure of data. The other variable of access,
NUME_MODE
, is equal to
the true modal position in the spectrum of the eigenvalue (determined by the test of
Sturm cf [§2.5], [§2.6] [R5.01.01]).
·
In buckling, the eigenvalues are stored by order ascending algebraic.
variables
NUM_ORDRE
and
NUM_MODE
take the same value equal to this command.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
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U4.5- booklet: Methods of resolution
HT-66/05/004/A
3.4 Word
key
METHOD
Three methods of resolution are available for the problem to the eigenvalues
·
The method WILL GO (known as of Sorensen), makes it possible to treat the two types of problems
generalized and quadratic. It is the method by defect and is based on:
-
obtaining a matrix of Hessenberg by using a factorization of the Arnoldi type
-
the calculation of the eigenvalues of this problem projected by a method QR
-
a certain number of restartings allowing to refine the sought eigenvalues
by the user, the other eigenvalues necessary to the method being used as values
auxiliaries.
·
The method of Lanczos, makes it possible to treat the two types of problems generalized and
quadratic. It is based on:
-
obtaining a matrix tridiagonale projected via the method of Lanczos,
-
the resolution of the system tridiagonal reduced by a method QR,
·
The iterative method of Bathe and valid Wilson only for the generalized problem, is
based on:
-
construction with each iteration of a projected generalized problem of smaller size,
-
the calculation of the eigenvalues of this problem projected by a method of Jacobi.
METHOD
=
/
“SORENSEN” [DEFECT]
One uses the method of Sorensen (cf [§5] [R5.01.01]) to calculate the values and vectors
clean of the generalized or quadratic problem. This option cannot be used for one
quadratic problem.
/
“TRI_DIAG”
One uses the method of Lanczos (then method QR on the projected system) to calculate them
values and clean vectors of the generalized or quadratic problem (cf [§4] [R5.01.01]).
/
“JACOBI”
One uses the method of Bathe & Wilson (cf [§6] [R5.01.01]) (then method of Jacobi on
system projected) to calculate the values and clean vectors of the generalized problem. This
option cannot be used for a quadratic problem.
3.5 Word
key
OPTION
OPTION =
/“MODE_RIGIDE”
/
“WITHOUT”
[DEFECT]
Key word usable only with the method of Lanczos for a generalized modal problem. It
allows to detect and calculate as a preliminary, by an algebraic method the modes of body
of rigid (modes associated with a null eigenvalue) (cf [§5.5.4] [R5.01.01]). They are used by
the continuation to calculate the other modes with the algorithm of Lanczos. They are provided to the user
only if they belong to the modes requested. If the modes of rigid body are calculated without
to use this option, the eigenvalues calculated by the algorithm of Lanczos are not null
but very close to zero.
3.6 Word
key
CALC_FREQ
CALC_FREQ
=_F (…
Key word factor for the definition of the parameters of calculation of the eigenvalues and their number.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
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Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3.6.1 Operand
OPTIONS
OPTION
=
“TAPE”
One seeks all the eigenvalues in a given tape. This
tape is defined by the argument of
FREQ: (F
1
F
2
)
or by that of
CHAR_CRIT: (
1
2
)
.
This option is not usable with a quadratic modal problem.
“CENTER”
They are sought
NMAX_FREQ
the eigenvalues closest to
frequency
F
(argument of the key word
FREQ: F)
or closest to
the critical load
(argument of the key word
CHAR_CRIT:
)
.
“PLUS_PETITE”
[DEFECT]
They are sought
NMAX_FREQ
smaller eigenvalues.
See [§2.9] and [§4.4] [R5.01.01].
3.6.2 Operand
APPROACH
APPROACH
=/
“REAL”
[DEFECT]
/
“IMAG”
/
“COMPLEX”
This key word defines the type of approach (real, imaginary or complex) for the choice of pseudo
scalar product of the quadratic problem (cf [§5.5.2] [R5.01.02]). In general the default value
(reality) is valid.
This operand does not have a direction that for the analysis of the free vibrations of a deadened structure (modes
proper complexes; the key word
MATR_C
must be well informed). In buckling, that does not have any
interest.
3.6.3 Operand
FREQ
FREQ = l_f
List frequencies (can be used only if
TYPE_RESU = “DYNAMIC”
): its use
depends on
OPTION
chosen.
OPTION
=
“TAPE”
One awaits two values (
F
1
F
2
) which defines the tape
of search,
OPTION
=
“CENTER”
Only one value of frequency is awaited,
The values stipulated under this key word must be positive.
3.6.4 Operand
AMOR_REDUIT
AMOR_REDUIT = l_a
Value of the reduced damping which makes it possible to define the eigenvalue complexes around
which one seeks the eigenvalues closest. (can be used only if
TYPE_RESU = “DYNAMIC”
and
MATR_C
informed).
OPTION
=
“CENTER”
One awaits only one value of reduced damping,
The value stipulated under this key word must be positive and lie between 0 and 1. In buckling, that
no interest has.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
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Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3.6.5 Operand
CHAR_CRIT
CHAR_CRIT = l_c
List critical loads (can be used only if
TYPE_RESU = “MODE_FLAMB”
): its
use depends on
OPTION
chosen.
OPTION
=
“TAPE”
One awaits two values (
1
2
) which defines the tape
of search,
OPTION
=
“CENTER”
One awaits only one value of critical load,
The values stipulated under this key word are positive or negative.
3.6.6 Operand
NMAX_FREQ
NMAX_FREQ
= nf
(10)
[DEFECT]
Numbers maximum eigenvalues to calculate.
This key word is ignored with the option “BANDAGES” because one calculates all the eigenvalues then
contained in the stipulated tape.
In the two cases, if nf is strictly higher than the number of “ddl active”, nactif (cf [§2.2]
[R5.01.01]), then one forces it to take this value ceiling.
3.6.7 Operand
DIM_SOUS_ESPACE
DIM_SOUS_ESPACE
= of
COEF_DIM_ESPACE
= mse
EXCLUDE (“DIM_SOUS_ESPACE”, “COEF_DIM_ESPACE”)
If key word DIM_SOUS_ESPACE is not indicated or is initialized with a value strictly
lower than the number of required frequencies nf, the operator calculates one automatically
acceptable dimension for the subspace of projection (cf [§5.2] of this document and [§4.3],
[§5.5.2], [§6.5.3], [§7.3.1] [R5.01.01]) with aid COEF_DIM_ESPACE.
Thanks to given of this multiplicative factor, mse, one can project on a space whose size is
proportional to the number of frequencies contained in the interval of study. In
the encapsulation of MODE_ITER_SIMULT, MACRO_MODE_MECA [U4.52.02], one can thus optimize
the size of the subspaces which remains proportional to the number of required frequencies:
subspaces rich in eigenvalues thus do not penalize poorest (in term of
CPU).
One can however arbitrarily fix the size of this subspace, via the value of the catch by
key word DIM_SOUS_ESPACE (which must be higher than nf to be taken into account).
In both cases, if the size of the subspace of projection ndim is strictly higher than
number “active ddl”, nactif (cf [§2.2] [R5.01.01]), then one forces it to take this value
ceiling.
Note:
·
If one uses the method of Sorensen (IRAM) and that ndim - nf < 2, of the requirements
numérico-data processing force to impose ndim = nf + 2.
·
Into quadratic one works on a real problem of double size: 2 * nf, 2 * ndim.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
11/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
3.6.8 Operands of IRAM (if
METHOD = “SORENSEN”
)
PREC_SOREN = pso
(0. )
[DEFECT]
Note:
The method considers whereas it must work with the smallest possible precision, it
“zero machine”. To have an order of magnitude of it, in double precision on the machines
standards, this value is close to 2.22 .10
- 16
)
NMAX_ITER_SOREN
= nso
(20)
[DEFECT]
PARA_ORTHO_SOREN
= porso
(0.717)
[DEFECT]
They are parameters of adjustment of the necessary precision on the modes (by defect, the precision
machine is selected), of the number of restartings authorized of the method of Sorensen
(cf [§5.4.2] and [§6.4] [R5.01.01]) and of the coefficient of orthogonalization of the IGSM of Kahan-Parlett
(cf [§11.4] [R5.01.01]).
Note:
At the time of the first passages, it is strongly advised not to modify these parameters which
the mysteries of the algorithm concern rather and which are initialized empirically with
values standards.
3.6.9 Operands of the method of Lanczos (if
METHOD = “TRI_DIAG”
)
PREC_ORTHO =
Po
(1.10
- 12
) [DEFECT]
NMAX_ITER_ORTHO
= nio
(5)
[DEFECT]
PREC_LANCZOS
=
pl
(1.10
- 8
)
[DEFECT]
NMAX_ITER_QR
= nim
(30)
[DEFECT]
The first two parameters make it possible, respectively, to adjust the precision
of orthogonalization and the number of réorthogonalisations in the method of Lanczos for
to obtain independent vectors generating the subspace (cf [§5.5.1] [R5.01.01]).
The third is a parameter of adjustment to determine the nullity of a term on
surdiagonale of the matrix tridiagonale characterizing the reduced problem obtained by the method of
Lanczos. It is right a criterion of deflation and not, as opposed to what could let believe
its name, a quality standard of the modes (cf [§5.4.1] [R5.01.01]).
The last fixes the maximum iteration count for the resolution of the system reduced by
method QR ([§5.5.2] and [§10] [R5.01.01]).
Note:
At the time of the first passages, it is strongly advised not to modify these parameters which
the mysteries of the algorithm concern rather and which are initialized empirically with
values standards.
3.6.10 Operands of the method of Bathe & Wilson (if
METHOD = “JACOBI”
)
PREC_BATHE = pbat
(1.10
- 10
) [DEFECT]
NMAX_ITER_BATHE
= nbat
(40)
[DEFECT]
PREC_JACOBI = pjaco
(1.10
- 2
)
[DEFECT]
NMAX_ITER_JACOBI
= njaco
(12)
[DEFECT]
The first two parameters make it possible, respectively, to adjust the precision of convergence
and the maximum number of allowed iterations of the method of Bathe & Wilson (cf [§7]
[R5.01.01]).
The two others make it possible to adjust the precision of convergence and the maximum number
iterations permitted by the method of JACOBI (cf [§12] [R5.01.01]) who allows to exhume them
clean modes of the matrix projected by the preceding method.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
12/16
Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
Note:
At the time of the first passages, it is strongly advised not to modify these parameters which
the mysteries of the algorithm concern rather and which are initialized empirically with
values standards.
3.6.11 Operands
SEUIL_FREQ
,
PREC_SHIFT
and
NMAX_ITER_SHIFT
PREC_SHIFT = PS
(0.05)
[DEFECT]
SEUIL_FREQ = sf
(0.01)
[DEFECT]
NMAX_ITER_SHIFT
= NS
(5)
[DEFECT]
For the three possible options
“PLUS_PETITE”
,
“TAPE”
or
“CENTER”
, one is carried out
factorization LDL
T
matrix
(
)
(
)
With
F
B
- 2
2
*
.
F
*
depends on the method used. If
F
*
is
detected as being an Eigen frequency or being located near Eigen frequencies
(loss of more than
ndeci=8
decimals during the factorization of the matrices), the frequency
F
*
is
then modified (cf [§2.6] and [§2.9] [R5.01.01]):
(
)
(
)
F
F
PS
F
F
PS
*
*
*
*
-
+
=
× -
=
× +
1
1
or
If
(
)
(
)
With
F
B
- 2
2
*
is not factorisable LDL
T
and
(
)
F
sf
*
, one carries out
following amendment:
F
sf
* -
= -
. It is considered whereas
F
*
is associated a mode of body
rigid. The amendment of this frequency makes it possible a priori to enter all the modes of
rigid body. One does not carry out more
NS
amendments of the value
F
*.
In the case of linear buckling, the transposition is immediate while replacing
F
* (frequency
of vibration) by
* (critical load),
(
)
2
2
F
*
by
* and
sf
by
(
)
2
2
sf
.
Note:
At the time of the first passages, it is strongly advised not to modify these parameters which
the mysteries of the algorithm concern rather and which are initialized empirically with
values standards.
3.6.12 Operand
NPREC_SOLVEUR
NPREC_SOLVEUR
= ndeci
(8)
[DEFECT]
ndeci
represent the number of decimals which one is authorized to lose during the factorization of
stamp shiftée
(
)
(
)
With
F
B
- 2
2
*
or
(
)
With
B
-
. If one loses more
ndeci
decimals, the matrix
is regarded as noninvertible (cf [§2.6] and [§2.9] [R5.01.01]).
Note:
At the time of the first passages, it is strongly advised not to modify this parameter which
rather relate to a mystery of the algorithm and which is initialized empirically with a value
standard.
3.7 Word
key
VERI_MODE
VERI_MODE = _F (…
Key word factor for the definition of the parameters of the checking of the clean modes ([§2.9]
[R5.01.01]).
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
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Author (S):
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Key
:
U4.52.03-G
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:
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Instruction manual
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HT-66/05/004/A
3.7.1 Operand
STOP_ERREUR
STOP_ERREUR =/
“YES”
[DEFECT]
/
“NOT”
Allows to indicate to the operator if it must stop (
“YES”
) or to continue (
“NOT”
) if
one of the criteria
THRESHOLD
or
STURM
is not checked.
By defect the concept of exit is not produced.
3.7.2 Operand
THRESHOLD
THRESHOLD = R
(1.10
- 6
)
[DEFECT]
Tolerance level for the standard of error relating of the mode to the top of which the mode is
regarded as forgery.
The standard of relative error of the mode is:
(
)
With
B X
Ax
,
-
2
2
for
0
for the generalized problem and
(
)
2
2
2
B
C
With X
Ax
,
+
-
for the quadratic problem
3.7.3 Operand
STURM
STURM
=/
“YES”
[DEFECT]
/
“NOT”
Checking known as of
STURM
(
“YES”
) allowing to make sure that the algorithm used in
the operator determined the exact number of eigenvalues in the interval of search
([§2.5] [§2.6] [R5.01.01]).
3.7.4 Operand
PREC_SHIFT
PREC_SHIFT = prs
(0.05)
[DEFECT]
This parameter (which is a percentage) makes it possible to define an interval containing the values
clean calculated, for which the checking of Sturm will be carried out ([§2.6] [R5.01.01]).
3.8 Operands
SENSIBLITE
SENSITIVITY =
Activate the calculation of derived from the modes compared to a significant parameter of the problem.
It is it should be noted that at present, the derivative of the multiple modes is not available, because it
pose theoretical and practical problems particular.
The document [U4.50.02] specifies the operation of the key word.
Code_Aster
®
Version
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Titrate:
Operator
MODE_ITER_SIMULT
Date:
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Key
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3.9 Operand
STOP_FREQ_VIDE
STOP_FREQ_VIDE =/
“YES”
[DEFECT]
/
“NOT”
“YES”
stop calculation if no eigenvalue is detected in the tape stipulated by
the user: an exception (named
BandeFrequenceVide
) is emitted. It can be treated
to continue the course of the study. One can find an example under the case SDLL11a: test
try:
MODE1=MODE_ITER_SIMULT (MATR_A=K_ASSE, MATR_B=M_ASSE,
CALC_FREQ=_F (
OPTION = “TAPE”,
FREQ = (100., 200. )))
except aster.BandeFrequenceVideError:
MODE1=MODE_ITER_SIMULT (MATR_A=K_ASSE, MATR_B=M_ASSE,
CALC_FREQ=_F (
OPTION = “TAPE”,
FREQ = (200., 3500.,)))
“NOT”
do not stop calculation (emission only of one
ALARM
) if no eigenvalue is
detected in the tape stipulated by the user.
This key word is used in the macro-control
MACRO_MODE_MECA
[U4.52.02] in order to allow
the absence of eigenvalues in a tape of search.
3.10 Operand
INFORMATION
INFORMATION
=/
1
[DEFECT]
/2
Indicate the level of impression in the file MESSAGE.
1:
Impression on the file
“MESSAGE”
eigenvalues, of their modal position, of
reduced damping, of the standard of error a posteriori and certain useful parameters
to follow the course of calculation (cf [§5.2])
2:
Impression rather reserved for the developers.
3.11 Operand
TITRATE
TITRATE = Ti
Titrate attached to the concept produced by this operator [U4.03.01].
4
Phase of checking
One checks according to options':
OPTION = “TAPE”
the argument of the key word
FREQ
or of the key word
CHAR_CRIT
must provide two values exactly,
OPTION = “CENTER”
the argument of the key word
FREQ
or of the key word
CHAR_CRIT
must provide only one value exactly,
OPTION = “PLUS_PETITE”
the argument of the key word
FREQ
or of the key word
CHAR_CRIT,
is ignored.
If the maximum precise details and numbers of iterations are unrealistic (for example precise details
lower than the precision machine or of the negative iteration counts), calculation is not carried out.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
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Key
:
U4.52.03-G
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:
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Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
5 Phase
of execution
5.1 Checking
Matrices
With
,
B
(and
C
) arguments of the key words
MATR_A
and
MATR_B
(
and
MATR_C)
, must be
coherent between them (i.e. to be based on the same classification and the same mode of
storage).
5.2
Actions by defect
If key word DIM_SOUS_ESPACE is not indicated or is initialized with a value strictly
lower than the number of required frequencies nf (operand NMAX_FREQ), the operator calculates
automatically an acceptable dimension for the subspace of projection via the formulas
empirical (cf [§3.6.7]):
METHOD =
“SORENSEN”
ndim = MIN (MAX (2+nf, mse * nf), nactif)
with
mse = 2
by defect.
METHOD = “TRI_DIAG”
ndim = MIN (MAX (7+nf, mse * nf), nactif)
with
mse = 4
by defect.
METHOD = “JACOBI”
ndim = MIN (MAX (7+nf, mse * nf), nactif)
with
mse = 2
by defect.
where nactif of ddl active (i.e. the total number of ddl less the number of ddls is the number of
LAGRANGE and less the number of linear relations which bind ddls between them, cf [§2.2] [R5.01.01])
and mse is the factor of proportionality fixed by COEF_DIM_ESPACE.
If one solves a quadratic problem with the eigenvalues, the dimension of the subspace is
doubled.
The values of these various parameters are printed in the file MESSAGE.
6
Modal parameters/Standard of the modes/modal Position
At exit of this operator, the real or complex clean modes are standardized with largest
components which is not a multiplier of LAGRANGE. To choose another standard, it is necessary
to use the control
NORM_MODE
[U4.52.11].
In the case of a dynamic calculation, the structure of data
mode_meca_ *
, contains, in addition to
frequencies of vibration and the associated modal deformations, the modal parameters (mass
generalized, generalized stiffness, factor of participation, mass effective). One will find the definition of
these parameters in [R5.01.03].
In the case of a linear calculation of buckling, the structure of data
mode_flamb
, only contains
critical loads and associated deformations.
In the case of a dynamic calculation, the modal position of the modes corresponds to the position of the mode
in the whole of the spectrum defined by the initial matrices.
In the case of a linear calculation of buckling, the modal positions of the critical loads are
allotted of 1 to
nf
(
nf
being the number of critical loads calculated) by classifying the loads
critical by order ascending algebraic. All the modal positions are thus positive.
Code_Aster
®
Version
7.4
Titrate:
Operator
MODE_ITER_SIMULT
Date:
31/01/05
Author (S):
E. BOYERE, O. BOITEAU
Key
:
U4.52.03-G
Page
:
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Instruction manual
U4.5- booklet: Methods of resolution
HT-66/05/004/A
7
Impression of the results
To display the modal parameters associated with each mode and the co-ordinates with the modes, it is necessary
to use operator IMPR_RESU [U4.91.01] in the following way:
· Display of the modal parameters only in the form of table:
IMPR_RESU
(
RESU
=
_F (
RESULT = mode,
TOUT_PARA
=
“YES”,
TOUT_CHAM
=
“NOT”))
;
· Display of the modal parameters and the clean vectors:
IMPR_RESU
(RESU =_F (RESULT = mode,
TOUT_PARA
=
“YES”,
TOUT_CHAM
=
“YES”))
;
8
Sorting of modes/Characterization of
mode_meca_ *
For example, at the time of seismic stresses in modal analysis, the modal base used must contain
the modes which have an important unit effective mass in the direction of the seism.
Control EXTR_MODE [U4.52.12] makes it possible to extract in a structure of data of the type
mode_meca_ * of the modes which check a certain criterion and of concaténer several structures of
data of the mode_meca_ type *.
A macro-control, allowing to connect the controls
MODE_ITER_SIMULT
,
NORM_MODE
and
EXTR_MODE
was created:
MACRO_MODE_MECA
[U4.52.02].
9 Examples
9.1 Calculation of the 5 clean modes closest to a frequency
data (100 Hz)
mode = MODE_ITER_SIMULT
(MATR_A = rigid,
MATR_B
=
mass,
CALC_FREQ
=_F (
OPTION
=
“CENTER”,
FREQ
=
100.,
NMAX_FREQ
=
5
)
);
9.2
Calculation of the critical loads contained in a tape
mode = MODE_ITER_SIMULT
(MATR_A = rigid,
MATR_B
=
riggeo,
TYPE_RESU
=
“MODE_FLAMB”,
CALC_FREQ
=_F (
OPTION
=
“TAPE”,
CHAR_CRIT
=
(- 1.E8
,
1.5E8))
);