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Code_Aster
®
Version
7.4
Titrate:
Estimator of error in residue
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R4.10.02-B
Page
:
1/6
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA















Manual of Reference
R4.10 booklet: Estimator of error a posteriori
Document: R4.10.02



Estimator of error in residue



Summary

The estimator of error in residue allows to estimate the error of discretization due to the finite element method
on the elements of a mesh 2D or 3D. It is an explicit estimator of error utilizing the residues of
equilibrium equations and jumps of the normal stresses to the interfaces, contrary to the estimator of
Zhu-Zienkiewicz, which uses a technique of smoothing of the stresses a posteriori [R4.10.01] and [bib5].

This estimator is established in Code_Aster in elastoplasticity 2D and 3D.
background image
Code_Aster
®
Version
7.4
Titrate:
Estimator of error in residue
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R4.10.02-B
Page
:
2/6
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HT-66/05/002/A
1 Introduction
The estimator of error in residue was developed in 1993 by Bernardi-Métivet-Verfurth [bib1]. It is one
explicit estimator of error utilizing residues of the equilibrium equations (from where its name). It
apply to elliptic problems (Poisson, Stokes, or linear elasticity) in dimension 2 or 3.
These problems are supposed to be discretized by finite elements associated a regular triangulation.
Historically, the first estimator of explicit error relating to the unbalances is due to Babuska
and Rheinbolt [bib2] for the problems 1D with linear elements. Gago extended this estimator
to the 2D and added to the formulas the jumps of traction to the interfaces of the elements [bib3] and [bib4]. Of
new estimators were then proposed, in whom defects of surface traction with
borders of the field were also taken into account as well as an improvement of the estimate of
jumps inter-elements giving of the more reliable results.
One is interested here in the estimator in residue applied to the case of linear elasticity. The set aim is, with
the exit of an elastic design, to if required determine the card of error on the mesh in sight
to adapt the aforementioned (by refinement and/or déraffinement) or simply for information. The adaptation
can be done by chaining with the software of Homard cutting.

2
Formulation of the estimator in residue
That is to say
open of
R
N
, N = 2 or 3, of border
, and T a regular triangulation of
.
In linear elasticity, the continuous problem is written:
to find
()
U,
such as:
div
F
U U
G
D
D
NR
NR
+ =
=
=



0
in
on
on

.n
D
is the border of Dirichlet of the mesh
U
D
is the displacement imposed on this border
NR
is the border of Neumann
N
the unit normal with
NR
G
NR
is the loading applied to this border; it can continuous or be discretized.
F
is force of a voluminal type (gravity, rotation); it can continuous or be discretized.
H
is the stress obtained by the resolution of the discrete problem:
div
F
U
U
G
H
H
D
D
H
NR
NR
+ =
=
=



0
in
on
on

.n
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Code_Aster
®
Version
7.4
Titrate:
Estimator of error in residue
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R4.10.02-B
Page
:
3/6
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HT-66/05/002/A
with the relation
H
H
U
= dB
where:
D
is the matrix of Hooke
B
is the linearized operator of the deformations
If
K
indicate an element running of the mesh, the estimator of error (noted
()
) is defined like
being the quadratic average of the site indicators of error, noted
()
K
:
()
()
=


K
K T
2
1 2
/
The indicator by local residue
The indicator is composed of three terms; the first represents the residue of the equilibrium equation on
each mesh, the second term the jump of the normal stresses on the interfaces, the third
term the variation enters the normal stresses and the loading imposed on
NR
if the element intersects
NR
.
V1
S (K)
V3
V2
K
K:
Element running where one wishes to calculate the error,
V1 with V3:
Elements having a common edge with the current element,
S (K):
Together edges of the element running having neighbors.
Appear 2-a: internal Elements in a mesh
·
the first term of the estimator is the standard
L
2
residue of the equilibrium equation on
net
K
, multiplied by
H
K
who is, that is to say the diameter of the circle circumscribed for a finite element
triangular, that is to say the maximum diagonal for a quadrangle,
·
the second term is the integral, on
()
S K
defined [Figure 2-a], of the jumps of stresses
normals integrated on each edge
F
element which has a neighbor, and multiplied by
root of
H
F
, which is the length of the edge
F
,
background image
Code_Aster
®
Version
7.4
Titrate:
Estimator of error in residue
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R4.10.02-B
Page
:
4/6
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HT-66/05/002/A
NR
G
NR
K
F
NR
: Border of Neumann
: Force applied to the border of Neumann
G
NR
Appear 2-b: Elements located on the border of a mesh
·
the third term is the integral, on the intersection of each edge
F
edges
K
of
the current element
K
with the border of Neumann
NR
, of the jumps between the stresses
normals of the element and the force of Neumann
G
NR
, multiplied by the root of
H
F
, length
edge
F
.
There is thus the following formula for the estimator in residue:
()
()
()
[]
()
()
K
H F
div
H
N
H
G
N
K
H L K
F
F S K
H
L F
F
F
K
NR
H
L F
NR
=
+
+
+
-
2
2
2
1
2
1 2
1 2
/
/
.
.
éq 2-1
For the choice of the various terms of [éq 2-1], one returns to [bib1].

3
Properties of the estimator in residue
One notes
()
EX
K
the exact error
()
U U
H H K
-
1
on the element
K
(unknown factor a priori)
and
()
EX
the total exact error
()
U U
H H
-
1
There are then the following properties ([bib1]):
·
some is the element
K
, the elementary error
()
K
is raised by the exact site error
(multiplied by a constant independent of the triangulation),
that is to say
()
()
×
K
K
C
K
EX
1
·
the exact total error is raised by the error considered total
()
(multiplied by one
constant independent of
T
)
that is to say
()
()
EX
C
×
2
background image
Code_Aster
®
Version
7.4
Titrate:
Estimator of error in residue
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R4.10.02-B
Page
:
5/6
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HT-66/05/002/A
Constants
C
C
1
2
and
depend a priori on the type of finite element and boundary conditions of
problem. Kelly and Gago [bib3] proposed in 2D a constant
C
2
depending only on the degree
p
polynomial of interpolation used:
2)
(degree
QUAD8
and
TRIA6
for
1)
(degree
QUAD4
and
TRIA3
for
that is to say
6
4
1
6
2
1
24
1
2
2
2
/
1
2
2
p
C
p
C
p
C
=
=




=
For the 3D, one does not have evaluation of the constant. One can nevertheless say that the error
estimated total the total exact error in all the cases over-estimates. This result is not inevitably
truth at the local level.

4 Establishment
in
Aster
The estimator in residue is established in 2D and 3D on all the types of elements (except the pyramids). It
is calculated by control CALC_ELEM by activating option “ERRE_ELGA_NORE”.
This option calculates on each element:
·
the absolute error
()
K
(see [éq 2-1]),
·
the standard of the tensor of the stresses
()
H L K
2
who is used to normalize the absolute error,
·
the relative error
()
()
()
()
rel
H L K
K
K
K
=
×
+
100
2
2
2
.
Note:
This definition of the relative error implies that in the areas where the stresses are very
weak, the relative error can be important and nonsignificant. It is then the absolute error
that it is necessary to consider.
It also calculates at the total level:
·
the absolute error
()
()
=


K
K T
2
1 2
/
,
·
the total standard of the tensor of the stresses
()
()
H L
H L K
K T
2
2
2
1 2
=


/
,
·
the relative error
()
()
()
()
rel
H L
=
×
+
100
2
2
2
.
background image
Code_Aster
®
Version
7.4
Titrate:
Estimator of error in residue
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R4.10.02-B
Page
:
6/6
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HT-66/05/002/A
According to the expression [éq 2-1], one sees that to calculate the indicator of error on the mesh
K
, one must
to know:
1) possible loadings
F
on
K
and
G
NR
on
K
NR
(or their discretization
F
G
H
Nh
and
),
2) them
quantities
H H
K
F
,
and
N
dependant on the geometry of the element,
3) the stress field
H
,
4) the list of the neighbors of
K
to recover the stresses on these elements, necessary to
calculation of the 2
ème
term of [éq 2-1].
1 and 2 can be calculated or recovered easily.
3 must be calculated as a preliminary by one of options “SIGM_ELNO_DEPL”, “SIEF_ELNO_ELGA”
or “SIRE_ELNO_DEPL” (option without smoothing of the stresses).
In the contrary case, a fatal error message is transmitted.
4 requires the calculation of a particular connectivity mesh-meshs, in addition to standard connectivity
mesh-nodes. This new object is stored in the structure of data of the mesh type.
For the detail of the establishment in Aster, to see [bib6].
For the validation of the estimator, to see [bib7].

5 Bibliography
[1]
BERNARDI, B. METIVET, R. VERFÜRTH: Working group adaptive mesh: analyze
numerical of indicators of error. Note HI-72/93/062
[2]
BABUSKA, RHEINBOLDT: A posteriori error estimates for the finite element method. Int. J.
Num. Meth. Eng., Flight 12, 1597-1659, 1978
[3]
J.P. of S.R. GAGO: A posteriori error analysis and adaptivity for the finite element method.
PH. D. Thesis, University off Wales, Swansea, the U.K., 1982
[4]
D.W. KELLY, J.P. of S.R. GAGO, O.C. ZIENKIEWICZ, I. BABUSKA: A posteriori error
analysis and adaptive processes in finite element method: I - error analysis leaves. Int. J.
Num. Meth. Eng., Flight 19, 1593-1619, 1983
[5]
X. DESROCHES: Estimators of error in linear elasticity. Note HI-75/93/118
[6]
A. CORBEL: Establishment of an estimator of error in residue in the Code of mechanics
Aster. Report/ratio of end of training course - June 94
[7]
V. NAVAB: Validation of an estimator of error in residue in elasticity Bi and three-dimensional.
Report/ratio of training course - Mars 95