Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
1/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Organization (S):
EDF/SINETICS
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
Document: R4.10.03
Indicator of space error in residue
for transitory thermics
Summary
During digital simulations by finite elements, obtaining a rough result is not sufficient any more.
The user is increasingly petitioning of space error analysis compared to his mesh. He has
need for support methodological and pointed tools “numériquo-data processing” to measure the quality of
its studies and to improve them.
To this end, the indicators of space error a posteriori make it possible to locate, on each element, one
cartography of error on which the tools of mending of meshes will be able to rest: the first calculation on one
coarse mesh makes it possible to exhume the card of error starting from the data and the solution discretized (from where it
term “a posteriori”), refinement is carried out then locally by treating on a hierarchical basis this information.
The new indicator a posteriori (known as “in pure residue”) which has been just established post-to treat them
thermal solveurs of Code_Aster is based on their local residues extracted the semi-discretizations in
time. Via the option `
ERTH_ELEM_TEMP
'of
CALC_ELEM
, it uses the thermal fields (
EVOL_THER
) emanating
of
THER_LINEAIRE
and of
THER_NON_LINE
. It thus supplements the offer of the code in term of advanced tools
allowing to improve quality of the studies, their mutualisations and their comparisons.
The goal of this note is to detail theoretical, numerical work and data processing which governed sound
establishment. With regard to the theoretical study we, initially, limited ourselves to
linear thermics of a motionless structure discretized by the finite elements isoparametric standards. But,
in practice, the perimeter of use of this option was partially extended to thermics not
linear.
One gives to the reader the properties and the theoretical and practical limitations of the exhumed indicator, all in
connecting these considerations, which can sometimes appear a little “éthérées”, to a precise parameter setting of
operators accused and with the choices of modeling of the code. One tried constantly to bind the different ones
items approached, while detailing, has minimum, of the a little technical demonstrations seldom clarified in
specialized literature.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
2/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Contents
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
3/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
1
Problems Description of the document
During digital simulations by finite elements obtaining a rough result is not sufficient any more.
The user is increasingly petitioning of space error analysis compared to sound
mesh. It has need for support methodological and pointed tools “numériquo-data processing”
to measure the quality of its studies and to improve them.
For example, the precision of the results is often degraded by local singularities (corners,
heterogeneities…). One then seeks the good strategy to identify these critical areas and for
to refine/déraffiner in order to optimize the compromise site/total error. And this, with largest
possible precision, in an automatic, reliable way (the error analysis must be itself less
approximate possible!) robust and at lower cost.
For each type of finite elements, one in general has estimates a priori of the space error
[bib1], [bib3]. But those are checked only asymptotically (when the size
H
elements
tends towards zero) and they require a certain level of regularity which is precisely not reached in
areas with problem. Moreover, these increases subtend two types of strategies for
to improve calculation:
·
“methods
p
” which consists in locally increasing the command of the finite elements,
·
“
methods
H
” which locally refines in order to decrease the characteristics
geometrical of the elements.
We are interested here in the second strategy, but through another class of indicators:
indicators of errors a posteriori
. Since work founders of I. BABUSKA and
W. RHEINBOLDT [bib18], the importance of this type of indicator is well established and they arouse an interest
growing, as well in pure numerical analysis [bib5], [bib6], [bib7] as in the field of
applications [bib4], [bib16]. They were in particular established and used in N3S, TRIFOU and it
Code_Aster (for linear mechanics cf [R4.10.01], [R4.10.02]). For a “review” of the string
indicators existing, one will be able to consult the reference work of R. VERFURTH [bib7] or, it
report/ratio of X. DESROCHES [bib16], for a vision plus mecanician of these projections.
To take again a sales leaflet of Mr. FORT (cf [bib17] pp468-469), the development of
the estimate a posteriori is justified mainly by three reasons:
·
the first is the need for establishing the precision of the results obtained by a calculation elements
finished: which credit to grant to them? All phenomena and all the data which
they intervene are well taken into account in modeling?
·
the second objective is to make it possible whoever to use a computer code without having with
to intervene in the construction of the mesh in order to obtain the necessary total precision,
·
finally, the third direction of study is more particularly directed towards the problems
three-dimensional for which the size of the mesh is limited by the place memory
available and the cost of the resolution.
These specifications reveal two duaux problems: to estimate the precision of the solution
obtained on the main parameters of simulation and to propose means of calculating one
new solution which respects a minimal precision. The first problem is truly that of
the estimate of error whereas the second relates to the associated adaptive methods
(refinement/déraffinement, mending of meshes, displacement of points, follow-up of border…).
Thus these indicators make it possible to locate on each element a cartography of error on which
the tools of mending of meshes will be able to rest.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
4/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Note:
One prefers the denomination to him of
“
indicator " with the usual terminology
of “estimator” (translation literal of English “error estimator”). Taking into account the fact
that it has the same theoretical limitations that those of the solvor finite elements (that it
“post-draft”), which it is him even often sullied with numerical approximations and which it
is exhumed via relations of equivalence utilizing the many ones
constants dependant on the problem… the information which it subtends does not give
truly “that an order of magnitude” of the required space error. In spite of these
restrictions, these cartographies of error a posteriori do not remain less important about it,
and in any case, they constitute the only type of accessible information in it
field.
The first calculation on a coarse mesh makes it possible to associate, with each element triangulation, one
indicator calculated starting from the discretized data and of the first discrete solution. Refinement
be carried out then locally by treating on a hierarchical basis this information.
In short, and in a nonexhaustive way, the use of an indicator possibly coupled with one
remaillor:
·
provides a certain estimate of the error of space discretization,
·
get a better frequency of errors due to the local singularities,
·
allows to improve modeling of the facts of the case (materials, loadings,
sources…),
·
allows to optimize (even precision at lower cost) and to make reliable the process of
convergence of the mesh,
·
to estimate and qualify a calculation for a class of mesh given.
These considerations show clearly that the calculation of these estimators (which is finally only one
postprocessing of the problem considered) must:
·
to be much less expensive than that of the solution,
·
to require only the discretized data and the calculated solution,
·
to be able to be located,
·
to be equivalent (in a particular form) to the exact error.
We will see, that with the indicators in residue, one can obtain only one total increase
exact error joined to a local decrease of this same error. But these hight delimiters
and lower of the error are supplemented because, the first ensures us to have obtained a solution with
a certain tolerance, while the second enables us to optimize the number of points locally
to respect this precision and not to over-estimate it. They utilize constants which
not depending on the discretizations space and temporal.
The goal of this note is to detail theoretical, numerical work and data processing which have
governed the indicator installation of of error a posteriori allowing “post-to treat” them
thermal solveurs of Code_Aster. It is about an indicator in pure residue initiated by the option
`
ERTH_ELEM_TEMP
'of
CALC_ELEM
.
With regard to the theoretical study we, initially, limited ourselves to thermics
linear of a motionless structure discretized by the finite elements isoparametric standards. But,
in practice, the perimeter of use of this option was partially extended to thermics not
linear. For more details on the perimeter of use and functional of the thermal indicator and one
example of use, one will be able to refer to [§6].
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
5/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
The indicator a posteriori that we propose is an indicator in pure residue based on
local residues of the strong equation semi-discretized in time. For certain elements of the study
theoretical (and in particular its skeleton) we took as a starting point the innovative work by
C. BERNARDI and B. METIVET [bib6]. They have extended they-even, of elliptic with parabolic, them
results of R. VERFURTH [bib7]. They in particular were interested in calculations of indicators on
case models equation of heat with homogeneous condition of Dirichlet, semi-discretized in
time by a diagram of implicit Euler. We extended these results to the problems really
treaties by the linear operator of thermics of the code,
THER_LINEAIRE
. They are problems with
limits mixed (Cauchy-Dirichlet-Neumann-Robin) inhomogenous, linear, with variable coefficients and
discretized by one
- method.
A basic work was thus undertaken for encircling the theoretical springs of the problem well
subjacent thermics and to extrapolate the results of the problem models preceding. This so
to try to approach modelings and the perimeter of the code while detailing subtleties
often induced mathematics in the articles of art. A particular effort was brought to put
in prospect choices led in Code_Aster compared to search, passed and current,
like clarifying the general philosophy of these indicators.
One gives to the reader the properties and the theoretical and practical limitations of the released indicators
while connecting these considerations, which can sometimes appear a little “éthérées”, to a parameter setting
precis of the operator
CALC_ELEM
accused in this postprocessing. One tried constantly to bind
different the items approached, to limit the recourse to long mathematical digressions, all in
retailer has minimum many “technical” demonstrations seldom clarified a little in
specialized literature.
This document is articulated around the following parts:
·
Initially, one leads a theoretical study in order to underline holding them and
outcomes of the subjacent thermal problem, and, their possible links with the choices
of modeling of the code. First of all, the Abstracted Variational Framework is determined (CVA)
minimum (cf [§2.1]) on which one will be able to rest to show the existence and unicity
of a field of temperature solution (cf [§2.2]). By recutting these pre-necessary theoretical one
few “éthérés” with the practical stresses users, one deduce some from the limitations
as for the types of geometry and the licit loadings. Then one studies the evolution of
properties of stability of the problem (cf [§3]) during the process of semi-discretization in
time and in space.
·
These results of controllability are very useful to create the standards, the techniques and them
inequalities which intervene in the genesis of the indicator in residue. After having introduced them
usual notations of this type of problems (cf [§4.1]), a formulation is exhumed
possible of the indicator as well as the increase of the total error (cf [§4.2]) and the decrease
site error associated (cf [§4.4]). Various types of space indicators (cf [§4.3]) are
evoked and one details several used strategies of construction of indicators in
parabolic (cf [§4.5]). In this same paragraph, the temporal aspect of the problem is too
examined through the contingencies of management of the space error with respect to that of the pitch
time.
·
In a third part (cf [§5]), main contributions of these theoretical chapters and theirs
links with the thermal solveurs of the code are summarized.
·
Finally, one concludes by approaching the practical difficulties from implementation (cf [§6.1]),
environment necessary (cf [§6.2]), the parameter setting (cf [§6.3]) and the perimeter
of use (cf [§6.4]) of the indicator actually established in the operator of post-
processing
CALC_ELEM
. An example of use extracted from a case official test (
TPLL01J
) is
also detailed (cf [§6.5]).
Warning:
The reader in a hurry and/or not very interested by the theoretical springs genesis indicator
of error and subjacent thermal problem can, from the start, to jump to [§5] which recapitulates them
main theoretical contributions of the preceding chapters.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
6/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
2
The problem in extreme cases
2.1 Context
One considers a limited open motionless body occupying related
of
Q
R
(
Q
=2 or 3) of
border
I
I
=
=
=
3
1
:
:
regular characterized by its voluminal heat with constant pressure
(
X
p
C
(
the vectorial variable
X
symbolize the couple here
()
X, y
(resp.
(
)
Z
X, y,
) for
Q
=2
(resp.
Q
=3))
)
and its coefficient of isotropic thermal conductivity
()
X
.
Note:
One will thus not take account of a possible displacement of the structure
(cf.
THER_NON_LINE_MO
[R5.02.01]).
These data materials are supposed to be independent of time (modeling
THER
of Code_Aster)
and constants by element (discretization
0
P
).
Note:
With modeling
THER_FO
these characteristics can depend on time. As of
first versions of the code and before the installation of
THER_NON_LINE
, it
allowed to simulate “pseudo” non-linearities. Taking into account its use
rather marginal, we will not be interested in this modeling.
One is interested in the changes of the temperature in any point
X
opened and at any moment
[[
(
)
0
,
0
>
T
, when the body is subjected to limiting conditions and loadings
independent of the temperature but being able to depend on time. It is about voluminal source
()
, T
S X
,
boundary conditions of imposed the temperature type
()
, T
F X
(on the portion of surface
external
1
), imposed normal flow
()
, T
G X
(on
2
) and exchanges convectif
()
, T
H X
and
()
, T
T
ext.
X
(on
3
).
One places oneself thus within the framework of application of the operator
THER_LINEAIRE
[R5.02.01] of
Code_Aster by retaining only the conductive aspects of this linear thermal problem.
Note:
Non-linearities pose serious theoretical problems [bib2] to show
the existence, the unicity and the stability of the possible solution. Some are still
completely open… But in practice, that by no means prevents from “stretching” it
perimeter of use of the estimator of error which will be exhumed rigorously for
linear thermics, with nonlinear thermics (operator
THER_NON_LINE
[R5.02.02]).
This problem in extreme cases interfered (type Cauchy-Dirichlet-Neumann-Robin (also called condition
of Fourier) inhomogenous, linear and with variable coefficients) is formulated
(
)
] [
] [
] [
] [
=
×
=
+
×
=
×
=
×
=
-
)
(
)
0
,
(
,
0
,
0
,
0
,
0
)
(
0
3
2
1
0
X
X
T
T
HT
HT
N
T
G
N
T
F
T
S
T
T
T
C
P
ext.
p
div
éq
2.1-1
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
7/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Note:
·
In this theoretical study of the mixed problem
()
0
P
, it is supposed that the border dissociates
in portions on which acts inevitably a condition limits nonhomogeneous. This
assumption is not in fact not of primary importance and one can suppose the existence of a portion
4
, such
that
-
=
=
I
I
3
1
4
:
, on which a condition of homogeneous Neumann intervenes
(thus, when one builds the variational formulation associated with the strong formulation
()
0
P
, them
terms of edges related to this area disappear. The problem remains well posed then since it
is thermically unconstrained in this area. By means of computer, it is well it besides
who does, since the terms of edges are initialized to zero). In practice, it is besides
often the case.
·
It will be supposed that the coefficient of exchange
()
X
T,
H
is positive what is the case in
Code_Aster (cf [U4.44.02 §4.7.3]). And that will facilitate a little the things to us in
demonstrations to come (cf for example property 5).
·
The condition of Robin modelizing the convectif exchange (key word
EXCHANGE
) on a portion of
edge of the field, can be duplicated to take account of exchanges between two under-parts
border in opposite (key word
ECHANGE_PAROI
). This limiting condition modelizes one
thermal resistance of interface
] [
] [
×
=
+
×
=
+
=
=
,
0
,
0
,
21
1
2
2
12
2
1
1
21
12
3
HT
HT
N
T
HT
HT
N
T
T
T
ij
I
has
one
With
éq 2.1-2
Not to weigh down the writing of the problem and insofar as this option is similar to
condition of Robin with the external medium, we will not mention it specifically
in calculations which will follow.
·
The condition of Dirichlet can spread in the form of linear relations between the ddls
(key word
LIAISON_ *
) to simulate, in particular, of geometrical symmetries of the structure.
()
() ()
] [
()
()
] [
)
(
)
(
L
LIAISON_DD
OUP
LIAISON_GR
,
0
,
,
,
0
,
,
,
,
1
1
2
2
1
1
21
12
1
×
=
×
=
+
=
=
on
simply
more
or
on
has
one
With
T
T
T
T
T
T
T
T
T
T
I
I
I
J
J
J
I
I
I
I
ij
X
X
X
X
X
éq
2.1-3
In the same way functionalities
LIAISON_UNIF
and
LAISON_CHAMNO
allow to impose one
even temperature (unknown) with a whole of nodes. They constitute a surcouche
preceding conditions by imposing couples
(
)
,
private individuals. Not to weigh down
the writing of the problem and insofar as these options are only particular cases of
generic condition of Dirichlet, we will not specifically mention them in
calculations which will follow.
·
When the material is anisotropic conductivity is modelized by a diagonal matrix
expressed in the reference mark of orthotropism of material. That does not change basically
following calculations which hold account only isotropic case. It is just necessary to take guard of
not to switch over more, under the conditions limit of Neumann and Robin, the scalar product
with the normal and the multiplication by conductivity.
·
For a transitory calculation, the initial temperature can be selected in three manners
different: by carrying out a stationary calculation over the first moment, by fixing it at one
uniform or unspecified value created by one
AFFE_CHAM_NO
and by carrying out a recovery with
to start from a preceding transitory calculation. This choice of the condition of Cauchy does not have any
angle of attack on the theoretical study which will follow.
·
We will not treat the case where (almost) all the loadings are multiplied by one
even function dependant on time (option
FONC_MULT
, this well adapted functionality
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
8/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
for certain mechanical problems is disadvised in thermics, because it can re-enter in
conflict with the temporal dependence of the loadings and, in addition, it applies
selectively with each one of them. It was not included besides in
THER_NON_LINE
).
It is shown that the functional framework most general and most convenient for “the catch in hand” of it
parabolic problem is as follows.
For the geometry:
opened locally limited the only one with dimensions one of its border,
(H
1
)
variety of dimension
1
-
Q
, lipschitzienne or
1
C
by piece
(H
2
)
For the data:
()
(
)
()
()
()
()
()
()
(
)
3
2
3
2
1
2
2
2
1
2
1
2
1
2
2
0
1
2
;
,
0
,
,
;
,
0
,
;
,
0
,
;
,
0
;
,
0
-
-
-
L
L
H
L
C
H
L
T
H
L
G
H
L
F
L
T
H
L
S
p
ext.
(H
3
)
who allows us to obtain a solution in the following intersection
()
(
)
()
(
)
2
0
1
2
;
,
0
;
,
0
L
C
H
L
T
éq
2.1-4
Note:
That is to say
(
)
X
X,
Banach, one notes
(
)
X
L
p
;
,
0
the space of the functions
()
T
v
T
strongly measurable for the measurement
dt
such as
()
+
<
=
p
p
X
X
p
dt
T
v
v
1
0
,
;
,
0
. It is Banach, therefore a space of Hilbert for
the associated standard.
The introduction as of these spaces of Hilbert particular “space times” comes from the need from
to separate the variables
X
and
T
. Any function
()
] [
()
×
=
T
U
Q
T
U
,
,
0
:
,
:
X
X
can in fact
to be identified (by using the theorem of Fubini) with another function
] [
()
() () ()
{
}
T
U
T
U
X
T
U
T
U
,
~
:
~
,
0
:
~
X
X
=
. The transformation
U
U
~
constituting one
isomorphism, one will simplify the expressions thereafter while noting
U
what should have been meant
u~
.
Note:
·
The fact of separating, in first, the time of the variable of space makes it possible to be strongly inspired
conceptual tools developed for the elliptic problems. It is besides completely
coherent with the sequence “
semi-discretization in time/total discretization in
space “which usually chairs the determination of a formulation usable in
practical.
·
The assumptions on the geometry ensure us of the property of 1-prolongation of the open one
. Thus one will be able to confuse the space of Hilbert
()
()
()
(
)
=
Q
L
L
U
2
2
1
/
:
U
H
on which it is convenient to work, with space
()
()
()
{
}
=
=
U
U
H
U
D
U
H
Q
with
1
1
/
'
:
for which standard theoretical results on the traces, the densities of space and them
equivalent standards are licit.
·
Taking into account the character lipschitzien of the border the theoretical results which will follow
will be able to apply to the structures comprising of the corners (outgoing or re-entering). By
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
9/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
against the processing of points or points of graining leaves this theoretical framework
General. In the same way, the fact that the open one must locally be located the same with dimensions one of its border,
prevent (theoretically) the processing of fissure. To treat this type rigorously of
problem, an approach consists in correcting the basic functions of the finite elements by one
suitable function centered on the internal end of the fissure (cf P. GRISVARD. School
from Numerical Analysis CEA-EDF-INRIA on the breaking process, pp183-192, 1982).
·
The indicator in residue using the solution of the problem in temperature, its limitations
theoretical are thus, at best, identical to those of the aforesaid problem.
Taking into account the formulation [éq 2.1-1] one thus will be interested in a solution belonging to space
functional according to:
Note:
This space comprises also the possible conditions of Dirichlet “generalized” of
linear relations type between ddls.
()
{
}
F
U
/
:
1
,
0
1
=
=
=
:
U
1
H
U
W
T
éq
2-1-5
Moreover, thanks to the geometrical assumptions (H
1
) and (H
2
), there is an operator of raising
(compound of the operator of usual raising and the operator of prolongation by zero apart from
1
)
()
()
1
1
2
1
:
H
H
R
linear, continuous and surjective such as:
()
1
2
1
1
,
0
=
H
F
F
RF
éq
2-1-6
One thus will not be able to make the problem initial homogeneous in Dirichlet while being interested any more but in
the solution
()
{
}
0
:
1
=
=
=
U
U
/
:
1
,
0
1
H
U
V
U
éq
2-1-7
resulting from the decomposition
RF
U
T
+
=
:
éq
2-1-8
Note:
That is to say
(
)
X
X,
Banach, one notes
(
)
X
L
p
;
,
0
the space of the functions
()
T
v
T
strongly measurable for the measurement
dt
such as
()
+
<
=
p
p
X
X
p
dt
T
v
v
1
0
,
;
,
0
. It is Banach, therefore a space of Hilbert for
the associated standard.
This change of variable produces the problem simplified in
U
(
)
] [
] [
] [
] [
=
×
=
+
×
=
×
=
×
=
-
0
3
2
1
1
)
0
(
,
0
^
,
0
^
,
0
0
,
0
^
)
(
U
U
H
hu
N
U
G
N
U
U
S
U
div
T
U
C
P
p
éq
2-1-9
Code_Aster
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Titrate:
Indicator of error in residue for transitory thermics
Date:
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O. BOITEAU
Key
:
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R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
with the new second member
(
)
()
(
)
+
-
=
- 1
2
;
,
0
:
^
H
L
RF
div
T
RF
C
S
S
p
, éq
2-1-10
new loadings
()
-
=
-
2
2
1
2
,
,
0
:
^
H
L
N
RF
G
G
éq
2-1-11
(
)
()
-
-
=
-
3
2
1
2
,
,
0
:
^
H
L
N
RF
RF
T
H
H
ext.
éq
2-1-12
and the new initial condition
()
()
()
()
-
=
2
0
0
0
.,
.
:
.
L
RF
T
U
éq
2-1-13
Note:
·
This theoretical raising, which can appear a little “éthéré”, has an anchoring completely
concrete in the digital techniques implemented to solve this type of problem.
It corresponds to a substitution (this technique is not used in Code_Aster,
one prefers to him the technique of double dualisation via ddls of Lagrange [R3.03.01])
conditions limit of Dirichlet. By renumbering the unknown factors so that these conditions
appear in the last, the comparison can be schematized in the matric form
following
-
=
=
=
>
F
F
has
S
S
RF
T
T
Id
With
J
J
J
ji
:
^
:
0
0
1
1
The assumptions of regularity on the border also ensure us of the good following properties
for the workspaces. One then will be able to place itself within the usual abstracted variational framework.
Lemma 1
Under the assumptions (H
1
) and (H
2
) workspaces
W
and
V
with Hilberts are fitted with the standard
induced by
()
1
H
.
Proof:
The result comes simply owing to the fact that the application traces
()
()
1
2
1
1
,
0
:
L
H
is
composed of the application traces usual
()
()
()
2
2
1
1
0
:
L
H
H
linear, continuous and
surjective (taking into account the assumptions selected) and of the operator of restriction on
1
he too
linear, continuous and surjective. From share their definition, one deduces from it that
W
and
V
are closed sev
()
1
H
. It of Hilberts is thus fitted with the standard
,
1
.
!
Lemma 2
Under the assumptions (H
1
) and (H
2
), the standard and the pseudo norm induced by
()
1
H
are equivalent on
functional space
V
. One will note
()
0
>
P
the constant of Poincaré relaying this equivalence
()
,
1
,
1
v
P
v
V
v
Code_Aster
®
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Titrate:
Indicator of error in residue for transitory thermics
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Key
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Note:
One will note thereafter
()
=
T
p
T
U
U
.
,
:
supess
and
()
()
(
)
()
(
)
()
()
()
=
=
=
=
m
L
m
m
L
m
m
L
m
m
U
U
U
v
U
v
U
H
v
U
U
2
2
,
2
2
,
,
2
2
2
2
:
:
,
,
:
,
,
and
.
Proof:
This result is a corollary of the inequality of Poincaré checked by the open ones called of “Nikodym” of which
fact part
taking into account the assumptions selected. There are however two cases of figures:
·
that is to say the problem is really mixed and comprises conditions limit others that those of
Dirichlet,
(
)
0
1
-
my
(see the demonstration [bib1] §III.7.2 pp922-925),
·
either one takes into account only conditions of imposed the temperature type,
(
)
0
1
=
-
my
,
()
=
1
0
H
V
and one finds the standard result of equivalence of the standard
and of the pseudo norm on this space (see for example the demonstration [bib3] pp18-19).
!
The compilation of the preceding results makes it possible to encircle the Variational Framework Abstracted (CVA) on
which will rest the weak formulation:
·
()
()
1
1
0
H
V
H
,
·
()
()
=
=
- 1
2
'
'
:
H
V
H
L
H
V
while identifying
H
and its dual,
·
there is a linear canonical injection continues
V
in
H
,
·
V
is dense in
H
and the injection is compact (it inherits in that the properties
()
1
H
with respect to
H
),
·
V
is fitted with the pseudo norm induced by
()
1
H
and
H
of its usual standard.
Note:
According to a formulation of the theorem of compactness of Rellich adapted to spaces of
Sobolev on open (for example, theorem 1.5.2 [bib3] pp29-30).
2.2
Strong formulation with weak
By multiplying the main equation of the problem in extreme cases [éq 2.1-1] by a function test
V
v
and in
using the theorems of Green and Reynolds (to switch over the integral in space and derivation
in time, with
fixed and from the characteristics materials independent of time), one obtains:
()
()
()
()
+
=
+
D
v
N
T
U
dx
v
T
S
dx
v
T
U
dx
v
T
U
C
dt
D
p
^
.
éq
2.2-1
Code_Aster
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Titrate:
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:
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HI-23/02/014/A
By introducing the conditions limit in [éq 2.2-1], it occurs the weak formulation (within the meaning of
distributions (within this general framework, the temporal derivative is thus to take with the weak direction)
temporal of
] [
(
)
,
0
'
D
) following:
The solution is sought
(
)
(
)
H
C
V
L
U
;
,
0
;
,
0
0
2
éq
2.2-2
checking the problem
] [
()
()
(
)
()
(
)
()
(
)
=
=
+
0
,
0
2
)
0
(
,
,
;
,
,
0
:
)
(
U
U
v
T
B
v
T
U
T
has
v
T
U
C
dt
D
V
v
V
T
U
T
U
P
p
that
such
To find
éq
2.2-3
with
()
(
)
()
()
()
()
(
)
()
()
()
3
2
3
,
2
1
2
1
3
,
0
,
2
1
2
1
2
,
0
,
1
1
3
,
0
3
,
0
,
^
,
^
,
^
:
,
.
:
,
;
×
-
×
-
×
-
+
+
=
+
=
v
T
H
v
T
G
v
T
S
v
T
B
D
v
T
U
T
H
dx
v
T
U
v
T
U
T
has
éq
2.2-4
while noting
×,
,
Q
p
the hook of duality enters spaces
()
p
H
and
()
Q
H
.
Note:
·
The unknown field and the function test belong to the same functional space, which is
more comfortable from a numerical and theoretical point of view.
·
The hooks of duality will not be able to be transformed into integrals with the conventional direction (like
for the surface term of
(T has;..))
that if the space of membership of the news is restricted
source and of the new loadings with
()
(
)
()
(
)
()
(
)
3
2
2
2
2
2
2
2
;
,
0
^
;
,
0
^
,
;
,
0
^
L
L
H
L
L
G
L
L
S
and
éq
2.2-5
According to [éq 2-1-10] [éq 2-1-12] this restriction can be translated on the initial loadings
in the form
()
()
(
)
()
(
)
()
(
)
3
2
2
2
2
2
2
2
1
2
3
2
;
,
0
;
,
0
,
;
,
0
,
;
,
0
L
L
T
L
L
G
L
L
S
H
L
F
ext.
and
éq 2.2-6
·
The formulation
()
2
P
a direction has well, because it is shown that
()
(
)
] [
(
)
] [
(
)
,
0
'
,
0
,
;
2
D
L
v
T
U
T
has
T
()
(
)
()
(
)
] [
(
)
] [
(
)
,
0
'
,
0
,
;
,
0
2
,
0
2
D
L
v
T
U
C
T
V
v
and
V
L
T
U
C
T
p
p
()
()
(
)
()
()
()
] [
(
)
] [
(
)
,
0
'
,
0
,
^
;
,
0
^
2
,
1
1
1
1
1
2
D
L
v
T
S
T
H
H
v
H
L
T
S
T
×
-
-
-
and
()
()
()
()
()
] [
(
)
] [
(
)
,
0
'
,
0
,
^
;
,
0
^
2
,
2
1
2
1
2
,
0
2
2
1
2
2
1
2
,
0
2
2
1
2
2
D
L
v
T
G
T
H
H
v
H
L
T
G
T
×
-
-
-
and
and one finds obviously the same thing for the term of exchange on
3
.
Code_Aster
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HI-23/02/014/A
·
In the surface integrals one will note henceforth
()
T
U
and
v
what should be noted (in
any rigor)
()
v
T
U
I
I
,
0
,
0
and
.
·
Membership of the solution with
(
)
V
L
;
,
0
2
rise from the assumptions on the data and of
properties of the differential operators and trace. The fact that it must also belong to
(
)
H
C
;
,
0
0
comes just from the necessary justification of the condition of Cauchy.
One can then be interested in the existence and unicity of the solution of the initial problem
()
0
P
in
showing its equivalence with
()
2
P
and by applying to this last a parabolic alternative of
theorem of Lax-Milgram.
Theorem 3
Within the abstracted variational framework (CVA) definite previously and by supposing that assumptions
(H
1
), (H
2
) and (H
3
) are checked, then the problem
()
2
P
admits a solution and only one
(
)
(
)
H
C
V
L
U
;
,
0
;
,
0
0
2
.
Proof:
This result comes from theorems 1 & 2 of the “Dautray-Lions” (cf [bib3], §XVIII pp615-627). For
to use it is necessary nevertheless to check
·
Mesurability of the bilinear form
()
(
)
()
(
)
] [
,
0
,
;
,
2
on
v
T
U
T
has
T
V
v
T
U
·
Its continuity on
V
V
×
] [
()
(
)
()
()
()
()
(
)
()
()
(
)
()
+
,
1
,
1
2
2
3
,
,
2
,
2
1
,
2
1
,
,
1
,
1
,
3
3
3
3
,
,
,
;
,
0
v
T
U
P
C
T
H
V
v
T
U
v
T
U
T
H
v
T
U
v
T
U
T
has
T
p
max
with
3
C
the constant of continuity of the operator of trace on
3
and
()
P
the constant of
Poincaré.
·
Its
V
- ellipticity compared to
H
] [
(
)
()
(
)
(
)
{
()
(
)
2
,
0
2
3
,
,
2
0
2
,
0
2
,
0
2
3
,
,
2
0
2
,
0
0
2
0
,
;
2
,
;
,
0
3
3
-
-
>
-
+
>
+
-
+
v
C
T
H
C
v
v
v
T
has
V
v
v
C
T
H
C
v
v
v
T
has
T
p
4
4
4
4
4
4
3
4
4
4
4
4
4
2
1
with
0
C
the constant of continuity of the canonical injection of
()
1
H
in
()
2
L
.
·
The continuity of the linear form
()
T
B
on
V
] [
()
(
)
()
()
()
()
()
()
()
-
-
-
-
-
-
+
+
,
1
3
,
2
1
2
,
2
1
,
1
,
2
1
,
2
1
,
2
1
,
2
1
,
1
,
1
3
2
3
3
2
2
3
^
,
^
,
^
^
^
^
,
,
0
v
C
T
H
C
T
G
T
S
P
V
v
v
T
H
v
T
G
v
T
S
v
T
B
T
p
max
with
2
C
the constant of continuity of the operator of trace on
2
.
!
Theorem 4
Problems
()
0
P
and
()
2
P
are equivalent and thus the initial problem admits a solution and one
only
(
)
(
)
H
C
V
L
U
;
,
0
;
,
0
0
2
.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Proof:
The existence and the unicity of the solution of the problem
()
0
P
result of course from the preceding theorem, one
time that the equivalence of the two problems was shown. It thus remains to prove the implication
opposite
() ()
0
2
P
P
who is very hard to exhume “not formally”. In particular conditions
limits of Neumann, Robin and the condition of Cauchy are difficult to obtain rigorously.
“Dautray-Lions” proposes a very technical demonstration ([bib1] §XVIII pp637-641). While adapting
his results it is shown that in our case of figure, the conditions limit on
I
in fact are checked,
not on
()
-
I
H
L
2
1
2
,
,
0
, but on space
()
()
I
I
H
B
- 21
00
'
(while noting
] [
,
0
:
×
=
I
I
)
defined as being the dual topological one of
(
)
()
(
)
{}
{}
=
=
=
=
×
×
W
v
v
v
V
L
v
L
H
W
B
I
I
I
and
with
0
;
,
0
/
:
0
2
2
2
1
!
Note:
·
Because of low regularity imposed on thermal conductivity,
()
L
, one cannot
not to claim with the “standard” regularity
()
2
H
U
. Indeed in the case, for example,
of a Bi-material (with
2
1
=
) from which the characteristics are distinct from share and
of other of the border
, [éq 2-1-9] and the theorem of the divergence imposes
1
1
2
2
Appear 2.2-a: Example of Bi-material
()
()
()
] [
,
0
2
1
00
2
2
1
1
=
-
T
p
H
N
T
U
N
T
U
in
However
2
1
, therefore the condition of transmission cannot be carried out on the internal border
()
()
] [
,
0
2
1
T
p
p
N
T
U
N
T
U
Thus
()
()
()
2
2
1
2
H
H
T
U
do not involve
()
()
2
H
T
U
. This restriction us
will not allow to exhume, as in [bib6], of increases of the “strong” type of the error
space total and of the site indicator of error. Within our framework of more general work one
will have to be satisfied with estimates of the “weak” type.
·
This type of problem also meets when one treats the open polyhedric ones not
convex (for example comprising a re-entering corner). Open polyhedric (known as polygonal in
two-dimensional) is a finished meeting of polyhedrons. A polyhedron is an intersection finished of
closed half spaces.
·
To obtain estimates of the “strong” type, it is necessary to concede more regularity on
geometry and on the loadings
variety of dimension
1
-
Q
,
2
C
by piece (property of 2-prolongation)
(H
4
)
Code_Aster
®
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Titrate:
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Key
:
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HI-23/02/014/A
()
(
)
()
()
()
()
()
()
(
)
3
2
3
2
1
2
2
2
1
2
1
2
3
2
1
0
2
2
;
,
0
,
,
;
,
0
,
;
,
0
,
;
,
0
;
,
0
L
L
H
L
C
H
L
T
H
L
G
H
L
F
H
T
L
L
S
p
ext.
(H
5
)
What allows obtaining a solution in the following intersection
()
(
)
()
(
)
1
0
2
2
;
,
0
;
,
0
H
C
H
L
U
éq
2.2-7
Now that we made sure of the existence and the unicity of the solution within the framework
functional required by the operators of Code_Aster, we semi-will discretize in time
(P
0
)
then to spatially discretize the whole by a method finite elements. In parallel, us
will study its properties of stability. They we will be very useful to create the standards, them
techniques and the inequalities which will intervene in the genesis of the indicator of error in residue.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
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O. BOITEAU
Key
:
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
3
Discretization and controllability
3.1
Controllability of the continuous problem
By not making any concession on the assumptions of regularity seen in the preceding paragraph, one
with increase known as “weak” (to take again a terminology in force in the article which served as
base with our study [bib6]) following.
Property 5
Within the abstracted variational framework (CVA) definite previously and by supposing that assumptions
(H
1
), (H
2
) and (H
3
) are checked, one with the “weak” controllability of the continuous problem (with
()
()
(
)
0
,
,
,
,
0
,
1
>
P
my
K
I
I
)
()
()
()
()
()
+
+
+
+
-
-
-
T
p
T
p
D
H
G
S
K
U
C
D
U
T
U
C
T
p
0
2
,
2
1
2
,
2
1
2
,
1
1
2
,
0
0
2
,
0
0
2
,
0
3
2
^
^
^
éq 3.1-1
Proof:
One here will detail this a little technical demonstration because, on the one hand, the specialized literature re-enters
seldom in this level of details and, in addition, one will re-use same methodology for
to exhume all increases which will follow one another in this theoretical part of the document. All
initially, by multiplying the equation of [éq 2.1-1] by
()
T
U
, while integrating spatially on
, then
temporally on
[]
[[
,
0
,
0
T
T
with
one obtains, like the characteristics materials are
presumedly independent of time,
() ()
(
)
()
(
) ()
(
)
() ()
×
-
=
-
T
T
p
T
D
U
S
D
U
U
div
D
U
U
C
T
0
,
1
1
0
,
0
,
0
0
,
^
,
,
2
1
éq
3.1-2
By using the formula of Green and the conditions limit [éq 2.1-1] one obtains
()
() ()
(
)
() ()
() ()
() ()
() ()
+
+
=
+
+
-
×
-
×
-
×
-
T
T
T
p
p
D
U
H
U
G
U
S
D
U
H
D
U
U
U
C
T
U
C
0
,
2
1
2
1
,
2
1
2
1
,
1
1
0
2
0
,
0
2
,
0
0
2
,
0
3
2
,
^
,
^
,
^
,
2
1
éq 3.1-3
One can évincer the term of exchange of [éq 3.1-3] because it is supposed that
()
T
T
H
p
0
. By using one
argument of duality, the inequality of Cauchy-Schwartz, lemma 2 and the relation
() (
)
0
2
2
2
>
+
B
has
ab
, one obtains
() ()
()
()
()
+
-
×
-
T
T
T
D
U
P
D
S
D
U
S
0
2
,
0
2
,
2
0
2
,
1
2
0
,
1
1
^
1
2
1
,
^
éq
3.1-4
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One carries out same work on the loadings, thus defining the parameters
and
in
taking again the notations of theorem 3 (for
C
I
…), then one inserts these inequalities in [éq 3.1-3]
()
()
(
)
()
()
()
()
+
+
+
+
+
-
+
×
-
×
-
×
-
T
p
T
p
D
H
G
S
U
C
D
U
C
C
P
T
U
C
0
2
2
,
2
1
2
1
2
2
,
2
1
2
1
2
2
,
1
1
2
,
0
0
0
2
,
0
2
2
3
2
2
2
2
,
2
2
,
0
3
2
^
^
^
2
éq
3.1-5
It now remains to seek a triplet of strictly positive realities
(
)
,
,
, not privileging any
particular term, in order to reveal a constant independent of the solution and parameter setting
in front of the term in gradient. One arbitrarily chooses to pose
()
(
)
1
2
2
2
3
2
2
2
2
,
2
=
+
+
-
C
C
P
éq
3.1-6
Maybe, for example,
()
(
)
()
()
(
)
()
(
)
()
()
(
)
()
(
)
()
()
(
)
+
+
=
+
+
=
+
+
=
3
my
1
my
3
my
1
my
3
my
1
my
2
2
3
3
,
2
2
2
2
2
,
2
2
1
,
2
P
C
P
C
P
éq
3.1-7
From where increase [éq 3.1-1] while taking
()
()
(
)
()
(
)
()
(
)
()
(
)
+
+
+
+
=
1
my
,
1
my
,
1
my
1
3
my
3
2
3
2
2
2
1
,
2
1
C
C
P
K
max
éq
3.1-8
!
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Note:
·
The recourse to the measurements of the external borders is an easy way allowing the inequality
to support the passage to limit (
0
I
) when one or more limiting conditions come to
to miss in this mixed problem.
·
While placing itself within the particular framework of a homogeneous problem of Cauchy-Dirichlet with
characteristics materials constants equal to the unit
S
S
C
p
=
=
=
=
=
^
,
1
3
2
and
éq
3.1-9
and by introducing particular standards on
()
=
1
0
H
V
and its dual
()
()
()
()
(
) ()
×
-
-
+
+
=
=
,
1
2
*
,
1
*
,
1
,
1
1
0
,
*
,
1
3
1
,
^
sup
^
v
P
my
my
v
v
v
T
S
T
S
v
V
v
with
éq
3.1-10
one finds well the inequality (2) pp427 of [bib6].
·
If one allows more regularity on the geometry (H
4
) and on the data (H
5
), one can
to exhume during, known as “extremely”, of the preceding property. The control of the solutions that it
operate is of course more precise than with [éq 3.1-1] because it is carried out via stronger standards.
Contrary to “weak” increase, it also utilizes directly the infinite standard
coefficient of convectif exchange. One will not detail his obtaining here because this family of
increase is not essential for the calculation of the required indicator.
3.2
Semi-discretization in time
A pitch of time is fixed
T
such as
T
that is to say an entirety
NR
and such as the temporal discretization is
regular:
T
N
T
T
T
T
T
T
N
=
=
=
=
L
2
,
,
0
2
1
0
.
Note:
This assumption of regularity does not have really importance, it allows just
to simplify the writing of the semi-discretized problem. To modelize a transient
unspecified at the moment
T
N
, it is just enough to replace
T
by
N
N
N
T
T
T
-
=
+1
.
Semi-discretization in times of [éq 2.1-1] by
- method leads to the following problem:
The continuation is sought
()
V
U
NR
N
N
0
éq
3.2-1
such as
()
(
)
(
)
(
)
(
)
=
-
=
+
-
=
-
=
-
-
+
=
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
0
0
3
1
1
1
1
2
1
1
1
1
1
1
1
1
1
(.)
1
0
^
1
0
^
1
0
0
1
0
^
1
^
div
1
div
U
U
NR
N
H
U
H
N
U
NR
N
G
N
U
NR
N
U
NR
N
S
S
U
U
T
U
U
C
P
N
N
N
N
N
N
N
N
N
N
N
N
N
p
N
éq 3.2-2
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while posing
{
}
NR
N
G
H
H
S
U
T
N
N
=
0
and
^
,
,
^
,
^
,
,
with
X
While multiplying [éq 3.2-2] by a function test
v
and while integrating on
, one finds (via the formula of
Green) of course the variational formulation [éq 2.2-3] semi-discretized in time
() (
)
(
) (
)
()
(
)
+
=
+
+
+
+
+
+
+
+
+
V
v
v
B
T
v
U
C
v
U
T
N
has
T
v
U
C
V
U
H
H
H
H
G
G
S
S
U
P
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
,
,
,
;
,
,
,
^
,
^
,
^
,
^
,
^
,
^
,
,
0
1
,
0
1
1
1
1
1
1
1
2
p
p
that
such
To calculate
given
Being
éq
3.2-3
with
(
)
{
}
(
)
()
(
)
3
2
3
,
2
1
2
1
3
,
0
1
,
2
1
2
1
2
,
0
1
,
1
1
1
1
1
1
1
1
1
,
^
,
^
,
^
:
,
.
:
,
;
^
,
^
,
^
,
,
,
1
:
×
-
+
×
-
+
×
-
+
+
+
+
+
+
+
+
+
=
+
=
-
+
=
v
H
v
G
v
S
v
B
D
v
hu
dx
v
U
v
U
T
N
has
H
G
S
B
hu
U
N
N
N
N
N
N
N
N
N
N
where
éq
3.2-4
This semi-discretization in time made it possible to transform our parabolic problem into one
elliptic problem to which one can apply the theorem of standard Lax-Milgram. Assumptions of
this theorem are checked easily thanks to the results of continuity and ellipticity of
demonstration of theorem 3. From where the existence and the unicity of the continuation
()
V
U
NR
N
N
0
sought.
Note:
· While posing
0
=
RF
one finds the variational formulation semi-discretized well of
Code_Aster (cf [R5.02.01 §5.1.3]). (Or them) the condition (S) of Dirichlet (generalized or not)
are checked in the workspace
W
which the solution must belong. Moreover, in
implicitant completely
- method (Euler retrogresses) one finds the formulation of the code
SYRTHES [bib9].
· To be able semi-to discretize by
- method one needs to restrict the membership of
the new source with
()
(
)
- 1
0
;
,
0
^
H
C
S
(to be able to take a value in one moment
given). In addition, the initialization of the iterative process [éq 3.2-3] necessarily involves
()
1
0
H
U
.
· To simplify the expressions one will not mention any more the temporal dependence of the form
bilinear
(T has;..)
(for the implicitation of the term of exchange), it will remain implied by
that of the solution.
Code_Aster
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As for the continuous problem, by not making any concession on the assumptions of regularity,
one with “weak” increase following:
Property 6
By supposing that the assumptions of property 5 are checked, that it
- diagram is
unconditionally stable (
2
1
), that
()
(
)
- 1
0
;
,
0
^
H
C
S
and
()
1
0
H
U
, one with
controllability “
weak
” of the problem semi-discretized in time (with
()
()
(
)
0
,
,
,
,
0
,
1
>
P
my
K
I
I
)
+
+
+
+
-
-
+
+
-
+
-
+
-
+
+
+
+
+
2
,
2
1
1
2
,
2
1
1
2
,
1
1
1
2
,
0
1
2
,
0
2
,
0
1
2
,
0
1
2
,
0
1
3
2
^
^
^
2
2
1
0
2
3
4
2
1
N
N
N
N
N
p
N
p
N
N
p
H
G
S
T
K
U
T
NR
N
U
C
U
C
U
T
U
C
éq 3.2-5
Proof:
This inequality is obtained easily by taking again the stages described in the demonstration of
property 5. It is necessary, on the other hand, to multiply [éq 3.2-2] by the particular function test
(
)
V
U
U
U
N
N
N
-
+
=
+
+
1
:
1
1
éq
3.2-6
and évincer the term of exchange by the argument
(
)
()
(
)
2
,
0
1
1
1
1
2
,
0
1
1
3
3
3
,
,
+
+
+
+
+
+
<
N
N
N
N
N
N
N
N
U
H
H
dx
U
hu
U
H
H
max
min
0
éq 3.2-7
In addition there are not this time the source term and the loadings which require the easy way
[éq 3.1-4], it should also be set up on the cross term
(
)
dx
U
U
C
N
N
p
1
1
2
+
-
. From where one
fourth parameter
checking a system of the type [éq 3.1-6]
()
(
)
1
2
1
2
1
2
2
2
2
3
2
2
2
2
,
2
=
-
-
=
+
+
-
C
C
P
éq
3.2-8
!
Note:
· If one does not place a conditionally stable diagram in the case of, in addition to
numerical problems which are likely to occur at the time of implementation the effective of
the operator, one will not be able to determine the parameters
(
,
)
governing the equation [éq 3.2-8].
· While placing themselves within the particular framework [éq 3.1-9] of the article [bib6] and by taking again the standards
equivalent [éq 3.1-10], like
2
1
2
3
4
<
-
, one finds well the inequality (5) pp428.
Code_Aster
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While stating [éq 3.2-5] for the values of
{
}
N
m
,
1
,
0 K
and by summoning these increases until
N
,
one obtains the “weak” increase following which takes account of the history of the solutions and of
data.
Corollary 7
Under the assumptions of property 6, one with increase
(
)
(
)
-
=
-
+
-
+
-
+
-
=
-
=
+
+
+
+
-
-
+
+
1
0
2
,
2
1
1
2
,
2
1
1
2
,
1
1
1
2
,
0
0
1
0
2
,
0
1
0
2
,
0
1
2
,
0
3
2
^
^
^
0
3
4
1
4
N
m
m
m
m
p
N
m
m
p
N
m
m
N
p
H
G
S
T
K
NR
N
U
C
U
C
U
T
U
C
éq 3.2-9
or more simply
-
=
-
+
-
+
-
+
-
=
+
+
+
+
+
1
0
2
,
2
1
1
2
,
2
1
1
2
,
1
1
1
2
,
0
0
1
0
2
,
0
1
2
,
0
3
2
^
^
^
0
N
m
m
m
m
p
N
m
m
N
p
H
G
S
T
K
NR
N
U
C
U
T
U
C
éq
3.2-10
Proof:
Obtaining [éq 3.2-9] being already explained, it remains to be shown [éq 3.2-10]. This inequality more
“coarse” comes simply owing to the fact that
(
)
(
)
2
,
0
0
2
,
0
0
1
0
2
,
0
3
4
0
1
4
-
=
-
-
U
C
U
C
U
C
p
p
N
m
m
p
éq
3.2-11
!
Note:
·
One can obviously make the same remark as [bib6] by noting that the last term of
[éq 3.2-9] is a sum of Riemann which tends towards the last term of [éq 3.1-1] when it
no time tends towards zero. In addition, if one introduces the function (with
(
)
[
]
T
N
T
N
+
1
,
temporal function characteristic of the interval
(
)
[
]
T
N
T
N
+
1
,
)
()
(
)
[
]
()
T
U
T
U
T
N
T
N
N
+
+
=
1
,
1
closely connected per pieces in [éq 3.1-1], one finds exactly [éq 3.2-9].
·
As for [éq 3.1-1], by adopting the less restrictive approaches (H
4
) and (H
5
), one finds
a “strong” version of properties 6 and 7.
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3.3
Error of temporal discretization
The preceding results on the continuous problem and its form semi-discretized in time are
re-used jointly to study the controllability of the error of temporal discretization
()
0
:
0
0
=
-
=
E
T
N
U
U
E
NR
N
N
N
éq
3.3-1
One starts by revealing this error by withdrawing from the equation [éq 3.2-2] the relations
()
(
)
(
)
(
) ()
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
() ()
()
(
) (
) ()
T
N
S
T
N
U
T
T
N
U
C
T
N
S
T
N
U
T
T
N
U
C
T
T
N
U
T
N
U
D
T
U
T
p
p
T
N
T
N
-
+
-
=
-
+
+
+
=
+
-
+
=
+
^
1
1
1
1
^
1
1
1
1
1
div
div
éq
3.3-2
that is to say
(
)
()
(
)
+
=
-
-
+
+
+
T
U
C
D
T
U
T
E
T
E
E
C
p
T
N
T
N
N
N
N
p
1
1
1
1
div
éq
3.3-3
while noting
(
)
(
)
(
) (
) ()
T
N
T
U
T
N
T
U
T
U
E
E
E
N
N
N
-
+
+
=
-
+
=
+
+
1
1
:
1
:
1
1
éq
3.3-4
From this expression one can describe, via the recourse to the formula of Taylor, controllability
“weak” of the error of temporal discretization. But to be able to use the derivative temporal of
the solution continues one needs a minimum of regularity in
T
, for example by conceding that
(
)
()
(
)
- 1
2
1
;
,
0
;
,
0
H
H
V
H
U
éq
3.3-5
Property 8
By supposing that the solution checks the additional assumption of temporal regularity [éq 3.3-5], one
with the “weak” controllability of the error of temporal discretization
()
(
)
(
)
(
)
(
)
(
)
-
=
-
=
+
+
-
-
+
1
0
2
2
2
2
2
3
1
1
0
2
,
0
1
2
,
0
1
1
4
0
N
m
p
N
m
m
N
p
T
m
T
U
T
m
T
U
C
T
K
E
T
E
C
NR
N
éq 3.3-6
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Proof:
While evaluating [éq 3.3-3] by a formula of Taylor to command 2, one utilizes the derivative second
temporal of the solution and one shows that the continuation of error
()
V
E
NR
N
N
0
check a similar problem
with [éq 3.2-2] (by supposing that the temporal discretization of the conditions limit are exact)
()
(
)
(
)
()
(
)
(
)
()
=
-
=
+
-
=
-
=
-
+
-
-
=
-
-
+
+
+
+
+
+
+
+
0
.
1
0
0
1
0
0
1
0
0
1
0
1
1
2
div
0
3
1
1
1
2
1
1
1
2
2
2
2
1
1
1
3
E
NR
N
E
H
N
E
NR
N
N
E
NR
N
E
NR
N
T
N
T
U
T
N
T
U
T
C
E
T
E
E
C
P
N
N
N
N
N
p
N
N
N
p
N
éq 3.3-7
One can then apply the second result of the corollary 7 to him from where [éq 3.3-6] (one could, of course,
just as easily to apply the rough result of this corollary or that of the property 6 from which it rises).
!
Note:
·
While placing itself within the particular framework [éq 3.1-9] of the article [bib6] with an implicit scheme
(
=
1) and by taking again the equivalent standards [éq 3.1-10] one finds well the inequality (8)
pp429. It is enough to make tend
0
T
and to approximate the integral by the sum of Riemann
what constitutes the second member of [éq 3.3-6].
·
The existence and the unicity of the continuation (
E
N
) rises of course from that of (
U
N
) but one also can
redémontrer by applying the theorem of Lax-Milgram to the weak formulation rising from
[éq 3.3-7].
3.4
Total discretization in time and space
It is supposed that the field
is polyhedric or not and that it is discretized spatially by one
regular family
(
H
)
H
triangulations. Because of this regularity finite element method
applied to
()
1
2
+
N
P
converge when the largeest diameter of the elements
K
of
(
H
)
H
tends towards zero
0
:
=
K
H
H
T
K
H
max
éq
3.4-1
Note:
·
Finite elements (
K
,
P
K
,
K
) equivalents with same elements of reference are closely connected, they
relations of compatibility on their common borders and the stresses check
geometrical [éq 3.4-1] and [éq 3.4-2].
·
It is reminded the meeting that the diameter of
K
is reality
()
2
,
:
K
H
K
=
-
y
X
y
X
max
.
Code_Aster
®
Version
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Titrate:
Indicator of error in residue for transitory thermics
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O. BOITEAU
Key
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While noting
K
the roundness (one recalls that the roundness of
K
is reality
{
}
K
K
=
spheres
diameter
max
:
) associated
K
, finite elements of
(
H
)
H
also satisfy
stress
>
K
K
H
/
0
éq
3.4-2
In the usual triplet (
K
,
P
K
,
K
) one defines polynomial space as being that of the polynomials of
total degree lower or equal to
K
on
K
()
K
P
K
K
P
=
:
éq
3.4-3
and approximation spaces it (with the “weak” direction) associated
()
{
}
V
K
v
T
K
V
v
V
K
K
H
H
H
H
=
P
/
:
éq
3.4-4
To conclude, one will note
H
, the operator of projection which associates the solution continues its
V
H
interpolated
H
H
H
v
v
V
V
:
éq
3.4-5
Note:
With a regular family of triangulations, this operator of interpolation is continuous and it can
to be written
()
=
I
I
I
H
NR
v
v
X
while noting
I
X
nodes of the mesh and
I
NR
their function of
form associated.
It will be of a very particular importance when it is necessary to describe the increase which will exhume
the indicator of error.
Note:
·
In practice the mesh is often polygonal, the approximation
H
of
becomes
then more rudimentary than in the polyhedric case. To preserve the convergence of
the method it is then necessary to resort to isoparametric elements (cf [bib3] pp113-123 or
P. GRISVARD. Behavior off the solutions off year elliptic boundary problem in A polygonal gold
polyhedral domain. Numerical solution off PDE, ED. Academic Near, 1976).
·
The indicator in residue was established in Code_Aster only for the elements
isoparametric (triangle, quadrangle, face, tetrahedron, pentahedron and hexahedron). Moreover,
as they are simplexes or parallélotopes, the associated triangulation is
regular (cf [bib3] pp108-112).
·
For the simplexes the relation [éq 3.4-2] results in the existence of a lower limit on
angles and, for the parallélotopes, by the existence of an upper limit controlling them
relationship between the height, the width and the length.
·
In the definition [éq 3.4-4] of
V
H
, they are the intrinsic relations of compatibility with
family of elements which assures us
()
()
(
)
K
H
v
K
H
K
v
K
H
H
K
K
H
=
:
,
1
1
P
éq
3.4-6
In the literature one often prefers the more regular definition to him
()
=
0
*
:
C
V
V
H
H
éq
3.4-7
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
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O. BOITEAU
Key
:
R4.10.03-A
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
By regaining the semi-discretized shape
()
1
2
+
N
P
with functions tests in
V
H
the problem is obtained
completely discretized in time and space (for one
H
fixed) according to:
The continuation is sought
()
H
NR
N
N
H
V
U
0
éq
3.4-8
initialized by
0
0
:
U
U
H
H
=
éq
3.4-9
checking the following problem
(
) (
)
(
) (
)
(
)
(
)
+
=
+
+
+
+
+
+
+
+
+
+
H
H
H
N
H
N
H
H
N
H
H
N
H
H
N
H
N
N
N
N
N
N
N
N
N
H
N
H
V
v
v
B
T
v
U
C
v
U
has
T
v
U
C
V
U
H
H
H
H
G
G
S
S
U
P
,
,
,
,
,
,
^
,
^
,
^
,
^
,
^
,
^
,
1
,
0
1
,
,
0
1
1
1
1
1
1
1
,
2
p
p
that
such
To calculate
given
Being
éq 3.4-10
Just as one supposed in the preceding paragraph as the temporal discretization of
loadings was exact
()
{
}
NR
N
G
H
H
S
T
N
E
N
N
=
-
=
0
and
^
,
,
^
,
^
0
:
with
(H
6
)
, one supposes here moreover than their space discretization is too
{
}
NR
N
G
H
H
S
H
N
N
H
N
H
=
=
0
and
^
,
,
^
,
^
:
with
(H
7
)
In Code_Aster, these assumptions can not be checked and it will be seen that they impact
the quality of the indicator in residue and its relations between equivalence and the exact error (cf [§4.3]). In
practical, even if one is obliged to compose with this approximation, it is not truly
problems as long as the loadings “are not chahutés too much” in time and space.
By applying the theorem of standard Lax-Milgram following the skeleton developed in
demonstration of theorem 3, one shows the existence and the unicity of the continuation
()
N
N
H
U
in the closed sev
(it is thus Hilbert, pre-necessary essential for the use of the famous theorem)
V
H
of Hilbert
V
.
Moreover, by applying the second result of corollary 7 (one could, of course, just as easily
to apply the rough result of this corollary or that of the property 6 from which it rises), controllability
“weak” of the completely discretized problem takes the following form:
Property 9
While being based on the triangulation defined previously and by supposing that the assumptions (H
6
) and
(H
7
) are checked, one with increase
-
=
-
+
-
+
-
+
-
=
+
+
+
+
+
1
0
2
,
2
1
1
2
,
2
1
1
2
,
1
1
1
2
,
0
0
1
0
2
,
0
1
,
2
,
0
3
2
^
^
^
0
N
m
m
m
m
H
p
N
m
m
H
N
H
p
H
G
S
T
K
NR
N
U
C
U
T
U
C
éq
3.4-11
while noting
(
)
m
H
m
H
m
H
U
U
U
-
+
=
+
+
1
:
1
1
,
.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Note:
·
While placing itself within the particular framework [éq 3.1-9] of the article [bib6] with an implicit scheme
(
=
1) and by taking again the equivalent standards [éq 3.1-10] one finds well the inequality (14)
pp430.
·
By adopting the less restrictive approaches (H
4
) and (H
5
), one finds a version “strong” of
this increase utilizing the standard
H
1
field result.
Now that we determined the functional framework ensuring us of the existence and the unicity of
continuation discrete solution and to study the evolution of the controllability of the problem during
discretizations, we go mutualiser these results a little “éthérés” to release increase where
the indicator will intervene.
4
Indicator in pure residue
4.1 Notations
To build the site indicator of error one will require the following notations:
·
The whole of the faces (resp. nodes) of the element
K
is indicated by
S (K)
(resp.
NR (K)
).
·
The whole of the nodes associated with one with its faces
F
(pertaining to
S (K)
) is noted
NR (F)
.
Note:
To make simple, one will indicate under the term “face”, the with dimensions one of a finite element in 2D or
one of its faces in 3D.
·
The diameter of the element
K
(resp. of one of its faces
F
) is noted
H
K
(resp.
H
F
).
·
The whole of the triangulation
(
H
)
breaks up in the form
3
,
2
,
1
,
,
:
H
H
H
H
H
T
T
T
T
T
=
while noting
(
H, I
)
the whole of the finite elements having a face contained in the border
I
.
·
With same logic, the whole of the faces of the triangulation
(
H
)
breaks up in the form
3
,
2
,
1
,
,
:
H
H
H
H
H
S
S
S
S
S
=
with
{}
{
}
()
I
H
I
H
I
H
T
K
K
S
K
T
K
K
S
I
,
,
/
:
3
,
2
,
1
=
=
·
In the same way, the whole of the nodes of the triangulation
(
H
)
breaks up in the form
3
,
2
,
1
,
,
:
H
H
H
H
H
NR
NR
NR
NR
NR
=
·
The function “bubble” associated with
K
(resp.
F
) is noted
K
(resp.
F
).
Note:
It is related to
D (
)
(together of the indefinitely derivable functions and with support
compact) resulting from the theorem of truncation on compact: its support is limited to
compact in question (here
K
or
F
) and it is worth between 0 and 1 on its interior (with the direction
topological of the term). It is thus null on the border of compact and outside
the aforementioned.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
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O. BOITEAU
Key
:
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·
One notes
F
P
the operator of raising on
K
traces on
F
, built starting from an operator of
raising fixed on the element of reference.
·
Union of the finite elements of the triangulation sharing at least a face with
K
is noted
()
()
=
'
'
:
K
S
K
S
K
K
·
Union of the finite elements of the triangulation containing
F
in their border is noted
()
'
'
:
K
S
F
K
F
=
·
Union of the finite elements of the triangulation which share at least a node with
K
(resp. with
F
) is noted
()
()
=
'
'
:
K
NR
K
NR
K
K
(resp.
()
()
=
'
'
:
K
NR
F
NR
K
F
).
K
T
H
K
H
K
F
F
F
Appear 4.1-a: Designation of the types of vicinities for
K
and
F
.
4.2
Increase of the total space error
We thus will see how to obtain a local indicator of calculable error from
data and of the discrete solution
()
N
N
H
U
. As the discretized workspace is included in
continuous space
V
V
H
, one can re-use [éq 3.2-3] with
v
H
. While withdrawing to him [éq 3.4-10] it occurs
(with
N
and
H
fixed and by supposing (H
6
) and (H
7
))
(
)
(
)
(
)
(
)
(
)
(
)
(
)
H
H
H
N
H
N
H
N
H
N
H
N
H
N
V
v
v
U
U
C
v
U
U
has
T
v
U
U
C
-
=
-
+
-
+
+
+
+
,
0
1
,
1
,
0
1
1
,
,
,
p
p
éq
4.2-1
Note:
·
This relation states the orthogonal character of the space error with respect to the elements of
V
H
.
·
It supposes in addition which the discretization is “consistent” i.e. there is not
additional errors introduced by the numerical integration of the integrals. In
practical it is of course not the case!
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
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O. BOITEAU
Key
:
R4.10.03-A
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R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Let us consider the following linear form
()
(
)
(
)
(
)
(
)
V
v
v
U
U
has
T
v
U
U
C
v
With
N
H
N
N
H
N
-
+
-
=
+
+
+
+
,
,
:
1
,
1
,
0
1
1
p
éq
4.2-2
who will be used to us as discussion thread during this demonstration. By packing it via [éq 4.2-1], one obtains
()
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
H
N
H
N
H
N
H
N
H
N
H
N
N
H
N
v
v
U
U
has
T
v
v
U
U
C
V
v
v
v
U
U
C
v
U
U
C
v
With
-
-
+
-
-
+
-
-
+
-
=
+
+
+
+
,
,
,
,
1
,
1
,
0
1
1
,
0
,
0
p
p
p
éq
4.2-3
While taking [éq 3.2-3] after having replaced
v
H
by
V
v
v
H
-
, one can build
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
+
+
+
+
+
+
+
-
-
-
-
-
-
=
-
-
+
-
+
-
-
,
0
1
1
,
,
0
1
,
0
1
,
1
,
0
1
1
,
,
,
,
,
H
N
H
N
H
H
N
H
H
N
H
N
H
N
H
N
H
N
N
H
N
v
v
U
U
C
v
v
U
has
T
v
v
B
T
V
v
v
v
U
U
has
T
v
v
U
U
U
U
C
p
p
éq 4.2-4
Then
With (v)
becomes
()
(
)
(
)
(
)
(
)
(
)
(
)
(
)
H
N
H
H
N
H
N
H
H
N
N
H
N
v
v
U
has
T
v
v
U
U
C
V
v
v
v
B
T
v
U
U
C
v
With
-
-
-
-
-
-
+
-
=
+
+
+
,
,
,
,
1
,
,
0
1
1
,
0
p
p
éq
4.2-5
Then one breaks up the last three terms on each element
K
triangulation and one apply,
with the last, the formula of Green
()
(
)
(
)
(
) (
)
(
)
(
)
()
(
)
-
-
-
+
-
-
+
-
-
-
-
+
-
-
+
-
=
+
+
+
+
+
+
+
+
+
3
,
1
1
,
1
2
,
1
,
1
,
1
,
1
,
1
1
,
0
^
^
2
div
^
,
H
F
H
N
H
N
H
N
H
F
H
N
H
N
H
H
F
H
N
H
H K
H
N
H
N
H
N
H
N
N
H
N
S
F
S
F
S
F
T
K
D
v
v
hu
N
U
H
T
D
v
v
N
U
G
T
V
v
v
D
v
v
N
U
T
dx
v
v
U
T
U
U
C
S
T
v
U
U
C
v
With
p
p
éq 4.2-6
Note:
·
One allowed oneself to replace the hooks of duality of [éq 3.2-4] by integrals and one
can apply the formula of Green bus to the compact one
K
the assumptions (H
4
) and (H
5
) are
checked (while replacing
by
K
and
I
by
I
K
). One thus has
()
()
()
(
)
(
)
3
2
2
2
2
2
1
^
^
,
^
,
,
-
K
L
H
K
L
G
K
L
S
K
H
U
K
H
v
v
H
H
and
éq 4.2-7
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Let us point out some properties of the operator
H
of projection
L
2
- local introduced by
P. CLEMENT [bib8]
()
H
H
H
v
v
V
L
V
2
:
éq
4.2-8
It checks in particular increases of errors of projection
()
F
F
F
H
F
H
H
K
K
K
H
K
H
v
H
C
v
v
v
v
K
S
F
T
K
v
H
C
v
v
v
v
V
v
,
1
5
,
0
,
0
,
1
4
,
0
,
0
:
,
:
-
=
-
-
=
-
éq
4.2-9
where constants
C
4
and
C
5
depend on the smallest angles of the triangulation. While taking this
operator of space projection and by applying the inequality of Cauchy-Schwartz to [éq 4.2-6] it occurs
thus:
()
(
)
(
)
(
)
()
+
+
+
+
+
+
+
+
+
-
-
+
-
+
+
+
-
-
-
-
3
,
,
1
,
0
1
1
,
1
5
2
,
,
1
,
0
1
,
1
5
,
,
1
,
0
1
,
5
,
1
,
0
1
,
1
1
4
,
0
^
^
2
div
^
,
H
F
F
N
H
N
H
N
F
H
F
F
N
H
N
F
H
F
F
N
H
F
H
K
K
N
H
N
H
N
H
N
K
N
H
N
S
F
S
F
S
F
T
K
v
hu
N
U
H
H
Tc
v
N
U
G
H
Tc
V
v
v
N
U
H
C
T
v
U
T
U
U
C
S
H
Tc
v
U
U
C
v
With
p
p
éq 4.2-10
Code_Aster
®
Version
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Titrate:
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O. BOITEAU
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This inequality clearly lets show through a possible formulation of the indicator in pure residue:
Definition 10
In the framework of the operator of transitory thermics linear of Code_Aster, the continuation
()
(
)
H
T
K
NR
N
N
K
0
theoretical local indicators can be written in the form
()
(
)
()
()
()
()
+
+
+
+
+
+
+
+
+
+
-
-
+
-
+
+
+
-
-
=
K
S
F
K
S
F
F
N
H
N
H
N
F
F
N
H
N
F
K
S
F
F
N
H
F
K
N
H
N
H
N
H
p
N
K
N
hu
N
U
H
H
N
U
G
H
N
U
H
U
div
T
U
U
C
S
H
K
2
3
,
0
1
1
,
1
,
0
1
,
1
,
0
1
,
,
0
1
,
1
1
1
^
^
2
1
^
:
éq 4.2-11
It is initialized by
()
(
)
()
()
()
-
-
+
-
+
+
+
=
K
S
F
K
S
F
F
H
H
F
F
H
F
K
S
F
F
H
F
K
H
K
U
H
N
U
H
H
N
U
G
H
N
U
H
U
div
S
H
K
2
3
,
0
0
0
0
0
,
0
0
0
,
0
0
,
0
0
0
0
^
^
2
1
^
:
éq 4.2-12
The continuation
()
(
)
NR
N
N
0
theoretical total indicators is defined as being
()
()
2
1
2
:
0
=
H
T
K
N
N
K
NR
N
éq 4.2-13
Code_Aster
®
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Titrate:
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O. BOITEAU
Key
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R4.10.03-A
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HI-23/02/014/A
Note:
·
While placing itself within the particular framework [éq 3.1-9] of the article [bib6] with an implicit scheme
(
=
1) one finds well the definition (24) pp432.
·
Whatever the initialization retained for thermal calculation, one starts the temporal continuation
of cartography of indicators of error as if one were in hover: no the term in
temporal finished difference,
N
+1=0 (in Code_Aster a transitory field of temperature
is initialized with index 0) and
=1.
·
It should be stressed that this indicator is composed of four terms: the term main,
named voluminal term of error, controlling the good checking of the equation of
heat, to which three secondary terms are added checking the good behavior of the conditions
limits (terms of jump, flow and exchange). In 2d-PLAN or 3D (resp. in 2D-AXI), if
the unit of the geometry is the meter, the unit of the first is it
W.m
(resp.
1
.
.
-
rad
m
W
) and that
other terms is it
2
1
.m
W
(resp.
1
2
1
.
.
-
rad
m
W
). Attention thus with the units taken in
count for the geometry when one is interested in the gross amount of the indicator and not in its
relative value!
·
While taking as a starting point the the increases developed by R. VERFURTH (cf [bib7] pp84-94) for
the Poisson's equation one could have taken as indicator the root of the sum of the squares
terms quoted above.
()
(
)
()
()
()
()
2
1
2
,
0
1
1
,
1
2
,
0
1
,
1
2
,
0
1
,
2
,
0
1
,
1
1
2
1
2
3
^
^
2
1
^
:
~
-
-
+
-
+
+
+
-
-
=
+
+
+
+
+
+
+
+
+
+
K
S
F
K
S
F
F
N
H
N
H
N
F
F
N
H
N
F
K
S
F
F
N
H
F
K
N
H
N
H
N
H
p
N
K
N
hu
N
U
H
H
N
U
G
H
N
U
H
U
div
T
U
U
C
S
H
K
éq 4.2-14
This definition leads to an increase of the total error which is more optimal than that which
will be released thereafter. But we preferred, to remain homogeneous with the writings
of B. METIVET [bib6] and with the estimator in linear mechanics already installation in
code, us to hold some with the version of definition 10.
Code_Aster
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While resting on [éq 4.2-10] and definition 10 one can then exhume the increase of the total error
following:
Property 11
Under the assumptions of properties 6, (H
6
) and by using definition 10, one has, at the total level,
the “weak” increase of the error (with
()
(
)
0
,
,
,
5
4
,
2
>
C
C
P
K
) via the history of
indicators
(
)
(
)
(
)
(
)
(
)
(
)
()
(
)
=
-
=
+
+
-
=
+
-
-
-
+
-
-
+
-
N
m
m
H
p
N
m
m
H
m
N
m
m
H
m
p
N
H
N
p
T
K
U
U
C
NR
N
U
U
T
U
U
C
U
U
C
0
2
2
2
,
0
0
0
1
0
2
,
0
1
,
1
1
0
2
,
0
2
,
0
3
4
0
1
4
éq 4.2-15
or more simply
(
)
(
)
(
)
()
(
)
=
-
=
+
+
+
-
-
+
-
N
m
m
H
p
N
m
m
H
m
N
H
N
p
T
K
U
U
C
NR
N
U
U
T
U
U
C
0
2
2
2
,
0
0
0
1
0
2
,
0
1
,
1
2
,
0
0
éq
4.2-16
Proof:
The estimates [éq 4.2-15] [éq 4.2-16] are obtained by reiterating the same process as for
properties 5, 6 and 7. One takes in [éq 4.2-10] the particular function test
1
,
1
:
+
+
-
=
N
H
N
U
U
v
éq
4.2-17
One évince the term of exchange by the usual argument
(
)
(
)
(
)
+
+
+
>
-
-
3
0
1
,
1
1
dx
U
U
U
U
H
N
H
N
N
H
éq
4.2-18
It is necessary to apply the easy way [éq 3.1-4] to the cross term
(
)
(
) (
)
dx
U
U
U
U
C
N
H
N
N
H
N
p
-
-
-
+
+
1
1
1
2
and on
the product utilizing the indicator. One has then to find the parameters
and
checking a system
type [éq 3.2-8]
()
1
2
1
2
1
2
2
2
,
2
=
-
-
=
-
P
éq
4.2-19
who admits solution only if the diagram is unconditionally stable (
2
1
). From where increase
[éq 4.2-15] [éq 4.2-16] while taking
()
(
)
2
5
2
4
,
2
2
, C
C
P
K
max
=
éq
4.2-20
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
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O. BOITEAU
Key
:
R4.10.03-A
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HI-23/02/014/A
The inequality [éq 4.2-16] more “coarse” results from the same sales leaflet as for corollary 7.
!
Note:
·
While placing itself within the particular framework [éq 3.1-9] of the article [bib6] with an implicit scheme
(
=
1) one finds well the inequality (25) pp432 (with
C
=max (1,
K
2
)).
·
By adopting the less restrictive approaches (H
4
) and (H
5
), one finds a version “strong” of
this property.
·
This property can be shown more quickly while noticing than the inequation [éq 4.2-10]
is similar to the equation of the problem semi-discretized in time [éq 3.2-3]: except for the inequality, in
changing
U
by
u-u
H
and while taking as term
()
v
B
N
,
the second member of [éq 4.2-10].
One can then directly apply the corollary 7 to him which is during estimate
sought!
·
Of [éq 4.2-15] [éq 4.2-16] it appears that, at one moment given, the error on the approximation
condition of Cauchy and the history of the total indicators intervenes on
total quality of the solution obtained. One will be able to thus minimize the error overall
of approximation due to the finite elements in the course of time while re-meshing “with good
knowledge “, via the continuation of indicators, the structure. Because, in practice, one realizes that it
refinement of the meshs makes it possible to decrease their error and thus cause to drop
temporal sum of the indicators. The total error will butt (and it is moral) against the value
floor of the error of approximation of the initial condition (which will have tendency it
also to drop of course!). The indicator “over-estimates” the space error overall.
·
With the other alternative of indicator [éq 4.2-14] one finds the same type of increase.
However the constant
K
2
change. It is is multiplied by the constant
C
6
checking
(cf [bib7] pp90)
2
,
1
6
2
,
1
2
,
1
+
v
C
v
v
H
F
H
K
S
F
T
K
éq
4.2-21
2
6
2
:
~
K
C
K
=
éq
4.2-22
According to the definitions [éq 2-1-8], [éq 2-1-10] with [éq 2-1-13] if the taking into account of the limiting conditions
of Dirichlet (generalized or not), via the ddls of Lagrange, is exact (what is the case in
Code_Aster)
()
NR
N
T
N
RF
RF
RF
RF
H
N
N
H
N
H
=
=
=
0
:
(H
8
)
the preceding property produces the following corollary then:
Corollary 11bis
Under the assumptions of property 11 by supposing (H
8
), one with the increase of the space error
total expressed in temperature
(
)
(
)
(
)
()
(
)
=
-
=
+
+
+
-
-
+
-
N
m
m
H
p
N
m
m
H
m
N
H
N
p
T
K
T
T
C
NR
N
T
T
T
T
T
C
0
2
2
2
,
0
0
0
1
0
2
,
0
1
,
1
2
,
0
0
éq
4.2-23
by using definition 10 of the indicator also expressed in temperature
ext.
HT
H
G
G
S
S
T
U
^
^
,
^
,
and
éq 4.2-24
Code_Aster
®
Version
6.0
Titrate:
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Date:
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O. BOITEAU
Key
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R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
4.3
Various types of possible indicators
By extrapolating a remark of [bib5] (pp194-195) it appears that increases of property 11
can exhume itself while taking as indicator
()
(
)
()
()
()
()
()
()
()
()
+
+
+
+
+
+
+
+
+
+
-
-
+
-
+
+
+
-
-
=
K
S
F
K
S
F
K
T
L
N
H
N
H
N
S
F
K
T
L
N
H
N
S
F
K
S
F
K
T
L
N
H
S
F
K
p
L
N
H
N
H
N
H
p
N
R
K
N
T
p
hu
N
U
H
H
N
U
G
H
N
U
H
U
div
T
U
U
C
S
H
K
2
3
1
1
,
1
1
,
1
1
,
1
,
1
1
1
,
^
^
2
1
^
:
éq
4.3-1
where constant the R and S are worth
(
)
(
)
()
()
T
Q
Q
Q
T
Q
S
p
Q
Q
Q
p
Q
R
Q
p
Q
p
T
1
2
1
2
1
:
,
2
1
:
,
3
5
6
2
1
,
1
-
-
-
-
-
=
-
-
+
=
=
=
>
3D
or
2D
éq
4.3-2
Note:
Just to introduce this generic shape of indicators, one passes from the notation hilbertienne
standards of spaces to the notation of Lebesgue
It is parameterized by the types of standards voluminal and surface which intervene for its obtaining.
Contrary to the indicator which we chose (
()
K
N 1
2
,
2
+
who corresponds to
p
=
T
=2), some
use the voluminal standard
L
1
(
p
=
T
=2) or on the contrary the infinite standard.
This last formulation, just like its simplified form of definition 10 (or [éq 4.2-14]),
constitute an indicator of error well a posteriori because its calculation requires only knowledge
geometrical materials, loadings, data, of
and of the approximate field solution
U
H
accused thermal problem. However the exact estimate of the indicator is not always
possible when one has intricate loadings. Two approaches are then
possible:
·
Either one approximates the integrals which re-enter in the composition of definition 10 by one
formulate quadrature.
·
Either one approximates the loadings by a linear combination of simpler functions
who will be able to allow an exact integration. Generally same architecture is used
that that which was installation for the finite elements modelizing the field of temperature.
Note:
·
In both cases the loadings are “prisoners of the selected vision finite elements”
to modelize the field solution.
Code_Aster
®
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These two strategies are equivalent and in Code_Aster it is the first which was
reserve: the voluminal integral is calculated by a formula of Gauss, those surface by one
formulate Newton-Dimensions.
Both introduce a skew into the calculation of the estimator who can be represented in
introducing the approximate versions of the loadings and source (into the initial problem in
T
and
in the problem transformed into
U
)
1
,
1
,
,
1
,
1
,
,
,
+
+
+
+
N
H
N
H
ext.
N
H
N
H
H
T
G
S
and
éq
4.3-3
1
,
1
,
1
,
1
,
^
,
^
,
^
+
+
+
+
N
H
N
H
N
H
N
H
H
H
G
S
and
éq
4.3-4
in spaces of voluminal approximation (for the source) and surface (for the loadings)
()
()
()
{
}
()
()
(
)
{
}
I
it
F
H
I
H
I
H
I
H
L
K
H
H
H
H
F
v
S
F
L
v
X
K
v
T
K
L
v
X
I
=
=
P
P
,
2
1
2
/
:
/
:
éq
4.3-5
In fact, one introduces two types of numerical errors during the calculation of the indicator: that inherent
with the formulas of quadrature (for polynomial loadings of a high nature) and that due to
voluminal term. Indeed, this last requires a double derivation which one carries out in three stages
because in Code_Aster one does not recommend the use of the derived seconds of the functions of forms.
Note:
They were recently introduced to treat the derivation of the rate of refund of energy
(cf [R7.02.01 § Annexe 1]).
On the one hand, one calculates (in the thermal operator) the heat flux at the points of gauss, then one
extrapolate the values with the nodes corresponding by smoothing local (cf [R3.06.03]
CALC_ELEM
with
OPTION
=
“FLUX_ELNO_TEMP”
) before calculating the divergence of the vectors flow at the points of Gauss.
With finite elements quadratic the intermediate operation is only approximate (one affects
like value with the median nodes the half the sum of their values to the extreme nodes). However
numerical tests (limited) showed that, even in
P
2
, this approach does not provide
results very different from those obtained by a direct calculation via good the derivative second.
Note:
· Indices
L
1
,
L
2
, L
3
of these polynomial spaces can be unspecified and different from
that of the approximate solution:
K
. However, to prevent that these terms do not become
prevalent (it is a question of rather estimating the error on the solution than that on
modeling of the loadings) one will tend to take
(
)
3
,
2
,
1
2
=
-
I
K
L
I
.
Definition 10 and the weak estimate 11 associated are rewritten then in the following form. This
new definition,
()
K
N
R
1
+
, is subscripted by one
R (
one takes again in that the usual notations of [bib6]
and [bib7]) (for “reality”) in order to notify well that it corresponds better to the values which are calculated
indeed in the code.
Code_Aster
®
Version
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Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
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O. BOITEAU
Key
:
R4.10.03-A
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:
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Definition 12
In the framework of the operator of transitory thermics linear of Code_Aster, the continuation
()
(
)
H
T
K
NR
N
N
R
K
0
real local indicators can be written in the form
()
(
)
()
()
()
()
+
+
+
+
+
+
+
+
+
+
-
-
+
-
+
+
+
-
-
=
K
S
F
K
S
F
F
N
H
H
N
H
N
H
F
F
N
H
N
H
F
K
S
F
F
N
H
F
K
N
H
N
H
N
H
p
N
H
K
N
R
U
H
N
U
H
H
N
U
G
H
N
U
H
U
div
T
U
U
C
S
H
K
2
3
,
0
1
1
,
1
,
,
0
1
,
1
,
,
0
1
,
,
0
1
,
1
1
,
1
^
^
2
1
^
:
éq 4.3-6
It is initialized by
()
(
)
()
()
()
-
-
+
-
+
+
+
=
K
S
F
K
S
F
F
H
H
H
H
F
F
H
H
F
K
S
F
F
H
F
K
H
H
K
R
U
H
N
U
H
H
N
U
G
H
N
U
H
U
div
S
H
K
2
3
,
0
0
0
0
0
,
0
0
0
,
0
0
,
0
0
0
0
^
^
2
1
^
:
éq 4.3-7
The continuation
()
(
)
NR
N
N
0
real total indicators is defined as being
()
()
2
1
2
:
0
=
H
T
K
N
R
N
R
K
NR
N
éq 4.3-8
Note:
· One can make the same remarks as for his alter “theoretical” ego. They is declined too
according to the formulations [éq 4.2-14]
()
K
N
R
~
and [éq 4.3-1], [éq 4.3-2]
()
K
N
T
p
R,
,
.
While being based on the results of property 11, definition 12 and the triangular inequality one can then
to exhume the increase of the following real total error (one began again that the simplified version):
Code_Aster
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Property 13
Under the assumptions of properties 6, (H
6
) and by using definition 12, one has, at the total level,
“weak” increase of the error (with
()
(
)
0
,
,
,
5
4
,
2
>
C
C
P
K
) via the history of
real indicators
(
)
(
)
(
)
()
(
)
()
(
)
{
}
()
(
)
()
()
-
=
+
+
+
+
+
+
-
=
+
+
+
-
=
+
+
+
-
-
+
-
+
-
+
+
+
-
-
+
-
H
H
T
K
N
m
K
S
F
F
m
H
m
H
H
m
m
H
F
K
S
F
F
m
m
H
F
T
K
N
m
K
m
m
H
K
m
R
R
H
p
N
m
m
H
m
N
H
N
p
hu
U
H
H
H
H
G
G
H
T
K
S
S
H
K
K
T
K
U
U
C
NR
N
U
U
T
U
U
C
1
0
2
,
0
1
1
1
1
,
2
,
0
1
1
,
2
1
0
2
,
0
1
1
,
2
2
1
2
0
2
2
,
0
0
0
1
0
2
,
0
1
,
1
2
,
0
3
2
^
^
^
^
^
^
0
éq 4.3-9
Under (H
8
), one with the same expression in temperature
(
)
(
)
(
)
()
(
)
()
(
)
{
}
()
(
)
(
)
(
)
(
)
()
-
=
+
+
+
+
-
=
+
+
+
-
=
+
+
-
-
-
+
-
+
-
+
+
+
-
-
+
-
H
H
T
K
N
m
K
S
F
F
m
H
ext.
m
H
H
ext.
H
F
K
S
F
F
m
m
H
F
T
K
N
m
K
m
m
H
K
m
R
R
H
p
N
m
m
H
m
N
H
N
p
T
T
H
T
T
H
H
G
G
H
T
K
S
S
H
K
K
T
K
T
T
C
NR
N
T
T
T
T
T
C
1
0
2
,
0
1
1
,
2
,
0
1
1
,
2
1
0
2
,
0
1
1
,
2
2
1
2
0
2
2
,
0
0
0
1
0
2
,
0
1
,
1
2
,
0
3
2
0
éq 4.3-10
by using definition 12 of the indicator also expressed in temperature
ext.
HT
H
G
G
S
S
T
U
^
^
,
^
,
and
éq 4.3-11
Note:
· As for the theoretical value there is a morals with the history because, when one will refine, the error
total will butt against the value floor due to the approximations of the initial condition, of
limiting conditions and of the source. One cannot obtain results of better quality
how data input of the problem!
4.4
Decrease of the local space error
Before exhuming the decrease of the space error, one will have to introduce some results
complementary:
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Lemma 14
It is shown that there are strictly positive constants
C
I
(
I
=6… 11) checking
{
}
()
{
}
()
F
F
F
v
H
C
v
v
C
v
v
P
H
C
F
v
v
H
C
v
v
C
v
v
C
K
v
K
F
K
F
F
F
F
K
F
L
L
L
K
K
K
K
K
K
K
K
K
K
K
L
L
L
K
-
-
-
,
0
1
11
,
0
,
0
2
1
10
,
0
,
0
2
1
9
,
,
,
sup
,
0
1
8
,
0
,
0
2
1
7
,
0
,
0
6
,
,
,
sup
3
2
1
3
2
1
P
P
éq 4.4-1
Proof:
One passes to the element of reference then one uses the fact that the standards are equivalent on
polynomial spaces considered, because they are of finished size (cf [bib5] pp196-98, [bib7] [§1]).
!
These preliminary results are crucial to determine a decrease of the site error by
the real indicator. But one will see that one will be able to obtain only one opposite room of [éq 4.3-9], [éq 4.3-10].
Code_Aster
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HI-23/02/014/A
Property 15
Under the assumptions of property 6, (H
6
) and while being based on definition 12 and the lemma 14, one
has, at the local level, the “weak” decrease of the error (with
(
)
0
11
6
,
3
>
= L
I
C
K
I
) via
the real indicator
()
(
)
()
(
)
1
0
^
^
^
^
^
^
3
2
,
0
1
1
1
,
1
2
1
,
0
1
,
1
2
1
,
0
1
,
1
,
0
1
,
1
,
0
1
1
3
1
-
+
-
-
+
-
+
-
+
-
+
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
NR
N
U
H
hu
H
H
H
G
G
H
S
S
H
U
U
T
U
U
U
U
C
H
K
K
K
K
K
K
K
N
H
H
N
H
N
H
N
F
N
H
N
F
N
H
N
K
N
H
N
N
H
N
N
H
N
p
K
N
R
éq 4.4-2
Under (H
8
), one with the same expression in temperature
()
(
)
(
)
(
)
1
0
3
2
,
0
1
,
1
,
,
1
,
1
1
,
1
2
1
,
0
1
,
1
2
1
,
0
1
,
1
,
0
1
,
1
,
0
1
1
3
1
-
-
-
-
+
-
+
-
+
-
+
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
NR
N
T
T
H
T
T
H
H
G
G
H
S
S
H
T
T
T
T
T
T
T
C
H
K
K
K
K
K
K
K
N
H
N
H
ext.
N
H
N
N
ext.
N
F
N
H
N
F
N
H
N
K
N
H
N
N
H
N
N
H
N
p
K
N
R
éq 4.4-3
by using definition 12 of the indicator also expressed in temperature
ext.
HT
H
G
G
S
S
T
U
^
^
,
^
,
and
éq 4.4-4
Proof:
This a little technical demonstration comprises three stages which will consist in raising
successively each term of the indicator [éq 4.3-6] (by using the inequalities of the property
14) and to gather increases obtained:
Firstly, one will replace in the equation [éq 4.2-6] the term in
H
v
v
-
by the product
W
K
making
to intervene the function “bubble” of
K
(
)
K
K
K
N
H
N
H
N
H
N
H
K
H
v
W
U
div
T
U
U
C
S
v
T
K
=
+
-
-
=
+
+
+
:
^
:
1
,
1
1
,
p
éq
4.4-5
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From where succession of increases, via [éq 4.4-1] and the inequality of Cauchy-Schwartz,
(
) (
)
(
)
(
)
(
)
(
)
-
+
-
+
-
-
-
-
+
-
+
-
-
-
-
-
-
+
-
-
-
+
+
+
+
+
+
+
+
+
+
-
+
+
+
+
+
+
+
+
K
N
H
N
K
N
H
N
K
N
H
N
N
H
N
K
K
K
K
K
N
H
N
K
N
H
N
K
K
N
H
N
N
H
N
K
N
H
N
K
N
H
N
K
N
H
N
N
H
N
K
K
K
K
K
S
S
U
U
T
U
U
U
U
C
C
C
v
W
S
S
U
U
H
T
U
U
U
U
C
C
W
S
S
W
U
U
has
W
T
U
U
U
U
C
dx
v
W
C
v
,
0
1
,
1
,
0
1
,
1
,
0
1
1
8
6
2
7
,
0
,
0
,
0
1
,
1
,
0
1
,
1
1
,
0
1
1
8
2
7
1
,
1
1
,
1
,
0
1
1
2
7
2
7
2
,
0
^
^
,
1
^
^
,
1
,
^
^
,
,
max
max
éq 4.4-6
Then, one reiterates the same process for the surface terms
W
F, I
()
1
,
1
,
1
,
1
,
,
:
:
K
F
K
F
N
H
F
H
v
P
W
N
U
v
S
K
S
F
=
=
+
éq
4.4-7
()
2
,
2
,
1
,
1
,
2
,
2
,
:
^
:
F
F
F
F
N
H
N
H
F
H
v
P
W
N
U
G
v
S
K
S
F
=
-
=
+
+
éq
4.4-8
()
(
)
3
,
3
,
1
1
,
1
,
3
,
3
,
:
^
:
F
F
F
F
N
H
H
N
H
N
H
F
H
v
P
W
U
H
N
U
H
v
S
K
S
F
=
-
-
=
+
+
+
éq
4.4-9
Code_Aster
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Titrate:
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R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Maybe, for example, for
I
=1 succession of increases, via [éq 4.4-1] and the inequality of
Cauchy-Schwartz,
(
)
(
)
(
)
(
)
(
)
(
)
(
)
+
-
+
-
+
-
-
-
+
-
+
-
+
-
-
-
-
-
-
-
+
-
-
-
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
+
+
F
F
F
F
F
F
F
F
F
K
F
N
H
N
F
N
H
N
F
N
H
N
N
H
N
F
F
F
F
K
N
H
N
N
H
N
F
N
H
N
N
H
N
F
K
F
N
H
N
F
N
H
N
F
N
H
N
N
H
N
F
F
F
F
F
v
H
S
S
H
U
U
H
T
U
U
U
U
H
C
C
C
v
W
v
S
S
U
U
H
T
U
U
U
U
C
C
W
v
W
S
S
W
U
U
has
W
T
U
U
U
U
C
D
v
W
C
v
,
0
2
1
,
0
1
,
1
2
1
,
0
1
,
1
2
1
,
0
1
1
2
1
11
9
2
10
,
0
1
,
,
0
1
,
,
0
,
0
1
,
1
,
0
1
,
1
1
,
0
1
1
11
2
10
,
0
1
,
,
0
1
,
1
,
1
1
,
1
,
1
,
0
1
,
1
1
2
10
1
,
1
,
2
10
2
,
0
1
,
^
^
,
1
^
^
,
1
,
,
^
^
,
,
max
max
éq 4.4-10
Finally it is enough to carry out the linear combination implying [éq 4.4-9] and [éq 4.4-10] to conclude
(bus
()
K
S
F
v
v
v
H
H
K
F
K
F
with
and
,
0
,
0
).
!
Note:
·
This local decrease of the error is also declined according to the formulations [éq 4.2-14]
()
K
N
R
~
and [éq 4.3-1], [éq 4.3-2]
()
K
N
T
p
R,
,
.
·
While placing itself within the particular framework [éq 3.1-9] of the article [bib6] with an implicit scheme
(
=
1) one finds well the inequality (29) pp432.
·
By adopting the less restrictive approaches (H
4
) and (H
5
), one finds a version “strong” of
this property.
·
This result provides only one opposite room of total increase [éq 4.3-9], [éq 4.3-10]
but within the framework of this type of indicator one will not be able to obtain better compromise.
These estimates are optimal within the meaning of [bib5]. They show the equivalence of
summon hilbertienne indicators with the space part of the total exact error.
constants of equivalence are independent of the parameters of discretizations in space and
in time, they depend only on the smallest angle of the triangulation.
·
This increase of the real indicator of error shows, which if one very locally refines
(around
K
) in order to decrease
()
K
N
R
, one is not ensured of a reduction in the error
in an immediate vicinity of the area concerned (in
K
). The indicator “under
consider locally “the error space and only a more macroscopic refinement realizes
theoretically a reduction in the error (cf property 13).
Code_Aster
®
Version
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Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
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Key
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R4.10.03-A
Page
:
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Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
4.5 Complements
The constant
K
3
just like its alter preceding ego,
K
2
, depends intrinsically on the type on
limiting conditions enriching the equation by initial heat as well as type by
temporal and space discretization. To try to free itself from this last stress,
SR. GAGO [bib10] proposes (on a problem 2D models) a dependence of the constant
K
2
in
function of the type of finite elements used. It is written
2
2
2
24
~
:
p
K
K
=
éq
4.5-1
where
p
is the degree of the polynomial of interpolation used (
p
=1 for
TRIA3
and
QUAD4
,
p
=2 for
TRIA6
and
QUAD8/9
). From where the idea, once the indicator of total error calculated, to multiply it by this
“corrective” constant
2
24
1
p
. This strategy was implicitly retained for the calculation of
the indicator of error in mechanics (option `
ERRE_ELGA_NORE'
of
CALC_ELEM
, cf [R4.10.02 §3]).
We however did not adopt it for thermics because this constant was not given
that empirically on the equation of Laplace 2D. We do not want to thus skew the values of
indicators.
It was question, until now, only of cards of indicators of space errors calculated with one
moment given of the transient of calculation. But, in fact, there are several channels to build one
indicators of error on a parabolic problem:
·
one can very well, first of all, semi-to discretize the strong formulation space some and to control sound
space error by indicator of error adapted a posteriori to the stationary case (in our
elliptic case). Then one applies a solvor, of pitch and command variables, treating the equations
ordinary differentials (for example [bib10] [bib11] [bib12]),
·
a second strategy consists in semi-discretizing in time then in space and determining
the indicator of one moment space error given (for example [bib4] [bib6] [bib13]) from
local residues of the semi-discretized form. One applies a linear solvor to the form
variational allowing to repeatedly build the solution at one moment given from
the solution of the previous moment,
·
another possibility consists in discretizing simultaneously in time and space on
suitable finite elements and to control their “space-time” errors in manner
coupled (for example [bib14] [bib15]).
This last scenario is most tempting from a theoretical point of view because it proposes a control
complete of the error and it allows to avoid unfortunate decouplings as for the possible ones
refinements/déraffinements controlled by a criterion with respect to the other (cf following paragraph). It is
however very heavy to set up in a large industrial code such as Code_Aster. It supposes
indeed, to be optimal, nothing less than one separate management pitch of time by finite elements. It
who from the point of view of architecture supporting the finite elements of the code is a true challenge!
One thus prefers the second scenario to him which has the large advantage of being able to be established directly
in a code D finite elements because this it is based above all on the resolution of the system completely
discretized. It is this type of indicators which was set up in N3S, TRIFOU and Code_Aster.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
43/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Within the framework of a true “space-time” discretization of the problem (scenario 3), one
obtains, in any rigor, a “space-time” indicator for each element of discretization
[
]
1
,
+
×
N
N
T
T
K
who is the balanced sum of three terms:
1) the residue of the calculated solution and the data discretized compared to the strong formulation
problem (
P
0
) evaluated on
[
]
1
,
+
×
N
N
T
T
K
,
2) the jump space through
[
]
1
,
+
×
N
N
T
T
K
of the operator traces associated (who connects naturally
formulations weak and strong via the formula of Green),
3) the temporal jump through
[
]
1
,
+
×
N
N
T
T
K
calculated solution.
The solution which was installation does not make it possible obviously to reveal explicitly it
term of temporal jump. It re-appears however implicitly, because of method of semi
particular temporal discretization, in all the terms in
definitions 10 and 12.
On the other hand, the fact of being interested mainly only in the space discretization and sound
possible refinement/déraffinement should not occult certain contigences with respect to
management of the pitch of time. Indeed, during transitory calculations comprising of abrupt
variations of loadings and/or sources in the course of time, for example of the shocks
thermics, fields of calculated temperatures
(
)
NR
N
T
N
<
0
can oscillate spatially and
temporally. Moreover, they can violate the “principle of the maximum” by taking values in
outside terminals imposed by the condition of Cauchy and the conditions limiting. To surmount it
numerical phenomenon parasitizes one shows, on a canonical case without condition of exchange
(cf [R3.06.07 §2]), that the pitch of time must remain between two terminals:
()
()
max
min
T
T
H
T
<
<
éq
4.5-2
In practice, it is difficult to have an order of magnitude of these terminals, one thus has evil, if one detects
oscillations, to modify the pitch of time in order to respect [éq 4.5-2]. In addition, this type
of operation is not always possible sometimes because it is necessary to take into account the abrupt ones precisely
variations of loadings (in particular when
T
is too small).
When
T
one is too tall can function as implicit Euler (
=1) what will cause of
to gum the upper limit.
On the other hand when it is too weak, two palliative strategies are offered to the user:
· diagonaliser the matrix of mass via the lumpés elements (cf [R3.06.07 §4] [§5])
proposed in the code (that requires installation to treat the elements
P
2
or
modeling
2d_AXI
),
· to decrease the size of the meshs (that increases complexities necessary calculation and memory).
It is from this point of view that the refinements/déraffinements practiced on the faith for our
indicator can have an angle of attack. The fact of refining will not pose any problem on the other hand in
déraffinant one can deteriorate very well the decrease of [éq 4.5-2]. It is necessary thus to be very circumspect if
one uses the option déraffinement software LOBSTER (encapsulated for Code_Aster in
MACR_ADAP_MAIL
option `
DERAFFINEMENT
'[U7.03.01]) on case test comprising a thermal shock.
We now will summarize the main contributions of the preceding theoretical chapters and theirs
holding and bordering with respect to the thermal calculation set up in Code_Aster.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
44/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
5
Summary of the theoretical study
That is to say (
P
0
) the problem in extreme cases interfered (inhomogenous Cauchy-Dirichlet-Neumann-Robin type
linear and with variable coefficients) solved by the operator
THER_LINEAIRE
(
)
] [
] [
] [
] [
=
×
=
+
×
=
×
=
×
=
-
)
(
)
0
,
(
,
0
,
0
,
0
,
0
div
)
(
0
3
2
1
0
X
X
T
T
HT
HT
N
T
G
N
T
F
T
S
T
T
T
C
P
ext.
p
éq
5-1
Taking into account the choices of modelings operated in Code_Aster (by
AFFE_MATERIAU
,
AFFE_CHAR_THER
…) one determines the Abstracted Variational Framework (CVA cf [§2]) minimal on which
one will be able to rest to show the existence and the unicity of a field of temperature solution
(cf [§2]). By recutting these pre-necessary theoretical a little “éthérés” with the practical stresses
users, one deduces some from the limitations as for the types of geometry and the licit loadings.
Then, while semi-discretizing in time and space by the usual methods of the code (of which
one ensures oneself of course of the cogency and owing to the fact that they preserve the existence and the unicity of the solution),
one studies the evolution of the properties of stability of the problem (cf [§3]). These results of
controllability are very useful for us to create the standards, the techniques and the inequalities which
intervene in the genesis of the indicator in residue. In these stages of discretization us
also briefly let us approach the influence of such or such theoretical assumption on the perimeter
functional of the operators of the code.
Before summarizing the main theoretical results concerning the indicator of error, we go
repréciser some notations:
·
a pitch of time is fixed
T
such as
T
either an entirety NR and that temporal discretization or
regular:
T
N
T
T
T
T
T
T
N
=
=
=
=
L
2
,
,
0
2
1
0
,
Note:
This assumption of regularity does not have really importance, it allows just
to simplify the writing of the semi-discretized problem. To modelize a transient
unspecified at the moment
T
N
, it is just enough to replace
T
by
T
N
=t
n+1
- T
N.
·
that is to say
the parameter of
- method semi-discretizing temporally (
P
0
),
·
are
N
T
and
N
H
T
fields of temperatures at the moment
(
)
NR
N
T
N
0
, exact solutions
initial problem (
P
0
), respectively semi-discretized in time and completely discretized in
time and in space.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
45/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Taking into account the modelings installation in the code, we can suppose that
temporal discretization of the loadings and the source is exact and that the taking into account,
via Lagranges, conditions limit (generalized or not) of Dirichlet is too. By
against, one of the approaches to modelize the numerical approximations carried out during calculations
integral of the indicator of error, consists in supposing inaccurate the space discretization of
loadings and of the source. Their approximate values are noted
1
,
1
,
,
1
,
1
,
,
,
+
+
+
+
N
H
N
H
ext.
N
H
N
H
H
T
G
S
and
éq
5-2
while posing
(
)
(
)
{
}
1
0
,
,
,
,
,
1
1
,
1
-
-
+
+
=
+
NR
N
H
G
T
S
T
T
N
T
N
ext.
N
and
with
X
X
éq
5-3
Note:
The establishment of this type of indicator (in mechanics as in thermics) is also sullied
of another type of numerical approximations related to calculations of the derived seconds of the term
voluminal (cf [§4.3]). Its effect can possibly feel when one is interested in
intrinsic value of the voluminal error for sources very chahutées on a mesh
coarse.
They exist two constants then
K
2
and
K
3
independent of the parameters of discretization in time
and spaces some, depending only on the smallest angle on the triangulation and the type of problem, which
allow to build:
·
An increase of the total space error (the history of the total real indicator
“over-estimates” the total space error)
(
)
(
)
(
)
()
(
)
()
(
)
{
}
()
(
)
(
)
(
)
(
)
()
-
=
+
+
+
+
-
=
+
+
+
-
=
+
+
-
-
-
+
-
+
-
+
+
+
-
-
+
-
H
H
T
K
N
m
K
S
F
F
m
H
ext.
m
H
H
ext.
H
F
K
S
F
F
m
m
H
F
T
K
N
m
K
m
m
H
K
m
R
R
H
p
N
m
m
H
m
N
H
N
p
T
T
H
T
T
H
H
G
G
H
T
K
S
S
H
K
K
T
K
T
T
C
NR
N
T
T
T
T
T
C
1
0
2
,
0
1
1
,
2
,
0
1
1
,
2
1
0
2
,
0
1
1
,
2
2
1
2
0
2
2
,
0
0
0
1
0
2
,
0
1
,
1
2
,
0
3
2
0
éq 5-4
·
A decrease of the local space error (it “underestimates” the local space error)
()
(
)
(
)
(
)
1
0
3
2
,
0
1
,
1
,
,
1
,
1
1
,
1
2
1
,
0
1
,
1
2
1
,
0
1
,
1
,
0
1
,
1
,
0
1
1
3
1
-
-
-
-
+
-
+
-
+
-
+
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
NR
N
T
T
H
T
T
H
H
G
G
H
S
S
H
T
T
T
T
T
T
T
C
H
K
K
K
K
K
K
K
N
H
N
H
ext.
N
H
N
N
ext.
N
F
N
H
N
F
N
H
N
K
N
H
N
N
H
N
N
H
N
p
K
N
R
éq 5-5
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
46/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
·
With the continuation
()
(
)
H
T
K
NR
N
N
R
K
0
local real indicators (by using the notations of
[§4.1])
()
()
()
()
()
(
)
()
(
)
(
)
()
()
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
+
-
+
+
+
-
-
=
+
+
+
=
K
S
F
K
S
F
F
N
H
N
H
ext.
F
F
N
H
N
H
F
K
S
F
F
N
H
F
K
N
H
N
H
N
H
p
N
H
K
N
éch
R
N
flow
R
N
jump
R
N
flight
R
N
R
N
T
T
T
H
H
N
T
G
H
N
T
H
T
div
T
T
T
C
S
H
K
K
K
K
K
2
3
,
0
1
,
1
,
,
0
1
,
1
,
,
0
1
,
,
0
1
,
1
1
,
1
,
1
,
1
,
1
,
1
2
1
:
:
éq 5-6
who is initialized by
()
(
)
()
(
)
(
)
()
()
-
-
+
-
+
+
+
=
K
S
F
K
S
F
F
H
H
ext.
F
F
H
H
F
K
S
F
F
H
F
K
H
H
K
R
N
T
T
T
H
H
N
T
G
H
N
T
H
T
div
S
H
K
2
3
,
0
0
0
,
0
0
0
,
0
0
,
0
0
0
0
2
1
:
éq 5-7
This local continuation makes it possible to build the continuation
()
(
)
NR
N
N
0
real indicators
total
()
()
2
1
2
:
0
=
H
T
K
N
R
N
R
K
NR
N
éq 5-8
Of [éq 5-4] (cf [§4.2]) it appears that, at one moment given, the error on the approximation of the condition of
Cauchy and the history of the total indicators intervenes on the total quality of the solution obtained.
One will be able to thus minimize overall the error of approximation due to the finite elements with the course
time while re-meshing “advisedly”, via the continuation of indicators, the structure. Because, in practice,
one realizes that the refinement of the meshs makes it possible to decrease their error and thus cause to drop
temporal sum of the indicators. The total error will butt (and it is moral) against the value floor
had with the approximations of the initial condition, the limiting conditions and the source (which will have
tendency it-also to drop of course!). One cannot obtain results of better quality
how data input of the problem!
The result [éq 5-5] (cf [§4.4]) provides only one opposite room of total increase [éq 5-4] (it
“must” would have been to reveal also an increase at the local level) but, within the framework of it
type of indicator, one will not be able to obtain better compromise. These estimates are optimal with
feel [bib5]. They illustrate the equivalence of the sum hilbertienne indicators with the part
space of the total exact error. The constants of equivalence are independent of the parameters
discretizations in space and time, they depend only on the smallest angle of the triangulation
and of the type of dealt with problem.
According to this increase of the indicator [éq 5-6], if one refines very locally (around the element
K
)
in order to decrease
()
K
N
R
, one is not ensured of a reduction in the error in an immediate vicinity
area concerned (in
K
). The indicator “locally underestimates” the error space and only
a more macroscopic refinement carries out a reduction in the error theoretically.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
47/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Only in pure residue, all a “zoology” of indicators of space error are permissible
(cf [§4.3]), we retained a type similar of it to that already set up for the mechanics of
Code_Aster. Being based on the solutions and the discrete loadings of the moment running and the moment
precedent (except with the first pitch of time), its theoretical limitations are thus, at best, those
inherent in the resolution of the problem in temperature: no areas comprising of points of
graining or of point, not of fissure, problem to the multi-material interfaces,
- diagram
unconditionally stable, regular family of triangulation, polygonal mesh discretized by
isoparametric finite elements, oscillations and violation of the principle of the maximum (cf [§4.5]). Of course,
in practice, one very often passes in addition to, and this without ties up, this perimeter of “theoretical” use.
But it is necessary well to keep in mind, that as a “simple postprocessing” of (
P
0
),
the indicator cannot unfortunately provide more reliable diagnosis in the areas where
resolution of the initial problem stumbles (close to fissure, shock…). Its denomination prudently
reserved of indicator (instead of the usual terminology of estimator) is in these particular cases
more that never of setting! But if, in these extreme cases, its gross amount perhaps prone to monetary bond,
its utility as an effective and convenient provider of cards of error for one
mending of meshes or a refinement/déraffinement remains completely justified.
In the same vein, even if the formulation [éq 5-6] were established only in the transitory linear case,
isotropic or not, defines by (
P
0
), one could also stretch his perimeter of use to the non-linear one
(operator
THER_NON_LINE)
, to conditions limit different (
ECHANGE_PAROI
for example) or with
other types of finite elements (lumpés isoparametric elements, elements of structure…) (cf.
[§2.1]). For more information on the “data-processing” perimeter corresponding to its establishment
effective in the code, one can refer to [§6.2] or the user's documentation of
CALC_ELEM
[U4.81.01].
It was question, until now, only of cards of indicators of space errors calculated with one
moment given of the transient of calculation. But, in fact, there are several channels to build one
indicators of error on a parabolic problem (cf [§4.5]). That which we retained
does not allow a complete control of the error and it always requires a certain vigilance when one
draft of the problems of the type shocks (the same one as for the problem post-treaty!). It does not make
to appear that implicitly the term of jump temporal in all the terms in
of [éq 5-6].
To finish, it should be stressed that this indicator is thus composed of four terms:
·
the main term, called term of voluminal error, controlling the good checking of
the equation of heat,
·
to which three secondary terms are added checking the good behavior of the space jumps and
limiting conditions: terms of flow and exchange.
In
2D-PLAN
or in
3D
(resp. in
2D-AXI
), if the unit of the geometry is the meter, the unit of first is
W.m
(resp.
1
.
.
-
rad
m
W
) and that of the other terms is it
2
1
.m
W
(resp.
1
2
1
.
.
-
rad
m
W
). Caution
thus with the units taken into account for the geometry when one is interested in the gross amount of
the indicator and not with its relative value!
We now will approach, after the practical difficulties of implementation in the code,
environment necessary and its perimeter of use. One will conclude for a drawn example of use
of a case official test.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
48/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
6
Implementation in Code_Aster
6.1 Difficulties
particular
To calculate this type of indicator it is necessary to compose with the vision “elementary calculation + assembly”
generally deployed in all the codes finite elements. However the estimate, at the local level, of
(K)
requires, not only the knowledge of its local fields, but also that of its meshs
neighbors. One thus needs to carry out a “total calculation” on a scale
K
, in calculation
room! A copied strategy on what had been set up for the estimator in mechanics
consist in transmitting this type of information in the components of wide cards which they
will be transmitted in argument of input of
CALCULATION
. It is this type of contingency which explains
the heterogeneity of processing at the time of the overloads of loadings between the thermal solveurs and it
calculation of our indicator (cf [§6.2]).
Another type of difficulty, numerical this time, relates to the calculation of the voluminal term.
Indeed, it requires a double derivation which one carries out in three stages, because in Code_Aster one
do not recommend the use of the derived seconds of the functions of forms.
Note:
They were recently introduced to treat the derivation of the rate of refund of energy
(cf [R7.02.01 § Annexe 1]).
On the one hand, one calculates (in the thermal operator) the vector flow at the points of gauss, then one
extrapolate the corresponding values with the nodes by local smoothing (cf [R3.06.03]
CALC_ELEM
with
OPTION
=
“FLUX_ELNO_TEMP”
and [§6.2]) in order to calculate its divergence at the points of Gauss. With
quadratic finite elements the intermediate operation is only approximate (one affects like
value with the median nodes the half the sum of their values to the extreme nodes). However
numerical tests (limited) showed that this approach does not provide results very different from
those obtained by a direct calculation via good the derivative second.
Lastly, it was necessary to determine various geometrical characteristics (diameters, normals, jacobiens…),
interfacings of the elements in opposite and to reach the data which they subtend in all
cases of figures envisaged by the code (started from mesh symmetrized and/or heterogeneous, loading
function or reality, non-linear material, all isoparametric elements 2D/3D and all them
thermal loadings).
Beyond these fastidious developments, a large effort of validation
“géométrico-data processing” was deployed to try to track possible bugs in this
entrelac of small formulas. These hard tests on small cases model tests (
TPLL01A/H
for
2d_PLAN
/
3D
and
TPNA01A
for
2d_AXI
) appeared profitable (including for the indicator in
mechanics and lumpés elements!) and essential. Because one does not lay out, to my knowledge,
theoretical values allowing to validate in certain situations these indicators: “nothing
resemble more one value of indicator… than another value of indicator! “. In the absence of other
thing and, although in a process of validation that is not the panacea, it are thus necessary
to try to release a maximum of confidence in all these components.
6.2 Environment
necessary/parameter setting
The calculation of this indicator is carried out, via the option `
ERTH_ELEM_TEMP
'of the operator of post-
processing
CALC_ELEM,
on one
EVOL_THER
(provides to the key word
RESULT)
coming from a calculation
former thermics (linear or not, transient or stationary, isotropic or orthotropic, via
THER_LINEAIRE
or
THER_NON_LINE,
cf more precise perimeter [§6.4]).
As one already underlined, it requires as a preliminary the recourse to the option `
FLUX_ELNO_TEMP
'of
CALC_ELEM
who determines the values of the vector heat flux to the nodes (cf example of use
[§6.5]).
This indicator consists of fifteen components by elements and for a given moment. In order to
to be able post-to treat them via
POST_RELEVE
or GIBI one needs to extrapolate these fields by element in
fields with the nodes by element. The addition of the option `
ERTH_ELNO_ELEM
'(after the call to
`
ERTH_ELEM_TEMP'
) allows to carry out this purely data-processing transformation. For one moment
and a given finite element, it does nothing but duplicate the fifteen components of the indicator on each
nodes of the element.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
49/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
For carrying out the integral postprocessing of desired thermal calculation well, it is necessary:
·
To carry out on all the geometry, TOUT=' OUI' (default value, if not calculation stops
in
ERREUR_FATALE
). This provisional choice was led by data-processing contingencies
and functional calculuses, bus thus all the finite elements are seen affecting a homogeneous indicator
calculated with the same number of terms (if not quid of the concept of term of jump and of
term of CL at the edge of the field considered
?). In addition the tool of
refinement/déraffinement of the code (software
LOBSTER
encapsulated in
MACR_ADAP_MAIL
),
natural outlet of our cartographies of error, does not make it possible to treat only parts of
mesh.
Note:
That poses problems of propagation of subdivisions to preserve conformity
triangulation. In fact, to divert this type of contingency, it would be necessary, that is to say to define
a buffer zone making the junction enters the “dead” area of the mesh and the area
“activates” to treat, that is to say in a way more optimal but also much more difficult
from a point of view structures, to reduce it to a layer of joined elements.
·
To provide the same temporal parameter setting: value of
(default value equalizes to 0.57)
provided to the key word
PARM_THETA
; if necessary if transitory problem is dealt with, it is necessary
to inform the usual fields
ALL/NUME/LIST_ORDRE
with licit values opposite
thermal calculation. The calculation of the history of the indicator can then be carried out from
any moment of a transient, knowing that with the first increment one carries out calculation
as in hover (
=1,
N
+1=0 and not of term of finished difference cf [éq 5-7]).
Moreover, in hover, if the user provides a value of
different from 1, one imposes to him
this last value after having informed some.
In a related way, one detects the request for supply of cards of errors between
noncontiguous sequence numbers (there is one
ALARM
) or the data of one
EVOL_THER
not comprising a field of temperature and vector flow to the nodes (calculation stops in
ERREUR_FATALE
). The value of
and numbers it sequence number taken into account are
layouts in the file message [§6.3]. The sequence number and the corresponding moment
accompany also each occurrence by indicator of error in the file result ([§6.3]).
·
To use the same loadings and by complying with the rules of particular overloads
with the options of error analyzes of this operator. Thus, in the thermal solveurs (and
mechanics) one incorporates the limiting conditions of the same type, whereas in calculations
errors of
CALC_ELEM
(and thus also with our indicator) one cannot take into account,
for a limiting type of condition given, that the last provided to the key word
EXCIT
. The command of
these loadings revêt thus a crucial importance!
Note:
This restriction finds its base in the first remark of the paragraph
precedent. For making well it would be necessary, that is to say concaténer on the elements of skin
concerned all the limiting conditions, is to provide to elementary calculations cards
variable sizes containing all the loadings exhaustively. The first
solution seems by far most optimal but also hardest to put in
work. It would then be necessary also to make the same thing for the indicator in residue of
mechanics (
OPTION=' ERRE_ELGA_NORE'
).
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
50/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
However, in the event of conflict between loadings of the same type, one often can and
easily to find a solution palliative via
AFFE_CHAR_THER
adequate. The user is
informed presence of several occurrence of the same type of loading by one
message of
ALARM
and lists it loadings actually taken into account is traced
in the file message ([§6.3]).
The code stops on the other hand in
ERREUR_FATALE
if the provided loadings pose some
problems (interpolation of loadings function, access to the components, presence of
CHAMPGD
coefficient of exchange and absence of
CHAMPGD
outside air temperature or
vice versa….),
·
Within the same general framework: value of the model (parameter
MODEL
), of necessary materials
(
CHAM_MATER
), of the structure
EVOL_THER
data (
RESULT
) and result (assignment of
CALC_ELEM
with possibly one “
reuse
” réentrant). They are traced in the file
message ([§6.3]).
If the user does not respect this necessary homogeneity of parameter setting (to the rules of
overload near) between the thermal solvor and the tool for postprocessing, it is exposed to
skewed results even completely false (without inevitably a message of
ALARM
or one
ERREUR_FATALE
stops, one cannot all control and/or prohibit!). There remains only judge then
relevance of its results.
Let us recapitulate all this parameter setting of the operator
CALC_ELEM
impacting the calculation directly of
the indicator of space error in thermics.
Key word factor
Key word
Default value
Value obligeatoire (O)
or advised (C)
MODEL
Idem thermal calculation
(O)
CHAM_MATER
Idem thermal calculation
(O)
ALL “YES”
“YES”
(O)
ALL/NUME/LIST_ORDRE “YES”
“YES”
(C)
PARM_THETA
0.57
Idem thermal calculation
(O)
RESULT
EVOL_THER
calculation
thermics (O)
reuse
EVOL_THER
calculation
thermics (C)
EXCIT CHARGES
Idem thermal calculation +
regulate of overload (O)
OPTION
`
FLUX_ELNO_TEMP
'
“ERTH_ELEM_TEMP”
“ERTH_ELNO_ELEM”
INFORMATION
1 1
(C)
Table 6.2-1: Summary of the parameter setting of
CALC_ELEM
impacting the calculation of the indicator
Note:
·
In transient, it (strongly) is advised to calculate the history of the indicator on
moments of calculations contiguous. If not, the postprocessing of the temporal semi-discretization
will be distorted, and according to the devoted formula… the user will become only judge of
relevance of its results.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
51/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
6.3
Presentation/analysis of the results of the error analysis
The option `
ERTH_ELEM_TEMP
'provides in fact, not one, but fifteen components by elements
finished
K
and by pitch of time
T
n+1
. Indeed, for each one of the four terms of [éq 5-6], _ the term
the voluminal main thing and the three surface secondary terms _, one calculates not only the error
absolute, but also a term of standardization (the theoretical value of the discretized loadings
that one would have had to find) and the associated relative error. By summoning these three families of four
contributions, one establishes also the total absolute error, the total term of standardization and the relative error
total. What makes the account well!
The fact of dissociating the contributions of each component of this indicator allows
to compare their relative importances and to target strategies of refinement/déraffinement
on a certain type of error. Even if the voluminal term (representing the good checking
equation of heat) and the term of jump (related to modeling finite elements) remain them
dominating terms, it can prove to be useful to measure the errors due to certain type of
loading in order to refine their modeling or to re-mesh the accused frontier zones.
Moreover this type of strategy can be easily diverted of its goal first in order to make
refinement/déraffinement by area: it is enough to impose, only in this area, a type of
fictitious limiting condition (with very bad value in order to cause a large error).
Mode of calculation of these components and the name of their component “of greeting” in the field
symbolic system `
ERTH_ELEM_TEMP
'of
the EVOL_THER
are recapitulated in the table below (in
being based on the nomenclature of [éq 5-6]).
Absolute error
Relative error
(in %)
Term of standardization
Term
voluminal
()
K
N
flight
R
1
,
+
TERMVO
()
()
.
100
1
,
1
,
×
+
+
K
NR
K
N
flight
R
N
flight
R
TERMV2
()
K
N
H
K
N
flight
R
S
H
K
NR
,
0
1
,
1
,
:
+
+
=
TERMV1
Term of
jump
()
K
N
jump
R
1
,
+
TERMSA
()
()
.
100
1
,
1
,
×
+
+
K
NR
K
N
jump
R
N
jump
R
TERMS2
()
F
N
H
F
N
jump
R
N
T
H
K
NR
,
0
1
,
2
1
1
,
2
:
=
+
+
TERMS1
Term of flow
()
K
N
flow
R
1
,
+
TERMFL
()
()
.
100
1
,
1
,
×
+
+
K
NR
K
N
flow
R
N
flow
R
TERMF2
()
F
N
H
F
N
flow
R
G
H
K
NR
,
0
1
,
2
1
1
,
:
+
+
=
TERMF1
Term
of exchange
()
K
N
éch
R
1
,
+
TERMEC
()
()
.
100
1
,
1
,
×
+
+
K
NR
K
N
éch
R
N
éch
R
TERME2
()
(
)
(
)
F
N
H
ext.
F
N
éch
R
T
T
H
H
K
NR
,
0
1
,
2
1
1
,
:
+
+
-
=
TERME1
Total
()
()
+
+
=
I
N
I
R
N
R
K
K
1
,
1
:
ERTABS
()
()
.
100
1
1
×
+
+
K
NR
K
N
R
N
R
ERTREL
()
()
+
+
=
I
N
I
R
N
R
K
NR
K
NR
1
,
1
:
TERMNO
Table 6.3-1: Components of the indicator of error.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
52/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
For the absolute error and the term of standardization, in
2D-PLAN
or in
3D
(resp. in
2D-AXI
), if the unit
geometry is the meter, the unit of the first term is it
W.m
(resp.
1
.
.
-
rad
m
W
) and that of
other terms is it
2
1
.m
W
(resp.
1
2
1
.
.
-
rad
m
W
).
Attention thus with the units taken into account for the geometry when one is interested in
gross amount of the indicator and not with its relative value!
This information is accessible in three forms:
·
For each moment of the transient, these fifteen values are summoned on all the mesh (one
fact the same thing as in the table [Table 6.3-1] while replacing
K
by
) and traced
in a table of the file result (.
RESU
).
**********************************************
THERMICS: ESTIMATOR Of ERROR IN RESIDUE
**********************************************
IMPRESSION OF THE TOTAL STANDARDS:
SD EVOL_THER RESU_1
SEQUENCE NUMBER 3
MOMENT 5.0000E+00
ABSOLUTE ERROR/RELATIVE/STANDARDIZATION
TOTAL 0.5863E-05 0.2005E- 04% 0.2923E+02
VOLUMINAL TERM 0.3539E-05 0.0000E+ 00% 0.0000E+00
TERM JUMP 0.2217E-05 0.1006E- 04% 0.2205E+02
TERM FLOW 0.4384E-06 0.3886E- 05% 0.1128E+02
TERM EXCHANGE 0.4591E-06 0.5755E- 05% 0.7977E+01
**********************************************
Example 6.3-1: Layout of the option `
ERTH_ELEM_TEMP
'in the file result
·
It is stored by means of computer in the fifteen components of the field symbolic system
`
ERTH_ELEM_TEMP
'of
SD_RESULTAT
thermics. The variables of access of this field are
, for each mesh (in our example
M1
), the sequence number (
NUME_ORDRE
) and the moment
(
INST
). With the option `
ERTH_ELNO_ELEM
'one with the same thing for each node of
the element considered.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
53/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
FIELD BY ELEMENT AT THE POINTS OF GAUSS OF REFERENCE SYMBOL
ERTH_ELEM_TEMP
SEQUENCE NUMBER: 3 INST: 5.00000E+00
M1
ERTABS ERTREL TERMNO
TERMVO TERMV2 TERMV1
TERMSA TERMS2 TERMS1
TERMFL TERMF2 TERMF1
TERMEC TERME2 TERME1
1 0.5863E-05 0.2005E-04 0.2923E+02
0.3539E-05 0.0000E+00 0.0000E+00
0.2217E-05 0.1006E-04 0.2205E+02
0.4384E-06 0.3886E-05 0.1128E+02
0.4591E-06 0.5755E-05 0.7977E+01
........
FIELD BY ELEMENT AT THE POINTS OF GAUSS OF REFERENCE SYMBOL
ERTH_ELNO_ELEM
SEQUENCE NUMBER: 3 INST: 5.00000E+00
M1
ERTABS ERTREL TERMNO
TERMVO TERMV2 TERMV1
TERMSA TERMS2 TERMS1
TERMFL TERMF2 TERMF1
TERMEC TERME2 TERME1
N1 0.5863E-05 0.2005E-04 0.2923E+02
0.3539E-05 0.0000E+00 0.0000E+00
0.2217E-05 0.1006E-04 0.2205E+02
0.4384E-06 0.3886E-05 0.1128E+02
0.4591E-06 0.5755E-05 0.7977E+01
N3 0.5863E-05 0.2005E-04 0.2923E+02
........
Example 6.3-2: Layouts, via
IMPR_RESU
, of the components of the field symbolic system
`
ERTH_ELEM_TEMP
“/”
ERTH_ELNO_ELEM'
in the file result
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
54/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
·
One can also trace the values of each one of these components in the file message
(.
MESS
) by initializing the key word
INFORMATION
to 2. However this functionality rather reserved for
developers (for maintenance or of the pointed diagnoses) also revealed
complementary impressions (documented but too exhaustive) on the elements
constituting the indicator and the characteristics of the finite elements and their vicinities.
TE0003 **********
NOMTE/L 2D THPLTR3/T
RHOCP 2.0000000000000
ORIENTATION NETS 1.0000000000000
…
---> TERMVO/TERMV1 1.2499997764824 1.2499997764826
>>>
CURRENT MESH <<< 3 TRIA3
DIAMETER 3.5355335898314D-02
EDGES OF THE TYPE SEG2
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
NUMBER Of EDGE/HF 1 2.4999997764826D-02
A NUMBER OF SUMMITS 2
INTERFACING 1 2
XN 0.59999992847442 0.59999992847442
YN - 0.80000005364418 - 0.80000005364418
JAC 1.2499998882413D-02 1.2499998882413D-02
<<< CLOSE MESH 2 QUAD4
IGREL/IEL 1 2
LOCAL/TOTAL INOV 2 5
….
*********************************************
TOTAL ON MESH 2
ABSOLUTE ERROR/RELATIVE/MAGNITUDE
TOTAL 0.5900D-03 0.1079D- 03% 0.5466D-03
VOLUMINAL TERM 0.1768D-01 0.1000D- 03% 0.1768D-01
TERM JUMP 0.5882D-03 0.1080D- 03% 0.5448D-03
TERM FLOW 0.0000D+00 0.0000D+ 00% 0.0000D+00
TERM EXCHANGE 0.0000D+00 0.0000D+ 00% 0.0000D+00
*********************************************
Example 6.3-3: Layout, via
INFO=2
, in the file message
Note:
·
When the term of standardization is null (a certain type of loading or source is null,
as it is the case in the examples [Example 6.3-1] and [Example 6.3-2] above with
voluminal term), one does not calculate the term of relative error associated. There remains initialized to zero.
·
Moreover, to calculate indeed the absolute error relating to a null limiting condition (one
flow or a condition of exchange) it should be imposed as a function via
AFFE_CHAR_THER_F
adhoc. And this for simple data-processing contingencies, which make
that with a constant loading, one cannot make the distinction between:
- null limiting condition
“
the user imposes zero on this portion of border and he wants
to test the associated absolute error,
- null limiting condition
“
there are no limiting conditions on this edges,
·
Tests of not-regression “numérico-data processing” showed that the manner of
to modelize the loadings and the source, as constants or functions, could
to especially influence notably the values of very small terms of error (in relative error well
sure) and to worry the user unnecessarily. This phenomenon is explained by differences of
codings of the discretized loadings [éq 5-2]. This type of behavior is found too
as soon as one changes linear solvor, preconditionnor, method of renumerotation,
of platform…
·
In hover, when one uses a nonnull source with linear finite elements, it
term main is very badly estimated since it requires a double derivation of the field of
temperature. One
ALARM
thus warns the user and the enjoint to pass into quadratic.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
55/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
6.4 Perimeter
of use
This indicator was developed, for the moment, only on the isoparametric elements
(
TRIA3/6
,
QUAD4/8/9
,
TETRA4/10
,
PENTA6/13/15
and
HEXA8/20/27
) and for modelings
PLAN
, PLAN_DIAG,
AXIS, AXIS_DIAG, 3D
and
3d_DIAG
. It thus does not calculate the contributions of
elements of structure of the hull type (
COQUE_PLAN, COQUE_AXIS, HULL
), of the pyramids
(PYRAM5 and PYRAM13) and of the modeling of Fourier (
AXIS_FOURIER
). It does not allow either
to calculate the terms of jumps of these elements with the authorized elements. However, if a mesh
comprise licit and illicit elements, calculation does not stop and, via
OPTION
2 in
the suitable catalogs of elements, one warns the user of not taken into of the aforesaid account
elements.
Indeed to carry out this postprocessing, it is necessary as a preliminary to call, explicitly, the option
`
FLUX_ELNO_TEMP
'(calculation of the vector heat flux to the nodes) and, implicitly,
`
INIT_MAIL_VOIS
'(determination of the characteristics of the vicinity
K
of an element
K
). One is
thus tributary of their respective perimeters of use.
It is also necessary to keep in mind some more minor rules but which can cover one
very particular importance for very precise studies:
1) The calculation of the indicator treats only the elements of the mesh pertaining to the model
indicated by the key word
MODEL
control
CALC_ELEM
. One can thus work with
mesh (not cleaned) comprising “meshs of outline” to which one allots one
different model.
2) In
one
mesh in dimension
Q
, one calculates the terms of jump and loading, only
on elements of skin of dimension
Q
- 1. Therefore, one treats the relations of
/QUAD SORTED
with
SEG
and relations
TETRA/PENTA/HEXA
with
FACE
. For example, in the event of
presence of segments in a three-dimensional mesh, the option will not stop but it
will not take into account their (possible) contributions.
3) The option `
ERTH_ELEM_TEMP
'and its preliminary options do not know them
PYRAM
.
Their contributions will be ignored. This gap comes from their introduction into
Code_Aster more recent than those of the already quoted preliminary options.
Note:
In any event these elements are minority in a mesh 3D and are not
generated that by the voluminal free maillor of GIBI, which creates some locally for
to supplement portions of mesh hexahedral.
4) In
2D,
one should not accidentally intercalate a segment between two triangles or
quadrangles, if not the term of jump of these elements will not be calculated and one will enquérira oneself with
wrong of the existence of a possible limiting condition. Calculation will not stop but with
this interface, the value of the indicator will be incomplete. However, for needs
private individuals (charging density internal and localized in a structure, fissures…), one
can of course allow this kind of situation. In 3D, the problem arises of course too
when one intercalates quadrangles or triangles between two
FACE
contiguous.
5) the same type of imbroglio occurs when two points of the mesh are superimposed
geometrically. There still, calculation should not stop, but the value of
the indicator will be incomplete on the level of this area of recovery,
6) If one works with a mesh which results from operations of symmetrization, it is necessary to test
not to be in the two preceding cases of figures. Moreover, on both sides of
the axis of symmetry, the close meshs do not have inevitably (with in particular the maillor
GIBI) of the orientations which meet the standard of Code_Aster (they should be
reversed). The calculation of the indicator, for which this information is crucial (to define them
external normals with each mesh and interfacings in opposite), detects the problem in
calculating the jacobien each mesh. In 2D, an algorithm of substitution allows
to circumvent the problem and to rebuild the tables of interfacing “nodes of the element
running/nodes of its neighbors “. In 3D, the problem is much more difficult and private individual with
each element, the code thus stops in
ERREUR_FATALE
in the event of problem.
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
56/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
7) If one wants to refine or déraffiner his mesh with
MACR_ADAP_MAIL
[U7.03.01], it
mesh should comprise only triangles or tetrahedrons. Concerning the loadings
surface or voluminal, the “good practice” consists in using only groups of
meshs. If groups of nodes are used, one must expect distorted calculations, because
after some refinements, other points will have probably fit
geometrically in the area concerned with
GROUP_NO
without seeing itself affecting one
unspecified loading (one cannot modify the composition of one
GROUP_NO
in the course of
session!).
For specific loadings or points of statement (on which go, for example,
to rest POST_RELEVE_T) it
GROUP_NO
is licit. On the other hand, it is not advised
to use meshs directly (
MA
) or of the nodes (
NO
) (apart from group), because in it
case, with the liking of the renumérotations,
LOBSTER
probably will lose their trace. It cannot
to preserve the memory of the meshs or the nodes that through a name of
GROUP_MA
or of
GROUP_NO
. Thanks to this mechanism, it can adopt a Lagrangian vision of becoming of
these meshs or of these points!
The calculation of the indicator takes place indifferently on one
EVOL_THER
coming from
THER_LINEAIRE
or of
THER_NON_LINE
, stationary or transitory, isotropic or orthotropic, and,
on a motionless structure with a grid by elements answering the preceding criteria.
Into non-linear one takes into account non-linearities of materials and the amendment of the problem in
enthalpy. However one does not treat the possible contributions of non-linear loadings
(
FLUX_NL
and
RADIATION
). The user is informed by it by one
ALARM
, just like it is informed
not taken into account of a limiting condition of type
ECHANGE_PAROI
. Indeed, into linear one
recognizes, for the moment, that the contributions of the loadings
SOURCE
,
FLUX_REP
and
EXCHANGE
. For
the taking into account of these loadings, of the particular rules of overload are applied
(cf [§6.2]).
6.5 Example
of use
To familiarize itself with the use of this indicator in thermics and its possible coupling with
the LOBSTER encapsulation
®
(for more information, one will be able to consult the site
http://www.code-aster.com/outils/homard
) via
MACR_ADAP_MAIL
[U7.03.01] one can
to take as a starting point this expurgée version of the case test
TPLL01J
[V4.02.01]. It is however only about one
case data-processing test of not-regression putting forward the use of certain functionalities of
new process control language PYTHON (loops, test…).
MATERI=DEFI_MATERIAU (THER=_F (LAMBDA = 0.75, RHO_CP = 2.0))
M= [Nun] * 5
MAIL=LIRE_MAILLAGE ()
# Initial Mesh
M [1] =DEFI_GROUP (reuse=MAIL, MAILLAGE=MAIL,
CREA_GROUP_NO=_F (TOUT_GROUP_MA = “YES”))
# Vecteurs results has each iteration
MODE= [Nun] * 4
MATE= [Nun] * 4
CHA1= [Nun] * 4
RESU= [Nun] * 4
# Loops indicating calculation/mending of meshes; PYTHON makes 3 iterations
for K in arranges (1,4):
# Assignment of materials/model/loading
SUBDUE [K] =AFFE_MATERIAU (MAILLAGE=M [K],
AFFE=_F (ALL = “YES”, MATER = MATERI))
MODE [K] =AFFE_MODELE (MAILLAGE=M [K],
AFFE=_F (ALL = “YES”, MODELING = “3D”,
PHENOMENON = “THERMAL”))
CHA1 [K] =AFFE_CHAR_THER (MODELE=MODE [K],
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
57/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
TEMP_IMPO=_F (GROUP_NO = “F1INF”, TEMP = 100.),
FLUX_REP=_F (GROUP_MA = “FLOW”, FLUN = - 1200.),
ECHANGE=_F (GROUP_MA = “ECHA”, COEF_H = 30.,
TEMP_EXT = 140.))
# Calculation thermal
RESU [K] =THER_LINEAIRE (MODELE=MODE [K],
CHAM_MATER=MATE [K],
EXCIT=_F (LOAD = CHA1 [K]))
# Calculation of the indicator of error
RESU [K] =CALC_ELEM (reuse=RESU [K], MODELE=MODE [K],
TOUT=' OUI',
CHAM_MATER=MATE [K], RESULTAT=RESU [K],
EXCIT= _F (CHARGE= CHA1 [K]),
PARM_THETA=0.57,
OPTION= (“FLUX_ELNO_TEMP”, “ERTH_ELEM_TEMP”, “ERTH_ELNO_ELEM”))
# Subtlety PYTHON to define the new mesh
M [k+1] =CO (“M_ % of % (k+1))
# Adaptation of the mesh while basing itself on component ERTABS of
# ERRE_ELEM_TEMP of RESU [K].
# Old mesh: M [K]. Mesh refines: M [k+1]
# MACR_ADAP_MAIL (ADAPTATION=_F (
MAILLAGE_N = M [K],
MAILLAGE_NP1 = M [k+1],
RESULTAT_N = RESU [K],
INDICATOR = “ERTH_ELEM_TEMP”,
NOM_CMP_INDICA = “ERTABS”))
Example 6.5-1: Expurgé of the command file of case-test TPLL01J
In this other example extracted Internet site of LOBSTER
®
, coupling
ERTH_ELEM_TEMP/
MACR_ADAP_MAIL
[U7.03.01] the circulation of a “hot” fluid of share simulates and
of other of a metal part bent (in top and bottom, via a condition of EXCHANGE depend on
time in AFFE_CHAR_THER_F). The circulation of the fluid is carried out left towards the line.
The precision is especially necessary at the ends of the structure, on the level of the propagation of the fluid:
thanks to the indicating coupling of error/tool of refinement-déraffinement, the mesh thus remains fine in
edge of part, in the area where concentrates the “hot” fluid. Finally it is déraffiné with the back, one
time that the fluid passed.
It is also noted that, as envisaged by the theory (cf remarks [§2.2]), the resolution of the problem
thermics “is blunted” in the re-entering corners and that the indicator of error (although it is him
also penalized in these areas) this established fact announces (even when the part cooled).
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
58/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Example 6.5-2: Use of the option `
ERTH_ELEM_TEMP
'coupled with LOBSTER
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
59/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
7
Conclusion Perspective
During digital simulations by finite elements, obtaining a rough result is not sufficient any more.
The user is increasingly petitioning of space error analysis compared to the mesh.
He has need for support methodological and pointed tools “numériquo-data processing” to measure
the quality of its studies and to improve them.
To this end, the indicators of space error a posteriori make it possible to locate, on each element,
a cartography of error on which the tools of mending of meshes will be able to rest: a first
calculation on a coarse mesh makes it possible to exhume the card of error starting from the data and of
solution discretized (from where the term “a posteriori”), refinement is carried out then locally in
treating on a hierarchical basis this information.
The new indicator a posteriori which has been just established post-to treat the thermal problems of
Code_Aster is based on their local residues extracted the semi-discretizations in time. Via the option
`
ERTH_ELEM_TEMP
'of
CALC_ELEM
, it uses the thermal fields (
EVOL_THER
) emanating from
THER_LINEAIRE
and of
THER_NON_LINE
.
This new indicator supplements the offer of the code in term of advanced tools making it possible to improve
quality of the studies, their mutualisations and their comparisons. Indeed, of the indicators of error in
mechanics and macro of refinement/déraffinement
MACR_ADAP_MAIL
[U7.03.02] are already
available. It remains to supplement the perimeter of use of these tools and, to pack them, in particular for
to better manage non-linearities and the interactions space error/temporal error.
Note:
Estimator by smoothing of stresses of Zhu & Zienkiewicz (
CALC_ELEM
+
OPTION
“ERRE_ELEM_NOZ1/2”
[R4.10.01]) and indicator in pure residue (
“ERRE_ELGA_NORE”
[R4.10.02]).
Thereafter, the prospects for this work are several commands:
·
From a functional point of view, the complétude of this indicator could also improve in
taking into account possible nonlinear limiting conditions (
FLUX_NL
and
RADIATION
)
and of the exchanges between walls (
ECHANGE_PAROI
). In the long term, it would also be necessary to be able to rest
on finite elements of structure (hull…), of the pyramids and capacity to deal with problems
of convection-dissemination (operator
THER_NON_LINE_MO
[R5.02.04]).
·
From a theoretical point of view, when new limiting conditions are used and/or when one
be based on new modelings (hull, beam…), a study
“numériquo-functional calculus” similar to that of this document, should be carried out to judge
theoretical and practical limitations (with respect to Code_Aster) of such an indicator and to exhume
its adhoc formulation.
·
Let us recall finally that a string of indicators of error a posteriori are available, and,
that enough little was tested and validated on industrial cases. In order to refine diagnoses,
to establish comparisons and to set up strategies of mending of meshes per class of
problem, it would be interesting to pack the list of the indicators available. Different
indicators in residue plus local problem thus appeared more effective (but also more
expensive) during numerical tests (into elliptic) in N3S [bib5].
Note:
The indicator is the standard of the solution of a local, of the same problem standard than it
problem initial, but discretized on spaces of higher degree and of which the second
member is the residue. According to the limiting conditions affixed with this local problem, one
in distinguishes from various types. They thus mix the vision “bases hierarchical” and them
aspects “residue” of the indicators of error a posteriori.
·
The ideal consists in discretizing simultaneously in time and space on finite elements
adapted and to control their “space-time” errors in a coupled way. This
“space-time” indicator gives access to a complete control of the error and it allows
to avoid unfortunate decouplings as for possible refinements/déraffinements
controlled by a criterion with respect to the other (cf discussion [§4.5]). It is however very heavy with
to set up in a large industrial code such as Code_Aster. It supposes indeed, for
to be optimal, nothing less than one separate management pitch of time by finite elements. What
from the point of view of architecture supporting the finite elements of the code is true
challenge!
Code_Aster
®
Version
6.0
Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU
Key
:
R4.10.03-A
Page
:
60/60
Manual of Reference
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
8 Bibliography
[1]
R. DAUTRAY & J. - L. LIONS and Al Analyzes mathematical and numerical calculation for
sciences and techniques. ED. Masson, 1985.
[2]
J. - L. LIONS. Some methods of resolution of the problems in extreme cases non-linear. ED.
Dunod, 1969.
[3]
P.A. RAVIART & J.M. THOMAS. Introduction to the numerical analysis of the equations to
derivative partial. ED. Masson, 1983.
[4]
C. BERNARDI, O. BONNIN, C. LANGOUET & B. METIVET. Residual error indicators for
linear problems. Extension to the Navier-Stokes equations. Proc. Int. Conf. Finite Elements in
Fluids, Venezia 95, p 347-356. Note HI72/95/018,1995.
[5]
C. BERNARDI, B. METIVET & R. VERFURTH. Working group “adaptive Mesh”:
analyze numerical indicators of errors. Note HI73/93/062, 1993.
[6]
C. BERNARDI & B. METIVET. Indicator of error for the equation of heat. Review
European of the finite elements, flight n°9, n°4, pp425-438, 2000.
[7]
R. VERFURTH. With review off a posteriori error adaptive and estimate mesh-refinement
techniques. ED. Wiley & Teubner, 1996.
[8]
P. CLEMENT. Approximation by finite local element functions using regularization. RAIRO
Numerical analysis, flight n°9, pp77-84, 1975.
[9]
I. RUUP & PENIGUEL. Code SYRTHES: conduction and radiation. Theoretical manual of
V3.1. Note HE41/98/048, 1998.
[10]
S. ADJERID & J.E. FLATHERTY. With room refinement finite element method for 2D parabolic
systems. SIAM J.Sci.Stat.Comput., 9, pp795-811, 1988.
[11]
Mr. BIETERMAN & I. BABUSKA. The finite element method for parabolic equations, has
posteriori error estimate. Numer. Maths. 40, pp339-371, 1982.
[12]
R.E. BIENNER & Al An adaptive finite element method for steady and transient problems.
SIAM J.Sci.Stat.Comput., 8, pp529-549, 1987.
[13]
F. BORNEMANN. Year adaptive multilevel approach to parabolic equations. 3 shares in
IMPACT off comp. In Sci. And Engrg. 2, pp279-317, 1990. 3, pp93-122, 1991. 4, pp1-45,
1992.
[14]
K. ERIKSSON & C. JOHNSON. Adaptive finite element methods for parabolic problems.
SIAM J.Nume.Anal., 28, pp43-77, 1991.
[15]
C. JOHNSON & V. THOMEE. Year a posteriori error estimate and adaptive timestep control
for has backward Euler discretization off has parabolic problem. SIAM J.Nume.Anal, 27, pp277-
291, 1990.
[16]
X. DESROCHES. Estimators of errors in linear elasticity. Note HI75/93/118, 1993.
[17]
Mr. FORT and Al Estimate a posteriori and adaptation of mesh. European review of
finite elements. Vol. 9, 4, 2000.
[18]
I. BABUSKA & W. RHEINBOLT. A posteriori error estimates for the finite element method.
International Newspaper for Numerical Methods in Engineering, vol. 12, pp.1597-1615, 1978.