Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
1/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
Organization (S):
EDF/IMA/MN, SAMTECH
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
Document: V2.03.007
SDLS07 - Clean modes of an envelope
spherical thin
Summary:
This test from guide VPCS makes it possible to validate the algorithm of search for eigenvalues
MODE_ITER_SIMULT
[U4.52.02] with the operators of rigidity and mass corresponding to modelings
following:
1) Three-dimensional hulls: finite elements
DKT
(mesh of one 1/8 of sphere),
2) Elements
finished
2D
axisymmetric
TRIA6
and
QUAD8
(mesh of a section),
3) Three-dimensional hulls: isoparametric axisymmetric finite elements
SEG3
(linear mesh of
the section),
4) Hulls
three-dimensional
COQUE_3D
: finite element
MEC3QU9H
(mesh 1/8 of sphere),
5) Hulls
three-dimensional
COQUE_3D
: finite element
MEC3TR7H
.
The results obtained are compared with the analytical solution (HAYEK) and reveal for the six first
modes of the lower deviations than:
·
0,45% for the axisymmetric elements continuous mediums,
·
0,20% for the elements of hull
DKT
,
·
0,17% for the elements of axisymmetric hulls isoparametric,
·
0,17% for the elements
COQUE_3D
.
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
2/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
1
Problem of reference
1.1 Geometry
T
R
m
It is about a thin sphere, of average radius R
m
= 2.5 m, and thickness T = 0.1 Mr.
1.2
Properties of materials
The material is homogeneous, isotropic, elastic linear. The elastic coefficients are:
E = 200.000 MPa and
= 0.3.
The density is constant and is worth:
= 7.800 kg/m
3
.
1.3
Boundary conditions and loadings
The structure is free in space.
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
3/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
2
Reference solution
2.1
Method of calculation used for the reference solution
For the thin spheres (
I T
R
.
<<
with
I
, command of the mode), clean modes with displacement
radial and tangential establish by a theory of membrane are given by [bib1] and [bib2]:
(
)
(
) (
)
F
R
E
B
B
I
I
B
I
I
I
I
I
=
-
=
±
-
-
+ -
= + + +
2
1
1
2
4 1
2
1 3
2
2
2
2
2
with
and
The theory presented by Hayek makes it possible to introduce a correction of the effect of bending (approximation of
general theory of Wilkinson) who leads to values of
I
function of
(
)
has
T
R
B
I I
=
=
+
2
2
12
1
/
and solution of:
(
) (
)
[
]
[
]
(
)
(
)
[
]
I
I
has
B
has
ab
ab B
B
B
has
4
2
2
2
2
1 3
1
1
4
5
1
2 1
0
-
+
-
- +
+
+
+
-
+ -
+ -
-
+
=
2.2
Results of reference
Eigen frequencies:
I
Eigen frequencies
2
237.25
3
282.85
4
305.24
5
324.17
6
346.76
7
376.68
8
416.
9
465.75
10
526.20
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Card-index VPCS SDLS 07/89 in the Guide of Validation of the Software packages of Calculation of
Structures/SFM AFNOR TECHNIQUE 1990.
[2]
S. HAYEK: “Vibrations off has spherical Shell in acoustic medium”, Journal off the Acoustical
Society off America, vol. 40, 2, 1996, p. 342-348
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
4/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
3 Modeling
With
3.1
Characteristics of modeling
Hulls
DKT
0
Z
X
y
2,5
2,5
2,5
(3 symmetry planes)
The discretized geometry is represented above. Elements
DKT
are plane breakages with
3 nodes. The number of the nodes on the meridian line and the equator is: 34.
The boundary conditions applied to the three borders correspond to the conditions of symmetry
(displacements and locked rotations).
3.2
Characteristics of the mesh
A number of nodes: 1128
A number of meshs and types: 2125 TRIA3
3.3 Functionalities
tested
Controls
Keys
AFFE_MODELE
AFFE
MODELING: “DKT”
[U4.22.01]
AFFE_CARA_ELEM
HULL
THICK: 0.10
[U4.24.01]
CALC_MATR_ELEM
OPTION
“RIGI_MECA”
[U4.41.01]
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ:(230., 530.)
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
5/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
4
Results of modeling A
4.1 Values
tested
(Frequencies in Hertz)
Value of parameter I of
the reference solution
Reference
Aster
% difference
237.25
237.24
237.24
0.005
0.003
2
3
282.85
not obtained [§4.2]
4
305.24
304.97
304.99
305.08
0.089
0.080
0.054
5
324.17
not obtained [§4.2]
6
346.76
346.11
346.12
346.30
346.38
0.186
0.185
0.133
0.108
7
376.68
not obtained [§4.2]
8
416.00
414.89
414.92
415.16
415.24
415.33
0.266
0.259
0.201
0.183
0.161
9
465.75
not obtained [§4.2]
10
526.20
524.34
524.43
524.71
524.94
524.97
525.12
0.353
0.337
0.283
0.240
0.234
0.205
4.2 Remarks
The reference solution does not give the multiplicity of the modes. One observes with calculations of
commands of multiplicity which grow with the value of the frequency.
Modes 3, 5, 7, 9 are not obtained because of the boundary conditions chosen for this model,
with the three symmetry planes.
4.3 Parameters
of execution
Version: 4.00.02
Machine: C90
System:
UNICOS 8.0
Overall dimension memory:
16 megawords
Time CPU To use:
393.7 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
6/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
5 Modeling
B
5.1
Characteristics of modeling
axisymmetric 2D
X
y
No boundary condition.
5.2
Characteristics of the mesh
A number of nodes: 365
A number of meshs and types: 40 QUAD8 and 80 TRIA6
5.3 Functionalities
tested
Controls
Keys
AFFE_MODELE
AFFE
MODELING: “AXIS”
[U4.22.01]
CALC_MATR_ELEM
OPTION
“RIGI_MECA”
“MASS_MECA”
[U4.41.01]
MODE_ITER_SIMULT
CALC_FREQ
“TAPE” FREQ:(230., 530.)
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
7/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
6
Results of modeling B
6.1 Values
tested
Frequencies in Hertz
Identification
n° mode
Reference
Aster
% difference
237.25
237.24
0.036
2
3
282.85
282.78
0.023
4
305.24
304.85
0.125
5
324.17
323.32
0.262
6
346.76
345.22
0.443
7
376.68
374.14
0.674
8
416.00
412.03
0.955
9
465.75
459.75
1.286
10
526.20
517.51
1.651
6.2 Parameters
of execution
Version: 4.00.02
Machine: C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
8.88 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
8/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
7 Modeling
C
7.1
Characteristics of modeling
Axisymmetric hulls 1D
X
y
+ 2,50
- 2,50
+ 2,50
No boundary condition.
One chooses the model of Coils-Kirchhoff to describe kinematics. With the element chosen, this
kinematics is obtained by penalization: one puts a great value for coefficient A_CIS. By
elsewhere, one neglects the correction of metric.
7.2
Characteristics of the mesh
A number of nodes: 81
A number of meshs and types: 40 SEG3
7.3 Functionalities
tested
Controls
Keys
AFFE_MODELE
AFFE
MODELING: “COQUE_AXIS”
[U4.22.01]
AFFE_CARA_ELEM
HULL
THICK: 0.10,
[U4.24.01]
A_CIS: 1.E6
MODI_METRIQUE: “NOT”
CALC_MATR_ELEM
OPTION
“RIGI_MECA”
[U4.41.01]
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
“TAPE” FREQ:(220., 530.)
[U4.52.02]
MODE_ITER_INV
CALC_FREQ
FREQ:(220., 530.)
[U4.52.01]
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
9/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
8
Results of modeling C
8.1 Values
tested
(Frequencies in Hertz)
Identification
n° mode
Reference
Aster
ITER_SIMULT
Aster
ITER_INV
% difference
2
237.25
237.31
237.32
0.025/0.029
3
282.85
282.77
282.78
0.028/ 0.025
4
305.24
304.95
304.95
0.096
5
324.17
323.68
323.68
0.150
6
346.76
346.23
346.23
0.154
8.2 Parameters
of execution
Version: 4.00.02
Machine: C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
8.0 seconds
8.3 Remarks
The purpose of this test with this modeling is only to test the matrix of mass. A satisfactory variation
being observed on the first six frequencies, one chose not to calculate the following ones.
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
10/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
9 Modeling
D
9.1
Characteristics of modeling
Hulls 3D
MEC3QU9H
0
Z
X
y
2,5
2,5
2,5
(3 symmetry planes)
The boundary conditions applied to the three borders correspond to the conditions of symmetry
(displacements and locked rotations).
9.2
Characteristics of the mesh
A number of nodes: 331
A number of meshs and types: 75 QUAD9
9.3 Functionalities
tested
Controls
Keys
AFFE_MODELE
AFFE
MODELING: “COQUE_3D”
[U4.22.01]
AFFE_CARA_ELEM
HULL
THICK: 0.10
[U4.24.01]
CALC_MATR_ELEM
OPTION
“RIGI_MECA”
[U4.41.01]
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ:(230., 530.)
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
11/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
10
Results of modeling D
10.1 Values
tested
(Frequencies in Hertz)
Identification
n° mode
Reference
Aster
% difference
237.25
237.25
237.26
0
0.004
2
3
282.85
not obtained [§10.2]
4
305.24
305.18
305.19
305.20
0.019
0.017
0.011
5
324.17
not obtained [§10.2]
6
346.76
346.17
346.19
346.25
346.36
0.169
0.165
0.147
0.114
7
376.68
not obtained [§10.2]
8
416.00
413.81
413.84
413.84
414.02
414.09
0.525
0.520
0.518
0.476
0.46
9
465.75
not obtained [§10.2]
10
526.20
520.57
520.62
520.64
521.28
521.29
521.31
1.071
1.06
1.056
0.935
0.933
0.929
10.2 Remarks
Modes 3, 5, 7, 9 are not obtained because of the boundary conditions chosen for this model,
with the three symmetry planes.
10.3 Parameters
of execution
Version: 4.00.14
Machine: C90
System:
UNICOS 8.0
Overall dimension memory:
16 megawords
Time CPU To use:
41.3 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
12/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
11 Modeling
E
11.1 Characteristics of modeling
Hulls 3D
MEC3TR7H
0
Z
X
y
2,5
2,5
2,5
(3 symmetry planes)
11.2 Characteristics of the mesh
A number of nodes: 925
A number of meshs and types: 294 TRIA7
11.3 Functionalities
tested
Controls
Keys
AFFE_MODELE
AFFE
MODELING: “COQUE_3D”
[U4.22.01]
AFFE_CARA_ELEM
HULL
THICK: 0.10
[U4.24.01]
CALC_MATR_ELEM
OPTION
“RIGI_MECA”
[U4.41.01]
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ:(230., 530.)
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
13/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
12
Results of modeling E
12.1 Values
tested
(Frequencies in Hertz)
Identification
n° mode
Reference
Aster
% difference
237.25
237.25
237.25
0.001
0.001
2
3
282.85
not obtained [§12.2]
4
305.24
305.20
305.22
305.22
0.011
0.008
0.005
5
324.17
not obtained [§12.2]
6
346.76
346.32
346.43
346.46
346.58
0.126
0.095
0.086
0.051
7
376.68
not obtained [§12.2]
8
416.00
413.91
414.33
414.36
414.99
415.14
0.502
0.402
0.394
0.241
0.206
9
465.75
not obtained [§12.2]
10
526.20
520.
521.02
521.43
522.32
523.03
523.77
1.176
0.985
0.907
0.738
0.602
0.461
12.2 Remarks
Modes 3, 5, 7, 9 are not obtained because of the boundary conditions chosen for this model,
with the three symmetry planes.
12.3 Parameters
of execution
Version: 4.00.14
Machine: C90
System:
UNICOS 8.0
Overall dimension memory:
16 megawords
Time CPU To use:
98.64 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLS07 clean Modes of a thin spherical envelope
Date:
07/12/98
Author (S):
P. MASSIN, B. QUINNEZ, A. LAULUSA
Key:
V2.03.007-C
Page:
14/14
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X
13
Summary of the results
·
Modeling hull
DKT
, here restricted with the modes having 3 symmetries compared to
plans X = 0, y = 0, Z = 0, provide the Eigen frequencies with an error lower than 0.4% on the 20
first modes.
·
Modeling continuous medium axisymmetric 2D provides the Eigen frequencies with an error
lower than 2%.
·
Modeling
COQUE_AXIS
(quadratic isoparametric elements) provides the frequencies
clean with an error lower than 0.2% on the first 5 modes (space discretization
identical to the trace of the axisymmetric mesh 2D).
·
Modeling
COQUE_3D
degenerated (elements of thick hull MEC3QU9H, MEC3TR7H)
to provide the Eigen frequencies with an error lower than 1.2% on the first 10 modes.