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
Organization (S): EDF/SINETICS
Handbook of Référence
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 grid. 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 grid 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 option “ERTH_ELEM_TEMP” of CALC_ELEM, it uses the thermal fields (EVOL_THER) emanating
THER_LINEAIRE and 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
Contents
1 Problems Description of the document ........................................................................................... 3
2 the problem in extreme cases ......................................................................................................................... 6
2.1 Context .......................................................................................................................................... 6
2.2 Strong formulation to weak ................................................................................................... the 11
3 Discretization and controllability .............................................................................................................. 16
3.1 Controllability of the continuous problem ................................................................................................ 16
3.2 Semi-discretization in time ........................................................................................................ 18
3.3 Error of temporal discretization ............................................................................................... 22
3.4 Total discretization in time and space ................................................................................. 23
4 Indicator in pure residue ....................................................................................................................... 26
4.1 Notations ....................................................................................................................................... 26
4.2 Increase of the total space error .......................................................................................... 27
4.3 Various types of possible indicators ........................................................................................ 34
4.4 Decrease of the local space error ............................................................................................ 37
4.5 Complements ................................................................................................................................ 42
5 Summary of the theoretical study ........................................................................................................ 44
6 Implementation in Code_Aster .................................................................................................... 48
6.1 Particular difficulties .................................................................................................................. 48
6.2 Environment necessary/parameter setting ...................................................................................... 48
6.3 Presentation/analysis of the results of the error analysis ............................................................ 51
6.4 Perimeter of use .................................................................................................................... 55
6.5 Example of use ...................................................................................................................... 56
7 Conclusion Perspective .................................................................................................................... 59
8 Bibliography ........................................................................................................................................ 60
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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
grid. 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 of the elements
tends towards zero) and they require a certain level of regularity which is precisely not reached in
zones with problem. Moreover, these increases subtend two types of strategies for
to improve calculation:
·
the “methods p” which consist 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. FORTIN (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 grid 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 grids 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 principal 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 grid 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 grid,
·
to estimate and qualify a calculation for a class of grid 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].
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 groundwork) 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 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, one determines Cadre Variationnel Abstrait (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 constraints 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 step
time.
·
In a third part (cf [§5]), principal 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
principal theoretical contributions of the preceding chapters.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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
3
border
=
: =
: I
regular characterized by its voluminal heat with constant pressure
i=1
C
(X
p
(the vectorial variable X symbolizes here the couple (X, y) (resp. (X, y, Z)) 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 item X of opened and at any moment
T [,
0 [(> 0), 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
S (X, T), of boundary conditions of imposed the temperature type F (X, T) (on the portion of surface
external 1
), normal flow imposed G (X, T) (on 2
) and exchanges convectif H (X, T) and Text (, T
X) (out of 3
).
One places oneself thus within the framework of application of 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
T
C
-
p
div (T) = S
×],
0 [
T
T = F
×
1
], 0 [
T
(P)
0
= G
×
2
], 0 [éq
2.1-1
N
T
+ HT = HT
×
ext.
3
], 0 [
N
T (,
X)
0 = T (X)
0
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
Note:
·
In this theoretical study of the problem mixed (0
P), one supposes 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
3
that = -
4:
, on which a condition of homogeneous Neumann intervenes
=
I
I 1
(thus, when one builds the variational formulation associated with the strong formulation (0
P), them
terms of edges related to this zone disappear. The problem remains well posed then since it
is thermically unconstrained in this zone. 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 H (T, X) 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 modelling the convectif exchange (key word ECHANGE) 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 models one
thermal resistance of interface
1
T + HT = HT
×
1
2
12
], 0 [
With =
, T =
N
3
12
21
T
I
has
one
éq 2.1-2
ij
2
T + HT = HT
×
2
1
21
], 0 [
N
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.
With =
,
1
12
21 T = T
I
(
has
one
)
OUP
LIAISON_GR
ij
I
J
1i 1
T (X, T) + 2 T
J
(X, T) = (X, T) on ×
2
1
], 0 [éq
2.1-3
I
J
simply
more
or
T
I I (,
X T) = (,
X T) on 1
×],
0 [(
)
L
LIAISON_DD
I
In the same way functionalities LIAISON_UNIF and LAISON_CHAMNO make it possible to impose one
even temperature (unknown) with a whole of nodes. They constitute a surcouche
preceding conditions by imposing couples (,) particular. 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 modelled 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 commutate 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 a 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
incidence 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
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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,
(H1)
variety of dimension Q - 1, lipschitzienne or 1
C per piece
(H2)
For the data:
2
S L (
1
,
0; -
H ())
2
0
T L ()
1
2
-
-
2
F L,
0; H
(1)
1
2
2
,
G L,
0; H
(2
)
1
2
2
,
T L
ext.
,
0; H
(3
)
(H
3)
, C,
L
p
()
2
H L (,
0;
L (3
))
who allows us to obtain a solution in the following intersection
T 2
L (
1
,
0; H ()) 0
C (
2
,
0; L ()) éq
2.1-4
Note:
That is to say (X,
) Banach, one notes LP (, 0;X) the space of the functions T v (T)
X
strongly measurable for dt measurement such as
1
v
=
v (T) dt
0,
. It is Banach, therefore a space of Hilbert for
; p, X
p
p < +
0
X
the associated standard.
The introduction as of these spaces of Hilbert particular “space times” comes from the need from
to separate variables X and T. Any function U: (X, T) Q =
: ×
], 0 [U (, xt) can in fact
to be identified (by using the theorem of Fubini) with another function
u~: T] [{u~
,
0
(T) X u~
:
(T) (X) = U (X, T)}. The transformation U u~
constituting one
isomorphism, one will simplify the expressions thereafter by 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
1
H ()
=
Q
: U L2 ()/U (L2 ())
on which it is convenient to work, with space
H 1 () =
: {U Of ()/U H1 (Q) with U
= U}
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
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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
of Analyze Numérique 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.
T W:= {
1
U H ()/U =
: U
= F
0 1
,
} éq
2-1-5
1
Moreover, thanks to the geometrical assumptions (H1) and (H2), there is an operator of raising
(compound of the operator of usual raising and the operator of prolongation by zero apart from
1
1
2
1
) R: H (1) H () linear, continuous and surjective such as:
1
2
01, RF = F
F H (1
) 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
U V =
: {U 1
H ()/
U:= U =
0 1
,
} 0 éq
2-1-7
1
resulting from the decomposition
T =
: U + RF
éq
2-1-8
Note:
That is to say (X,
) Banach, one notes LP (, 0;X) the space of the functions T v (T)
X
strongly measurable for dt measurement such as
1
v
=
v (T) dt
0,
. It is Banach, therefore a space of Hilbert for
; p, X
p
p < +
0
X
the associated standard.
This change of variable produces the problem simplified out of U
U
C
- div
p
(U) = ^s
×],
0 [
T
U = 0
×
1
], 0 [
U
(1
P)
= ^g
×
2
], 0 [éq
2-1-9
N
U
+ hu = ^h
×
3
], 0 [
N
U ()
0 = U
0
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 10/60
with the new second member
RF
^s =
: S - C
+ div
p
(RF) 2
L (
- 1
,
0; H ()), éq
2-1-10
T
new loadings
RF
2
1
^g =
: G -
- 2
L
,
0, H
(2)
éq
2-1-11
N
1
^h =
RF
: H (T
- RF
ext.
)
2
-
- 2
L
,
0, H
(3)
éq
2-1-12
N
and the new initial condition
u0 (). =
: T (). - RF (0
.,) 2
0
L ()
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
With
0
T
s^ =
: S -
F has
ji J
=
j>J
0 Id
T
=
: RF
1
1
F
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 (H1) and (H2) the workspaces W and V of Hilberts are provided with the standard
induced by
1
H ().
Proof:
The result comes simply owing to the fact that the application traces
1
01: H ()
2
L (1
) is
1
composed of the application traces usual
1
0: H () H () 2
2
L () 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 provided with the standard
.
,
1
!
Lemma 2
Under the assumptions (H
1
H
1) and (H2), the standard and the pseudo norm induced by
() are equivalent on
functional space V. One will note P () > the 0 constant of Poincaré relaying this equivalence
v V
v
P () v
,
1
,
1
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Indicator of error in residue for transitory thermics
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:
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Note:
U thereafter will be noted
=
: supess U (T) and
,
pp. T
(
2
2
2
U,)
v (m
H ()
) (U,)
v
2
2
m =
:
(
U,) vL2 (), U
=
:
U
and U
=
:
U
.
m
,
2
m
,
L ()
L2 ()
m
m
=
m
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, my (-) 0 (see the demonstration [bib1] §III.7.2 pp922-925),
1
·
either one takes into account only conditions of imposed the temperature type,
my (-) = 0, V =
1
H
and one finds the standard result of equivalence of the standard
0 ()
1
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 Cadre Variationnel Abstrait (CVA) on
which will rest the weak formulation:
·
1
H
V
H
,
0 ()
1 ()
·
V H = 2
: L () = H “V”
- 1
H () by identifying H and its dual,
·
there is a continuous linear canonical injection of 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 provided 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 principal 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 commutate the integral in space and derivation
in time, with fixed and from the characteristics materials independent of time), one obtains:
D C U
.
^
éq
2.2-1
p (T) v dx +
U (T) v dx = S (T)
U (T)
vdx +
v
D
dt
N
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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 [)) following:
The solution is sought
U L2 (,
0;V) C 0 (,
0; H)
éq
2.2-2
checking the problem
To find U: T],
0 [U (T) V
that
such
D
(P)
v V
C U T, v
T has;U T, v
B T, v
2
(p ()) + (()) = (()) éq
2.2-3
0,
dt
U ()
0 = 0
U
with
(T has;U (T), v):= U (T) .v dx + H (T)
U T vd
0,3 () 0,3
3
éq
2.2-4
(B (T), v):= ^s (T), v
+ ^g (T)
^
, v 1 1
+ H T, v
0,2
() 0,3 1 1
1
- ×,
1
- ×, 2
- ×, 3
2 2
2 2
while noting,
the hook of duality enters spaces
p
H () and Q
H ().
p×q,
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 traditional direction (like
for the surface term of A (T;..)) that if the space of membership of the news is restricted
source and of the new loadings with
2
^s L (
2
,
0; L ()
2
, ^g L (
2
,
0 L ()
2
^
;
and H L
,
0; L éq
2.2-5
2
(
2 (3)
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
F L
,
0; H ()
2
, S L
,
0; L, G L
,
0; L
and T
L,
0; L
1
(
2 ()
2 (
2 ()
2
2
ext.
(
2 (3)
éq 2.2-6
·
The formulation (P has a direction well, because it is shown that
2)
T has (T;U (T), v)
2
L] (,
0 [) Of]
(, 0 [)
T C U
p (T)
2
L (,
0;V) and v V
T
(C up (T), v)
2
L]
(, 0 [) Of] (, 0 [)
0,
T ^s (T)
2
L (,
0;
1
H - ()
1
and v H ()
1
H - ()
T ^s (T),
2
v
L] (,
0 [) Of]
(, 0 [)
1
- ×,
1
1
1
1
-
-
T ^g (T)
2
L
,
0;
2
H (
and
v H H
2)
2
0,2
(2)
2 (2)
T ^g (T),
2
v 1 1 L
D
0,2
] (, 0 [)
']
(, 0 [)
- ×,
2 2 2
and one finds obviously the same thing for the term of exchange on 3.
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·
In the surface integrals one will henceforth note U (T) and v what should be noted (in
any rigor) U
and
.
0, I (T)
v
0, I
·
Membership of the solution with L2 (,
0;V) rises from the assumptions on the data and of
properties of the differential operators and trace. The fact that it must also belong to
C 0 (,
0; H) 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 (P in
0)
showing its equivalence with (P and by applying to this last a parabolic alternative of
2)
theorem of Lax-Milgram.
Theorem 3
Within the abstracted variational framework (CVA) definite previously and by supposing that assumptions
(H1), (H2) and (H3) are checked, then the problem (P admits a solution and only one
2)
U L2 (,
0;V) C 0 (,
0; H).
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 (
U (T), v)
2
V
T has (T;U (T), v) on],
0 [
·
Its continuity on V ×V
p T],
0 [have (T;U (T), v)
U (T)
v
+ H (T)
U (T)
v
,
,
1
,
1
1
,
1
,
,
3
2 3
2 3
(
U (T), v) 2
V
max (
, H (T)
2
2
C P
U T
v
3
()
,
,
) (), 1, 1
3
with C the constant of continuity of the operator of trace on and P () the constant of
3
3
Poincaré.
·
Its V - ellipticity compared to H
p T],
0 [have (T;v, v)
2
- 2
+
v
C
- H T
C v
0
0,
(
()
2
3
,
,
) 20,
2
3
v V has (
T;v, v)
2
- 2
+ v
+ C
- H T
C
v
0
0,
(
()
2
3
,
,
) 2
{
0,
2
3
> 0
1
4
4
4
4
4
4
2
4
4
4
4
4
4
3
> 0
with C the constant of continuity of the canonical injection of
1
H () in 2
L ().
0
·
The continuity of the linear form B (T) on V
p T],
0 [(B (T), v) ^s (T)
v
+ G (T)
^
^
1
v 1
+ H (T)
v
-,
1
,
1
-,
1
1
,
,
3
2
2
-,
2
2
3
2 3
2
v V
P ()
max ^s (T)
, ^g (T)
^
1
C, H T
C
v
2
()
-,
1
-,
1
3
2
,
,
1
-
2
2 3
with C the constant of continuity of the operator of trace on.
2
2
!
Theorem 4
The problems (P and (P are equivalent and thus the initial problem admits a solution and one
2)
0)
only
U L2 (,
0;V) C 0 (,
0; H).
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Proof:
The existence and the unicity of the solution of the problem (P result of course from the preceding theorem, one
0)
time that the equivalence of the two problems was shown. It thus remains to prove the implication
opposite (P P which is very hard to exhume “not formally”. In particular conditions
2)
(0)
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 one shows that in our case of figure, the limiting conditions on I in fact are checked,
- 1
- 1
not on L2
,
0, H2 (
, but on space (B) 'H2
(by noting I:= ×
)
I
]
,
0 [
00 (I
I
)
I)
defined as being the dual topological one of
1
B =
: W H (
) L2
2
(I/
2
,
0;
with
0 and
I
) v L (
V)
v {
v
v
W
×
0} =
{
×
} =
=
I
!
Note:
·
Because of low regularity imposed on thermal conductivity,
L (), one cannot
not to claim with the “standard” regularity U
2
H (). Indeed in the case, for example,
of a Bi-material (with =) from which the characteristics are distinct from share and
1
2
of other of the border, [éq 2-1-9] and the theorem of the divergence imposes
1
2
1
2
Appear 2.2-a: Exemple of Bi-material
U (T)
U (T)
1
-
=
2
in H
p T
1
2
00 ()
], 0 [
N
1
N
2
However, therefore the condition of transmission cannot be carried out on the internal border
1
2
U (T)
U (T)
p p T],
0 [
N
1
N
2
Thus U (T)
2
H ()
2
H
do not involve U (T)
2
H (). This restriction us
1
(2)
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 Q - 1, C per piece (property of 2-prolongation)
(H4)
2
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2
S L (
2
,
0; L ()
1
T H
0
()
3
1
1
2
2
F L
,
0; H (, G L
,
0; H
, H T
,
0; H
(H5)
1)
2
2 (2)
2
2
ext.
(3)
, C,
L
p
()
2
H L (,
0;
L (3)
What allows obtaining a solution in the following intersection
U 2
L (
2
,
0; H () 0
C (
1
,
0; H () 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 (P0)
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.
Handbook of Référence
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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
(H1), (H2) and (H3) are checked, one with the “weak” controllability of the continuous problem (with
K
my
P >)
1 (
,
(I),
0, I
()
) 0
,
T
2
2
2
p T
C U T
U
D
C U
p
() +
()
+
0,
p
0 0,
0
0,
éq 3.1-1
T
2
K
^
^
^
1
S () 2
+ G () 21 + H ()
D
-,
1
-,
1
2
-,
0
2
2 3
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 U (T), while integrating spatially on, then
temporally on [,
0 T] with T [,
0 [one obtain, like the characteristics materials are
presumedly independent of time,
T
T
T
1 (C U, U D div U, U D s^, U
D éq 3.1-2
p ()
()
- ((() ()
=
0,
0,
() ()
- ×
1,
1
2 T
0
0
0
By using the formula of Green and the conditions limit [éq 2.1-1] one obtains
1
T
T
2
2
C U T
C U
U, U
D
H U 2D
p
() -
p
0
+ (() ()
+
0,
() ()
0,
0,
=
2
0
0
éq 3.1-3
T s^ (), U ()
+ g^ () U ()
^
,
1 1
+ H (), U ()
D
- ×
1,
1
- ×,
1 1
2
,
0
- ×
2 2
2 2 3
One can évincer the term of exchange of [éq 3.1-3] because it is supposed that H (T) 0 p T. By using one
argument of duality, the inequality of Cauchy-Schwartz, lemma 2 and the relation
2
has
2ab + (
b)2 (> 0), one obtains
T
T
2
T
2
1 1
s^ (), U ()
D
s^ () 2
P ()
D +
2
U ()
D
- ×
1,
1
2 2
-,
1
0,
éq
3.1-4
0
0
,
0
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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 Ci…), then one inserts these inequalities in [éq 3.1-3]
2
2
T
P
2
C U T
2
2
C 2 2
C 2 2
U
D
p
()
()
+
-
(+
+
2
3
)
()
0,
0,
,
0
2
2
^
éq
3.1-5
2
^
H
T
2
s^ ()
G () 1 1
()
- ×,
- 1×1,
C U
+
1,
1
2 2 2
2 2 3
D
p
0 0,
- ×
+
+
2
2
2
0
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
2
P ()
2 -
(2 2 2 2 2
+ C + C =
éq
3.1-6
2
3
) 1
,
Maybe, for example,
((
my 1) +)
1
2
=
,
2
P () (
(
my) +)
3
((
my 2) +)
1
2
=
,
éq
3.1-7
2
2
C P
2
() ((
my) +)
3
((
my 3) +)
1
2
=
,
2
2
C P
3
() ((
my) +)
3
From where increase [éq 3.1-1] while taking
2
P () (
(
my) +)
3
1
2
2
C
C
K =
2
3
max
éq
3.1-8
1
,
,
my
1
my
1
my
1
,
(
(1) +) ((2) +) ((3) +)
!
Handbook of Référence
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Note:
·
The recourse to measurements of the external borders is an easy way allowing the inequality
to support the passage to limit (0) when one or more limiting conditions come to
I
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
= C = 1, = = and s^ = S
éq
3.1-9
p
2
3
and by introducing particular standards on V =
1
H
and its dual
0 ()
^s (T) v
my
^s (T)
,
*
- ×
1,
1
*
() +
= sup
with
v
=
1
v
éq
3.1-10
-,
1
*
,
1
2
,
1
v V, v0
v
(my () + 3) P ()
,
1
one finds well the inequality (2) pp427 of [bib6].
·
If one allows more regularity on the geometry (H4) and the data (H5), 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
One fixes a step of time T
such as
either an entirety NR and such as the temporal discretization or
T
regular: T =,
0 T = T
, T = 2 T
T = N T
.
0
1
2
L N
Note:
This assumption of regularity does not have really importance, it allows just
to simplify the writing of the semi-discretized problem. To model a transient
unspecified at the moment T
=
-
N, it is just enough to replace
T
by T
T
T.
N
n+1
N
Semi-discretization in times of [éq 2.1-1] by - method leads to the following problem:
The continuation is sought
(one)
V
éq
3.2-1
0nN
such as
n+1
U
- one
C
- div
p
(n+1
U
) - (1 -) div (one) = n+1
^s
+ (1 -) ^sn 0 N NR - 1
T
n+
1
U
= 0
0 N NR - 1
1
(
N 1
n+1
U
P
^ N
1
) +
=
+1
G
0 N NR - 1
2
N
n+1
U
n+1
n+1
n+1
^
+ H U
= H
0 N NR - 1
3
N
0
U (.) = U
0
éq 3.2-2
Handbook of Référence
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Titrate:
Indicator of error in residue for transitory thermics
Date:
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Author (S):
O. BOITEAU Key
:
R4.10.03-A Page
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while posing
N
=,
X N
with {U, S h^
,
^, H, g^} and 0 N NR
T
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
N
N
n+1
N
n+1
N
n+1
N
n+
Being gift
be U, s^, s^
, g^, g^
h^
,
h^
,
, H, H 1
(n+
P 1 To calculate
that
such
2
)
n+
U 1 V
éq
3.2-3
(
n+
C U 1, v
p
) + T has (N
n+
T; U 1, v
) = (C one, v
p
) + T (bn, v
) (vV)
0,
0,
with
N 1
+
N 1
:
+
=
+
(1 -) N where {U, hu, B, S, ^ ^g} ^
, H
has (
N 1
N T; +
U
, v
)
N 1
:
+
= U .v dx +
(hu) N 1+
v D
éq
3.2-4
3
(N 1+
B, v
)
N 1
+
N 1
+
N 1
^
:= ^s, v
+ ^g,
+
v
+ H, v
0,2
1 1
0,3
1 1
1
- ×,
1
- ×, 2
- ×, 3
2 2
2 2
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 (U N)
V required.
0nN
Note:
· While posing RF = 0 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 to which the solution must belong. Moreover, in
implicitant it 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 ^s 0
C (
- 1
,
0; H () (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
U 1
H
.
0
()
· To simplify the expressions one will not mention any more the temporal dependence of the form
bilinear A (T;..) (for the implicitation of the term of exchange), it will remain implied by
that of the solution.
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Indicator of error in residue for transitory thermics
Date:
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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
1
unconditionally stable (
), that ^s 0
C (
- 1
,
0; H () and U 1
H
, one with
0
()
2
controllability “
weak
” of the problem semi-discretized in time (with
K
my
P >)
1 (
,
(I),
0, I
()
) 0
,
n+
2
N
1
N
4
3
1
+ 2
1
+ 2
1
-
2
C U
+ T
U
C U
+
N
C U
p
p
p
0,
0,
0,
0,
2
2
T
K T
n+
2
1
2
2
2
0 N NR - 1
+
U
+ 1
n+1
n+1
n+1
^
^s
^g
1
H
0,
+
+
-,
1
-, 2
- 1,
2
2
2
2 3
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
U n+1 =
: U n+1 +
(1 -) a V éq
3.2-6
and évincer the term of exchange by the argument
0 < min (N N 1
H, +
H
)
2
N 1
+
U
(hu) N 1+ N 1+
U dx
max (N N 1
H, +
H
)
2
N 1
+
U
éq 3.2-7
0,
0,
3
3
3
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 (2 -) C U N 1
1
+ undx
. From where one
p
fourth parameter checking a system of the type [éq 3.1-6]
2
P ()
2 -
(2 2 2 2 2
+ C + C =
2
3
) 1
,
éq
3.2-8
2
2
- 1 - 2 = 1
!
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
4 - 3
1
equivalent [éq 3.1-10], like
<, one finds well the inequality (5) pp428.
2
2
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Indicator of error in residue for transitory thermics
Date:
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:
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While stating [éq 3.2-5] for the values of m {1
,
0
,
K}
N and by summoning these increases to 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
N 1
N 1
2
2
2
2
N
C U
+ T
U
C U
C U
p
-
m+
1
+ 4
(1 -) -
m
p
(4 - 3)
p
0,
0
0,
0,
0,
m=
0
m=
0
N 1
2
2
2
0 N NR
+ K
m
m
^
T
^s
^
m
G
H
1
-
+1
+1
+1
+
1
+
-,
1
-,
1
2
,
m=
-
0
2
2 3
éq 3.2-9
or more simply
N 1
2
2
2
N
C U
+ T
U
C U
p
-
m+
1
p
0,
0
0,
0,
m=
0
éq
3.2-10
N 1
2
2
2
0 N NR
+ K
m
m
^
T
^s
^
m
G
H
1
-
+1
+1
+1
+
1
+
-,
1
-,
1
2
,
m=
-
0
2
2 3
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
(
N
4 1 -) 1
2
C um 0
p
0,
m=
0
éq
3.2-11
(4 -)
2
2
3
C U
C U
p
0
p
0
0,
0,
!
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 [
N T
, (n+1) T
]
temporal function characteristic of interval [N T
, (N +)
1 T
]) U (T) un+
=
1
[
N T
, (n+1) T
] (T)
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 (H4) and (H5), one finds
a “strong” version of properties 6 and 7.
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Indicator of error in residue for transitory thermics
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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
N NR
in:= U N - U (NT)
éq
3.3-1
0
E = 0
One starts by revealing this error by withdrawing from the equation [éq 3.2-2] the relations
(n+) T
1
1
U
()
U (N +)
1 T
) - U (N T
)
D =
T
T
T
N T
U
(N +)
1 T
)
C
= div
+1 + ^
+1
éq
3.3-2
p
(U (N) T)
S (N
) T)
T
(1 -)
U
(N T
)
C
= 1 - div
+ 1 - ^
p
(
) (U (N T) (
) S (N T)
T
that is to say
n+1
N
n+1 T
E
- E
1
C
-
1
div
éq
3.3-3
p
(
n+
U
E
=
D +
U
)
(
)
()
C p
T
T
T
T
NT
while noting
en+1 =
: en+1 +
(1 -) in
U
U
éq
3.3-4
=
:
(
N +)
1) + (1 -) U
T
(N T
)
T
T
T
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
U 1
H (,
0;V) 2
H (
- 1
,
0; H ()
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
N 1
2
2
0 N
N
NR
C E
+ T
E
p
-
m+
1
0,
m=
0,
0
K (T) 3
N
C
1
(p) 2 - 1 (
U
U
1) 2
2
-
(
m T)
-
(m +) 1t)
2
2
4
m=0
T
T
éq 3.3-6
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Indicator of error in residue for transitory thermics
Date:
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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 (in)
V checks a similar problem
0nN
with [éq 3.2-2] (by supposing that the temporal discretization of the conditions limit are exact)
n+1
E
- in
C
- div
p
(n+1
E) =
T
C
T
p
2u
2u
(1 -)
(NT) -
(N +) 1t)
0 N NR -
2
2
1
2
T
T
(n+1
P
in
N
NR
3
) +1
= 0
0
- 1
1
n+1
E
= 0
0 N NR - 1
2
nn+1
E
+ n+1 n+1
H
E
= 0
0 N NR - 1
3
N
0
E (). = 0
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 tighten T 0 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 (in) rise of course from that of (one) 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 of triangulations. Because of this regularity finite element method
applied to (N 1
+
P
converge when the largeest diameter of the elements K of (
2
)
H) H tends towards zero
H:= max H 0
éq
3.4-1
K
K
Th
Note:
·
The finite elements (K, km No, K) are closely connected equivalents with same elements of reference, they
relations of compatibility on their common borders and the constraints check
geometrical [éq 3.4-1] and [éq 3.4-2].
·
It is pointed out that the diameter of K is real H:= max X - Y.
K
(X y)
2
, K
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By noting K the roundness (one recalls that the roundness of K is reality
=
: max {diameter
sph
eras K) associated K, finite elements of (
K
}
H) H also satisfy
constraint
>
HK
0/
éq
3.4-2
K
In the usual triplet (K, km No, K) one defines polynomial space as being that of the polynomials of
total degree lower or equal to K on K
P =
: P
éq
3.4-3
K
K (K)
and approximation spaces it (with the “weak” direction) associated
V =
:
/
P
éq
3.4-4
H
{v V K T v
H
H
H K
K (K)}
V
To conclude, one will note H, the operator of projection which associates the solution continues its Vh
interpolated
:V V
H
H
éq
3.4-5
v vh
Note:
With a regular family of triangulations, this operator of interpolation is continuous and it can
to be written v =
H
v (xi) Neither by noting the xi nodes of the grid and Ni their function of
I
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 grids are 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 Vh, they are the intrinsic relations of compatibility with
family of elements which assures us
H
, K v
1
P
1 =
:
éq
3.4-6
H K
K (K)
H (K)
v
H
H
(
K)
In the literature one often prefers the more regular definition to him
*
V =
: V 0
C
éq
3.4-7
H
H
()
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By regaining semi-discretized shape (N 1
+
P
with functions tests in V
2
)
H one obtains the problem
completely discretized in time and space (for a H fixed) according to:
The continuation is sought
(naked
V
éq
3.4-8
H) 0nN
H
initialized by
0
U:= U
éq
3.4-9
H
H 0
checking the following problem
N
N
n+1
N
n+1
N
n+1
N
n+
Being gift
be U, s^, s^
, g^, g^
h^
,
h^
,
, H, H 1
(
H
H, n+
P
1
To calculate
that
such
2
)
n+
U 1 V
(
H
H
n+
C U 1, v
T has U 1, v
C U, v
T B 1, v
v
V
p H
H)
+ (n+
, H
H) = (
N
p H
H)
+ (n+
H)
(
H
H)
0,
0,
éq 3.4-10
Just as one supposed in the preceding paragraph as the temporal discretization of
loadings was exact
in
N
=
: -
(N T
) = 0 with {S H, ^
,
^
H, g^} and 0 N NR
(H6)
, one supposes here moreover than their space discretization is too
H
N
N
N
=
: = with
,
^
,
^
, ^ and 0
(H7)
H
H
{S H H G}
N
NR
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 groundwork developed in
demonstration of theorem 3, one shows the existence and the unicity of continuation (N
U
in the closed sev
H) N
(it is thus Hilbert, pre-necessary essential for the use of the famous theorem) Vh 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 (H6) and
(H7) are checked, one with increase
N 1
2
2
2
N
C U
+ T
U
C
U
p
H
-
m+
1
H
p
H
0,
,
0
0,
0,
m=
0
éq
3.4-11
N 1
2
2
2
0 N NR
+ K
m
m
^
T
^s
^
m
G
H
1
-
+1
+1
+1
+
1
+
-,
1
-,
1
2
,
m=
-
0
2
2 3
by noting m+1
m
U
=
: U +1 + 1 - U
.
, H
H
(
) mh
Handbook of Référence
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Version
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Indicator of error in residue for transitory thermics
Date:
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:
R4.10.03-A Page
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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 (H4) and (H5), one finds a version “strong” of
this increase utilizing the H1 standard of the 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)) NR (F) is noted.
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) HK (resp is noted. HF).
·
The whole of the triangulation (H) breaks up in the form
T:= T
T T T
H
H,
H 1
,
H, 2
H, 3
by noting (H, I) the whole of the finite elements having a face contained in border I.
·
With same logic, the whole of the faces of the triangulation (H) breaks up in the form
S:= S
S S S
H
H,
H 1
,
H, 2
H, 3
with
I
{,
1
}
3
,
2
S:= K
/K T
K
= S K
H, I
{
H
I}
()
K Th, I
·
In the same way, the whole of the nodes of the triangulation (H) breaks up in the form
NR:= NR
NR NR NR
H
H,
H 1
,
H, 2
H, 3
·
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
this one.
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:
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·
One notes P the operator of raising on K of traces on F, built starting from an operator of
F
raising fixed on the element of reference.
·
The union of the finite elements of the triangulation sharing at least a face with K is noted
=
:
K'
K
S (K) S (K')
·
The union of the finite elements of the triangulation containing F in their border is noted
:=
K'
F
F S (K')
·
Union of the finite elements of the triangulation which share at least a node with K (resp. with
F) is noted
=
:
K'
(resp.
=
:
K'
).
K
F
NR (K) NR (K')
NR (F) NR (K')
Th
H
F
K
F
K
F
K
Appear 4.1-a: Désignation 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
U
. As the discretized workspace is included in
H) N
continuous space V V, one can re-use [éq 3.2-3] with v
H
h. While withdrawing to him [éq 3.4-10] it occurs
(with N and H fixed and while supposing (H6) and (H7))
(C
éq
4.2-1
p (n+1
N
U
- U +1, v
+ T
U +1 - U +1, v has =
C U - U, v
v
V
H
) H)
((N
N
, H) H)
(p (N nh) H) (H H)
0,
0,
Note:
·
This relation states the orthogonal character of the space error with respect to the elements of
Vh.
·
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!
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 28/60
Let us consider the following linear form
(
To v) =
: (C
éq
4.2-2
p (U n+1 - U n+1,
+
+1 - +1,
H
) v)
T has (one
U N v
, H
) (v V)
0,
who will be used to us as discussion thread during this demonstration. By packing it via [éq 4.2-1], one obtains
(
To v) = (CP (N
N
U - U, v
+ C U - U, v - v
+
H)
)
(p (N nh) (H)
0,
0,
(
éq
4.2-3
v
V)
(CP (n+1 N
U
- U +1, v - v
+ T
U +1 - U +1, v - v has
H
) (H)
((N
N
, H)
H)
0,
While taking [éq 3.2-3] after having replaced v
-
H by v
v
V, one can build
H
(CP (n+1 n+
U
-
1
U
- N
U + N
U
v v
T has U
U
v v
H
H),
- H) + (n+1
n+
-
1, -
, H
H)
=
0,
0,
(
éq 4.2-4
v V)
T (n+1
B, v - v
T has U
v v
C U
U
v v
H)
-
N
,
N
N,
0,
(+1 - -
1 -
-
, H
H)
(p (+h H)
H) 0,
Then A (v) becomes
(
To v) = (CP (N
N
U - U, v
+ T
B +1, v - v -
H)
)
(N
H)
0,
(
éq
4.2-5
v
V)
(CP (n+1 N
U
- U, v - v
- T
U +1, v - v has
H
H)
H)
(N, H
H)
0,
Then one breaks up the last three terms on each element K of the triangulation and one applies,
with the last, the formula of Green
n+
(
To v)
1
N
= (CP (
U
U
N
U - N
U
v
T
S
C
U
v v dx
H),)
+
N 1
H
H
N 1
^
div
0,
+
-
-
+
p
(+, H) (- H)
T
K
K
Th
n+1
T
U
-
H,
(v - v D
H)
2
N
F
F
S
H,
n+1
U
v - v V
+ T
G
v v D
H
n+1
^
-
H,
(- H)
N
F
F
S
H, 2
n+1
U
+ T
n+1
^h -
H,
-
(hu
v v D
H)
n+1
(- H)
N
F
F
S
H, 3
éq 4.2-6
Note:
·
One allowed oneself to replace the hooks of duality of [éq 3.2-4] by integrals and one
the formula of Green bus to compact K the assumptions (H4 can apply) and (H5) are
checked (while replacing by K and
I by
K
). One thus has
I
1
v - v H
and
éq 4.2-7
H
(K)
2
, U H
H
(K)
2
, ^S L (K)
2
, G L (K)
2
^
^
H L K
2
(
3)
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 29/60
Let us point out some properties of the operator H of L2-local projection introduced by
P. CLEMENT [bib8]
V L2
:
V
H
()
H
éq
4.2-8
v vh
It checks in particular increases of errors of projection
v
V
v - v
:= v - v
C H v
H
H
4
K
0, K
0, K
,
1 K
éq
4.2-9
K
T,
F
S K
v - v
:= v - v
C
H v
H
()
H
H
5
F
0, F
0, F
,
1 F
where the constants C4 and C5 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:
n+
(
To v)
1
N
- (CP (
U
U
N
U - N
U
v
Tc
H S
C
U
v
H),)
N 1
-
^
-
H
H + div N 1
0
4
,
+
K
p
(
+
, H)
,
1
T
K
K
0, K
Th
n+
1
T
U
+
C
H
v
F
H,
5
2
,
1
N
F
F
S
H,
0, F
n+1
U
v V
+ Tc
H
^ N
H
G
v
5
+1 -
,
F
,
1
N
F
F S
0, F
H, 2
n+1
U
+
^n
H
N
Tc
H H
hu
v
5
+1 -
,
-
F
(H) +1
,
1
N
F
F S
0, F
H, 3
éq 4.2-10
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Code_Aster ®
Version
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Titrate:
Indicator of error in residue for transitory thermics
Date:
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Author (S):
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:
R4.10.03-A Page
: 30/60
This inequality clearly lets show through a possible formulation of the indicator in pure residue:
Definition 10
K
In the framework of the operator of transitory thermics linear of Code_Aster, continuation (N
(K)
H
T
0nN
theoretical local indicators can be written in the form
n+1
N
n+
U
- U
1
U 1
n+1
(K)
n+
=
: H S 1
^
-
H
H
C
+ div U 1
H
,
K
p
(
n+
H,)
+
H
F
+
T
2 FS
N
(K)
0, K
0, F
éq 4.2-11
n+1
n+1
U
U
n+
H G 1 -
H,
+
H H 1
,
^
^
hu
1
F
n+ -
H
-
F
(H) n+
FS2 (K)
N
FS3 (K)
N
0, F
0, F
It is initialized by
1
0
(
0
K)
U
=
: H S 0
^ + div U 0
H
K
(H) +
0, K
H
F
+
2 FS
N
(K)
0, F
éq 4.2-12
0
0
U
U
H G 0 -
H
+
H h0
^
^
h0u 0
F
-
H -
F
H
FS2 (K)
N
FS3 (K)
N
0, F
0, F
Continuation (N
()
theoretical total indicators is defined as being
0nN
1
0
N
N
NR () = N
(K) 2
2
:
éq 4.2-13
K HT
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 31/60
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 principal,
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 W.m (resp.
1
W.m.
-
rad) and that
1
1
other terms is it
2
W.m (resp.
1
2
W.m.
-
rad). 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.
1
2
2
N 1
+
N
n+
U
- U
U
n+
H
H
n+
1
2
1
H ^s - C
+ div U
H
K
p
(
1
H
+
H
F
+
,
)
2
1
,
T
2 FS K
N
K
~
N 1
+
(K)
()
0,
0, F
:=
2
2
N 1
+
N 1
U
+
U
N 1
+
H,
N 1
+
H,
^
H ^
G
-
+
H H
hu
F
-
-
F
(H) N 1+
FS K
N
FS K
N
2 (
)
3 (
)
0, F
0, F
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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 32/60
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, of (H6) and by using definition 10, one has, at the total level,
the “weak” increase of the error (with K
PC C >) via the history of
2 (
, (),
,
4
5
) 0
,
indicators
1
1
2
2
2
C U
U
4 1
C U
U
T
U 1 U 1
p (
N
N
N - nh)
+ (-) -
p (m -
m
H)
+ -
(m+
m+
-
, H)
0,
0,
0,
m=
0
m=
0
N
2
0 N NR
(4 -)
3
C U
u0
K T
2
p (
-
0
) +
2
(m
H
()
0,
m=0
éq 4.2-15
or more simply
1
2
2
C U
U
T
U 1 U 1
p (
N
N - nh)
+ -
(m+
m+
-
, H)
0,
0,
m=
0
N
2
0 N NR
C U
u0
K T
2
éq
4.2-16
p (
-
0
) +
2
(m
H
()
0,
m=0
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
N 1
+
N 1
v:
+
= U
- U
éq
4.2-17
, H
One évince the term of exchange by the usual argument
(H (U - U
éq
4.2-18
H)
n+1
(n+1 n+
U
-
1
U
dx
, H)
> 0
3
It is necessary to apply the easy way [éq 3.1-4] to the cross term (2 -) C
+1
1
- +
-
1
and on
p (U N
U nh) (a unh) dx
the product utilizing the indicator. One has then to find the parameters and checking a system
type [éq 3.2-8]
2
P ()
2
2
-
= 1
,
éq
4.2-19
2
2
- 1 - 2 = 1
1
who admits solution only if the diagram is unconditionally stable (
). From where increase
2
[éq 4.2-15] [éq 4.2-16] while taking
2
P ()
K =
max C, C
éq
4.2-20
2
(2 2
4
5)
,
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 33/60
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, K2)).
·
By adopting the less restrictive approaches (H4) and (H5), 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-uh and while taking as term (bn, v
) 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 K2 constant changes. It is is multiplied by checking the C6 constant
(cf [bib7] pp90)
2
2
2
v
+
éq
4.2-21
K
v
C v
6
,
1
,
1
,
1
KT
F
F
H
HS
~
K:= C K
éq
4.2-22
2
6
2
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)
H
RF N =
: RF N = RF N = RF
0
(H8)
H
H
(N T)
N
NR
the preceding property produces the following corollary then:
Corollary 11bis
Under the assumptions of property 11 while supposing (H8), one with the increase of the space error
total expressed in temperature
1
2
2
C T
T
T
T1 T1
p (
N
N - nh)
+ -
(m+
m+
-
, H)
0,
0,
m=
0
N
2
0 N NR
C T
T 0
K T
2
éq
4.2-23
p (
-
0
) +
2
(m
H
()
0,
m=0
by using definition 10 of the indicator also expressed in temperature
U T, s^ S, G G and h^
^
HT
éq 4.2-24
ext.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Code_Aster ®
Version
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Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
O. BOITEAU Key
:
R4.10.03-A Page
: 34/60
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
n+1
N
n+
U
- U
1
U 1
n+1
K: H S 1
^
C
div
U 1
H
,
p, T (
)
R
n+
=
-
H
H +
K
p
(
n+
H,)
+
sF
H
+
T
p
2 F S K
N
L (K)
()
Tl (K)
éq
4.3-1
n+1
n+1
U
U
S
n+
H G 1 -
H,
+
H H 1
,
^
^
hu
1
F
S n+ -
H
-
F
(H) n+
FS2 (K)
N
N
T
F S K
L (K)
3 ()
T
L (K)
where constant the R and S are worth
6
T,
1
p > 1 (
2D Q = 2) or
p
(
3D Q =)
3
5
R (Q, p)
Q
Q
:= Q +1- -
éq
4.3-2
2
p
1
- 1
- 1
S (Q, T)
Q
Q
:= Q - -
-
2
2
T
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 (N 1
+
who corresponds to p=t=2), some
2,2 (K)
use the voluminal standard L1 (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 approximate field solution uh
accused thermal problem. However the exact estimate of the indicator is not always
possible when one has complicated 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 modelling the field of temperature.
Note:
·
In both cases the loadings are “prisoners of the selected vision finite elements”
to model the field solution.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Code_Aster ®
Version
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Titrate:
Indicator of error in residue for transitory thermics
Date:
03/06/02
Author (S):
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:
R4.10.03-A Page
: 35/60
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-Cotes.
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)
N 1
+
N 1
+
N 1
+
N 1
S
, G
,
+
T
H
and
éq
4.3-3
, H
, H
ext., H
, H
N 1
+
N 1
+
N 1
+
N 1
^
^s, ^g,
+
H
H
and
éq
4.3-4
, H
, H
, H
, H
in spaces of voluminal approximation (for the source) and surface (for the loadings)
X =
: v L2/K
T
v
P K
H (
) {H
()
H
H K
L (
)}
1
éq
4.3-5
X =
: v L2/F
S
v
,
P F
H (I)
{H (I)
H I
H
(
F
I)}
I
it
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 P2, this approach does not provide
results very different from those obtained by a direct calculation via good the derivative second.
Note:
· The indices l1, l2, l3 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 L K - 2
.
I
(I =, 1)
3
,
2
Definition 10 and the weak estimate 11 associated are rewritten then in the following form. This
new definition, N 1
+
, by a R (one is subscripted takes again in that the usual notations of [bib6]
R
(K)
and [bib7]) (for “reality”) in order to notify well that it corresponds better to the values which are calculated
indeed in the code.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 36/60
Definition 12
K
In the framework of the operator of transitory thermics linear of Code_Aster, continuation (N
K
R (
) HT
0nN
real local indicators can be written in the form
n+1
N
n+
U
- U
1
U 1
n+1
K: H S 1
^
C
div
U 1
H
,
R
()
n+
=
-
H
H +
K
, H
p
(
n+
H,)
+
H
F
+
T
2 FS
N
(K)
0, K
0, F
n+1
n+1
U
U
n+
H G 1 -
H,
+
H H 1
,
^
^
H U
1
F
, H
n+ -
H
-
F
, H
(H H) n+
FS2 (K)
N
FS3 (K)
N
0, F
0, F
éq 4.3-6
It is initialized by
1
U 0
0
K: H s0
^
div
U 0
H
R (
) =
+
K
H
(H) +
0, K
H
F
+
2 FS
N
(K)
0, F
éq 4.3-7
0
0
U
U
H G 0 -
H
+
H h0
^
^
h0u 0
F
H
-
H -
F
H
H
H
FS2 (K)
N
FS3 (K)
N
0, F
0, F
Continuation (N
()
real total indicators is defined as being
0nN
1
0
N
N
NR
K
éq 4.3-8
R () =
NR () 2
2
:
K HT
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] N
~
and [éq 4.3-1], [éq 4.3-2] N
.
R, p, T (K)
R (K)
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):
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 37/60
Property 13
Under the assumptions of properties 6, of (H6) and by using definition 12, one has, at the total level,
“weak” increase of the error (with K
PC C >) via the history of
2 (
, (),
,
4
5
) 0
,
real indicators
1
2
2
C U
U
T
U 1 U 1
p (
N
N - nh)
+ -
(m+
m+
-
, H)
0,
0,
m=
0
N 1
2
0
2
2
2
N NR
C U
u0
K T
0
K
1
K
H2S 1
^
S 1
^
p (
-
0
H)
+
2
(R () + - ({m+R ()
m+
m+
+
-
K
, H
} +
0,
0, K
KT
m=
H
0
N 1
2
2
K T
^
^
^
^
2
-
m+1
m+
H G
- G 1
+
H H 1. 1
H U
1
hu
1
F
, H
0, F
m+
m+
-
-
F
, H
(H H) m+ + (
H) m+
0, F
KT M=0 F
H
S2 (K)
FS3 (K)
éq 4.3-9
Under (H8), one with the same expression in temperature
1
2
2
C T
T
T
T1 T1
p (
N
N - nh)
+ -
(m+
m+
-
, H)
0,
0,
m=
0
N 1
2
0
2
2
2
N NR
C T T 0
K T
0
K
1
K
H2S 1 S 1
p (
-
0
H)
+
2
(R () + - ({m+R ()
m+
m+
+
-
K
, H
} +
0,
0, K
KT
m=
H
0
N 1
2
2
K T
2
-
m+1
m+
H G
- G 1
+
H
H T
T
1
H T
T
1
F
, H
0, F
F (H (
-
ext., H
H) m+ - ((
-
ext.
H) m+
0, F
KT M=0 F
H
S2 (K)
FS3 (K)
éq 4.3-10
by using definition 12 of the indicator also expressed in temperature
U T, s^ S, G G and h^
^
HT
éq 4.3-11
ext.
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:
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 38/60
Lemma 14
It is shown that there are constants strictly positive Ci (i=6… 11) checking
1
v
P
sup K, L, L, L (K)
{1 2 3}
C
v
v
C
2 v
6
K
7
K
0, K
0, K
0, K
v
C h-1 v
K
8 K
K
0, K
0, K
- 1
1
v
2
P
éq 4.4-1
sup K, L, L, L (F)
{1 2 3}
C H
Statement
v
C
2 v
9 F
K
F
10
F
0, F
0, F
0, F
v
C h-1 v
K
11 F
K
0,
0,
F
F
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].
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 39/60
Property 15
Under the assumptions of property 6, of (H6) and while being based on definition 12 and the lemma 14, one
has, at the local level, the “weak” decrease of the error (with K C I =
>) via
3 (
,
6
I
L)
11
0
the real indicator
N 1
+
N 1
U
- +
U
- one - N
U
H
C
H
H
+
K
p
(
N 1
+
N 1
U
- +
U
+
, H) 0,
T
K
0,
K
1
n+
n+
n+
n+
n+
1
R (K) K
H ^ 1
S
- ^ 1
S
+ H ^ 1
2 G
- ^ 1
G
+
3
K
, H
F
, H
0,
0,
K
K
2
1
^n+
n+
n+
n+
1
2
^ 1
H H
- H
- hu
+ H U
F
, H
(H) 1 (
H H) 1
0,
K
3
0
N NR - 1
éq 4.4-2
Under (H8), one with the same expression in temperature
N 1
+
N 1
T
-
+
T
- T N - N
T
H
C
H
H
+
K
p
(
N 1
+
N 1
T
-
+
T
+
, H) 0,
T
K
0,
K
1
n+
n+
n+
n+
n+
1
R (K)
1
1
1
1
2
K
H S
- S
+ H G
- G
+
3
K
, H
F
, H
0,
0,
K
K
2
1
N 1
2
+
H H
F
(N 1+ N 1
T
- +
T
- n+
n+
H
T
- n+
T
ext.,
)
1
, H (
1
1
ext., H
, H)
0,
K
3
0
N NR - 1
éq 4.4-3
by using definition 12 of the indicator also expressed in temperature
U T, s^ S, G G and h^
^
HT
éq 4.4-4
ext.
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 v - v by the product W
H
K making
to intervene the function “bubble” of K
n+1
N
+1
U
- U
N
H
H
K
T
v =
: s^
- C
+ div U +1
H
K
, H
p
(
N
, H)
T
éq
4.4-5
W =
: v
K
K K
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 40/60
From where succession of increases, via [éq 4.4-1] and the inequality of Cauchy-Schwartz,
2
2
n+
U 1
2
- n+
U 1 - N
U - N
U
v
C W v dx C
, W
U 1 has U 1, W
S 1
^
S 1
^, W
K 0, K
7 K K
7
H
H
+
K
(n+ - n+
, H
K) - (n+ -
n+
, H
K)
T
K
0,
n+1
n+1
N
N
C 2max,
1
^
^
7
(C8) U - U - U - U
H
H
+ -
H 1
U 1 U 1
S 1 S 1
W
K
(n+ - n+
, H)
n+
n+
+
-
, H
0, K
T
0,
0,
0, K
K
K
K
2
n+1
n+1
N
N
v
C
U
U
U
U
7 max,
1 C
U 1 U 1
S 1
^
S 1
^
K
(8)
-
-
-
H
H
+
(n+ - n+
, H)
n+
n+
+
-
, H
0, K
C6
0, K
0, K
T
0, K
éq 4.4-6
Then, one reiterates the same process for the surface terms WF, I
N 1
+
F
S (K)
uh,
S
v
:=
H,
F 1
,
N
éq
4.4-7
W
:= statement
F 1
,
K
F K 1
,
N 1
+
F
S (K)
U
N 1
+
H,
S
v
:= ^g
-
H, 2
F, 2
H,
N
éq
4.4-8
W
:= statement
F, 2
F
F F, 2
N 1
+
F
S (K)
U
N 1
+
H,
^
S
v
:
N
= H
-
- H U +
H, 3
F, 3
H,
(H H) 1
N
éq
4.4-9
W
:= statement
F, 3
F
F F, 3
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 41/60
Maybe, for example, for i=1 succession of increases, via [éq 4.4-1] and the inequality of
Cauchy-Schwartz,
n+
U 1 - n+
U 1 - N
U - N
U
H
H, W
+ U 1 U 1 has, W
2
F 1
,
(n+ - n+
, H
F 1
,)
2
2
v
C
W v D
T
C
F 1
,
0, F
10 F 1, F 1,
10
0,
F
- (n+
S 1
^
- n+
S 1
^, W
v, W
, H
F 1
,)
- (K F 1,)
0,
0,
n+
U 1 - n+
U 1 - N
U - N
U
H
H
+ -
H 1 U 1 U 1
F
(n+ - n+
, H)
C 2 max,
1
10
(C11)
0,
T
F
0,
W
F
F 1,
0, F
n+1
n+1
+ s^
- s^
+ v
, H 0,
K 0, F
F
1 n+
U 1 - n+
U 1 - N
U - N
U
- 1
2
2
H
H
H
+ H
U 1
2
U 1
F
F
(n+ - n+
, H)
0,
C
T
v
10 max,
1 C
0,
F 1
,
(11)
F
F
0, F
C9
1
1
+
n+
H S 1
2 ^
- n+
S 1
^
+ H2 v
F
, H
F
K
0,
0, F
F
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 H H and
v
v
v
with F S
).
F
K
(K)
0,
0,
F
K
!
Note:
·
This local decrease of the error is also declined according to the formulations [éq 4.2-14]
N
~
and [éq 4.3-1], [éq 4.3-2] N
.
R, p, T (K)
R (K)
·
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 (H4) and (H5), 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 N
, one is not ensured of a reduction in the error
R (K)
in an immediate vicinity of the zone 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).
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
: 42/60
4.5 Complements
The K3 constant just like its alter preceding ego, K2, 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 constraint,
SR. GAGO [bib10] proposes (on a problem 2D models) a dependence of the K2 constant in
function of the type of finite elements used. It is written
~
K2
K:=
éq
4.5-1
2
2
24 p
where p is the degree of the polynomial of interpolation used (p=1 for the 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
1
“corrective” constant
. This strategy was implicitly retained for the calculation of
2
24 p
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 ways 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 step 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 step 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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
K × [T, T
who is the balanced sum of three terms:
N
N 1
+]
1) the residue of the calculated solution and the data discretized compared to the strong formulation
problem (P
K ×
0) evaluated on
[T, T,
N
N 1
+]
2) the jump space through K × [T, T
of the operator traces associated (who connects naturally
N
N 1
+]
formulations weak and strong via the formula of Green),
3) the temporal jump through K × [T, T
calculated solution.
N
N 1
+]
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 step 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, the fields of calculated temperatures T N (0 < N NR) 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 step of time must remain between two terminals:
T
H < T
< T
éq
4.5-2
min ()
(
max
)
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 step 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 is too large one 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 P2 elements 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 incidence. 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 HOMARD (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 principal contributions of the preceding theoretical chapters and theirs
holding and bordering with respect to the thermal calculation set up in Code_Aster.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
5
Summary of the theoretical study
That is to say (P0) the problem in extreme cases interfered (inhomogenous Cauchy-Dirichlet-Neumann-Robin type
linear and with variable coefficients) solved by operator THER_LINEAIRE
T
C
- div
p
(T) = S
×],
0 [
T
T = F
×
1
], 0 [
T
(P)
G
0
=
×
2
], 0 [éq
5-1
N
T
+ HT = HT
×
ext.
3
], 0 [
N
T (X)
0
, = 0
T (X)
Taking into account the choices of modelings operated in Code_Aster (by AFFE_MATERIAU,
AFFE_CHAR_THER…) one determines Cadre Variationnel Abstrait (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 constraints
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 principal theoretical results concerning the indicator of error, we go
repréciser some notations:
·
one fixes a step of time T
such as
either an entirety NR and that temporal discretization or
T
regular: T =,
0 T = T
, T = 2 T
T = N T
,
0
1
2
L N
Note:
This assumption of regularity does not have really importance, it allows just
to simplify the writing of the semi-discretized problem. To model a transient
unspecified at the moment tn, it is just enough to replace T by tn=tn+1-tn.
·
that is to say the parameter of - method semi-discretizing temporally (P0),
·
are N
T and N
T fields of temperatures at the moment T 0
, exact solutions
N (
N
NR)
H
initial problem (P0), respectively semi-discretized in time and completely discretized in
time and in space.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 model 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
N 1
+
N 1
+
N 1
+
N 1
S, G
,
+
T
H
and
éq
5-2
, H
, H
ext., H
, H
while posing
N 1
+
=
,
X (N +)
1
+ (1 -),
X N
with {T, S, T, G,
and
éq
5-3
ext.
}
H
0 N NR - 1
T
T
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 grid
coarse.
They exist then two constants K2 and K3 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)
1
2
2
C T
T
T
T1 T1
p (
N
N - nh)
+ -
(m+
m+
-
, H)
0,
0,
m=
0
N 1
2
0
2
2
2
N NR
C (T - H
T
K T
0
K
1
K
H2S 1 S 1
p
0
0)
+
2
(R () + - ({m+R ()
m+
m+
+
-
K
, H
} +
0,
0, K
KT
m=
H
0
N 1
2
2
K T
2
-
m+1
m+
H G
- G 1
+
H
H T
T
1
H T
T
1
F
, H
0, F
F (H (
-
ext., H
H) m+ - ((
-
ext.
H) m+
0, F
KT M=0 F
H
S2 (K)
FS3 (K)
éq 5-4
·
A decrease of the local space error (it “underestimates” the local space error)
N 1
+
N 1
T
-
+
T
- T N - N
T
H
C
H
H
+
K
p
(
N 1
+
N 1
T
-
+
T
+
, H) 0,
T
K
0,
K
1
n+
n+
n+
n+
n+
1
R (K)
1
1
1
1
2
K
H S
- S
+ H G
- G
+
3
K
, H
F
, H
0,
0,
K
K
2
1
N 1
2
+
H H
F
(N 1+ N 1
T
- +
T
- n+
n+
H
T
- n+
T
ext.,
)
1
, H (
1
1
ext., H
, H)
0,
K
3
0
N NR - 1
éq 5-5
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
K
·
With continuation (N
K of local real indicators (by using the notations of
R (
) HT
0nN
[§4.1])
n+1
K
1
:
K
1
K
1
K
1
K
R
()
n+
= R, flight ()
n+
+ R, jump ()
n+
+ R, flow ()
n+
+ R, éch ()
n+
T1 - N
T
1
N
T1
éq 5-6
n+
=
: H S 1 -
H
H
C
+ div T1
H
,
K
, H
p
(nh,)
+
+
+
H
F
+
T
2 FS
N
(K)
0, K
0, F
n+1
n+1
T
T
n+
H G 1 -
H,
+
H
H T
T
1
,
F
, H
F ((
-
ext.
) n+ -
H
, H
FS2 (K)
N
FS3 (K)
N
0, F
0, F
who is initialized by
1
T 0
0
K: H s0 div T 0
H
R (
) =
+
K
H
(H) +
0, K
H
F
+
2 FS
N
(K)
0, F
0
0
T
0
0
T
H G -
H
+
H
H T
T
F
H
F ((
-
ext.
)
-
H
H
FS2 (K)
N
FS3 (K)
N
0, F
0, F
éq 5-7
This local continuation makes it possible to build continuation (N
()
real indicators
0nN
total
1
0
N
N
NR
K
éq 5-8
R () =
NR () 2
2
:
K HT
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 N
, one is not ensured of a reduction in the error in an immediate vicinity
R (K)
zone concerned (in K). The indicator “locally underestimates” the error space and only
a more macroscopic refinement carries out a reduction in the error theoretically.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 step of time), its theoretical limitations are thus, at best, those
inherent in the resolution of the problem in temperature: no zones 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 grid 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 encumbers, this perimeter of “theoretical” use.
But it is necessary well to keep in mind, that as a “simple postprocessing” of (P0),
the indicator cannot unfortunately provide more reliable diagnosis in the zones 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 guarantee,
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 (P0), 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 to
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 ways 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 [éq 5-6].
To finish, it should be stressed that this indicator is thus composed of four terms:
·
the principal 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 3D (resp. in 2D-AXI), if the unit of the geometry is the meter, the unit of first is
1
1
W.m (resp.
1
W.m.
-
rad) and that of the other terms are it
2
W.m (resp.
1
2
W.m.
-
rad). 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 K scale, 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 CALCUL. 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 grid 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 tests model (TPLL01A/H for
2d_PLAN/3D and TPNA01A for the 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 option “ERTH_ELEM_TEMP” of the operator of post-
processing CALC_ELEM, on a EVOL_THER (provides to key word RESULTAT) 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 option “FLUX_ELNO_TEMP” of
CALC_ELEM which 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 option “ERTH_ELNO_ELEM” (after the call to
“ERTH_ELEM_TEMP”) makes it possible 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 HOMARD encapsulated in MACR_ADAP_MAIL),
natural outlet of our cartographies of error, does not make it possible to treat only parts of
grids.
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” zone of the grid and the zone
“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 key word PARM_THETA; if necessary if transitory problem is dealt with, it is necessary
to inform usual fields TOUT/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 are a ALARME) or the data of a 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 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').
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
However, in the event of conflict between loadings of the same type, one often can and
easily to find a solution palliative via the adequate AFFE_CHAR_THER. The user is
informed presence of several occurrence of the same type of loading by one
message of ALARME and the list of the 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 of the coefficient of exchange and absence of the CHAMPGD of the outside temperature or
vice versa….),
·
Within the same general framework: value of the model (parameter MODELE), necessary materials
(CHAM_MATER), of the structure EVOL_THER given (RESULTAT) and result (assignment of
Réentrant CALC_ELEM with possibly “reuse”). 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 ALARME or one
ERREUR_FATALE stops it, one cannot all control and/or prohibit!). There remains only judge then
relevance of its results.
Let us recapitulate all this parameter setting of 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)
MODELE
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)
RESULTAT
EVOL_THER of calculation
thermics (O)
reuse
EVOL_THER of calculation
thermics (C)
EXCIT CHARGES
Idem thermal calculation +
regulate of overload (O)
OPTION
“FLUX_ELNO_TEMP”
“ERTH_ELEM_TEMP”
“ERTH_ELNO_ELEM”
INFO
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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
6.3
Presentation/analysis of the results of the error analysis
Option “ERTH_ELEM_TEMP” provides in fact, not one, but fifteen components by elements
stop K and by step of time tn+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 zone: it is enough to impose, only in this zone, 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
Term of standardization
(in %)
Term
N 1
+
N 1
+
K
N
NR 1
+
K:= H S 1
+
R, flight (
)
N
R, flight (
)
R, flight (K)
voluminal
×100
K
, H 0, K
TERMVO
N 1
+
NR
K
R, flight (
)
.
TERMV1
TERMV2
Term of
N 1
+
N 1
+
K
1
R, jump (
)
R, jump (K)
jump
×
N
+
1
+
H2
T1
N
F
, H
TERMSA
n+
NR
NR
K:=
R, jump (
)
R jump (K)
.
100
1
,
2
N
TERMS2
0, F
TERMS1
Term of flow
N 1
+
N 1
+
K
1
R, flow (
)
R, flow (K)
×
N
NR 1
+
K:= H2 G 1
+
R, flow (
)
N
TERMFL
n+
NR
F
, H 0, F
R flow (K)
.
100
1
,
TERMF1
TERMF2
Term
N 1
+
N 1
+
K
1
R, éch (
)
R, éch (K)
of exchange
×100
N
NR 1
+
K:= H2 H T - T
1
+
R, éch (
)
F ((ext.
) N
TERMEC
N 1
+
NR
K
, H 0, F
R, éch (
)
.
TERME2
TERME1
Total
n+1
K
1
:
K
N 1
+
n+
NR 1 K:
NR 1 K
R
() = n+R, I ()
R (K)
R (
) = n+R, I ()
×
I
n+
NR
I
R
(K)
.
100
1
ERTABS
TERMNO
ERTREL
Table 6.3-1: Components of the indicator of error.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
For the absolute error and the term of standardization, in 2D-PLAN or 3D (resp. in 2D-AXI), if the unit
geometry is the meter, the unit of the first term is W.m (resp.
1
W.m.
-
rad) and that of
1
1
other terms is it
2
W.m (resp.
1
2
W.m.
-
rad).
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 grid (one
fact the same thing as in table [Table 6.3-1] by replacing K by) and traced
in a table of the file result (.RESU).
**********************************************
THERMIQUE: ESTIMATOR Of ERROR IN RESIDUE
**********************************************
IMPRESSION OF THE TOTAL STANDARDS:
SD EVOL_THER RESU_1
SEQUENCE NUMBER 3
INSTANT 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 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 the thermal SD_RESULTAT. The variables of access of this field are
, for each mesh (in our M1 example), the sequence number (NUME_ORDRE) and the moment
(INST). With option “ERTH_ELNO_ELEM” one with the same thing for each node of
the element considered.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
·
One can also trace the values of each one of these components in the file message
(.MESS) by initializing key word INFO 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
DIAMETRE 3.5355335898314D-02
EDGES OF THE TYPE SEG2
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
NUMBER Of EDGE/HF 1 2.4999997764826D-02
A NUMBER OF SUMMITS 2
CONNECTIQUE 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 [Exemple 6.3-1] and [Exemple 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
the 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 “it does not have there limiting conditions on this edges,
·
Tests of not-regression “numérico-data processing” showed that the manner of
to model 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 principal is very badly estimated since it requires a double derivation of the field of
temperature. A ALARME thus warns the user and the enjoint to pass into quadratic.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 type hull (COQUE_PLAN, COQUE_AXIS, COQUE), 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 grid
comprise licit and illicit elements, calculation does not stop and, via the 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 grid pertaining to the model
indicated by key word MODELE of command CALC_ELEM. One can thus work with
grids (not cleaned) comprising “meshs of outline” to which one allots one
different model.
2) In
one
grid 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 the TRIA/QUAD
with the SEG and relations TETRA/PENTA/HEXA with the FACE. For example, in the event of
presence of segments in a three-dimensional grid, the option will not stop but it
will not take into account their (possible) contributions.
3) Preliminary option “ERTH_ELEM_TEMP” and its options do not know the 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 grid 3D and are not
generated that by the voluminal free maillor of GIBI, which creates some locally for
to supplement portions of grids 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 contiguous FACE.
5) the same type of imbroglio occurs when two points of the grid are superimposed
geometrically. There still, calculation should not stop, but the value of
the indicator will be incomplete on the level of this zone of recovery,
6) If one works with a grid 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.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
7) If one wants to refine or déraffiner his grid with MACR_ADAP_MAIL [U7.03.01], it
grid 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 zone concerned with the GROUP_NO without seeing itself affecting one
unspecified loading (one cannot modify the composition of a GROUP_NO in the course of
session!).
For specific loadings or points of statement (on which go, for example,
to rest POST_RELEVE_T) the GROUP_NO is licit. On the other hand, it is not advised
to directly use meshs (MA) or nodes (NO) (apart from group), because in it
case, to the liking of the renumérotations, HOMARD 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. Grâce à 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 a 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 modification of the problem in
enthalpy. However one does not treat the possible contributions of non-linear loadings
(FLUX_NL and RAYONNEMENT). The user is informed by a ALARME, just like it by 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 loadings SOURCE, FLUX_REP and ECHANGE. 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 encapsulation of HOMARD® (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 Grid
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= [None] * 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
MATE [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],
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 (CHARGE = 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 grid
M [k+1] =CO (“M_ % of % (k+1))
# Adaptation of the grid while basing itself on component ERTABS of
# ERRE_ELEM_TEMP of RESU [K].
# Old grid: M [K]. Grid 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 HOMARD®, coupling
ERTH_ELEM_TEMP/MACR_ADAP_MAIL [U7.03.01] simulates the circulation of a “hot” fluid of share and
of other of a metal part bent (in top and bottom, via a condition of ECHANGE 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 grid thus remains fine in
edge of part, in the zone 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 zones) this established fact announces (even when the part cooled).
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
Example 6.5-2: Use of option “ERTH_ELEM_TEMP” coupled with HOMARD
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
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 grid.
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 grid 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] is 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 constraints 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 RAYONNEMENT)
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-diffusion (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 step of time by finite elements. What
from the point of view of architecture supporting the finite elements of the code is true
challenge!
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
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
8 Bibliography
[1]
R. DAUTRAY & J. - L. LIONS and mathematical Al Analyze 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 Maillage”:
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 handbook 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. FORTIN and Al Estimation a posteriori and adaptation of grids. European review of
finite elements. Vol. 9, 4, 2000.
[18]
I. BABUSKA & W. RHEINBOLT. A posteriori error estimates for the finite element method.
International Journal for Numerical Methods in Engineering, vol. 12, pp.1597-1615, 1978.
Handbook of Référence
R4.10 booklet: Estimator of error a posteriori
HI-23/02/014/A
Outline document