Code_Aster ®
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
1/16
Organization (S): EDF/IMA/MN
Handbook 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 constraints, 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 constraints 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.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
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Author (S):
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Key:
V3.04.111-A Page:
2/16
1
Problem of reference
1.1 Geometry
y
= 1.0
= 1.0
B
C
X
With
D
E
has
()
y
B
C
With
R
X
E
D
1.0
3.0
(H)
1.2
Material properties
E = 1.000 MPa
= 0.3.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
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Author (S):
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Key:
V3.04.111-A Page:
3/16
1.3
Boundary conditions and loadings
On AB, U
=
X
0
On ED, U =
y
0
F

= (X = 4)
X
xx
.
On CD tractions F = (X = 4)
y
xy


.
F =
=
X
xy (y
4).
On BC tractions F =
=
y
yy (y
4)


.
According to the analytical solution [2].
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
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Key:
V3.04.111-A Page:
4/16
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 = 1 E E
X
X.
The analytical solution of this problem is [bib1]:
2
4

has 3
3 A
= 1
cos +
xx
cos (
4)
cos (
4)
r2

2

+ 2 r4
2
4

has
= -
1 cos (
3 A
2
-
yy
) cos (
4)
cos (
4)
r2

2

- 2 r4
2
4

has
= -
1 sin (
3 A
2
+
xy
) sin (
4)
sin (
4)
r2

2

+ 2 r4
where:
·
A 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)).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
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Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
5/16
3 Modeling
With
3.1
Characteristics of modeling
3.2
Characteristics of the grid
A number of nodes: 357.
A number of meshs and types: 640 TRIA3.
3.3 Functionalities
tested
Commands
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
6/16
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
To 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.
­ 7.4 10­3
-
0.1
xy ZZ2
0.
­ 6.1 10­2
-
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 eabs ZZ1
1.33 10­4
1.10­3
eabs ZZ2
8.13 10­5
1.10­3
erel ZZ1
6.63%
0.1
erel ZZ2
4.05%
0.1
eabs ZZ1
0.445 10­2
0.424 10­2
­ 4.76
eabs ZZ2
0.445 10­2
0.451 10­2
+1.31
erel ZZ1
3.44%
3.28%
erel ZZ2
3.44%
3.49%
ZZ1
0.952
ZZ2
1.013
4.2 Remarks
= estimated error is an indication of effectivity of the estimator.
exact error
Contents of the files results:
· absolute errors and relative total by the 2 methods,
· maximum and minimal values of the constraints 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
Obstruction memory:
8 megawords
Time CPU To use:
19.0 seconds
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
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Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
7/16
5 Modeling
B
5.1
Characteristics of modeling
5.2
Characteristics of the grid
A number of nodes: 357.
A number of meshs and types: 320 QUAD4.
5.3 Functionalities
tested
Commands
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
8/16
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
To 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.
2.8 10­3
-
0.1
xy ZZ2
0.
­ 1.04 10­2
-
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 eabs ZZ1
1.57 10­4
1.10­3
eabs ZZ2
2.40 10­4
1.10­3
erel ZZ1
5.79%
0.1
erel ZZ2
8.83%
0.1
eabs ZZ1
0.320 10­2
0.294 10­2
­ 8.1
eabs ZZ2
0.320 10­2
0.307 10­2
­ 4.2
erel ZZ1
2.48%
2.28%
erel ZZ2
2.48%
2.37%
ZZ1
0.919
ZZ2
0.958
6.2 Remarks
= estimated error is an indication of effectivity of the estimator.
exact error
Contents of the files results:
· absolute errors and relative total by the 2 methods,
· maximum and minimal values of the constraints 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
Obstruction memory:
8 megawords
Time CPU To use:
13.4 seconds
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
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Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
9/16
7 Modeling
C
7.1
Characteristics of modeling
7.2
Characteristics of the grid
A number of nodes: 357.
A number of meshs and types: 160 TRIA6.
7.3 Functionalities
tested
Commands
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
10/16
8
Results of modeling C
8.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
To xx ZZ1
3.
2.975
­ 0.83
0.1
xx ZZ2
3.
2.957
­ 1.43
0.1
yy ZZ1
0.
6.86 10­2
-
0.3
yy ZZ2
0.
7.52 10­2
-
0.3
xy ZZ1
0.
­ 3.12 10­2
-
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 eabs ZZ1
1.81 10­4
1.10­3
eabs ZZ2
2.92 10­4
1.10­3
erel ZZ1
4.69%
0.1
erel ZZ2
7.56%
0.1
eabs ZZ1
0.152 10­2
0.123 10­2
­ 19
eabs ZZ2
0.152 10­2
0.167 10­2
+9.9
erel ZZ1
1.17%
0.95%
erel ZZ2
1.17%
1.29%
ZZ1
0.810
ZZ2
1.099
8.2 Remarks
= estimated error is an indication of effectivity of the estimator.
exact error
Contents of the files results:
· absolute errors and relative total by the 2 methods,
· maximum and minimal values of the constraints 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
Obstruction memory:
8 megawords
Time CPU To use:
10.4 seconds
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
11/16
9 Modeling
D
9.1
Characteristics of modeling
9.2
Characteristics of the grid
A number of nodes: 277.
A number of meshs and types: 80 QUAD8.
9.3 Functionalities
tested
Commands
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
12/16
10
Results of modeling D
10.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
To 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.
2.47 10­2
-
0.3
xy ZZ1
0.
­ 5.8 10­3
-
0.3
xy ZZ2
0.
­ 2.41 10­2
-
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 eabs ZZ1
6.1 10­5
1.10­3
eabs ZZ2
2.1 10­4
1.10­3
erel ZZ1
1.45%
0.1
erel ZZ2
5.01%
0.1
eabs ZZ1
9.01 10­4
2.90 10­4
+67.9 (!)
eabs ZZ2
9.01 10­4
8.88 10­4
­ 1.5
erel ZZ1
0.697%
0.22%
erel ZZ2
0.697%
0.687%
ZZ1
0.321
ZZ2
0.985
10.2 Remarks
= estimated error is an indication of effectivity of the estimator
exact error
Contents of the files results:
· absolute errors and relative total by the 2 methods,
· maximum and minimal values of the constraints 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
Obstruction memory:
8 megawords
Time CPU To use:
8.6 seconds
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
13/16
11 Modeling
E
11.1 Characteristics of modeling
11.2 Characteristics of the grid
A number of nodes: 357.
A number of meshs and types: 80 QUAD9.
11.3 Functionalities
tested
Commands
Keys
MECA_STATIQUE
OPTION
“SIEF_ELGA_DEPL”
[U4.31.01]
CALC_ELEM
OPTION
“ERRE_ELEM_NOZ1”
[U4.61.02]
OPTION
“ERRE_ELEM_NOZ2”
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
14/16
12
Results of modeling E
12.1 Values
tested
Identification
Reference
Aster
% difference
tolerance
To 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.
­ 1.4 10­3
-
0.1
xy ZZ2
0.
1.89 10­2
-
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 eabs ZZ1
6.1 10­5
1.10­3
eabs ZZ2
2.1 10­4
1.10­3
erel ZZ1
1.45%
0.1
erel ZZ2
5.01%
0.1
eabs ZZ1
8.99 10­4
2.75 10­4
+69.4 (!)
eabs ZZ2
8.99 10­4
8.55 10­4
­ 4.9
erel ZZ1
0.695%
0.21%
erel ZZ2
0.695%
0.66%
ZZ1
0.306
ZZ2
0.951
12.2 Remarks
= estimated error is an indication of effectivity of the estimator
exact error
Contents of the files results:
· absolute errors and relative total by the 2 methods,
· maximum and minimal values of the constraints 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
Obstruction memory:
8 megawords
Time CPU To use:
8.8 seconds
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/96/017 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
V3.04.111-A Page:
15/16
13
Summary of the results
TRIA3
QUAD4
TRIA6
QUAD8
QUAD9
E
exact
3.44%
2.48%
1.17%
0.697%
0.695%
rel
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 constraints with the nodes, as a whole, are approximated better with ZZ2, especially for the elements
ex
*
of command 2. If one makes tighten H towards 0, the rates of convergence with H of
- are higher by
method ZZ2 for all the types of elements to those of method ZZ1 (* is the smoothed constraint).
Estimator ZZ1 is not reliable for the elements of command 2, the nodal constraints 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).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
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Code_Aster ®
Version
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Titrate:
SSLV111 Estimateur of error on a perforated plate
Date:
26/01/98
Author (S):
X. DESROCHES
Key:
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16/16
Intentionally white left page.
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