Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
Organization (S): EDF/IMA/MN, SAMTECH
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
Document: V2.03.007
SDLS07 - Modes clean 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 (grid of one 1/8 of sphere),
2) Elements
finished
2D axisymmetric TRIA6 and QUAD8 (grid of a section),
3) Three-dimensional hulls: axisymmetric finite elements isoparametric SEG3 (linear grid of
the section),
4) Hulls
three-dimensional
COQUE_3D: finite element MEC3QU9H (grid 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.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
1
Problem of reference
1.1 Geometry
Rm
T
It is of a thin sphere, average radius Rm = 2.5 m, and about 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/m3.
1.3
Boundary conditions and loadings
The structure is free in space.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
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), the clean modes with displacement
radial and tangential establish by a theory of membrane are given by [bib1] and [bib2]:

E
F
I
=
I
2 R
(1-2)
= 1 B ± b2 - 4 1-2
2 + - 2
= 2
with
and

+ +1+
I
(
) (I I) B I I 3
2
The theory presented by Hayek makes it possible to introduce a correction of the effect of inflection (approximation of
general theory of Wilkinson) who leads to values of I function of
= T 2 has/
R2
12
B = I (I +)
1
and solution of:
4
2
2
2
2
-
+
-
- +
+
+
+
-
+ -
+ -
-
+
=
I
I [1 3 A (1) B (1 A ab)] ab [B
B
4
5] (1) [B 2 (1 A)] 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 Guide de Validation of Progiciels de Calcul 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
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
3 Modeling
With
3.1
Characteristics of modeling
Hulls DKT
Z
2,5
(3 symmetry planes)
0
2,5
2,5
X
y
The discretized geometry is represented above. Elements DKT are plane facets 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 blocked rotations).
3.2
Characteristics of the grid
A number of nodes: 1128
A number of meshs and types: 2125 TRIA3
3.3 Functionalities
tested
Commands
Keys
AFFE_MODELE
AFFE
MODELISATION: “DKT”
[U4.22.01]
AFFE_CARA_ELEM
COQUE
EPAIS: 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]
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
4
Results of modeling A
4.1 Values
tested
(Frequencies as Hertz)
Value of parameter I of
Reference
Aster
% difference
the reference solution
237.25
237.24
­ 0.005
237.24
­ 0.003
2
3
282.85
not obtained [§4.2]
4
305.24
304.97
­ 0.089
304.99
­ 0.080
305.08
­ 0.054
5
324.17
not obtained [§4.2]
6
346.76
346.11
­ 0.186
346.12
­ 0.185
346.30
­ 0.133
346.38
­ 0.108
7
376.68
not obtained [§4.2]
8
416.00
414.89
­ 0.266
414.92
­ 0.259
415.16
­ 0.201
415.24
­ 0.183
415.33
­ 0.161
9
465.75
not obtained [§4.2]
10
526.20
524.34
­ 0.353
524.43
­ 0.337
524.71
­ 0.283
524.94
­ 0.240
524.97
­ 0.234
525.12
­ 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
Obstruction memory:
16 megawords
Time CPU To use:
393.7 seconds
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
5 Modeling
B
5.1
Characteristics of modeling
axisymmetric 2D
y
X
No boundary condition.
5.2
Characteristics of the grid
A number of nodes: 365
A number of meshs and types: 40 QUAD8 and 80 TRIA6
5.3 Functionalities
tested
Commands
Keys
AFFE_MODELE
AFFE
MODELISATION: “AXIS”
[U4.22.01]
CALC_MATR_ELEM
OPTION
“RIGI_MECA”
[U4.41.01]
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
“TAPE” FREQ:(230., 530.)
[U4.52.02]
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
6
Results of modeling B
6.1 Values
tested
Frequencies as Hertz
Identification
Reference
Aster
% difference
n° mode
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
Obstruction memory:
8 megawords
Time CPU To use:
8.88 seconds
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
7 Modeling
C
7.1
Characteristics of modeling
Axisymmetric hulls 1D
y
+ 2,50
+ 2,50
X
- 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 grid
A number of nodes: 81
A number of meshs and types: 40 SEG3
7.3 Functionalities
tested
Commands
Keys
AFFE_MODELE
AFFE
MODELISATION: “COQUE_AXIS”
[U4.22.01]
AFFE_CARA_ELEM
COQUE
EPAIS: 0.10,
[U4.24.01]
A_CIS: 1.E6
MODI_METRIQUE: “NON”
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]
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
8
Results of modeling C
8.1 Values
tested
(Frequencies as Hertz)
Identification
Reference
Aster
Aster
% difference
n° mode
ITER_SIMULT
ITER_INV
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
Obstruction 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.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
9 Modeling
D
9.1
Characteristics of modeling
Hulls 3D MEC3QU9H
Z
2,5
(3 symmetry planes)
0
2,5
2,5
X
y
The boundary conditions applied to the three borders correspond to the conditions of symmetry
(displacements and blocked rotations).
9.2
Characteristics of the grid
A number of nodes: 331
A number of meshs and types: 75 QUAD9
9.3 Functionalities
tested
Commands
Keys
AFFE_MODELE
AFFE
MODELISATION: “COQUE_3D”
[U4.22.01]
AFFE_CARA_ELEM
COQUE
EPAIS: 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]
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
10
Results of modeling D
10.1 Values
tested
(Frequencies as Hertz)
Identification
Reference
Aster
% difference
n° mode
237.25
237.25
0
237.26
0.004
2
3
282.85
not obtained [§10.2]
4
305.24
305.18
­ 0.019
305.19
­ 0.017
305.20
­ 0.011
5
324.17
not obtained [§10.2]
6
346.76
346.17
­ 0.169
346.19
­ 0.165
346.25
­ 0.147
346.36
­ 0.114
7
376.68
not obtained [§10.2]
8
416.00
413.81
­ 0.525
413.84
­ 0.520
413.84
­ 0.518
414.02
­ 0.476
414.09
­ 0.46
9
465.75
not obtained [§10.2]
10
526.20
520.57
­ 1.071
520.62
­ 1.06
520.64
­ 1.056
521.28
­ 0.935
521.29
­ 0.933
521.31
­ 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
Obstruction memory:
16 megawords
Time CPU To use:
41.3 seconds
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
11 Modeling
E
11.1 Characteristics of modeling
Hulls 3D MEC3TR7H
Z
2,5
(3 symmetry planes)
0
2,5
2,5
X
y
11.2 Characteristics of the grid
A number of nodes: 925
A number of meshs and types: 294 TRIA7
11.3 Functionalities
tested
Commands
Keys
AFFE_MODELE
AFFE
MODELISATION: “COQUE_3D”
[U4.22.01]
AFFE_CARA_ELEM
COQUE
EPAIS: 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]
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
12
Results of modeling E
12.1 Values
tested
(Frequencies as Hertz)
Identification
Reference
Aster
% difference
n° mode
237.25
237.25
­ 0.001
237.25
­ 0.001
2
3
282.85
not obtained [§12.2]
4
305.24
305.20
­ 0.011
305.22
­ 0.008
305.22
­ 0.005
5
324.17
not obtained [§12.2]
6
346.76
346.32
­ 0.126
346.43
­ 0.095
346.46
­ 0.086
346.58
­ 0.051
7
376.68
not obtained [§12.2]
8
416.00
413.91
­ 0.502
414.33
­ 0.402
414.36
­ 0.394
414.99
­ 0.241
415.14
­ 0.206
9
465.75
not obtained [§12.2]
10
526.20
520.
­ 1.176
521.02
­ 0.985
521.43
­ 0.907
522.32
­ 0.738
523.03
­ 0.602
523.77
­ 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
Obstruction memory:
16 megawords
Time CPU To use:
98.64 seconds
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X

Code_Aster ®
Version
4.0
Titrate:
Clean SDLS07 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
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 grid 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.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/98/040 - Ind X