Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 1/8
Organization (S): EDF-R & D/AMA
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
Document: U2.07.01
Note of use of FOURIER modeling
1 Goal
The analysis of Fourier is intended to calculate the response of structures for axisymmetric geometry
solicited by nonaxisymmetric loadings broken up into Fourier series.
Limitations:
· the decomposition of the loading in Fourier series is supposed to be made by
the user,
· the Aster establishment relates to only isotropic or orthotropic materials,
· in thermics, there is not total command making it possible to solve a problem on
several harmonics. Calculation must be done harmonic by harmonic.
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 2/8
2 Notations
All the fields considered (forces, displacements, strains, stresses, flow) are expressed in
cylindrical co-ordinates with following convention on the command of the components:
Z
radial component according to R
axial component according to Z
component tangential (or circumferential) according to
Example: (ur, uz, U)
(Fr, fz, F)
R
uz
U
ur
The grid is localized in plan (R, Z), the symmetry of revolution being done around axis OZ.
trihedron (R, Z,) is directed in the direct direction.
Z
R
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 3/8
3 Modeling,
loadings
To affect the axisymmetric finite elements Fourier on the grid, the operator is used
AFFE_MODELE in the following way:
Mo = AFFE_MODELE (MAILLAGE = my,
AFFE
=
_F
(
TOUT
=
“OUI”,
PHENOMENE
=
“MECANIQUE”
or
“THERMIQUE”,
MODELISATION
=
“AXIS_FOURIER”
)
);
The decomposition in Fourier series of the loading must be made as a preliminary by the user
cos
0
L
sin
0
L
NR
S
has
that is to say F =
cos L
F (R, Z) +
sin L
F (R, Z)
L
L
l=0
0
- sin
0
cos
L
L
S (A)
S (A)
S (A)
S (A)
with F
=
,
,
L
(Fr fz F
L
L
L
)
The loads S
has
L
F and L
F are introduced harmonic by harmonic and type by type by the operator
AFFE_CHAR_MECA. One does not specify the mode nor the type on this level.
Example: one supposes a loading in pressure distributed symmetrical mode 1 and pure torsion
(antisymmetric mode 0).
One will write:
ch1sym = AFFE_CHAR_MECA
(Model = Mo,
PRES_REP
=
_F
(
GROUP_MA
=
“grma”,
PRES = p));
ch0anti = AFFE_CHAR_MECA
(Model = Mo,
FORCE_NODALE
=
_F
(
FZ
=
F,
NOEUD = “N1”));
The boundary conditions of the Dirichlet type will be introduced into a load with share:
to chdir = AFFE_CHAR_MECA (
Model = Mo,
DDL_IMPO=
_F (
GROUP_NO
=
“grno”,
DX
=
0.,
DY
=
0.,
DZ
=
0.,)
)
;
The acceptable loadings by the elements of Fourier are:
in elasticity:
Elements
Nature of the loading
Key word AFFE_CHAR_MECA
Temperature
TEMP_CALCULEE
TRIA3 - TRIA6
Forces of volume
FORCE_INTERN
QUAD4 - QUAD8 - QUAD9
Rotation
ROTATION
Gravity
PESANTEUR
Specific forces
FORCE_NODALE
SEG2 - SEG3
Pressure
PRES_REP
Surface forces
FORCE_CONTOUR
in thermics:
Elements
Nature of the loading
Key word AFFE_CHAR_THER
Surface Source
of
heat
SOURCE
Edge
Imposed normal flow
FLUX_REP
Exchange
ECHANGE
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 4/8
4 Resolution
with
Aster
Once the loading broken up into harmonics of Fourier, harmonics being uncoupled them
from/to each other (with a number of different Fourier), it is necessary to assemble and solve as many systems
linear that there are harmonics.
Moreover, the nonanisotropic material being supposed, for the same number of harmonic, the modes
symmetrical and antisymmetric are uncoupled. One will have to thus make as many resolutions (with
corresponding load) that there are couples (harmonic, mode) different.
The establishment in Aster is different according to whether the phenomenon is thermal or mechanical.
4.1 Thermics
In thermics, there is no total command making it possible to calculate several directly
harmonics. One must thus proceed harmonic by harmonic. Moreover, calculations of matrix and
second elementary members can be done only with commands CALC_MATR_ELEM and
CALC_VECT_ELEM (and not by command THER_LINEAIRE).
The mode of Fourier is to be introduced into CALC_MATR_ELEM by single-ended spanner word MODE_FOURIER.
type of the harmonic is not necessary, the matrices (and vectors) being independent of the type.
type is only taken into account with the recombination of Fourier.
It is important to assemble the matrices and vectors corresponding to the various harmonics with
same classification in order to be able to recombine the fields results. The operator NUME_DDL who
built classification is thus used once for the first harmonic, classification thus
created being re-used for all the other harmonics. This is possible if they were differentiated
loads of Dirichlet of the loadings themselves (see example [§6.1]).
4.2 Mechanics
The command making it possible to treat several harmonics is MACRO_ELAS_MULT [U4.51.02]. In
this macro, the harmonics is regarded as loading cases and one thus does as much of
resolutions that there are harmonics. As in thermics, it is necessary to differentiate the loads from Dirichlet,
who must be identical for all the harmonics, of the loadings themselves, which
can vary.
One obtains a structure of data RESULTAT containing all the fields corresponding to
calculated harmonics (see example [§6.2]).
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 5/8
5 Postprocessings
5.1 Thermics
· The calculation of flows is done by operator CALC_CHAM_ELEM by specifying the number of
the harmonic by single-ended spanner word MODE_FOURIER.
The options of calculation of flow remain the standard options:
FLUX_ELNO_TEMP to calculate flows with the nodes by element
FLUX_ELGA_TEMP to calculate flows at the points of Gauss
The command of the components of the vector flow is (,
R,
Z
).
· The recombination of Fourier on the temperatures is done starting from operator COMB_CHAM_NO
[U4.72.02]. it makes it possible to obtain the temperatures in various angular sections introduced by
the user.
The recombination of Fourier on flows is made in COMB_CHAM_ELEM [U4.72.03] according to
even principle.
5.2 Mechanics
· The calculation of the strains and the stresses is done by operator CALC_CHAM_ELEM in
specifying the number of the harmonic by single-ended spanner word MODE_FOURIER.
The options of calculation remain the standard options:
EPSI_ELNO_DEPL to calculate the deformations with the nodes by element
SIEF_ELGA_DEPL to calculate the constraints at the points of Gauss
SIGM_ELNO_DEPL to calculate the constraints with the nodes by element
The command of the components of the tensor of the deformations (resp. constraints) is
(rr, zz, rz, R, Z) (resp. rr, zz, rz, R, Z).
· The recombination of Fourier can be done either by fields, or starting from a structure of data
RESULTAT.
- by fields: in a way similar to thermics, recombination of Fourier on
displacements is done in operator COMB_CHAM_NO [U4.72.02], that on the deformations
and forced in COMB_CHAM_ELEM [U4.72.03],
- starting from a result: operator COMB_FOURIER [U4.83.31] allows to recombine all them
harmonics of the fields appearing in the structure of data RESULTAT. This
recombination can be done on a list of angles.
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 6/8
6 Examples
6.1
Thermics: calculation on 2 harmonics
% CAS-TEST THERMAL HARMONIC FOURIER 1 AND 2
% MODELISATION: ELEMENTS THERMICS AXIS_FOURIER (QUAD4)
%
DEBUT ();
mall = LIRE_MAILLAGE ();
to subdue = DEFI_MATERIAU (
THER = _F (LAMBDA =1.
, RHO_CP =1. ));
chmat = AFFE_MATERIAU (
MAILLAGE = mall,
AFFE
=
_F
(
TOUT
=
“OUI”,
MATER
=
to subdue
)
)
;
moth = AFFE_MODELE
(MAILLAGE = mall,
AFFE
=
_F
(
TOUT
=
“OUI”,
PHENOMENE
=
“THERMIQUE”,
MODELISATION
=
“AXIS_FOURIER”,
)
)
;
%
% boundary conditions of Dirichlet
% -----------------------------------
%
to chdir = AFFE_CHAR_THER (MODELE = moth,
TEMP_IMPO
=
_F
(
GROUP_NO
=
“noe_cyl”,
TEMP=0.
)
)
;
%
% loading harmonic 1
% -----------------------
%
chth1 = AFFE_CHAR_THER (MODELE = moth,
SOURCE
=
_F
(
TOUT
=
“YES”, SOUR
=
- 3.
)
)
;
%
% loading harmonic 2
% -----------------------
%
chth2 = AFFE_CHAR_THER (MODELE = moth,
SOURCE
=
_F
(
TOUT
=
“YES”, SOUR
=
- 1.
)
)
;
%
% Resolution harmonic 1
% -----------------------
%
MTRE1 = CALC_MATR_ELEM (OPTION = “RIGI_THER”,
MODELE
=
moth,
CHAM_MATER
=
chmat,
MODE_FOURIER
=
1,
CHARGE
=
(to chdir,
chth1)
)
;
VCTER1 = CALC_VECT_ELEM (OPTION = “CHAR_THER”,
CHARGE
=
(to chdir,
chth1)
)
;
naked = NUME_DDL (MATR_RIGI = mtre1,
METHODE
=
“LDLT”
,
RENUM
=
“RCMK”
)
;
mtra1 = ASSE_MATRICE (
MATR_ELEM = mtre1,
NUME_DDL
=
naked
)
;
vcta1 = ASSE_VECTEUR (
VECT_ELEM = vcter1,
NUME_DDL
=
naked
)
;
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 7/8
&MTRA1 = FACT_LDLT (MATR_ASSE = MTRA1);
TMOD1 = RESO_LDLT (MATR_FACT = MTRA1,
CHAM_NO
=
vcta1
)
;
%
% Resolution harmonic 2
% -----------------------
%
MTRE2 = CALC_MATR_ELEM (OPTION = “RIGI_THER”,
MODELE = moth,
CHAM_MATER
=
chmat,
MODE_FOURIER
=
2,
CHARGE
=
(to chdir,
chth2)
)
;
VCTER2 = CALC_VECT_ELEM (OPTION = “CHAR_THER”,
CHARGE
=
(to chdir,
chth2)
)
;
mtra2 = ASSE_MATRICE (
MATR_ELEM = mtre2,
NUME_DDL
=
naked
)
;
vcta2 = ASSE_VECTEUR (
VECT_ELEM = vcter2,
NUME_DDL
=
naked
)
;
&MTRA2 = FACT_LDLT (MATR_ASSE = MTRA2);
tmod2 = RESO_LDLT
(MATR_FACT = mtra2,
CHAM_NO
=
vcta2
)
;
%
% Recombination of Fourier section 0.
% -----------------------------------
%
TPR00 = COMB_CHAM_NO (COMB_FOURIER = _F (CHAM_NO = TMOD1,
NUME_MODE
=
1,
TYPE_MODE
=
“SYME”),
(
CHAM_NO
=
tmod2,
NUME_MODE
=
2,
TYPE_MODE
=
“SYME”),
ANGL = 0. );
%
% Recombination of Fourier section 45.
% ------------------------------------
%
TPR45 = COMB_CHAM_NO (COMB_FOURIER = _F (CHAM_NO = TMOD1,
NUME_MODE
=
1,
TYPE_MODE
=
“SYME”),
(
CHAM_NO
=
tmod2,
NUME_MODE
=
2,
TYPE_MODE
=
“SYME”),
ANGL
=
45.
)
;
FIN ();
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Code_Aster ®
Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES Key
:
U2.07.01-A Page
: 8/8
6.2
Mechanics: calculation and recombination of Fourier on 2 harmonics
DEBUT ();
m = LIRE_MAILLAGE ();
Mo = AFFE_MODELE
(MAILLAGE = m,
AFFE =
_F
(
TOUT =
“OUI”,
PHENOMENE
= “MECANIQUE”,
MODELISATION
=
“axis_fourier”
));
my = DEFI_MATERIAU (ELAS
=_F (E =
72.,
NAKED
= 0.3,
RHO
= 0.
));
cm = AFFE_MATERIAU (MAILLAGE = m,
AFFE =
_F
(
TOUT =
“OUI”,
MATER
= my
));
bloqu = AFFE_CHAR_MECA_F (
MODELE
= Mo,
DDL_IMPO = _F (NODE
= “N1”,
DX = 0., DY = 0., DZ = 0. )
(
NOEUD
=
“N2”, DY
=
0.,
)
(
NOEUD
=
“N3”, DY
=
0.,
)
);
CH = AFFE_CHAR_MECA
(MODELE
= Mo,
PRES_REP = _F (GROUP_MA = “end”, PRES = 100. ));
%
% FOURIER CALCULATION ON THE 2 FIRST SYMMETRICAL HARMONICS
resu = MACRO_ELAS_MULT (MODELE
= Mo,
CHAM_MATER
=
cm,
CHAR_MECA_GLOBAL
=
bloqu,
CAS_CHARGE= (
_F
(
MODE_FOURIER
=
1,
TYPE_MODE
=
“SYME”,
CHAR_MECA
= CH,
OPTION
=
“SIGM_ELNO_DEPL”,
SOUS_TITER = “mode Fourier 1 SYME”),
_F
(
MODE_FOURIER
=
2,
TYPE_MODE
=
“SYME”,
CHAR_MECA
= CH,
OPTION
=
“SIGM_ELNO_DEPL”,
SOUS_TITER = “Fourier mode 2 SYME”),
);
%
% CALCULATION OF THE NODAL REACTIONS BY CALC_NO
%
&resu = CALC_NO (RESULTAT = resu,
EXCIT
=
_F (
CHARGE
=
CH
),
OPTION
=
“REAC_NODA”,
CHAM_MATER=
cm
);
angl1 = 45.
;
angl2 = 135.
;
%
% RECOMBINATION OF FOURIER ON DISPLACEMENTS, REACTIONS AND FORCED
%
% co_four = COMB_FOURIER (
RESULTAT = resu,
NOM_CHAM
=
(
“DEPL”,
“REAC_NODA”,
“SIGM_ELNO_DEPL”,)
ANGL =
(
angl1,
angl2
),
);
FIN ();
Handbook of Utilization
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
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