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:
1/12
Organization (S): EDF/IMA/MN
Handbook of Référence
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 constraints allowing to obtain the best
precision on the nodal constraints 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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
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
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
3/12
1 Introduction
Search on estimators of error on the solutions obtained by calculations finite elements and
their coupling with procedures of adaptive grid 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 grid 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 a modification 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
4/12
2
Principle of the method
2.1
Equations to be solved
One considers the solution (U,) of a linear elastic problem checking:
· equilibrium equations:
Lu
= Q in
ijnj =
T on
I
T
with L = T BDB operator of elasticity
· equations of compatibility:
= Drunk
U =
U on U
with = U
T
!
· the law of behavior:
= D
The problem discretized by finite elements consists in finding (uh, H) solution of:
U = NR U
H
H
éq 2.1-1
checking K U
F
H =
with K = T (BN) D (BN) D
F =
NR Q + T
D
NR
T
T D
T
where:
uh represents the nodal unknown factors of displacement
NR associated functions of form
The constraints are calculated starting from displacements by the relation:
H = dB uh
éq 2.1-2
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
5/12
2.2
Estimator of error and index of effectivity
One notes
E = U - U
the error on displacements
H
E = -
the error on the constraints
H
The standard of the energy of the error E is written:
1/2
E =
E L E
T
D
in the case of elasticity
1/2
= you D 1 E D
-
éq
2.2-1
The total error above breaks up into a sum of elementary errors according to:
NR
E 2 = E 2i
I = 1
where
NR is the total number of elements.
E I represents the local indicator of error on element I.
The goal is to consider the error exact starting from the equation [éq 2.2-1] formulated in constraints. The idea of
base method is to build a new stress field noted * from H and such
that:
E E *
*
= - H
The estimator of error will be written then:
1/2
0
T *
1
E = E D E * 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 this one 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]).
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
6/12
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 previously defined is asymptotically exact
E *
if
0 when
E 0
E
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 (HP)
and the error of the solution rebuilt in
E * = (+
0 H p) with > 0
then a simple calculation gives:
1 - 0 (H) 1 + 0 (H
)
and thus 1 when H 0
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
7/12
3
Construction of the stress field recomputed (*)
3.1 Version
1987
The solution uh resulting from the equation [éq 2.1-1] being C0 on (because of the choice of functions from
form C0), 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 constraints, 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 that uh, is:
*
*
= NR
éq 3.1-1
and one carries out a total smoothing by least squares of H, which amounts minimizing the functional calculus
T
J () = (- H) (- H) D 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
T H D
This total system is to be solved on each component of the tensor of the constraints.
stamp M is calculated and reversed only once.
3.2 Version
1992
The constraint of the field * differs 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 a node node interns S joint.
S is noted
=
K
K
! this unit called patch.
S K
For each component of *, one writes:
*
= Pa
S
S
éq 3.2-1
K
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
8/12
where
P contains the suitable polynomial terms
have the unknown coefficients of the corresponding students'rag processions
Example: 2D
P1 P = [
1, X, y] have = T [has, has, has
1
2
3]
Q1 P = [
1, X, y, xy] have = T [has, has, has, has
1
2
3
4]
The determination of the coefficients of the polynomial have is done by minimizing the functional calculus:
NR
2
F (A) =
*
H (X, y
I
I) -
(X, y)
S
I
I
I
K
=1
NR
= (
2
H (X, y
I
I) - P (X, y
I
I) have)
I =1
(discrete local smoothing of H by least squares)
where
(X, y
I
I) are the co-ordinates of the points of GAUSS on SK.
NR is the total number of points of GAUSS on all the elements of the patch
The solution ace checks:
NR
NR
T
P (
T
I
X, iy) P (ix, iy) has = P
S
(ix, iy) H (ix, iy)
I =1
I =1
NR
from where has
With
B
S =
- 1 with A = tP (ix, iy) P (ix, iy)
I =1
A 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 = - 1 + 2
min
xmax - xmin
y - y
y = - 1 + 2
min
ymax - ymin
where X
, X
, y
, y
min
max
min
max represent the values minimum and maximum X and y on the patch.
This method notably improves conditioning of A and removes the problem completely
precedent.
Once 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
9/12
Patchs internal:
QUAD4
QUAD8
QUAD9
TRIA3
TRIA6
points of GAUSS where are calculated the constraints 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
10/12
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 ace 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 grids 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 the 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
SSLV110 Manuel de Validation), from where a better estimate of the error.
One will find an illustration of these rates of convergence in the reference [bib 5].
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
11/12
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 RESULTAT.
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 outputs.
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.
· output-listing comprising same information at the total level (on all the structure)
All the fields obtained are displayable by IDEAS via command 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 grid must comprise one type of elements.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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:
12/12
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 - Part 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 - Part 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 - Some
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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-75/94/031/A
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