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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
1/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
Document: R4.10.01
Estimator of error of ZHU-ZIENKIEWICZ
in elasticity 2D
Summary:
One exposes in this document the method of estimate of the error of discretization suggested by
ZHU-ZIENKIEWICZ and applied to the system of linear elasticity 2D.
This estimator is based on a continuous smoothing of the calculated stresses allowing to obtain the best
precision on the nodal stresses compared to the methods standards.
Two successive versions of this estimator are described, corresponding each one to a different smoothing. These
two versions are available in Aster.
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Code_Aster
®
Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
2/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Principle of the method .......................................................................................................................... 4
2.1 Equations to solve ...................................................................................................................... 4
2.2 Estimator of error and index of effectivity ......................................................................................... 5
2.3 Construction of an estimator asymptotically exact ................................................................... 6
3 Construction of the stress field recomputed
()



*
..................................................................... 7
3.1 Version 1987 ................................................................................................................................... 7
3.2 Version 1992 ................................................................................................................................... 7
4 Establishment in Aster and current limits of use ...................................................................... 11
4.1 Establishment in Aster ................................................................................................................. 11
4.2 Operational limits ........................................................................................................................ 11
5 Bibliography ........................................................................................................................................ 12
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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
3/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
1 Introduction
Search on estimators of error on the solutions obtained by calculations finite elements and
their coupling with procedures of adaptive mesh made these last years great strides
considerable. The set aim is to mitigate the possible inadequacy of a modeling while adapting of one
automatic way mesh with the solution sought according to certain criteria (equal distribution of the error
of discretization, minimization of the number of nodes to reach a given, less precision
cost).
One introduces here an estimator of error of the type a posteriori within the framework of linear elasticity and
homogeneous 2D. Historically, this estimator, proposed by ZHU-ZIENKIEWICZ [bib1] in 1987, was
largely used because of its facility of establishment in its the existing weak and computer codes
cost. Nevertheless, the bad reliability of this estimator for the elements of even degree was
noted empirically (undervaluation of the error) and led the authors to an amendment of
their method in 1992 [bib2], [bib3] with numerical checking of the asymptotic convergence of
the estimator on all the types of elements.
Nevertheless, the applicability of version 92 being for the moment more reduced (see [ß3.2]), them
two versions of this estimator were established in Aster and are the subject of this note.
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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
4/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
2
Principle of the method
2.1
Equations to be solved
The solution is considered
()
U,
of a linear elastic problem checking:
·
equilibrium equations:
Lu
Q
=
=


in
on

ij J
I
T
N
T
with
L
BDB
=
T
operator of elasticity
·
equations of compatibility:



=
=


Drunk
U
U
U
on
with
=
U
T
!
·
the law of behavior:






=
D
The problem discretized by finite elements consists in finding
(
)
U
H
H
,



solution of:
U
U
H
H
NR
=
éq 2.1-1
checking
K U
F
H
=
with
() ()
K
B NR D B NR
=
T
D
F
NR Q
NR
=
+
T
T
D
T D
T
where:
U
H
represent the nodal unknown factors of displacement
NR
associated functions of form
The stresses are calculated starting from displacements by the relation:



H
=
D B U
H
éq 2.1-2
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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
5/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
2.2
Estimator of error and index of effectivity
One notes
E
U
U
H
= -
the error on displacements
E
H






=
-
the error on the stresses
The standard of the energy of the error
E
is written:
E
E L E
=


T
D
1 2
/
in the case of elasticity
=


-
T
D
E D E






1
1 2
/
éq
2.2-1
The total error above breaks up into a sum of elementary errors according to:
E
E
2
2
1
=
=
I
I
NR
where
NR
is the total number of elements.
E
I
represent the local indicator of error on the element
I
.
The goal is to consider the error exact starting from the equation [éq 2.2-1] formulated in stresses. The idea of
base method is to build a new noted stress field



*
from



H
and such
that:
E
E
H












=
-
*
*
The estimator of error will be written then:
0
1
1 2
E
E D E
=


-
T
D






*
*
/
The quality of the estimator is measured by the quantity
, called index of effectivity of the estimator:
=
0
E
E
An estimator of error is known as asymptotically exact if
1
when
E
0
(or when
H
0
),
what wants to say that the estimated error will always converge towards the exact error when the aforementioned decreases.
In an obvious way, the reliability of
0
E
depends on “quality” on



*
.
The two versions of the estimator of ZHU-ZIENKIEWICZ are different on this level (see [§3]).
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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
6/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
2.3
Construction of an estimator asymptotically exact
The characterization of such an estimator is provided by the following theorem (see [bib 2]).
Theorem
If
E
U
U
*
*
=
-
is the standard of error of the rebuilt solution, then the estimator of error
0
E
defined previously is asymptotically exact
if
E
E
*
0
when
E
0
This condition is carried out if the rate of convergence with
H
of
E
*
is higher than that of
E
.
Typically, if it is supposed that the exact error of the approximation finite element converges in
()
E
=
0 H
p
and the error of the solution rebuilt in
()
E
*
=
+
0 H
p
with
>
0
then a simple calculation gives:
()
()
1
0
1
0
-
+
H
H
and thus
1
when
H
0
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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
7/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
3
Construction of the stress field recomputed
()



*
3.1 Version
1987
The solution
U
H
resulting from the equation [éq 2.1-1] being
C
0
on
(because of the choice of functions of
form
C
0
), it follows that



H
calculated by [éq 2.1-2] is discontinuous with the interfaces of the elements.
To obtain acceptable results on the nodal stresses, one generally resorts to one
average with the nodes or a method of projection. It is this last method which is adopted
here.
It is supposed that



*
is interpolated by the same functions of form as
U
H
, that is to say:






*
*
=
NR
éq 3.1-1
and one carries out a total smoothing by least squares of
H
, which amounts minimizing the functional calculus
()
(
) (
)
J
D
T









=
-
-
H
H
in the space generated by
NR
.
By derivation,



*
must check
(
)
T
NR
D






*
-
=
H
0
by using the equation [éq 3.1-1], one obtains the linear system:
{}
{}
M
B



*
=
with
M
NR NR
=
T
D
and
{}
B
NR
H
=
T
D



This total system is to be solved on each component of the tensor of the stresses.
stamp
M
is calculated and reversed only once.
3.2 Version
1992
The stress of the field



*
differ compared to the version 1987 in the following way:
one supposes



*
polynomial of the same degree than displacements on the whole of the elements
having an internal node node
S
jointly.
One notes
S
K
K
S K
=
!
this unit called patch.
For each component of



*
, one writes:



*
S
K
=
AP
S
éq 3.2-1
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Code_Aster
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Version
2.6
Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
8/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
where
P
contains the suitable polynomial terms
has
S
unknown coefficients of the corresponding students'rag processions
Example:
2D
[
]
P
X y
1
1
,
P
=
[
]
has
S
=
T
has has
has
1
2
3
,
,
[
]
Q
X y xy
1
1
,
P
=
[
]
has
S
=
T
has has has has
1
2
3
4
,
,
,
Determination of the coefficients of the polynomial
has
S
is done by minimizing the functional calculus:
()
(
)
(
)
(
)
(
)
(
)
F
X y
X y
X y
X y
I
I
S
I
I
I
NR
I
I
I
I
I
NR
K
has
P
has
H
H
S
=
-




=
-
=
=









,
,
,
,
*
2
1
2
1
(discrete local smoothing of



H
by least squares)
where
(
)
X y
I
I
,
are the co-ordinates of the points of GAUSS on
S
K
.
NR
is the total number of points of GAUSS on all the elements of the patch
The solution
has
S
check:
(
) (
)
(
) (
)
T
I
I
I
I
T
I
I
I
NR
I
I
I
NR
X y
X y
X y
X y
P
P
has
P
S
H
,
,
,
,
=
=
=
1
1



from where
has
With
B
S
=
-
1
with
(
) (
)
With
P
P
=
=
T
I
I
I
NR
I
I
X y
X y
,
,
1
With
can very badly be conditioned (in particular on the elements of high degree) and consequently, impossible
to reverse in this form. To cure this problem, the authors [bib4] proposed one
standardization of the co-ordinates on each patch, which amounts carrying out the change of
variables:
X
X
X
X
X
y
y
y
y
y
= - +
-
-
= - +
-
-
1
2
1
2
min
max
min
min
max
min
where
X
X
y
y
min
max
min
max
,
,
,
represent the values minimum and maximum of
X
and
y
on the patch.
This method notably improves conditioning of
With
and the problem removes completely
precedent.
Once
has
S
determined, the nodal values are deduced according to the equation [éq 3.2-1] only on
the nodes intern with the patch, except in the case of patchs having nodes of edge.
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Code_Aster
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Version
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Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
9/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
Patchs internal:
QUAD4
QUAD8
QUAD9
TRIA3
TRIA6
points of GAUSS where the stresses are calculated



H
according to the equation [éq 2.1-2]
nodes of calculation of



*



*
internal node defining the patch
Patchs edges:
The nodal values with the nodes mediums belonging to 2 patchs are realized, in the same way for
nodes intern the QUAD9 in the case of.
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Code_Aster
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Version
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Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
10/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
Note:
In the case of finite elements of different type, the choice of
P
in the equation [éq 3.2 - 1] is delicate
(problems of validity of
has
S
if space is too rich, loss of super-convergence if it is not it
enough). A thorough study seems essential.
This is why estimator ZZ2 is limited for the moment to mesh comprising only one
only type of element. This restriction does not exist for ZZ1.
The authors showed numerically [bib3] that with this choice of



*
, their estimator was
asymptotically exact for elastic materials of which the characteristics are independent
field and for all the types of elements and that rates of convergence with
H
of
E
*
were
improved compared to the preceding version (especially for the elements of degree 2: to see case test
Manual SSLV110 of Validation), from where a better estimate of the error.
One will find an illustration of these rates of convergence in the reference [bib 5].
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Code_Aster
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Version
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Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
11/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
4 Establishment
in
Aster and current limits of use
4.1 Establishment
in
Aster
The two preceding estimators are established in Aster in the ordering of postprocessing
CALC_ELEM
[U4.61.02]. They are activated starting from options (
ERRE_ELEM_NOZ1
for ZZ1 and
ERRE_ELEM_NOZ2
for ZZ2) and enrich a structure of data
RESULT
.
Moreover, the calculation of the stress field smoothed by one or the other of the methods described with [ß3]
can be started separately of the calculation of estimate of the error (option
SIGM_NOZ1_ELGA
for ZZ1
and
SIGM_NOZ2_ELGA
for ZZ2). It should be noted that this field is discretized directly with the nodes and
not by element with the nodes, which reduces the volume of exits.
The estimator of error provides:
·
a field by element comprising 3 components:
-
the estimate of the relative error on the element,
-
the estimate of the absolute error on the element,
-
the standard of the energy of the calculated solution



H
.
·
exit-listing comprising same information at the total level (on all the structure)
All the fields obtained are displayable by IDEAS via the control
IMPR_RESU
.
4.2 Limits
of use
Linear elasticity and homogeneous 2D (forced and plane deformations, axisymmetric),
Types of elements:
triangles with 3 and 6 nodes,
quadrangles with 4, 8 and 9 nodes.
For ZZ2, the mesh must comprise one type of elements.
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Code_Aster
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Version
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Titrate:
Estimator of error of ZHU-ZIENKIEWICZ in elasticity 2D
Date:
11/04/94
Author (S):
X. DESROCHES
Key:
R4.10.01-A
Page:
12/12
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
5 Bibliography
[1]
ZIENKIEWICZ O.C., ZHU J.Z. : “A simple error estimator and adaptive procedure for practical
engineering analysis " - Int. Newspaper for Num. Puts. in Eng., flight 24 (1987).
[2]
ZIENKIEWICZ O.C., ZHU J.Z. : “The superconvergent patch recovery and a posteriori error
estimates - Share 1: the technical recovery " - Int. Newspaper for Num. Puts. in Eng., flight 33,
1331-1364 (1992)
[3]
ZIENKIEWICZ O.C., ZHU J.Z. : “The superconvergent patch recovery and a posteriori error
estimates - Share 2: error estimates and adaptivity " - Int. Newspaper for Num. Puts. in Eng., flight 33,
1365-1382 (1992)
[4]
ZIENKIEWICZ O.C., ZHU J.Z., WU J.: “Superconvergent patch recovery techniques - Somme
further tests " - Com. in Num. Puts. in Eng., flight 9, 251-258 (1993)
[5]
DESROCHES X.: “Estimators of error in linear elasticity” - Note HI-75/93/118.