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Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
1/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
Organization (S):
EDF/IMA/MN
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.111
SSLV111 - Estimator of error on a plate
perforated in linear elasticity
Summary:
This test validates and compares the 2 versions of the estimator of error of Zhu-Zienkiewicz (version 1 of 1987, noted
ZZ1, and version 2 of 1992, noted ZZ2) applied to the system of linear elasticity, in statics.
It comprises 5 modelings in plane stresses, corresponding each one to a type of finite element (TRIA3,
QUAD4, TRIA6, QUAD8, QUAD9).
The analytical solution is known and makes it possible to compare the errors estimated with the exact error.
The interest of the test resides:
·
the comparison enters the stresses smoothed with ZZ1 (continuous total smoothing) and ZZ2 (smoothing
room with patchs of elements),
·
in the comparison of the estimators between them,
·
in the qualitative and quantitative analysis of the results (relative errors total and local).
The test highlights the good behavior of ZZ2 on all the types of elements and the bad results of
ZZ1 on quadratic elements when the solution does not present a strong singularity, which is the case.
Code_Aster
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Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
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X. DESROCHES
Key:
V3.04.111-A
Page:
2/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
1
Problem of reference
1.1 Geometry
With
B
C
D
E
has
()
y
X
=
1.0
=
1.0
With
B
C
D
E
y
X
1.0
3.0
R
(H)
1.2
Material properties
E = 1.000 MPa
= 0.3.
Code_Aster
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4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
3/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
1.3
Boundary conditions and loadings
On AB,
U
X
=
0
On ED,
U
y
=
0
On CD tractions
(
)
(
)
F
X
F
X
X
xx
y
xy
=
=
=
=
4
4
.
.
On BC tractions
(
)
(
)
F
y
F
y
X
xy
y
yy
=
=
=
=
4
4
.
.
According to the analytical solution [2].
Code_Aster
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Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
4/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
One considers a portion of an infinite plate with a circular central hole, subjected to a loading
unit one-way in the direction
OX
E
E
X
X
=
1 &
&
.
The analytical solution of this problem is [bib1]:
()
()
()
()
()
()
()
()
xx
yy
xy
has
R
has
R
has
R
has
R
has
R
has
R
=
-
+
+
= -
-
-
= -
+
+
1
3
2
4
3
2
4
1
2
2
4
3
2
4
1
2
2
4
3
2
4
2
2
4
4
2
2
4
4
2
2
4
4
cos
cos
cos
cos
cos
cos
sin
sin
sin
where:
·
has
is the radius of the hole,
·
()
R,
polar co-ordinates.
2.2
Uncertainty on the solution
Analytical solution.
2.3 References
bibliographical
[1]
Zhu-Zienkiewicz: The superconvergent patch recovery and a posteriori error estimates -
1 leaves: the technical recovery (Int. J. for Num. Methods in Engineering vol. 33, p. 1355
(May 1992)).
Code_Aster
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Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
5/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
3.2
Characteristics of the mesh
A number of nodes: 357.
A number of meshs and types: 640 TRIA3.
3.3 Functionalities
tested
Controls
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
6/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
With
xx
ZZ1
3.
2.823
5.91
0.1
xx
ZZ2
3.
2.884
3.85
0.1
yy
ZZ1
0.
0.261
-
0.3
yy
ZZ2
0.
0.207
-
0.3
xy
ZZ1
0.
7.4 10
3
-
0.1
xy
ZZ2
0.
6.1 10
2
-
0.1
P
xx
ZZ1
1.15625
1.152
0.37
0.1
xx
ZZ2
1.15625
1.145
0.98
0.1
yy
ZZ1
0.15625
0.150
3.81
0.1
yy
ZZ2
0.15625
0.145
7.00
0.1
xy
ZZ1
0.125
0.117
6.11
0.1
xy
ZZ2
0.125
0.124
0.68
0.1
Net M1
E
ABS
ZZ1
1.33 10
4
1.10
3
E
ABS
ZZ2
8.13 10
5
1.10
3
E
rel
ZZ1
6.63%
0.1
E
rel
ZZ2
4.05%
0.1
E
ABS
ZZ1
0.445 10
2
0.424 10
2
4.76
E
ABS
ZZ2
0.445 10
2
0.451 10
2
+1.31
E
rel
ZZ1
3.44%
3.28%
E
rel
ZZ2
3.44%
3.49%
ZZ1
0.952
ZZ2
1.013
4.2 Remarks
=
estimated error
exact error is an indication of effectivity of the estimator.
Contents of the files results:
·
absolute errors and relative total by the 2 methods,
·
maximum and minimal values of the stresses and the errors,
·
lists of the meshs where the relative error is higher than 10%.
4.3 Parameters
of execution
Version: 3.02.11
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
19.0 seconds
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
7/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
5 Modeling
B
5.1
Characteristics of modeling
5.2
Characteristics of the mesh
A number of nodes: 357.
A number of meshs and types: 320 QUAD4.
5.3 Functionalities
tested
Controls
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
8/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
With
xx
ZZ1
3.
3.017
0.57
0.1
xx
ZZ2
3.
2.971
- 0.95
0.1
yy
ZZ1
0.
0.17
-
0.3
yy
ZZ2
0.
0.136
-
0.3
xy
ZZ1
0.
2.8 10
3
-
0.1
xy
ZZ2
0.
1.04 10
2
-
0.1
P
xx
ZZ1
1.15625
1.168
0.98
0.1
xx
ZZ2
1.15625
1.153
0.26
0.1
yy
ZZ1
0.15625
0.158
1.28
0.1
yy
ZZ2
0.15625
0.152
2.83
0.1
xy
ZZ1
0.125
0.121
2.99
0.1
xy
ZZ2
0.125
0.124
0.94
0.1
Net M1
E
ABS
ZZ1
1.57 10
4
1.10
3
E
ABS
ZZ2
2.40 10
4
1.10
3
E
rel
ZZ1
5.79%
0.1
E
rel
ZZ2
8.83%
0.1
E
ABS
ZZ1
0.320 10
2
0.294 10
2
8.1
E
ABS
ZZ2
0.320 10
2
0.307 10
2
4.2
E
rel
ZZ1
2.48%
2.28%
E
rel
ZZ2
2.48%
2.37%
ZZ1
0.919
ZZ2
0.958
6.2 Remarks
=
estimated error
exact error is an indication of effectivity of the estimator.
Contents of the files results:
·
absolute errors and relative total by the 2 methods,
·
maximum and minimal values of the stresses and the errors,
·
lists of the meshs where the relative error is higher than 10%.
6.3 Parameters
of execution
Version: 3.02.11
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
13.4 seconds
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
9/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
7 Modeling
C
7.1
Characteristics of modeling
7.2
Characteristics of the mesh
A number of nodes: 357.
A number of meshs and types: 160 TRIA6.
7.3 Functionalities
tested
Controls
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
10/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
8
Results of modeling C
8.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
With
xx
ZZ1
3.
2.975
0.83
0.1
xx
ZZ2
3.
2.957
1.43
0.1
yy
ZZ1
0.
6.86 10
2
-
0.3
yy
ZZ2
0.
7.52 10
2
-
0.3
xy
ZZ1
0.
3.12 10
2
-
0.3
xy
ZZ2
0.
0.155
-
0.3
P
xx
ZZ1
1.15625
1.166
0.85
0.1
xx
ZZ2
1.15625
1.153
0.25
0.1
yy
ZZ1
0.15625
0.167
6.92
0.1
yy
ZZ2
0.15625
0.153
1.87
0.1
xy
ZZ1
0.125
0.127
1.52
0.1
xy
ZZ2
0.125
0.124
0.58
0.1
Net M1
E
ABS
ZZ1
1.81 10
4
1.10
3
E
ABS
ZZ2
2.92 10
4
1.10
3
E
rel
ZZ1
4.69%
0.1
E
rel
ZZ2
7.56%
0.1
E
ABS
ZZ1
0.152 10
2
0.123 10
2
19
E
ABS
ZZ2
0.152 10
2
0.167 10
2
+9.9
E
rel
ZZ1
1.17%
0.95%
E
rel
ZZ2
1.17%
1.29%
ZZ1
0.810
ZZ2
1.099
8.2 Remarks
=
estimated error
exact error is an indication of effectivity of the estimator.
Contents of the files results:
·
absolute errors and relative total by the 2 methods,
·
maximum and minimal values of the stresses and the errors,
·
lists of the meshs where the relative error is higher than 10%.
8.3 Parameters
of execution
Version: 3.02.11
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
10.4 seconds
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
11/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
9 Modeling
D
9.1
Characteristics of modeling
9.2
Characteristics of the mesh
A number of nodes: 277.
A number of meshs and types: 80 QUAD8.
9.3 Functionalities
tested
Controls
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
12/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
10
Results of modeling D
10.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
With
xx
ZZ1
3.
3.063
2.11
0.1
xx
ZZ2
3.
3.037
1.24
0.1
yy
ZZ1
0.
0.101
-
0.3
yy
ZZ2
0.
2.47 10
2
-
0.3
xy
ZZ1
0.
5.8 10
3
-
0.3
xy
ZZ2
0.
2.41 10
2
-
0.3
P
xx
ZZ1
1.15625
1.170
1.19
0.1
xx
ZZ2
1.15625
1.153
0.29
0.1
yy
ZZ1
0.15625
0.162
3.54
0.1
yy
ZZ2
0.15625
0.153
1.87
0.1
xy
ZZ1
0.125
0.124
1.09
0.1
xy
ZZ2
0.125
0.124
0.84
0.1
Net M1
E
ABS
ZZ1
6.1 10
5
1.10
3
E
ABS
ZZ2
2.1 10
4
1.10
3
E
rel
ZZ1
1.45%
0.1
E
rel
ZZ2
5.01%
0.1
E
ABS
ZZ1
9.01 10
4
2.90 10
4
+67.9 (!)
E
ABS
ZZ2
9.01 10
4
8.88 10
4
1.5
E
rel
ZZ1
0.697%
0.22%
E
rel
ZZ2
0.697%
0.687%
ZZ1
0.321
ZZ2
0.985
10.2 Remarks
=
estimated error
exact error is an indication of effectivity of the estimator
Contents of the files results:
·
absolute errors and relative total by the 2 methods,
·
maximum and minimal values of the stresses and the errors,
·
lists of the meshs where the relative error is higher than 10%.
10.3 Parameters
of execution
Version: 3.02.11
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
8.6 seconds
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
13/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
11 Modeling
E
11.1 Characteristics of modeling
11.2 Characteristics of the mesh
A number of nodes: 357.
A number of meshs and types: 80 QUAD9.
11.3 Functionalities
tested
Controls
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
14/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
12
Results of modeling E
12.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
With
xx
ZZ1
3.
3.070
2.33
0.1
xx
ZZ2
3.
3.004
0.14
0.1
yy
ZZ1
0.
0.113
-
0.3
yy
ZZ2
0.
0.04
-
0.3
xy
ZZ1
0.
1.4 10
3
-
0.1
xy
ZZ2
0.
1.89 10
2
-
0.1
P
xx
ZZ1
1.15625
1.170
1.19
0.1
xx
ZZ2
1.15625
1.152
0.33
0.1
yy
ZZ1
0.15625
0.162
3.44
0.1
yy
ZZ2
0.15625
0.153
2.11
0.1
xy
ZZ1
0.125
0.124
1.06
0.1
xy
ZZ2
0.125
0.124
0.94
0.1
Net M1
E
ABS
ZZ1
6.1 10
5
1.10
3
E
ABS
ZZ2
2.1 10
4
1.10
3
E
rel
ZZ1
1.45%
0.1
E
rel
ZZ2
5.01%
0.1
E
ABS
ZZ1
8.99 10
4
2.75 10
4
+69.4 (!)
E
ABS
ZZ2
8.99 10
4
8.55 10
4
4.9
E
rel
ZZ1
0.695%
0.21%
E
rel
ZZ2
0.695%
0.66%
ZZ1
0.306
ZZ2
0.951
12.2 Remarks
=
estimated error
exact error is an indication of effectivity of the estimator
Contents of the files results:
·
absolute errors and relative total by the 2 methods,
·
maximum and minimal values of the stresses and the errors,
·
lists of the meshs where the relative error is higher than 10%.
12.3 Parameters
of execution
Version: 3.02.11
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
8.8 seconds
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
15/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
13
Summary of the results
TRIA3
QUAD4
TRIA6
QUAD8
QUAD9
E
rel
exact
3.44%
2.48%
1.17%
0.697%
0.695%
ZZ1
3.28%
2.28%
0.95%
0.22%
0.21%
ZZ2
3.49%
2.37%
1.29%
0.687%
0.66%
ZZ1
0.952
0.919
0.810
0.321
0.306
ZZ2
1.013
0.958
1.099
0.985
0.951
The stresses with the nodes, as a whole, are approximated better with ZZ2, especially for the elements
of command 2. If one makes tend
H
towards 0, rates of convergence with H of
ex
-
*
are higher by
method ZZ2 for all the types of elements to those of method ZZ1 (
*
is the smoothed stress).
Estimator ZZ1 is not reliable for the elements of command 2, the nodal stresses remain correct. One can
to check in this particular case that
0
when
H
0
, which shows that continuous total smoothing proves
insufficient to estimate the error in the case of a solution without singularities (case of this test).
ZZ2 is on the other hand reliable and asymptotically exact (
1
when
H
0
).
Code_Aster
®
Version
4.0
Titrate:
SSLV111 Estimator of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A
Page:
16/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A
Intentionally white left page.