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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
Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
Organization (S):
EDF-R & D/AMA















Instruction manual
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 control making it possible to solve a problem on
several harmonics. Calculation must be done harmonic by harmonic.
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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
Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
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:

radial component according to R
axial component according to Z
component tangential (or circumferential) according to

Example: (U
R
, U
Z
, U
)
(F
R
, F
Z
, F
)
U
Z
U
U
R
R
Z

The mesh is localized in plan (R, Z), the symmetry of revolution being done around axis OZ.
trihedron (R, Z,
) is directed in the direct direction.

R
Z
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Code_Aster
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Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
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07/05/04
Author (S):
X. DESROCHES
Key
:
U2.07.01-A
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Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
3 Modeling,
loadings
To affect the axisymmetric finite elements Fourier on the mesh, the operator is used
AFFE_MODELE
in the following way:
Mo = AFFE_MODELE (MESH = my,
AFFE
=
_F
(
ALL
=
“YES”,
PHENOMENON
=
“MECHANICAL”
or
“THERMAL”,
MODELING
=
“AXIS_FOURIER”
)
);
The decomposition in Fourier series of the loading must be made as a preliminary by the user
that is to say
()
()
=








+




-
=
NR
has
S
Z
R
Z
R
0
,
cos
0
sin
0
sin
,
sin
0
cos
0
cos
L
L
L
L
L
L
L
L
L
F
F
F
with
()
()
()
()
(
)
has
S
has
S
Z
has
S
R
has
S
F
F
F
L
L
L
L
,
,
=
F
Loads
has
S
L
L
F
F
and
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”,
CLOSE = p));
ch0anti = AFFE_CHAR_MECA
(Model = Mo,
FORCE_NODALE
=
_F
(
FZ
=
F,
NODE = “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
TRIA3 - TRIA6
QUAD4 - QUAD8 - QUAD9
Temperature
Forces of volume
Rotation
Gravity
Specific forces
TEMP_CALCULEE
FORCE_INTERN
ROTATION
GRAVITY
FORCE_NODALE
SEG2 - SEG3
Pressure
Surface forces
PRES_REP
FORCE_CONTOUR
in thermics:
Elements
Nature of the loading
Key word
AFFE_CHAR_THER
Surface Source
of
heat
SOURCE
Edge
Imposed normal flow
Exchange
FLUX_REP
EXCHANGE
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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
:
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Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
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 control 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 the controls
CALC_MATR_ELEM
and
CALC_VECT_ELEM
(and not by the control
THER_LINEAIRE
).
The mode of Fourier is to be introduced into
CALC_MATR_ELEM
by the 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 control 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
RESULT
containing all the fields corresponding to
calculated harmonics (see example [§6.2]).
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Code_Aster
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Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
:
07/05/04
Author (S):
X. DESROCHES
Key
:
U2.07.01-A
Page
:
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Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
5 Postprocessings
5.1 Thermics
·
The calculation of flows is done by the operator
CALC_CHAM_ELEM
by specifying the number of
the harmonic by the 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
(
)
,
,
Z
R
.
·
The recombination of Fourier on the temperatures is done starting from the 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 the operator
CALC_CHAM_ELEM
in
specifying the number of the harmonic by the 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 stresses at the points of Gauss
SIGM_ELNO_DEPL
to calculate the stresses with the nodes by element
The command of the components of the tensor of the deformations (resp. stresses) is
(
) (
)
Z
R
rz
zz
rr
Z
R
rz
zz
rr
,
,
,
,
,
,
,
,
,
,
resp.
.
·
The recombination of Fourier can be done either by fields, or starting from a structure of data
RESULT
.
-
by fields: in a way similar to thermics, recombination of Fourier on
displacements is done in the operator
COMB_CHAM_NO
[U4.72.02], that on the deformations
and forced in
COMB_CHAM_ELEM
[U4.72.03],
-
starting from a result: the operator
COMB_FOURIER
[U4.83.31] allows to recombine all them
harmonics of the fields appearing in the structure of data
RESULT
. This
recombination can be done on a list of angles.
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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
:
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Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
6 Examples
6.1
Thermics: calculation on 2 harmonics
% CAS-TEST THERMAL HARMONIC FOURIER 1 AND 2
% MODELING: ELEMENTS THERMICS AXIS_FOURIER (QUAD4)
%
BEGINNING ();
mall = LIRE_MAILLAGE ();
to subdue = DEFI_MATERIAU (
THER = _F (LAMBDA =1.
, RHO_CP =1. ));
chmat = AFFE_MATERIAU (
MESH = mall,
AFFE
=
_F
(
ALL
=
“YES”,
MATER
=
to subdue
)
)
;

moth = AFFE_MODELE
(MESH = mall,
AFFE
=
_F
(
ALL
=
“YES”,
PHENOMENON
=
“THERMAL”,
MODELING
=
“AXIS_FOURIER”,
)
)
;
%
% boundary conditions of Dirichlet
% -----------------------------------
%
to chdir = AFFE_CHAR_THER (MODEL = moth,
TEMP_IMPO
=
_F
(
GROUP_NO
=
“noe_cyl”,
TEMP=0.
)
)
;
%
% loading harmonic 1
% -----------------------
%
chth1 = AFFE_CHAR_THER (MODEL = moth,
SOURCE
=
_F
(
ALL
=
“YES”, SOUR
=
- 3.
)
)
;
%
% loading harmonic 2
% -----------------------
%
chth2 = AFFE_CHAR_THER (MODEL = moth,
SOURCE
=
_F
(
ALL
=
“YES”, SOUR
=
- 1.
)
)
;
%
% Resolution harmonic 1
% -----------------------
%
MTRE1 = CALC_MATR_ELEM (OPTION = “RIGI_THER”,
MODEL
=
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,
METHOD
=
“LDLT”
,
RENUM
=
“RCMK”
)
;

mtra1 = ASSE_MATRICE (
MATR_ELEM = mtre1,
NUME_DDL
=
naked
)
;

vcta1 = ASSE_VECTEUR (
VECT_ELEM = vcter1,
NUME_DDL
=
naked
)
;
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Code_Aster
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Version
6.4
Titrate:
Note of use of FOURIER modeling
Date
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07/05/04
Author (S):
X. DESROCHES
Key
:
U2.07.01-A
Page
:
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Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A

&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”,
MODEL = 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”),
ENG = 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”),
ENG
=
45.
)
;

END ();
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Code_Aster
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Titrate:
Note of use of FOURIER modeling
Date
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Instruction manual
U2.07 booklet: Method to reduce the size of modeling
HT-66/04/004/A
6.2
Mechanics: calculation and recombination of Fourier on 2 harmonics
BEGINNING ();
m = LIRE_MAILLAGE ();
Mo = AFFE_MODELE
(MESH = m,
AFFE =
_F
(
ALL =
“YES”,
PHENOMENON
= “MECHANICAL”,
MODELING
=
“axis_fourier”
));
my = DEFI_MATERIAU (ELAS
=_F (E =
72.,
NAKED
= 0.3,
RHO
= 0.
));
cm = AFFE_MATERIAU (MESH = m,
AFFE =
_F
(
ALL =
“YES”,
MATER
= my
));
bloqu = AFFE_CHAR_MECA_F (
MODEL
= Mo,
DDL_IMPO = _F (NODE
= “N1”,
DX = 0., DY = 0., DZ = 0. )
(
NODE
=
“N2”, DY
=
0.,
)
(
NODE
=
“N3”, DY
=
0.,
)
);
CH = AFFE_CHAR_MECA
(MODEL
= Mo,
PRES_REP = _F (GROUP_MA = “end”, CLOSE = 100. ));
%
% FOURIER CALCULATION ON THE 2 FIRST SYMMETRICAL HARMONICS
resu = MACRO_ELAS_MULT (MODEL
= 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 (RESULT = 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 (
RESULT = resu,
NOM_CHAM
=
(
“DEPL”,
“REAC_NODA”,
“SIGM_ELNO_DEPL”,)
ENG =
(
angl1,
angl2
),
);
END ();