Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 1/66
Organization (S): EDF-R & D/SINETICS
Handbook of Référence
R4.03 booklet: Analyze sensitivity
Document: R4.03.02
Calculation of sensitivities in thermics
Summary:
During digital simulations obtaining a rough result is not sufficient any more. The user is moreover in
petitioning of calculation of sensitivity compared to the data input of the problem. That allows him
to estimate the uncertainty which the field result according to the law of variation of the data answers. This
derived is also the basic substrate of problems opposite (retiming of parameters…) and of problems
of optimization.
This sensitivity can be obtained “manually”, but the experiment shows that these parametric studies
are often expensive, little mutualisables and less reliable than an analytical calculation established in the software of
calculation.
In this note, one places oneself in the perimeter of use of the standard thermal operators of
Code_Aster and one are interested in this analytical sensitivity of the field of temperature and its flow by
report/ratio with the characteristics material and the loadings. One described the process allowing there to exhume
the linear system which this derivative checks. In order to minimize the overcost calculation, a particular effort was
brought to bind its resolution to that of the initial problem.
One details theoretical, numerical work and data processing which governed the establishment of these
calculations of sensitivity in the code. One specifies their properties and their limitations while connecting these
considerations with a precise parameter setting of the accused operators and with the choices of modeling of the code. One has
tried constantly to bind different the items approached while detailing, has minimum, the demonstrations a little
techniques.
Required environment, the parameter setting and the perimeter of use of this new functionality
are described. An example extracted from an official case-test is clarified.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 2/66
Contents
1 Problems ....................................................................................................................................... 4
2 linear Thermics ................................................................................................................................. 8
2.1 Derived compared to voluminal heat .................................................................................. 10
2.1.1 Theoretical elements ............................................................................................................ 10
2.1.2 Establishment in Code_Aster ......................................................................................... 13
2.2 Derived compared to thermal conductivity ........................................................................... 15
2.2.1 Theoretical elements ............................................................................................................ 15
2.2.2 Establishment in Code_Aster ......................................................................................... 16
2.3 Derived compared to the source ..................................................................................................... 18
2.3.1 Theoretical elements ............................................................................................................ 18
2.3.2 Establishment in Code_Aster ......................................................................................... 18
2.4 Derived compared to the imposed temperature ............................................................................. 19
2.4.1 Theoretical elements ............................................................................................................ 19
2.4.2 Establishment in Code_Aster ......................................................................................... 20
2.5 Derived compared to normal flow imposed ................................................................................... 22
2.5.1 Theoretical elements ............................................................................................................ 22
2.5.2 Establishment in Code_Aster ......................................................................................... 22
2.6 Derived compared to the coefficient from convectif exchange ................................................................ 24
2.6.1 Theoretical elements ............................................................................................................ 24
2.6.2 Establishment in Code_Aster ......................................................................................... 25
2.7 Derived compared to the outside temperature ........................................................................... 26
2.7.1 Theoretical elements ............................................................................................................ 26
2.7.2 Establishment in Code_Aster ......................................................................................... 27
3 nonlinear Thermics ........................................................................................................................ 28
3.1 Derived compared to voluminal heat .................................................................................. 29
3.1.1 Theoretical elements ............................................................................................................ 29
3.1.2 Establishment in Code_Aster ......................................................................................... 34
3.2 Derived compared to thermal conductivity ............................................................................. 37
3.2.1 Theoretical elements ............................................................................................................ 37
3.2.2 Establishment in Code_Aster ......................................................................................... 38
3.3 Derived compared to the source ..................................................................................................... 38
3.3.1 Theoretical elements ............................................................................................................ 38
3.3.2 Establishment in Code_Aster ......................................................................................... 39
3.4 Derived compared to the imposed temperature ............................................................................. 39
3.4.1 Theoretical elements ............................................................................................................ 39
3.4.2 Establishment in Code_Aster ......................................................................................... 40
3.5 Derived compared to linear normal flow imposed ...................................................................... 41
3.5.1 Theoretical elements ............................................................................................................ 41
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
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6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
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3.5.2 Establishment in Code_Aster ......................................................................................... 42
3.6 Derived compared to non-linear normal flow imposed ............................................................... 42
3.6.1 Theoretical elements ............................................................................................................ 42
3.6.2 Establishment in Code_Aster ......................................................................................... 43
3.7 Derived compared to the coefficient from convectif exchange ................................................................ 43
3.7.1 Theoretical elements ............................................................................................................ 43
3.7.2 Establishment in Code_Aster ......................................................................................... 44
3.8 Derived compared to the outside temperature ........................................................................... 45
3.8.1 Theoretical elements ............................................................................................................ 45
3.8.2 Establishment in Code_Aster ......................................................................................... 46
3.9 Derived compared to emissivity/constant of Stefan-Boltzmann .............................................. 46
3.9.1 Theoretical elements ............................................................................................................ 46
3.9.2 Establishment in Code_Aster ......................................................................................... 47
3.10
Derived compared to the temperature ad infinitum ...................................................................... 48
3.10.1
Theoretical elements ................................................................................................ 48
3.10.2
Establishment in Code_Aster ............................................................................ 49
4 Summary of the sensitivities of the temperature .................................................................................. 50
5 Sensitivity of the heat flow ............................................................................................................... 54
6 Implementation in Code_Aster .................................................................................................... 56
6.1 Particular difficulties .................................................................................................................. 56
6.2 Environments necessary/parameter settings ................................................................................. 56
6.3 Perimeter of use .................................................................................................................... 60
6.4 Example of use ...................................................................................................................... 61
7 Perspective Conclusion/....................................................................................................................... 63
8 Bibliography ........................................................................................................................................ 64
Appendix 1
Concept of derived “within the meaning of the distributions” ...................................................... 65
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 4/66
1 Problems
During digital simulation, obtaining a result starting from a data file is not sufficient any more.
Taking into account uncertainties which weigh on the evaluation of the loadings, the geometries and of
characteristics material, taking into account also the numerical approximations due to modelings
employees, with their space-time discretizations and the algorithms of resolution, the user is
increasingly petitioning of calculations of sensitivity [bib5]. One then seeks to evaluate
sensitivity of a variable compared to an input datum of the problem. It makes it possible to estimate
the uncertainty (the probabilistic function) to which answers the field result according to the law of
variation of certain data.
This derivative can be estimated “manually”, but these of parametric studies are often
expensive, little mutualisables and less reliable than an analytical calculation established in the code.
Note:
· The sensitivities by finished differences are of course dependant on the parameters on
shift and of the grid, but into non-linear, another worsening factor is superimposed:
degree of convergence of the solution. In any rigor, this one also intervenes on quality
analytical sensitivities, because one uses the field of temperature solution for
to assemble the linear system “derived”.
· The third way gathers the techniques of automatic differentiation (ODYSSEY [bib7],
[bib9]…) but they are not plantable in Code_Aster because of sound
software architecture (transmission of arguments between the routines by pointer…). Of
any way these products are still “relatively embryonic” and their use
seem fixed quotas for with model problems or parts of software well
specific. The ideal would be of course to incorporate these problems as of the compiler…
Recently, the introduction of calculations of sensitivity of mechanical thermo fields [R4.03.01] and of
rate of refund of energy [R7.02.01] compared to a variation of field, showed
relevance and the feasibility of this type of approach in Code_Aster. By coupling this last with
software PROBAN, one can thus know the probability of starting of the rupture for a distribution
of variation of field given. This type of studies mechanic-reliability engineers, for example, was carried out in
the framework of project PROMETE [bib4] to determine the probability of rupture of a tank REP in
considering the variability the thickness of its lining.
These sensitivities can also intervene in a crucial way in the resolution of problems opposite
(retiming of parameters…) and in many problems of optimization.
In this document, one restricts oneself with the linear and non-linear thermal problems
Code_Aster and thus with the analytical sensitivities of the field of temperature T (and its flow) by
report/ratio with the characteristics material and the loadings. One places oneself in the perimeter
of use of the standard thermal operators (the loadings are supposed to be fixed, one
do not thus be interested in the phenomena of convection-diffusion in pointer of
THER_NON_LINE_MO [R5.02.04]) for isoparametric finite elements (one does not treat it
thermal problem for the thin hulls [R3.11.01] (modeling COQUE_ *) and for
elements of Fourier (resp. AXIS-FOURIER)) (THER_LINEAIRE [R5.02.01] and THER_NON_LINE
[R5.02.02]) and also in that of the operators of preprocessing of data (DEFI_MATERIAU
[U4.44.01], AFFE_MATERIAU [U4.44.03] and AFFE_CHAR_THER [U4.44.02]).
One is interested only in derivation of T and his flow, fields depending on the variables of space
X and of time T and parameters materials and loadings, compared to one of these parameters
(which must be a constant scalar by geometrical zones (these under-parts are supposed to be distinct
and motionless, one neglects thus in particular the phenomena of dilation)). Thus let us consider, by
example, a Bi-material whose isotropic thermal conductivity is a constant reality by zones: 1 on
1 and 2 out of 2
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
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: 5/66
1
2
1
2
Appear 1-a: Désignation of thermal conductivities for a Bi-material
One thus models total thermal conductivity in the form:
(X):= I X + I X
,
éq
1-1
1 1 ()
2 2 ()
(1)
2
2
1 if X
with I
I
I the indicating function of the ième part (I X
). One is interested in the sensitivity
I () =
: 0 if not
T
field of temperature compared to one of the two parameters
(, xt)
. The problem
I
0
i= I
formulate same manner for a loading or a limiting condition.
By parameterizing advisedly the “derived loadings and materials” (cf [§6.2], [§6.4]) calculations
developed thereafter and the data-processing developments that they subtend can too
to take into account the modelings more sophisticated with several space dependences and
temporal. For example, considering a thermal source,
S (X, T):= S I X X +
X X
éq
1-2
1 1 ()
1 (, T)
S I
2 2 () 2 (, T)
T
S
on
1
one can calculate
1
(, xt) while parameterizing
= I
and
1
1 =
S
S
0 on
1
I
S
0
2
I =si
S
0 on
= I
in the definition of the derived sources. In the remainder of the document,
2
2 =
1
S
on
2
2
2
we restrict ourselves with the first modeling [éq 1-1], in order not to overload them
later theoretical developments, but also because it appears closer to the real needs
users. On a case-by-case basis, we specify however more sophisticated derivations which are
accessible taking into account the new introduced functionalities and the perimeter from use of the code.
In order to more easily be able to commutate space or temporal derivation with derivation by
report/ratio with one of the parameters, one works with a derivation “within the meaning of the distributions” on
initial parabolic problem (derivation clarified in appendix 1). But the same exercise could
to be carried out starting from its version semi-discretized in time, of the variational formulation or of
linear system (one then takes the “discrete derivative” i.e. compared to the components of
discretized parameter) resulting from its discretization. In linear thermics, one shows that these
derivations, to each stage of the numerical process, lead to the same result. The problem
derived discretized being identical to the discrete derived problem, theoretical results exhumed on
continuous problem can apply to the problem actually implemented.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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The approach adopted for the calculation of the sensitivity is thus purely analytical. Him could
to prefer an semi-analytical method (it is the step partly retained by the codes
MSC/Nastran, FEMtools and ABAQUS (cf [§9] [bib8])) : broadly analytical for the determination
derived problem and using differences finished at the local level to determine the derivative of
matrices and of the second elementary derived members. But the latter, although easier with
to establish and to maintain, is also more expensive and it introduces a dependence with respect to
parameter of shift.
For linear thermics, the “derived” problem is very similar to the initial problem.
first member of the resulting linear system is preserved. He does not have thus to be reassembled,
only the second member is to be packed by suitable a source term. Theoretical results of existence,
of unicity and convergence of the solution are not appreciably modified. Obtaining the derivative
in temperature require the same processes numerical (dualisation and inversion of the system
linear resulting).
On the other hand, in non-linear thermics, the derived problem is metamorphosed: the operator
parabolic is modified. He became linear, just like the limiting conditions. These last
are more than of two types: Dirichlet or Robin, exit conditions of Neumann and radiation.
Because of its linear character, the usual theoretical results are thus much easier with
to exhume. In addition the resolution of the derived problem is faster and more robust than that of
initial problem. One does not need to have recourse to an algorithm of Newton-Raphson for
to determine the increment of temperature between two contiguous moments. A linear solvor is enough: no one is not
need to assemble a tangent matrix with each under-iteration.
This time the two members of this equation are fundamentally different from those of the problem in
temperature. However, after each step of time, once determined T+ starting from T, one does not have
to reassemble all the matrix of the linear system and its second associate member. It is enough to
to supplement the first tangent matrix of the step of time following by the term due to the non-linearity of
thermal conductivity. One also leaves the second member of the problem in temperature for
to constitute that which interests us: one packs it by the terms of implicitation of non-linearities of
thermal conductivity and of the limiting conditions.
One details theoretical, numerical work and data processing which governed the establishment of these
calculations of sensitivities in the code. One specifies their properties and their limitations while connecting these
considerations with a precise parameter setting of the accused operators and with the choices of modeling of
code. One tried constantly to bind different the items approached while detailing, has minimum, them
a little technical demonstrations.
In short, the perimeter of use of this functionality gathers it or not thermal, linear,
isotropic or anisotropic, stationary or transitory, being pressed on isoparametric finite elements
lumpés or not. Within this framework there, it covers the same perimeter as that with the operators
accused thermics.
The request for one or more sensitivities does nothing but enrich the structure of data
thermics (EVOL_THER) and provides also the thermal field of which they are the derivative. In term
of performance, the calculation of an analytical sensitivity is much less expensive than a calculation
standard since the same factorized matrix is re-used.
In addition to the calculation of sensitivities in thermics, Code_Aster proposes their hanging in mechanics
statics or quasi-static [R4.03.03] and in dynamics [R4.03.04]. All these functionalities and theirs
postprocessings associated (impressions, tests…) are included in the user's documentation
[U4.50.02] and belong to deliverable project “Incertitudes of numerical calculations” [bib5].
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 7/66
This document is articulated around the following parts:
· initially, one is interested in the various problems resulting from
derivation of the linear problem of thermics (THER_LINEAIRE) compared to the parameters
characteristics material and loadings,
· then one reiterates this process on the nonlinear problem of thermics (THER_NON_LINE),
· various linear systems “derived”, - directly plantable in Code_Aster
to determine such or such sensitivity -, are recapitulated in the third part,
· in the following paragraph, one describes necessary postprocessings to obtain the sensitivities
heat flow (CALC_ELEM/CALC_NO),
· one concludes by approaching the practical difficulties from implementation, the environment, it
parameter setting and the perimeter of use. An example of use extracted from an official case-test
(SENST04A) is also detailed.
Warning:
The reader in a hurry and/or not very interested by the theoretical springs genesis of these
sensitivities and the details of modeling of the code can, from the start, to jump to [§4] and [§6] which
the principal theoretical and practical contributions recapitulate preceding chapters.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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: 8/66
2 Thermics
linear
One considers a limited open body occupying related of Rq (q=2 or 3) of border
lipschitzienne characterized by its voluminal heat with pressure constant CP (X) (the variable
vectorial 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). These data materials are supposed to be independent of
time (modeling THER of Code_Aster) and constants by element (P0 discretization).
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 rather marginal use, us
we will not interest, initially, with its derivation.
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),
boundary conditions of imposed the temperature type F (X, T) (on the external portion of surface),
1
normal flow imposed G (X, T) (on) and exchanges convectif H (X, T) and T
).
2
ext. (X, T) (on 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.
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
= G
éq
2-1
2 ×],
0 [
N
T
+ HT = HT
ext.
3 ×],
0 [
N
T (X)
0
, = 0
T (X)
Note:
· The condition of Robin modelling the convectif exchange (key word ECHANGE) on a portion of
edges of the field, can be duplicated to take account of exchanges between two pennies
parts of the border in opposite (key word ECHANGE_PAROI). This limiting condition
model a thermal resistance of interface
T
1 + HT
HT
1 =
2
12 ×],
0 [
With
N
T
T
has
one
éq
2-2
3 = 12
,
21
I =
ij
T2 + HT
HT
2 =
1
21 ×],
0 [
N
· 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
structure.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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With =, T
has
one
I = T
ROUP)
(LIAISON_G
1
12
21
ij
T I X T
X
X
éq
2-3
I
+
T J
T
J
=
T
on ×
1
1 (,)
2 2 (,) (,)
1
], 0 [
I
J
or
simply
more
T X
X
I
I (, T)
= (, T) on ×
1
], 0 [
DLL)
(LIAISON_D
I
· One will not speak about functionalities LIAISON_UNIF and LAISON_CHAMNO which allow
to impose the same temperature (unknown) on a whole of nodes, because they are not
that a surcouche of the preceding conditions imposing of the couples (,) particular.
· When the material is anisotropic (modeling THER_ORTH), conductivity is
modelled by a diagonal matrix expressed in the reference mark of orthotropism of material.
That basically does not change calculations according to which only hold account
isotropic case. Guard should just be taken not to commutate more, under the conditions limit
Neumann and of Robin, the scalar product with the normal and multiplication by
conductivity. In practice, in elementary calculations, one is not interested in the derivative
normals. The problem thus arises only in the preliminary theoretical part.
The sensitivity compared to one of the components of anisotropic conductivity is not
not yet available. These calculations were set up in the accused subroutines
(elementary calculations YOU.), they await nothing any more but the software evolution consisting with
to extend the taking into account of the anisotropy to the functions (a modeling
THER_ORTH_FO). Indeed, from a point of view structures (cf DEFI_PARA_SENSI [bib6]),
variable ASTER representing the significant parameter must be a data-processing object of
function type.
· In all following calculations of sensitivities, one calculates only it derived by report/ratio
with a constant parameter by zone. If not, it would be necessary to introduce a concept of derivative
directional!
This does not exclude a temporal or space dependence from characteristics material or
loadings. By parameterizing advisedly the “derived” loadings and materials
in the command file, one can also have access to some derivatives made up
(cf [§6.2]/[§6.4]).
· 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 will affect the initialization of
derived problem.
· 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
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)).
In order to be able to consider different the derivative from the temperature in configurations
multimatériau and multichargement one introduces the following notations:
=
U I
I
I = ij (I = 1, 2ou 3)
Ui
Handbook of Référence
R4.03 booklet: Analyze sensitivity
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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The characteristics material and the loadings will be subscripted to notify their membership of such
or such opened or portion of border. Thus, if one is interested in a Bi-material, 1 models heat
voluminal of material occupying the open first and
1
2 that of material occupying the second
opened.
2
1
2
1
2
Appear 2-a: Désignation of voluminal heats for a Bi-material
2.1
Derived compared to voluminal heat
2.1.1 Elements
theoretical
One models total voluminal heat penny the form C X
I X
. All them
p () = I
I ()
(I)
I
C
opened being solidified, one has
p = I the indicating function of the ième part. The derivation of
I
I
I
[éq 2-1] then leads us “trivialement” (cf Annexe 1) to the new problem in extreme cases of which is
T
solution sought sensitivity, noted U =
,
I
U
~
C
p
- div (U) = S
×],
0 [
~ T
U = F
,
0
1 ×]
[
U
= ~g
,
0
éq
2.1.1-1
2 ×]
[
N
U
+ hu = ~h
,
0
3 ×]
[
N
U (X
= ~
)
0
,
u0
with the new source voluminal and the new limiting conditions and initial
~
~
~
~
T
~
F = G = H =,
0
S = - I
and
U
éq
2.1.1-2
I
= 0
0
T
One is thus brought to solve a homogeneous problem out of U similar to that which T answers. One fixes
a step of time T
such as
that is to say an entirety NR. Semi-discretization in times of [éq 2.1.1-1],
T
[éq 2.1.1-2] by - method leads to the following problem: to find a continuation.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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Note:
By using an adaptation of the theorem of Lax-Milgram to the parabolic problems
([R4.10.03 §1] or [bib1] theorems 1 & 2 chap XVIII pp615-620 or [bib3] pp220-241) one
show, under certain conditions of regularity on opened, materials, the loadings and
the initial condition, that this problem admits a single solution.
(U
éq
2.1.1-3
N)
V = U H U =
0nN
{1
0
()/1}
0
This space comprises also the conditions of “generalized” Dirichlet of linear relations type
between ddls when they exist.
such as:
n+1
U
- N
~n+1 ~
U
N
n+1
N
S
- S
C
p
- div (U) - (1 -) div (U) =
0 N NR - 1
T
N
~
T
+1
U
= n+1
F
0 N NR 1
1
-
n+1
U
= ~n+1
G
0 N NR 1
2
-
N
n+1
U
n+
N
~
1
+1
n+1
+ H U = H
0 N NR 1
3
-
N
0
U (X = ~
) u0
éq 2.1.1-4
while posing:
~
~
~
~
N
C
U = U,
X N
, S N = -
p (X)
N
N
N
N
T,
X N
, F = G = H = 0 and H =
H,
X N
NR
I
NR
NR
éq 2.1.1-5
By applying the theorem of Green to [éq 2.1.1-3], [éq 2.1.1-4], [éq 2.1.1-5] and by introducing them
following notations
+
-
=,
X (N +)
1
and =,
X N
with {U, T,}
H and 0 N NR - 1
NR
NR
one is brought to solve the following variational problem:
±
±
-
Being gift
be H
, T and U
+
To calculate U V
that
such
éq
2.1.1-6
0
v
+
V
has
0
(+u, v) = ±l (v)
with the bilinear form depending on the current moment (via h+)
+
has (+
U v) = 1
,
+
C U v dx
éq
2.1.1-7
p
+
+
U v dx + + +
H U v
D
T
3
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 12/66
and the linear form parameterized by the moments running and precedent (via H, U and T)
±
1
L (v) =
-
C U v dx 1
1
p
+ (-)
-
U v dx + (-) - -
H U v D +
T
3
éq 2.1.1-8
1 II (- +
T - T) v dx
T
Note:
· Contrary to the initial problem, the unknown field and the function test belong to same
functional space, which is more comfortable from a theoretical and numerical point of view.
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined T+ starting from T, one rests
also on H, h+ and U to determine u+. The matrix of the linear system corresponding does not have
not with being reassembled. Only the second member is to be packed by suitable the source term.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8].
· For a condition of exchange between walls, the term of usual exchange is of course replaced
by (by taking again the notations of [éq 2-2]) ±
H (±
±
U
.
I - U J) v D
ij
· These problems of suitable initialization of the problem derived are recurring. It was found
in the calculation of the derivative Lagrangian of the field of temperature compared to one
variation of field (cf [R4.03.01] operand DEUL_INIT).
· The reader interested by a theoretical study of the thermal problem actually put in
place in the code, which underlines these holding and bordering and their links with the choices of
modeling, will be able to refer to [§1] Doc. R: “Indicating of error in residue for
transitory thermics “[R4.10.03]. It relates to a related field of improvement and of
calibration of the studies, that of the space errors due to the grids finite elements.
To solve this problem numerically one spatially discretizes it by considering one
subspace H
V of V of dimension finished
0
0
N
U ± = u± NR V H
0
= ±
:
/
0
±
H
I
I
{U V K
U
H
H
H K
K (K)}
i=1
by noting (Th) H a regular family of triangulations of the polygonal or polyhedric field discretized
, P
H
K (K) the space of the polynomials of degree < (k+1) on K and Ni the function of form associated with
node n°i. From where the discretized variational problem
Note:
This property of total continuity of the elements and maximization of their characteristics
geometrical (which ensures the convergence of the finite element method) is checked for
all isoparametric elements of the code: segment, triangle, quadrangle, tetrahedron,
pentahedron and hexahedron.
± ±
-
Being gift
be H, T and U
H
H
H
+
To calculate U H
V
that
such
éq
2.1.1-9
H
0
v V has U, v L v
H
H
+
0
H (+
H
H) = ±
H (H)
leading to the linear system of command N.
In U+ = L
éq
2.1.1-10
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 13/66
The p boundary conditions of the Dirichlet type are taken into account in Code_Aster by one
technique of double dualisation [R3.03.01] via “ddls (degrees of freedom)” of Lagrange
= (
. That is to say J the whole of the nodes belonging to the portion of border on which
I) I,
1
= p
apply the condition of Dirichlet (card (J) = p), one considers the new unknown vector
~
U [+
= U
] T
and the operator B (of command p X N) checking
(+
B U) = u+ with I J
I
I
The homogeneous condition of Dirichlet of [éq 2.1.1-1], [éq 2.1.1-2] is realized while imposing
B U+ = C = 0
The dualized problem then consists in reversing the system of command n+2p
With LT
LT +
U
L
~ ~ +
~
WITH U = L
B - Id
Id
=
0
éq
2.1.1-11
B Id - Id 0
Note:
· One carries out the possible taking into account of limiting conditions of Dirichlet generalized like
in the problem in temperature (here p=1) but with a second null member
+
B U =
+
U
with J
J and C
J
J
=
= 0
J
I
We now will see how these calculations are declined in the code.
2.1.2 Establishment in Code_Aster
The matrix of this system results from the assembly of the following elementary terms, due to
contribution of the nodes (I, J) to the point of gauss (of weight (this weight gathers in fact “truth
G
G
weight “of the formula of quadrature multiplied by the jacobien of the element considered and possibly
by the radius of the point of gauss rg (in modeling AXIS or AXIS_DIAG))) element running K
(in Code_Aster the conditions limit are affected on particular elements of skin of
dimension q-1. No confusion not being really possible, one will make formal distinction here
between those and the elements of volume which support them).
WITH K
WITH K
éq
2.1.2-1
ij (
, G)
3
= iij (, G)
i=1
with
· the thermal term of mass (calculated by option MASS_THER)
A1 K, =
C K NR NR
ij (
) G
G
p (
) J (G) I (G)
T
· the thermal term of rigidity (RIGI_THER)
A2 K, = K NR. NR
ij (
G)
G
() J (G) I (G)
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 14/66
· the term of rigidity due to the conditions limit of exchange (RIGI_THER_COEF_F/R)
A3 K
,
=
+
3
H
K
NR
3
NR
ij (
G)
G
(
) J (G) I (G)
In the event of exchange between walls this term is replaced by (RIGI_THER_PARO_F/R)
A3 K
,
3
= h+ K
NR
3
- NR
NR
ij (
G)
G
(
) (J (G) F J (G) I (G)
()
by noting F the bijection putting in opposite the two walls.
The second member is written, with the same notations,
L K
L K
éq
2.1.2-2
J (
, G)
3
= ij (, G)
i=1
where
· the term resulting from the implicitation of the matrix of rigidity and mass (new option
CHAR_SENS_EVOL, copy of CHAR_THER_EVOL with U instead of T, the field material
derived and standard and the new source term)
L1 K, =
C K U NR + - 1 K U
. NR
J (
) G
G
p (
) (G) J (G) () G () (G)
J (G)
T
· the term resulting from the implicitation of the conditions limit of exchange
(CHAR_THER_TEXT_F/R with Text=0 and U instead of T)
L2 K
,
1
3
= - H K
U
3
NR
J (
G)
G (
) (
) (G) J (G)
In the event of exchange between walls this term is replaced by (CHAR_THER_PARO_F/R with U
instead of T)
L2 K
,
1
3
= - H K
U
3
- U F
NR
J (
G)
G (
) (
) ((G) ((G)) J (G)
· the term due to the “new source” comprising the field derived material (cf.
CHAR_SENS_EVOL above)
L3 K, = -
I K T + - T -
NR
J (
)
G
G
I (
) ((G)
(G) J (G)
T
As one already specified all the elementary terms of the matrix are the subject of an option
of calculation and will already have been evaluated for the calculation of T+. It thus remains to estimate the second member
by re-using (with a different parameter setting) the existing options of calculation or by introducing one
news (CHAR_SENS_EVOL). This new option is common with the other derived material
(thermal conductivity) and it redirects towards the same routine of elementary calculation (TE.). The chain
of character power station (SENS instead of THER) joined to a detection of the nullity of the field material
derived, allows to parameterize this routine towards one of its three possible orientations: calculation of
term of implicit mass and standard rigidity, idem of sensitivity compared to one of both
characteristics material which thus adds a new source term.
In accordance with the principles of architecture set up in the code to treat calculations
of sensitivity [bib6], the assembly and the resolution of [éq 2.1.1-11] are started by the analysis of
table of correspondence associated with the significant variable. It was seen that this calculation is very close to one
standard linear thermal calculation, only the initial condition and the loadings are modified
F =
T
,
0
G =,
0 T
and
ext. =
,
0
0
S = - I
U
I
= 0
T
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 15/66
This feedback is carried out via the succession of commands
v = DEFI_PARA_SENSI (VALE = < value of I >)
my = DEFI_MATERIAU (THER = _F (RHO_CP = v))
affe = AFFE_MATERIAU (AFFE = _F (GROUP_MA = < definition of I >,
MATER = my))
…
one = DEFI_CONSTANTE (VALE = 1. )
MEMO_NON_SENSI (NOM =_F (NOM_SD = “my”, PARA_SENSI = “v”,
NOM_COMPOSE = “ma_v”))
ma_v = DEFI_MATERIAU (THER = _F (RHO_CP = one))
MEMO_NON_SENSI (NOM = _F (NOM_SD = “affe”, PARA_SENSI = “v”,
NOM_COMPOSE = “affe_v”))
affe_v = AFFE_MATERIAU (AFFE = _F (GROUP_MA = < I >, MATER = ma_v))
…
resu = THER_LINEAIRE (CHAM_MATER = affe,
SENSIBILITE
=
(
v)
…)
Note:
· In his command file, the user will not have soon any more but to specify the first and it
third blocks of instruction. The block of the medium will be generated automatically by
supervisor thanks to the tree of dependence which it builds between the various commands.
C
· The essential data of this calculation, derived voluminal heat
p
I =
, is provided by
I
I
ma_v.
· To take into account a more sophisticated modeling of voluminal heat
C X
I X X
p () = I
I ()
I ()
(I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
C
source
p = I in the definition of the field derived material
I
I
ma_v.
I
We will unroll the same process for the various sensitivities, to start with that
concerning the other characteristic material: thermal conductivity.
2.2
Derived compared to thermal conductivity
2.2.1 Elements
theoretical
T
One poses (X) = I X
and U =
required sensitivity. All the open ones
I
I ()
(
I
)
I
I
being solidified, one has
= I the indicating function of the ième part. The derivation of [éq 2-1] us
I
I
I
conduit with a problem in extreme cases identical to [éq 2.1.1-1] but with a nonnull voluminal source
and of the new conditions of Neumann and Robin
~
T
~
G = H = - I
and
s~ = div
éq
2.2.1-1
I
(I T
I
)
N
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 16/66
It is a homogeneous problem of Cauchy-Dirichlet and inhomogenous Neumann-Robin similar to that
which T answers. Its semi-discretization in time led to seek a continuation (U
N)
V
0nN
0
checking a system similar to [éq 2.1.1-4] whose first relation is rewritten
N 1
+
U
- one
C
p
- div (
N 1
+
U
) - (1 -) (one) ~
div
N 1
=
+
S
+ (-) ~
1
S N
0 N NR - 1
T
éq 2.2.1-2
with the new source term
~sm = div (I T m
éq
2.2.1-3
I
) m {, nn+} 1
and new limiting conditions
+
~
~ 1
+1
N
N
T N
G
= H
= - I
éq
2.2.1-4
I
N
From where a variational problem identical to [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] comprising
even bilinear form [éq 2.1.1-8] joined to the linear form [éq 2.1.1-7] of which only the fourth
integral is modified to adapt to the new source
±
L (v) =
- I
1
L
éq
2.2.1-5
I (
+
T + (-) -
T) vdx
Note:
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined T+ starting from T, one rests on
H, h+ and U to determine u+. The matrix of the linear system corresponding does not have to be
reassembled. Only the second member is to be packed by suitable the source term.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.2.1-5].
· In stationary regime this complementary source term is tiny room to
L (v) =
- I
.
L
I T
v dx
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
2.2.2 Establishment in Code_Aster
Compared to [§2.1.2], only the term due to the new source is to be modified
L3 K, =
- I K T + + 1 - T -
NR
éq
2.2.2-1
J (
G)
G I (
) (
(G) () (G)
J (G)
What is made in the new option of calculation CHAR_SENS_EVOL with the field derived material and
standard.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. It was seen that this calculation is very close to a calculation
standard linear thermics, only the initial condition and the loadings are modified
F =
T
,
0
G = HT
and
ext. = - I
, S
I
= div (IiT)
0
U = 0
N
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 17/66
This feedback is carried out via the same succession of commands as with [§2.1.2] in
substituent LAMBDA (or LAMBDA_L/T/NR into orthotropic) with RHO_CP in the DEFI_MATERIAU.
Note:
· The essential data of this calculation, derived thermal conductivity I =
, is provided
I
I
by ma_v.
· When the material is anisotropic the thermal matrix of conductivity is expressed in
locate orthotropism of material: it is thus diagonal (in our q=2 example)
0
1
=
0 2
One can then dissociate derivation compared to a value of one of his diagonal terms
derivation compared to a value of this diagonal. The formulas detailed here are
identical whatever the configuration selected. Only the evaluation of
ij
ij
or
must
K
kl
to take account of these characteristics.
· In practice, one has access to the sensitivity compared to an isotropic conductivity
constant by zone. Sensitivity compared to a component of conductivity
anisotropic is not yet available. These calculations were set up in
subroutines accused (elementary calculations YOU.), they await nothing any more but the evolution
software consisting in extending the taking into account of the anisotropy to the functions (one
modeling
THER_ORTH_FO). Indeed, from a point of view structures
(cf DEFI_PARA_SENSI [bib6]), variable ASTER representing the significant parameter
must be a data-processing object of function type.
· It was reasoned here as if the condition of initial Cauchy of the problem had been fixed
uniform or unspecified. If it is given by carrying out a stationary calculation on
first moment it is necessary to reiterate this process with the derived problem. On the other hand, if it results
of a recovery starting from a preceding transitory calculation, the derived problem must be initialized with
to leave the value of the same derivative at the same moment of recovery.
In short, two initializations (that of the problem in temperature and that of the problem
derived) must be homogeneous. On the other hand, contrary to a calculation of thermics
standard, one cannot thus modify the conditions limit and it is necessary to carry out a recovery with
to start from a calculation of comparable nature.
· To take into account a more sophisticated modeling of thermal conductivity
(X) = I X X
I
I ()
I ()
(I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new one
source term
= I in the definition of the field derived material
I
I
ma_v.
I
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 18/66
2.3
Derived compared to the source
2.3.1 Elements
theoretical
T
One poses S (X) = S I X
S
and U =
required sensitivity. All the open ones being
I
I ()
(I)
I
if
S
solidified, one has
= I the indicating function of the ième part. The derivation of [éq 2-1] leads us
I
S
I
I
with a problem in extreme cases identical to [éq 2.1.1-1] but with another voluminal source
s~ = I
éq
2.3.1-1
I
It is a homogeneous problem similar to that which T answers. Its semi-discretization in time
conduit to seek a continuation (U
checking a system similar to [éq 2.1.1-4] of which
N)
V
0nN
0
first relation is rewritten
N 1
+
U
- one
C
éq
2.3.1-2
p
- div (
N 1
+
U
) - (1 -) (one) ~
div
= S
0 N NR - 1
T
From where a variational problem identical to [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] comprising
even bilinear form [éq 2.1.1-7] joined to the linear form [éq 2.1.1-8] of which only the fourth
integral is modified to adapt to the new source
±
L (v) = L + I v dx
éq
2.3.1-3
I
Note:
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined T+ starting from T, one rests
also on H, h+ and U to determine u+. The matrix of the linear system corresponding does not have
not with being reassembled. Only the second member is to be packed by suitable the source term.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.3.1-3].
· In stationary regime this complementary source term is not modified.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
2.3.2 Establishment in Code_Aster
Compared to [§2.1.2], only the term due to the new source is to be modified
L3 K, = I K NR
éq
2.3.2-1
J (
G)
G I (
) J (G)
It is enough to re-use standard option CHAR_THER_SOUR_F/R with the derived field source.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. It was seen that this calculation is very close to a calculation
standard linear thermics, only the initial condition and the loadings are modified
F = G = T
and
ext. =
,
0
0
S = I
U
I
= 0
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 19/66
This feedback is carried out via the succession of commands
v = DEFI_PARA_SENSI (VALE = < value of if >)
chth = AFFE_CHAR_THER_F (SOURCE = _F (GROUP_MA = < definition of I >,
SOUR = v))
…
one = DEFI_CONSTANTE (VALE = 1. )
MEMO_NON_SENSI (NOM =_F (NOM_SD = “chth”, PARA_SENSI = “v”,
NOM_COMPOSE = “chth_v”))
chth_v = AFFE_CHAR_THER_F (SOURCE = _F (GROUP_MA = < I >, SOUR = one))
…
resu = THER_LINEAIRE (EXCIT = chth,
SENSIBILITE
=
(
v)
…)
Note:
S
· The essential data of this calculation, the derived field source I =
, is provided by
I
chth_v.
S
I
· This calculation is independent of the three types of modeling of the source: constant by mesh
(AFFE_CHAR_THER + SOUR), constant by point of Gauss (AFFE_CHAR_THER +
SOUR_CALCULEE) and constant by mesh and dependant on time (AFFE_CHAR_THER_F +
SOUR). These considerations do not even re-enter in line of account during the effective calculation of
S
in chth_v, because this size represents the derivation of a function parameterized by
S
I
one of its constant parameters. One is not interested here in derivative of the type
S
S
X,
O
X.
S
X,
I (jg) (
T) U if (tj) (T)
· To take into account a more sophisticated modeling of the source
S (,
X T) = S I X,
X T
S
I
I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
S
source
= I in the definition of the derived loading
I
I
chth_v.
if
2.4
Derived compared to the imposed temperature
2.4.1 Elements
theoretical
T
One poses F (X) = F I X
F
and U =
required sensitivity. Portions of
I
I ()
(I)
I
fi
F
external border being solidified, one has
= I the indicating function of the ième portion.
1 J
I
F
I
1
I
derivation of [éq 2-1] leads us to a problem in extreme cases identical to [éq 2.1.1-1] but with one
another voluminal source and a new condition of Dirichlet
~
~
F = I
and
S
éq
2.4.1-1
I
= 0
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 20/66
It is an inhomogenous problem of Dirichlet and Cauchy- Neumann-Robin homogeneous similar to that
which T answers. Its semi-discretization in time led to seek a continuation (this space can
to comprise, if necessary, also conditions of “generalized” Dirichlet of linear relations type
between ddls)
(U
V = U H1
éq
2.4.1-2
0
1
/U = I
N)
{
() I
1
}
checking a system similar to [éq 2.1.1-3] whose first relation is rewritten
N 1
+
U
- one
C
éq
2.4.1-3
p
- div (
N 1
+
U
) - (1 -) div (one) = 0
0 N NR - 1
T
with the new limiting condition
~n
F +1 = I
éq
2.4.1-4
I
From where a variational problem identical to [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] with the same form
bilinear [éq 2.1.1-7] joined to the linear form [éq 2.1.1-8] whose fourth integral is null.
Note:
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined T+ starting from T, one rests
also on H, h+ and U to determine u+. The matrix of the linear system corresponding does not have
not with being reassembled. On the other hand this time it is necessary to constitute the Lagrangian part of
second member dualized in order to approximate new functional space V1.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.4.1-3], [éq 2.4.1-4].
Space discretization in a subspace H
V and the taking into account of the condition of Dirichlet
1
inhomogenous lead to a linear system dualized similar to [éq 2.1.1-11]
With LT
LT +
U
L
~ ~ +
~
WITH U = L
B - Id
Id
=
C
éq
2.4.1-5
B Id - Id C
with
F
J
C =
= éq
2.4.1-6
K
ij
F
I
by noting fj the value of the condition of Dirichlet to the node n°k (classification room) of 1.
2.4.2 Establishment in Code_Aster
Compared to [§2.1.2], only the second member is modified since
3
L K
éq
2.4.2-1
J (
, G) = 0
There is not thus an option of calculation particular to envisage, it is just necessary to assemble the linear system
dualized associated the derived imposed temperatures.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. It was seen that this calculation is very close to a calculation
standard linear thermics, only the initial condition and the loadings are modified
F = I, G
and
I
= Text = S = 0
0
U = 0
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 21/66
This feedback is carried out via the succession of commands
v = DEFI_PARA_SENSI (VALE = < value of fi >)
chth = AFFE_CHAR_THER_F (TEMP_IMPO = _F (GROUP_MA = < definition of i1 >,
TEMP = v))
…
one = DEFI_CONSTANTE (VALE = 1. )
MEMO_NON_SENSI (NOM =_F (NOM_SD = “chth”, PARA_SENSI = “v”,
NOM_COMPOSE = “chth_v”))
chth_v = AFFE_CHAR_THER_F (TEMP_IMPO = _F (GROUP_MA = < i1 >, TEMP = one))
…
resu = THER_LINEAIRE (EXCIT = chth,
SENSIBILITE
=
(
v)
…)
Note:
F
· The essential data of this calculation, the field imposed temperature derived I =
, is
I
F
I
provided by chth_v.
· This calculation is independent of the three types of modeling of the imposed temperature:
constant by mesh (AFFE_CHAR_THER + TEMP), plus the dependence in time provided by
a function (AFFE_CHAR_THER_F + TEMP) or provided by a structure of data
“EVOL_THER” (AFFE_CHAR_THER_F + EVOL_THER + “TEMP”). These considerations
F
do not even re-enter in line of account during the effective calculation of
in chth_v, because this
F
I
size represents the derivation of a function parameterized by one of its parameters
F
constant. One is not interested here in the derivative
X.
F
X,
I (
T J) (T)
· The calculation of derived from T compared to one of the parameters of the relations of Dirichlet
generalized [éq 2-3] same manner would be carried out. A distinction only appears
on the level of the Lagrangian components of the dualized system:
T
+
With U =
B U =
+
U
with J
J and C
T éq
2.4.2-2
J
J
= - J +
J
I
J
J
I
T
+
+
With U =
B U = U
with J J and C =
éq
2.4.2-3
J
J
I
J
I
The taking into data-processing account of these calculations would be carried out, like above, via the word
keys COEF_MULT_1/2 and COEF_IMPO of the key words factors LAISON_GROUP and
LIAISON_CHAMNO.
· Sensitivity compared to a multiplying coefficient of this condition of Dirichlet
generalized [éq 2.4.2-2 is not available] because it has little direction with coefficients
often discrete. One does not have access, by parameterizing advisedly the condition of Dirichlet
generalized derived, that with derivation compared to the total coefficient [éq 2.4.2-3].
· To take into account a more sophisticated modeling of a condition of Dirichlet
F (,
X T) = F I X,
X T
F
I
I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new one
F
source term
= I in the definition of the derived loading
I
I
chth_v.
fi
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 22/66
2.5
Derived compared to imposed normal flow
2.5.1 Elements
theoretical
T
One poses G (X) = G I X
G
and U =
required sensitivity. Portions
I
I ()
(I)
2 J
I
gi
G
being solidified, one has
= I the indicating function of the ième portion. The derivation of [éq 2-1]
I
G
2i
I
us leads to a problem in extreme cases identical to [éq 2.1.1-1] but with another voluminal source
and a new condition of Neumann
~
~
G = I
and
S
éq
2.5.1-1
I
= 0
It is a problem of inhomogenous Neumann and homogeneous Cauchy-Dirichlet-Robin similar to that
which T answers. Its semi-discretization in time led to seek a continuation (U
N)
V
0nN
0
checking a system similar to [éq 2.1.1-3] whose first relation is rewritten
N 1
+
U
- one
C
éq
2.5.1-2
p
- div (
N 1
+
U
) - (1 -) div (one) = 0
0 N NR - 1
T
with the new limiting condition
~n
G +1 = I
éq
2.5.1-3
I
From where a variational problem identical to [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] comprising
even bilinear form [éq 2.1.1-7] joined to the linear form [éq 2.1.1-8] of which only the fourth
integral is modified to adapt to the new “surface” source
±
L (v) = L + I v dx
éq
2.5.1-4
I
2
Note:
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined T+ starting from T, one rests
also on H, h+ and U to determine u+. The matrix of the linear system corresponding does not have
not with being reassembled. Only the second member is to be packed by suitable the source term.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.5.1-4]
· In stationary regime this source term is not modified.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
2.5.2 Establishment in Code_Aster
Compared to [§2.1.2], only the term due to the new source is to be modified
L3 K
,
=
éq
2.5.2-1
2
I K
NR
2
J (
G)
G I (
) J (G)
It is enough to re-use standard option CHAR_THER_FLUN_F/R with the field derived flow.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. It was seen that this calculation is very close to a calculation
standard linear thermics, only the initial condition and the loadings are modified
F = S = T
and
ext. =
,
0
0
G = I
U
I
= 0
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 23/66
This feedback is carried out via the succession of commands
v = DEFI_PARA_SENSI (VALE = < value of gi >)
chth = AFFE_CHAR_THER_F (FLUX_REP = _F (GROUP_MA = < definition of 2i >,
FLUN = v))
…
one = DEFI_CONSTANTE (VALE = 1. )
MEMO_NON_SENSI (NOM =_F (NOM_SD = “chth”, PARA_SENSI = “v”,
NOM_COMPOSE = “chth_v”))
chth_v = AFFE_CHAR_THER_F (FLUX_REP = _F (GROUP_MA = < 2i >, FLUN = one))
…
resu = THER_LINEAIRE (EXCIT = chth,
SENSIBILITE
=
(
v)
…)
Note:
G
· The essential data of this calculation, the field normal flow derived I =
, is provided by
I
G
I
chth_v.
· This calculation is independent of the three types of modeling of the condition of Neumann:
constant by mesh (AFFE_CHAR_THER + FLUN), plus the dependence in time provided by
a function (AFFE_CHAR_THER_F + FLUN) or components of vectorial flow
dependant on time and constants by meshs (AFFE_CHAR_THER_F + FLUN_X/Y/Z).
One can then dissociate derivation compared to the q-uplet components of derivation
compared to one of its components. The formulas detailed here are identical whatever
that is to say configuration selected. Only possibly the evaluation of
G L
G L
I
I
or
,
must take account of these characteristics.
K
(L {X y Z})
G
G
J
J
· The dependence in time of the condition of Neumann is not taken into account at the time of
G
calculation of
because this size represents the derivation of a function parameterized by one
G
I
G
of its constant parameters. One is not interested here in the derivative
X.
G
X,
I (
T J) (T)
· In practice, there is thus access to the sensitivity compared to a scalar flow or vector,
constant by zone. By parameterizing advisedly the vector derived flow, one can too
T
to obtain the sensitivity compared to one of its components
,
X
.
J (
T)
G
I
· To take into account a more sophisticated modeling of a normal flow
G (,
X T) = G I X,
X T
G
I I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new one
G
source term
= I in the definition of the derived loading
I
I
chth_v.
gi
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 24/66
2.6
Derived compared to the coefficient from convectif exchange
2.6.1 Elements
theoretical
T
One poses H (X) = H I X
H
and U =
required sensitivity. Portions being
I
I ()
(I)
3 J
I
hi
H
solidified, one has
= I the indicating function of the ième portion. The derivation of [éq 2-1] us
I
H
3i
I
conduit with a problem in extreme cases identical to [éq 2.1.1-1] but with another voluminal source and
a new condition of Robin
~
H = I
and
éq
2.6.1-1
I (Text - T)
~s = 0
It is an inhomogenous problem of Robin and homogeneous Cauchy-Dirichlet-Neumann similar to that
which T answers. Its semi-discretization in time led to seek a continuation (U
N)
V
0nN
0
checking a system similar to [éq 2.1.1-3] whose first relation is rewritten
N 1
+
U
- one
C
éq
2.6.1-2
p
- div (
N 1
+
U
) - (1 -) div (one) = 0
0 N NR - 1
T
with the new limiting condition
~n 1+
H
= I T
T
éq
2.6.1-3
I (N 1
+
N 1
+
ext.
-
)
From where a variational problem identical to [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] comprising
even bilinear form [éq 2.1.1-7] joined to the linear form [éq 2.1.1-8] of which only the fourth
integral is modified to adapt to the new “surface” source
±
L (v) = L + I
éq
2.6.1-4
I {(+
+
Text - T) + (1 -) (-
-
Text - T)}vdx
3
Note:
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined +
T from -
T, one rests
also on -
H, +
H and -
U to determine +
U. The matrix of the linear system corresponding
does not have to be reassembled. Only the second member is to be packed by suitable the source term.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.6.1-4].
· These calculations spread without sorrow in the condition of exchange between walls of [éq 2-2]. It
is enough to replace the ± terms
(±
±
T
by
ext. - T
) v D
3i
± (u± - u± v D with H, I, 1 - I.
I
J)
{
I (
) I}
ij
· In hover this complementary source term is tiny room to
L (v) = L + I
I (Text - T) v dx
3
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 25/66
2.6.2 Establishment in Code_Aster
Compared to [§2.1.2], only the term due to the new source is to be modified
T + K
3 - T
3
+ +
L
K
,
éq2.6.2-1
3
= I K
3
NR
J (
G)
(ext. (
)
(G)
G I (
)
-
-
(1 -) (T K
3 - T
ext. (
)
(G) J (G)
What is made in the new option of calculation CHAR_SENS_TEXT_F with the fields coefficient
of standard exchange and derived.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. It was seen that this calculation is very close to a calculation
standard linear thermics, only the initial condition and the loadings are modified
F = S = G =,
0
HT
and
ext. = I I (Text - T)
0
U = 0
This feedback is carried out via the succession of commands
v = DEFI_PARA_SENSI (VALE = < value of hi >)
chth = AFFE_CHAR_THER_F (ECHANGE = _F (GROUP_MA = < definition of 3i >,
COEF_H = v, TEMP_EXT = W))
…
one = DEFI_CONSTANTE (VALE = 1. )
zero = DEFI_CONSTANTE (VALE = 0. )
MEMO_NON_SENSI (NOM =_F (NOM_SD = “chth”, PARA_SENSI = “v”,
NOM_COMPOSE = “chth_v”))
chth_v = AFFE_CHAR_THER_F (ECHANGE = _F (GROUP_MA = < 3i >,
COEF_H = one, TEMP_EXT = zero))
…
resu = THER_LINEAIRE (EXCIT = chth,
SENSIBILITE
=
(
v)
…)
Note:
H
· The essential data of this calculation, the field coefficient of derived exchange I =
, is
I
H
I
provided by chth_v.
· To take into account a more sophisticated modeling of a coefficient of exchange
H (,
X T) = H I X,
X T
H
I I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute, with the indicating function, the new one
H
source term
= I in the definition of the derived loading
I
I
chth_v.
hi
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 26/66
2.7
Derived compared to the outside temperature
2.7.1 Elements
theoretical
In the case of convectif exchange with the external medium, one poses T
X
T
I X
T
ext. () = iext
I ()
(iext)
I
T
T
and U =
required sensitivity. As previously
ext. = I the indicating function of
I
T
I
I
T
ext.
ext.
the ième portion. The derivation of [éq 2-1] leads us to a problem in extreme cases identical to
3i
[éq 2.1.1-1] but with another voluminal source and a new condition of Robin
~
~
H = H I
and
S
éq
2.7.1-1
I
= 0
It is an inhomogenous problem of Robin and homogeneous Cauchy-Dirichlet-Neumann similar to that
which T answers. Its semi-discretization in time led to seek a continuation (U
N)
V
0nN
0
checking a system similar to [éq 2.1.1-3] whose first relation is rewritten
N 1
+
U
- one
C
éq
2.7.1-2
p
- div (
N 1
+
U
) - (1 -) div (one) = 0
0 N NR - 1
T
with the new limiting condition
~n 1+
N
H
= H 1+ I
éq
2.7.1-3
I
From where a variational problem identical to [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] comprising
even bilinear form [éq 2.1.1-7] joined to the linear form [éq 2.1.1-8] of which only the fourth
integral is modified to adapt to the new “surface” source
±
L (v) = L + I
éq
2.7.1-4
I {
+
H + (1 -) -
H} vdx
3
Note:
· The first member of this equation is formally identical to that of the equation in
temperature. After each step of time, once determined +
T from -
T, one
also rest on -
H, +
H and -
U to determine +
U. The matrix of the linear system
corresponding does not have to be reassembled. Only the second member is to be packed by the term
suitable source.
· By deriving the variational formulation (cf [§5.1.3]) from the problem in temperature [R5.01.02]
one finds well [éq 2.7.1-4].
· In stationary regime this source term is tiny room to
L (v) = L + I H vdx
I
3
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 27/66
2.7.2 Establishment in Code_Aster
Compared to [§2.1.2], only the term due to the new source is to be modified
L3 K
,
=
+ + 1
-
éq
2.7.2-1
3
I K
3
H K
3
H K
NR
3
J (
G)
G I (
) {(
) (
) (
)} J (G)
It is enough to re-use standard option CHAR_THER_TEXT_F with the fields coefficient of exchange
-
standard and derived, and while it “bluffant” with a field T =0.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. It was seen that this calculation is very close to a calculation
standard linear thermics, only the initial condition and the loadings are modified
F = S = G =,
0
0
T
and
ext. = I
U
I
= 0
This feedback is carried out via the same succession of commands as with [§2.6.2] in
substituent TEMP_EXT with COEF_H in the AFFE_CHAR_THER_F.
Note:
T
· The essential data of this calculation, the field derived outside temperature
ext.
I =
, is
I
I
T
ext.
provided by chth_v.
· To take into account a more sophisticated modeling of the outside temperature
T
,
X T
T I X
,
X T
T
ext. (
) = iext I () I () (iext)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new one
T
source term
ext. = I in the definition of the derived loading chth_v.
I
I
I
Text
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 28/66
3
Nonlinear thermics
In nonlinear thermics the characteristics materials CP (X, T) and (X, T) can depend on
temperature (in non-linear thermics, one cannot define anisotropic materials. There does not exist
of modeling THER_NL_ORTH. Modeling THER_NL). The body is subjected to the same types of
limiting conditions and of loadings which the linear problem to which are added two conditions
non-linear: normal flow imposed I (X, T) (on) and radiation ad infinitum of a gray body (on).
4
5
This last condition is modelled (P0) by its emissivity (X, T), the constant of Stefan-
Boltzmann (X, T) and the temperature ad infinitum T (X, T).
The operator of Code_Aster dedicated to this type of problem is THER_NON_LINE [R5.02.02]. It
allows to solve the problem in extreme cases interfered according to (Cauchy-Dirichlet-Neumann-Robin type
Inhomogenous, non-linear radiation and with variable coefficients)
C
p () T
T
- div ((T) T)
= S
×],
0 [
T
T = F
1 ×],
0 [
() T
T
= G
2 ×],
0 [
N
() T
T
+ HT = HT
ext.
3 ×],
0 [
N
éq
3-1
() T
T
= I (T)
4 ×],
0 [
N
() T
T
= ([
4
4
T +
.
273 15) - (T +
15
.
273
)] 5 ×], 0 [
N
T (X)
0
, = 0
T (X)
Non-linearities pose theoretical problems to show the existence and the unicity of
the solution [bib2]. They can be also prejudicial with the numerical resolution itself.
Thus, with regard to the modeling of voluminal heat CP (X, T), during an iteration,
either because the thermal transient is violent, or because the range of phase shift is
very small (for example, for a pure substance), two the reiterated successive ones of the temperature can
to locate on both sides one of its discontinuities. One then missed a large part of information
relating to the phase shift.
To free itself from this type of problem one rewrites the first equation of [éq 3-1] while introducing
a voluminal function enthalpy which will smooth these non-linearities (dependant on T (X, T) and while noting
T * =T (X, T *) the value of the temperature at one moment T * < T arbitrary)
(T) - div ((T) T) = S
×],
0 [
T
T
éq
3-2
with
(T) = C D
p ()
T *
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 29/66
Note:
· Of share its definition (difference in a primitive (in temperature) of voluminal heat
between the temperature considered and a temperature T * at one arbitrary moment), the function
enthalpy is known except for a constant of integration. It will thus have to be taken care that this
constant is eliminated in all the handled expressions.
· The generalized conditions of type exchanges linear walls [éq 2-2] or relations between ddls
[éq 2-3] are also usable. As in [§2] one will not be interested in derived by
report/ratio with the parameters of functionalities LIAISON_ *.
· For a transitory calculation, as the problem of thermics linear, three strategies
can govern the choice of the field of initial temperature and they have an incidence on
the initialization of the derived problem.
· Implicitly, THER_NON_LINE must tolerate conductivities rather badly strongly
non-linear. Because the tangent matrices and the initial predictive phase does not comprise it
term representing their derivative compared to the temperature. Moreover this term
complementary would have the bad taste to make nonsymmetrical the tangent matrix of
standard system and the matrix of the derived system! What is problematic to take in
count by means of computer in the handling of the structures of data.
In an any state of cause, with respect to the non-linear characteristics material
generally used, the perimeter of use of calculations of sensitivity is the same one
that that of the standard problem. It does not take into account rigorously
nonlinear thermal conductivity.
· In addition, as in linear thermics, one has calculates only the sensitivity by report/ratio
with a constant parameter by geometrical zone. What does not exclude a dependence
temporal, space or not-linear of characteristics material or loadings
nonconcerned by derivation. By parameterizing advisedly the loadings and
“derived” materials in the command file, one can also have access to
some derivatives made up (cf [§6.4]).
· Into non-linear, obtaining a sensitivity by finished differences is even less reliable
that into linear, because it can be very sensitive to the degree of convergence of the solution. In
any rigor, that also influences the quality of the analytical sensitivity, the field of
temperature solution intervening in the assembly of the linear system “derived”.
· Within the framework of non-linear thermics, the derivative compared to the enthalpy will not thus have
no the direction (a enthalpy is not constant!).
· On the other hand, knowing the sensitivity U of the field of temperature compared to a parameter,
one reaches easily that of enthalpy v compared to this same parameter, via the formula
(T)
T
T
v =
:
=
= C
= C U
T
p
p
3.1
Derived compared to voluminal heat
3.1.1 Elements
theoretical
That is to say C X
I X
, one will derive compared to the parameter the formulation
p () = I
I ()
(I)
I
I
T
[éq 3-2]. The required sensitivity is noted U =
. By noticing that the enthalpy can
I
to model like a function of time T and J (considered makes some like functions
indicatrixes of the type of material)
T (T, X, J)
: (T, X
=
-
J)
C
p (,
J) D
C p (
*
T T)
*
T (X, J)
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Titrate:
Calculation of sensitivities in thermics
Date:
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:
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its derivative is written
T
C
*
p
T
T
=
D +
CP
-
*
I
T
I
I
I
*
=
*
T
T
I T T
C
I (
-)
+
p
-
I
I
C
Indeed quantity
p
I =
is independent of the temperature (and thus time) because one
I
I
suppose that the portions are fixed (the under-parts of the body are supposed to be motionless, one
I
neglect in particular the phenomena of dilation). The derivative in time of the first term in enthalpy
of [éq 3-2] is thus worth
(T)
U
I
T
T
= I
+
éq
3.1.1-1
T
I
T
T
In addition one has
(
T (T, X,
X
J)
T T
T
=
+ T
J
+
=
U with {,}
I
T
T
I
I
I
J
T
{
{
J {I
0
0
ij
(
T + 273 15
. ) 4
X
= 4 (T + 273.15) 3 T T
T
+ T
J
+
= 4 (T + 273 15
. ) 3u
T
I
{I
{I
J
J {I
0
0
ij
éq 3.1.1-2
The derivation of [éq 3-2] leads us to the new problem in extreme cases out of U
U
T
-
C + U = ~
div
S
×],
0 [
T
T
U = ~f
,
0
1 ×]
[
+
U
U
= ~g
,
0
2 ×]
[
N
N
U ~
+ hu +
= H
,
0 éq
3.1.1-3
3 ×]
[
N
N
I
U ~
-
U +
= I
,
0
4 ×]
[
N T
N
U
+ 4 (T +
) 3
U +
= ~
15
.
273
,
0
5 ×]
[
N
N
U (X
= ~
)
0
,
u0
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:
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with the new source voluminal and the new limiting conditions and initial
T ~
~
~
~
~
~
~
S = - I
, F
and
éq
3.1.1-4
I
= G = H = I = = 0
U = 0
0
T
Note:
· The derivation of the formulation in voluminal heat [éq 3-1] led of course to same
result because voluminal heat is a function of the J (considered in fact
indicating functions of the type of material) and the temperature (depending itself on
time, of the variable of space and the J!)
C, T T,
X
p (J
(
J)
Its derivative compared to I is thus written
C
C
C
X
p
p
J
p T
T
T
=
+
+ T
J
+
= I +
I
J
T
T
I
J
{I
{I
{I
J
J {I
0
0
ij
ij
from where
U
C
p
T
U
T
CP
U
T
T
+ C
= I
+
U + C
=
+ I
T
p
T
I
T
T
p
T
T
I
T
I
one finds well the formulation [éq 3.1.1-1] of the only term which distinguishes these two
modelings.
Contrary to linear thermics, the derived problem is completely metamorphosed.
The parabolic operator is modified and it became linear out of U. All the problem is besides
become linear because the conditions limit underwent the same processing. The condition of Dirichlet
is from now on homogeneous and those of Neumann and radiation left room to conditions
of Robin. The condition of Cauchy became homogeneous. Contrary to the problem in
temperature, the theoretical results of existence and unicity of the solution U are thus easier with
to exhume.
From a practical point of view, one does not need to have recourse, as in THER_NON_LINE, with one
algorithm of crossbred Newton-Raphson of a predictive phase to determine the increment of
temperature between two contiguous moments. A linear solvor is enough. No one is not need to assemble one
stamp tangent with each under-iteration. The resolution of the “derived” problem is thus more
rapid and more robust than that of the initial problem.
Semi-discretization in times of [éq 3.1.1-3] by - method leads to the following problem:
to find a continuation (U
such as (with the notations of [§2.1.1])
N)
V
0nN
0
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n+1
N
n+1
U
-
U N
T
T
n+1
-
n+1
div
U n+1
T
+ n+1
n+1
U
T
T
N
~ N
~
+1
N
- (
N
N
N
N
S
S
1 -)
-
div
U T + U
=
0 N NR -
1
T
T
N
~
+1
U
= n+1
F
0 N NR 1
1
-
n+1
N 1
n+
N
U
1
+1
+
~
U
+
= n+1
G
0 N NR 1
2
-
N
N
n+1
N 1
n+
N
N
U
~
1
+1
1
+
+
n+1
+ H U +
= H
0 N NR 1
3
-
N
N
n+1
n+1
I
N 1
n+
N
U
1
+1
+
~
-
U
+
= n+1
I
0 N NR 1
4
-
N
T
N
n+1
n+1
3
N
N
N
N
N
U
+
+1
+1
4
(+1
T
+ 273.15) +1
+1
U
+
= ~n+1
0 N NR 1
5
-
N
N
0
U (0 = ~
) u0
éq 3.1.1-5
while posing
~
~
N
N
~
~
~
U = U,
X N
, T = T,
X N
, S N = - I X
,
~
,
0
I () T N
F N = G N = H N = I N = N =
NR
NR
in = I (,
X T N)
N
, = (,
X T N)
N
, = (,
X T N)
N
, H =
H,
X N
NR
N
N
N
=,
X N
, = X, N,
T =
T X, N
NR
NR
NR
éq 3.1.1-6
By applying the theorem of Green to [éq 3.1.1-5], [éq 3.1.1-6] and by introducing the notations
following
+
-
=
X, (N +)
1
and = X, N with {U, T, H, T, T
0
- 1
ext.
}
N
NR
NR
NR
+
-
=
X, T X,
(N +) 1 and =, xT, xn
with
{, I}
NR
NR
one is brought to solve the following variational problem
±
± ± ± ±
±
±
-
Being gift
be, I, H, T and U
+
To calculate U V
that
such
éq
3.1.1-7
0
v
+
V
has
0
(+u, v) = ±l (v)
Handbook of Référence
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Date:
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with the bilinear form depending on the current moment
1
+
has (+
U, v)
+
+
=
+
U v dx +
+
+ +
+
T U + U
. v dx +
T T
T
+
+ +
H U v D - I +
U v
D + 4 + +
(+
T +
15
.
273
) 3 +uv
D
T
3
4
4
éq 3.1.1-8
and the linear form parameterized by the moments running and precedent
1
±
L (v)
-
-
=
-
U v dx + (-)
1
- -
-
-
T U + U
. v dx - 1
II (+
-
T - T) v dx
T T
T
T
+ (-)
1 - -
H U v D
3
+ (
I
1 -)
-
-
U v D + (
4 -)
1 - -
(-
T +
15
.
273
) 3 - U v
D
T
4
5
éq 3.1.1-9
Note:
· While posing in [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] of the linear problem.
In addition, by deriving the variational formulation [éq 4.2-1] from the problem in temperature
[R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9].
· Contrary to the initial problem, the unknown field and the function test belong to
even functional space, which is more comfortable from a theoretical point of view and
numerical.
· This time the two members of this equation are fundamentally different from those from
problem in temperature. However, after each step of time, once determined T+ with
to start from T, one does not have to reassemble all the matrix of the linear system and his second
associate member. It is enough to supplement the first tangent matrix (by taking them again
notations of [R5.02.02]) of the step of next time (allowing to pass from +
++
T = T with ++
T)
1
2
by the term due to the non-linearity of thermal conductivity. One also leaves the second
member of the problem in temperature L (- ±
T, T) to constitute that which interests us: one
fabric by the terms of implicitation of non-linearities of the thermal conductivity and of
limiting conditions.
· Concerning the initialization of the problem derived the remarks from the linear case apply in
extenso (cf [§2.2.2].
· Contrary to calculation in temperature (at the time of the predictive phase of THER_NON_LINE,
the elimination of this constant imposes a suitable reformulation of the elementary term
CHAR_THER_EVOLNI), the constant of integration of the enthalpy does not appear here because one
handle that its derivative in temperature.
· For a condition of exchange between walls, the term of usual exchange is of course replaced
by (by taking again the notations of [éq 2-2]) ±
H (±
±
U
.
I - U J) v D
ij
Handbook of Référence
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Calculation of sensitivities in thermics
Date:
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:
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Space discretization of [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] provides the variational problem
discretized
±
± ± ± ±
±
±
-
Being gift
be, I, H, T and U
H
H
H
H
H
H
H
H
+
To calculate U H
V
that
such
éq
3.1.1-10
H
0
v V
has U, v
L v
H
H
+
0
H (+
H
H) = ±
H (H)
The taking into account of the conditions of Dirichlet leads then to the dualized linear system [éq 2.1.1-11].
We now will see how these calculations are declined in the code.
3.1.2 Establishment in Code_Aster
With the same notations that with [§2.1.1] one can break up the elementary matrix into
WITH K
WITH K
éq
3.1.2-1
ij (
, G)
5
= iij (, G)
i=1
with
· the term of mass and thermal rigidity (option of calculation MTAN_RIGI_MASS by estimating them
-
characteristics material in T)
1
+
WITH K, =
NR NR + + NR
. NR
ij (
) G
G
(G) J (G) I (G) G (G) J (G) I (G)
T
T
· a term of thermal rigidity due to the non-linearity of thermal conductivity (not introduced
because one supposes, in the perimeter of use of the sensitivities, which is independent of T)
2
+
WITH K, =
T + NR. NR
ij (
G)
G
(G)
(G) J (G) I (G)
T
· the term of rigidity due to the conditions limit of exchange (MTAN_THER_COEF_F/R)
A3 K
,
=
+
3
H
K
NR
3
NR
ij (
G)
G
(
) J (G) I (G)
In the event of exchange between walls this term is replaced by (RIGI_THER_PARO_F/R)
A3 K
,
3
= h+ K
NR
3
- NR
NR
ij (
G)
G
(
) (J (G) F J (G) I (G)
()
by noting F the bijection putting in opposite the two walls.
· the term of rigidity due to the condition of non-linear Neumann (MTAN_THER_FLUXNL in
-
considering flow non-linear in T)
4
I
+
With
K
,
4
= -
NR NR
ij (
G)
G
(G) J (G) I (G)
T
· the term of rigidity due to the condition of radiation (MTAN_THER_RAYO_F/R)
A5 K
,
= 4
+
+
+ 273.15
5
K
T
5
3 NR NR
ij (
G)
G
() (
) ((G)
) J (G) I (G)
The second member is written, with the same notations,
L K
L K
éq
3.1.2-2
J (
, G) 6
= ij (, G)
i=1
Handbook of Référence
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where
· the term resulting from the implicitation of the matrix of rigidity and mass (new option
CHAR_SENS_EVOLNI copies CHAR_THER_EVOLNI with U instead of T, the fields
derived and standard material and the new source term)
1
-
L K, =
U NR + - 1 - U
. NR
J (
) G
G
(G) (G) J (G) G () (G) (G) J (G)
T
T
· a term resulting from the implicitation of the non-linearity of thermal conductivity (not
introduced because one supposes, in the perimeter of use of the sensitivities, which is independent
T)
2
-
L K, = - 1
U T -. NR
J (
G)
G (
)
(G) (G)
(G) I (G)
T
· the term due to the “new source” (cf CHAR_SENS_EVOLNI)
L3 K, = -
I K T + - T -
NR
J (
)
G
G
I (
) ((G)
(G) J (G)
T
· the term resulting from the implicitation of the conditions limit of exchange (CHAR_THER_TEXT_F/R
with Text=0 and U instead of T)
L4 K
,
1
3
= - H K
U
3
NR
J (
G)
G (
) (
) (G) J (G)
In the event of exchange between walls this term is replaced by (CHAR_THER_PARO_F/R with U with
place of T)
L4 K
,
1
3
= - H K
U
3
- U F
NR
J (
G)
G (
) (
) ((G) ((G)) J (G)
· the term resulting from the implicitation of the condition of non-linear Neumann (new option
CHAR_SENS_FLUNL)
5
I
L
K
,
4
= 1
U NR
J (
G)
G (
)
(G) (G) J (G)
T
· the term resulting from the implicitation of the condition of radiation (new option
CHAR_SENS_RAYO_F)
L6 K
,
4
1
273.15
5
= -
- K
T -
5
+
3 U NR
J (
G)
G (
) () (
) ((G)
) (G) J (G)
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Note:
· The elementary terms of the matrix of the “derived” problem are identical to those of
stamp tangent initial problem. Indeed the problem in temperature can be written in
revealing a vector residue R [R5.02.02] and a vectorial function test V
T
V. R (+ +
T, T) = T
V. L (- ±
T, T)
While deriving compared to a parameter from calculation from the residue (for example a characteristic
material) noted this vectorial relation and by translating it in indicielle form, one has
(+
V R =
V L±
K
K)
(K K)
R
+ T
+
L
±
K
J
K
V
= V
K
K
T
{J
Kkj
T
+
+
+
L
+
K T
=
with V =
ij (
) J
I
(
K
ki)
4 4
3
2
1
With
{{
ij
U +
L
J
I
One thus finds well the indicielle formulation of the derived linear system [éq 2.1.1-10].
· Compared to the “derived” problem linear, the elementary matrix and the second member are
supplemented by the terms incorporating non-linearities of the thermal conductivity and of
limiting conditions. On the other hand, the new source is identical in both cases (one will be able
thus mutualiser the option of calculation).
· To take into account a more sophisticated modeling of voluminal heat
C
,
X T
I X,
X T
p (
) = I I () I () (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
C
source
p = I in the definition of the field derived material
I
I
ma_v.
I
As one already specified, all the elementary terms (except the second) of the matrix make
the object of an option of calculation and will already have been evaluated for the calculation of ++
T. It thus remains with
2
to estimate the second member by re-using (with a different parameter setting) the options of calculation
existing or by introducing news (CHAR_SENS_EVOLNI/FLUNL/RAYO_F). These last
redirect towards the same routine of elementary calculation (TE.) that their associated options standards
(CHAR_THER_EVOLNI/FLUNL/RAYO_F).
The central character string (SENS instead of THER) joined to a detection of the nullity of the field
derived material, makes it possible to parameterize these routines towards one or the other their orientations
possible: calculation of an elementary term derived (new source term) or of a term “spectator”
had with the presence of a condition of exchange, radiation or a non-linear flow.
In accordance with the principles of architecture set up in the code to treat calculations
of sensitivity [bib6], the assembly and the resolution of [éq 2.1.1-11] are started by the analysis of
table of correspondence associated with the significant variable. This feedback is carried out via
even succession of commands that with [§2.1.2] in substituent THER_NL with THER in
DEFI_MATERIAU and by replacing of course operator THER_LINEAIRE by THER_NON_LINE.
We will unroll the same process for the various sensitivities, to start with that
concerning the other characteristic material.
Handbook of Référence
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Calculation of sensitivities in thermics
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3.2
Derived compared to thermal conductivity
3.2.1 Elements
theoretical
By taking again the notations of [§2.2.1], one carries out derivation compared to the parameter, with
I
T
U =
sought sensitivity,
I
T
C
p
T
C
p
T
U
(C up)
=
U
+ C
=
T
T
p
T
T
I
(T)
(T)
U
T
T
=
T
I
The derivation of [éq 3-2] delivers to us a problem in extreme cases identical to [éq 3.1.1-1] but with one
another voluminal source and of new conditions of Robin
~
~
~
~
S = div (I T,
= = = ~ = -
éq
3.2.1-1
I
)
T
G H
I
II N
It is a homogeneous problem of Dirichlet-Cauchy and inhomogenous Robin. One can thus take back
same remarks concerning the linear character of the derived problem and simplifications
theoretical and numerical that that implies. Its semi-discretization in time results in seeking one
continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which the first relation
N)
V
0nN
0
rewrites itself
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.2.1-2
N
- (
1 -)
N
N
N
N
~
div
N 1
U T + U =
+
S +
N
(-) ~
1
S
0 N NR - 1
T
with the new source term
~sm = div (I T m
éq
3.2.1-3
I
) m {, nn+} 1
and new limiting conditions
1
+
~
~ 1
+1
~ +1
~ +1
+
N
N
N
N
T N
G
= H
= I
=
= - I
éq
3.2.1-4
I
N
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new source
±
L (v) = L - I
1
éq
3.2.1-5
I (
+
T + (-) -
T) vdxL
Handbook of Référence
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Date:
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:
R4.03.02-A Page
: 38/66
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.2.1-5]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.2.1-5] of
linear problem. In addition, by deriving the variational formulation [éq 4.2-1] from
problem in temperature [R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.2.1-5].
The new source term is identical into linear and non-linear.
· For the moment, operator DEFI_MATERIAU does not allow to model a conductivity
orthotropic non-linear thermics. There are not thus the particular cases of the linear problem.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.2.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
L3 K, that for the linear case (cf [éq 2.2.2-1]). What is made in the new option of calculation
J (
G)
CHAR_SENS_EVOLNI with the fields derived and standard material.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.1.2] in substituent THER_NL/LAMBDA with THER/RHO_CP in
DEFI_MATERIAU and by replacing of course operator THER_LINEAIRE by THER_NON_LINE.
Note:
· To take into account a more sophisticated modeling of thermal conductivity
(,
X T) = I X,
X T
I
I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
source
= I in the definition of the field derived material
I
I
ma_v.
I
3.3
Derived compared to the source
3.3.1 Elements
theoretical
By applying the formulas of the preceding paragraphs, the derivation of [éq 3-2] compared to
parameter S (cf [§2.3.1]) delivers to us a problem in extreme cases identical to [éq 3.1.1-1] but with one
I
another voluminal source
s~ = I
éq
3.3.1-1
I
One can thus renew the same remarks concerning the linear character of the derived problem and
theoretical and numerical simplifications that that implies. Its semi-discretization in time
conduit to seek a continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6]
N)
V
0nN
0
T
whose first relation is rewritten, by noting U =
sought sensitivity,
S
I
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq 3.3.1-2
N
- (
1 -) div
unT N + N
one = II 0 N NR - 1
T
The application of the theorem of Green to [éq 3.3.1-2] led to solve a variational problem
identical to [éq 3.1.1-7] comprising the same bilinear form [éq 3.1.1-8] joined to the linear form
[éq 3.1.1-9] whose only third integral is modified to adapt to the new source
±
L (v) = L + I vdx
éq
3.3.1-3
I
L
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.3.1-3]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.3.1-3] of
linear problem. In addition, by deriving the variational formulation [éq 4.2-1] from
problem in temperature [R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.3.1-3].
· The new source term is identical into linear and non-linear.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.3.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
L3 K, that for the linear case (cf [éq 2.3.2-1]) and the same option of calculation.
J (
G)
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.3.2] by replacing operator THER_LINEAIRE by
THER_NON_LINE.
Note:
The taking into account of more sophisticated modeling of the source term is carried out as in
linear.
3.4
Derived compared to the imposed temperature
3.4.1 Elements
theoretical
By applying the formulas of the preceding paragraphs, the derivation of [éq 3-2] compared to
parameter F (cf [§2.4.1]) delivers to us a problem in extreme cases identical to [éq 3.1.1-1] but with one
I
null voluminal source and another condition of Dirichlet
s~ =
~
0 and
F = I
éq
3.4.1-1
I
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 40/66
It is an inhomogenous problem of Dirichlet-Robin and homogeneous Cauchy. One can thus take back
same remarks concerning the linear character of the derived problem and simplifications
theoretical and numerical that that implies. Its semi-discretization in time results in seeking one
continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which the first relation
N)
V
0nN
1
T
rewrites itself, by noting U =
sought sensitivity,
F
I
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.4.1-2
N
- (
1 -) div
unT N + N
a =0 0 N NR - 1
T
with the new limiting condition
~n
F +1 = I
éq
3.4.1-3
I
The application of the theorem of Green to [éq 3.4.1-2], [éq 3.4.1-3] resulted in solving a problem
variational identical to [éq 3.1.1-7] comprising the same bilinear form [éq 3.1.1-8] joined to the form
linear [éq 3.1.1-9] whose only third integral is modified to adapt to the news
source: here this integral is null.
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
“null source” + [éq 3.4.1-3]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.4.1-2],
[éq 2.4.1-3], [éq 2.4.1-4] of the linear problem. In addition, by deriving the formulation
variational [éq 4.2-1] of the problem in temperature [R5.02.02] one finds [éq 3.1.1- well
7], [éq 3.1.1-8], [éq 3.1.1-9] + “null source” + [éq 3.4.1-3].
· The new source term is identical into linear and non-linear.
· One finds the same linear system not dualized as for the derivative in enthalpy. Only
taken into account of the condition of Dirichlet via of Lagranges will make the difference.
The space discretization and the taking into account of the inhomogenous condition of Dirichlet lead to
dualized linear system [éq 2.4.1-5]. Components of its second associate member with
Lagranges are nonnull.
3.4.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
3
L K
that for the linear case (cf [§2.4.2]) and the same option of calculation to assemble it
J (
, G) = 0
dualized linear system associated the “derived” conditions of Dirichlet.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.4.2] by replacing operator THER_LINEAIRE by
THER_NON_LINE.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 41/66
Note:
· Concerning various modelings of the imposed temperature and the conditions of
Dirichlet generalized, there is no difference between the non-linear one and the linear one. For more
information one can thus refer to [§2.4.2].
· The taking into account of more sophisticated modeling of the imposed temperature is carried out
as into linear.
3.5
Derived compared to linear imposed normal flow
3.5.1 Elements
theoretical
By applying the formulas of the preceding paragraphs, the derivation of [éq 3-2] compared to
parameter G (cf [§2.5.1]) delivers to us a problem in extreme cases identical to [éq 3.1.1-1] but with one
I
null voluminal source and another condition of Robin on
2
s~ =
and
g~
0
= I
éq
3.5.1-1
I
It is a homogeneous problem of Dirichlet-Cauchy and inhomogenous Robin. One can thus take back
same remarks concerning the linear character of the derived problem and simplifications
theoretical and numerical that that implies. Its semi-discretization in time results in seeking one
continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which the first relation
N)
V
0nN
0
T
rewrites itself, by noting U =
sought sensitivity,
G
I
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.5.1-2
N
- (
1 -) div
unT N + N
a =0 0 N NR - 1
T
with the new limiting condition
~n
G +1 = I
éq
3.5.1-3
I
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new “surface” source
±
L (v) = L + I vdx
éq
3.5.1-4
I
L
2
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.5.1-4]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.5.1-4] of
linear problem. In addition, by deriving the variational formulation [éq 4.2-1] from
problem in temperature [R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.5.1-4].
· The new source term is identical into linear and non-linear.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 42/66
3.5.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
L3 K
,
that for the linear case (cf [§2.5.2]) and the same option of calculation.
2
J (
G)
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.5.2] by replacing operator THER_LINEAIRE by
THER_NON_LINE.
Note:
· Concerning various modelings of this condition of linear Neumann there is not
no difference between the linear and non-linear derived problem. For more information one
can thus refer to [§2.5.2].
· The taking into account of more sophisticated modeling of imposed normal flow is carried out
as into linear.
3.6
Derived compared to non-linear imposed normal flow
3.6.1 Elements
theoretical
T
One poses I (X) = I I X
I
and U =
required sensitivity. Portions of border
I
I ()
(I)
I
II
I
external being solidified, one has
= I the indicating function of the ième portion. By applying them
4 J
I
I
4i
I
formulas of the preceding paragraphs, the derivation of [éq 3-2] compared to parameter I delivers to us
I
a problem in extreme cases identical to [éq 3.1.1-1] but with a null voluminal source and another
condition of Robin on
4
s~ =
~
0 and I = I
éq
3.6.1-1
I
It is a homogeneous problem of Dirichlet-Cauchy and inhomogenous Robin. One can thus take back
same remarks concerning the linear character of the derived problem and simplifications
theoretical and numerical that that implies. Its semi-discretization in time results in seeking one
continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which the first relation
N)
V
0nN
0
rewrites itself
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.6.1-2
N
- (
1 -) div
unT N + N
a =0 0 N NR - 1
T
with the new limiting condition
~n
I +1 = I
éq
3.6.1-3
I
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 43/66
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new “surface” source
±
L (v) = L + I vdx
éq
3.6.1-4
I
L
4
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.6.1-4]
±
±
±
±
=
I
G
C,
= 0, =,
=
±
0, I =,
and
4 = 2
5 =
T
p
T
T
2
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.5.1-4] of
linear problem. In addition, by deriving the variational formulation [éq 4.2-1] from
problem in temperature [R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.6.1-4].
· This new source term is identical, except for the borders, at the end source of the derivative
compared to linear normal flow.
· In stationary regime, it is of course unchanged.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.6.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
L3 K
,
that for the linear case by replacing the border by (cf [éq 2.5.2-1]). What
4
J (
G)
2
4
is made in the new option of calculation CHAR_SENS_FLUNL with the field derived non-linear flow.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.5.2] in substituent FLUX_NL with FLUX_REP in
AFFE_CHAR_THER_F and by replacing of course operator THER_LINEAIRE by THER_NON_LINE.
Note:
To take into account a truly non-linear modeling
I (,
X T) = I I X,
X T
I
I
I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
I
source
= I in the definition of the derived loading
I
I
chth_v.
II
3.7
Derived compared to the coefficient from convectif exchange
3.7.1 Elements
theoretical
By applying the formulas of the preceding paragraphs, the derivation of [éq 3-2] compared to
parameter H (cf [§2.6.1]) delivers to us a problem in extreme cases identical to [éq 3.1.1-1] but with one
I
null voluminal source and another condition of Robin on
3
s~ =
~
0 and H = I
-
éq
3.7.1-1
I (T
T
ext.
)
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 44/66
It is a homogeneous problem of Dirichlet-Cauchy and inhomogenous Robin. One can thus take back
same remarks concerning the linear character of the derived problem and simplifications
theoretical and numerical that that implies. Its semi-discretization in time results in seeking one
continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which the first relation
N)
V
0nN
0
T
rewrites itself, by noting U =
sought sensitivity,
H
I
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.7.1-2
N
- (
1 -) div
unT N + N
a =0 0 N NR - 1
T
with the new limiting condition
~n 1+
H
= I T
T
éq
3.7.1-3
I (N 1
+
N 1
+
ext.
-
)
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new source
±
L (v) = L + I
éq
3.7.1-4
I {(+
+
Text - T) + (1 -) (-
-
Text - T)}vdx L
3
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.7.1-4]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.6.1-4] of
linear problem. In addition, by deriving the variational formulation [éq 4.2-1] from
problem in temperature [R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.7.1-4].
· The new source term is identical into linear and non-linear.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.7.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
L3 K
,
that for the linear case (cf [éq 2.6.2-1]) and the same option of calculation.
3
J (
G)
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.6.2] by replacing operator THER_LINEAIRE by
THER_NON_LINE.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 45/66
Note:
· The taking into account of more sophisticated modeling of the coefficient of exchange is carried out
as into linear.
· The condition of imposed normal flow non-linear (cf [§3.6]) makes it possible to model an exchange
convectif with a non-linear coefficient of exchange via an adequate DEFI_FONCTION
I (,
X T) = J (,
X T) (T
,
X
-
,
X
ext. (
T) T (T)
T
Same manner, knowing U =
one could have access “easily” to
I
I
T
W =
. Indeed,
J
I
T T I
I T
=
+
J
I J
T J
{
{
T
T
- J
ext. -
from where
U (T - T)
W
ext.
= (
1+ ju)
3.8
Derived compared to the outside temperature
3.8.1 Elements
theoretical
By applying the formulas of the preceding paragraphs, the derivation of [éq 3-2] compared to
parameter I
T (cf [§2.7.1]) delivers to us a problem in extreme cases identical to [éq 3.1.1-1] but with one
ext.
null voluminal source and another condition of Robin on
3
s~ =
~
0 and H = hI
éq
3.8.1-1
I
It is a homogeneous problem of Dirichlet-Cauchy and inhomogenous Robin. One can thus take back
same remarks concerning the linear character of the derived problem and simplifications
theoretical and numerical that that implies. Its semi-discretization in time results in seeking one
continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which the first relation
N)
V
0nN
0
T
rewrites itself, by noting U =
sought sensitivity,
I
T
ext.
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.8.1-2
N
- (
1 -) div
unT N + N
a =0 0 N NR - 1
T
with the new limiting condition
~n 1+
N
H
= H 1+I
éq
3.8.1-3
I
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 46/66
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new “surface” source
±
L (v) = L + I
éq
3.8.1-4
I {
+
H + (1 -) -
H} vdx L
3
Note:
· While posing in the new variational problem [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.8.1-4]
±
±
=
C,
=
±
0, = and
4 = 5 =
T
p
T
one finds well the formulation [éq 2.1.1-6], [éq 2.1.1-7], [éq 2.1.1-8] + [éq 2.6.1-4] of
linear problem. In addition, by deriving the variational formulation [éq 4.2-1] from
problem in temperature [R5.02.02] one finds well [éq 3.1.1-7], [éq 3.1.1-8], [éq 3.1.1-9] +
[éq 3.8.1-4].
· The new source term is identical into linear and non-linear.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.8.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified. The same one is taken
L3 K
,
that for the linear case (cf [éq 2.7.2-1]) and the same option of calculation.
3
J (
G)
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§2.6.2] in substituent TEMP_EXT with COEF_H in
AFFE_CHAR_THER_F and while replacing, of course, operator THER_LINEAIRE by THER_NON_LINE.
Note:
The taking into account of a more sophisticated modeling of the outside temperature
be carried out as into linear.
3.9
Derived compared to emissivity/constant of Stefan-Boltzmann
3.9.1 Elements
theoretical
One poses (X) = I X
(resp. (X) = I X
) the parameter and
I
I ()
(I)
I
I ()
(I)
I
I
T
T
U =
(resp. U =
) required sensitivity. The portions being solidified, one has
= I
5 J
I
I
I
I
(resp.
= I) the indicating function of the ième portion. By applying the formulas of
I
5i
I
preceding paragraphs, the derivation of [éq 3-2] delivers a problem in extreme cases identical to us to
[éq 3.1.1-3] but with another voluminal source and another condition of Robin on (resp. in
5
inverting the role of and of)
~
~
S = 0 and I = I
éq
3.9.1-1
I
({
T + 273.15) 4 - (T + 273.15) 4}
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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One can thus renew the same remarks concerning the linear character of the derived problem and
theoretical and numerical simplifications that that implies. Its semi-discretization in led time
to seek a continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6] of which
N)
V
0nN
0
first relation is rewritten
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.9.1-2
N
- (
1 -) div
unT N + N
one = 0 0 N NR - 1
T
with the new condition (resp limits. by inverting the role of and of)
~ N 1
+
N 1
I
=
+
I
T
T
éq
3.9.1-3
I
({n+
N
+
15
.
273
) 4
1
- (+ +
15
.
273
) 4
1
}
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new “surface” source (resp. by inverting the role of and of)
L ± (v) = L + I
+
v
4
4
+
+ 273.15
- + + 273.15 -
I
({T
) (T
)}
5
éq 3.9.1-4
I
4
4
- 1 -
-
+ 273.15
- - + 273.15
I (
) v ({T
) (T
)} dx
Note:
· Derivation compared to the constant of Stefan-Boltzmann has certainly only one interest
practical miner. But its overcost of numerical establishment was moderate and it allows
cross validations.
· In stationary regime, this source term is reduced to (resp. by inverting the role of and of
)
L (v) = L + I
I
({T +
4
15
.
273
- T +
4
) (
15
.
273
)} vdxL
5
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.9.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified (resp. by inverting it
role of and of)
L3 K
,
=
+
+
+
15
.
273
- +
+
15
.
273
+
5
I
K
T
K
4
T
5
5
4 NR
J (
G)
G (I
) (
) ({(
)
) ((G)
)} J (G)
(1 -) I - K
T - K
4
+
15
.
273
- T -
+
15
.
273
5
5
4 NR
G (I
) (
) ({(
)
) ((G)
)} J (G)
éq 3.9.2-1
What is made in the new option of calculation CHAR_SENS_RAYO_F with the fields of
standard and derived radiation.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via
succession of commands.
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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v = DEFI_VALEUR_SENSI (VALE = < value of I >) (resp. I)
chth = AFFE_CHAR_THER_F (RAYONNEMENT= _F (GROUP_MA = < definition of 5i >,
SIGMA = v, EPSILON = W, TEMP_EXT = Z))
…
one = DEFI_CONSTANTE (VALE = 1. )
zero = DEFI_CONSTANTE (VALE = 0. )
MEMO_NON_SENSI (NOM =_F (NOM_SD = “chth”, PARA_SENSI = “v”,
NOM_COMPOSE = “chth_v”))
chth_v = AFFE_CHAR_THER_F (RAYONNEMENT = _F (GROUP_MA = < 5i >,
SIGMA = one, EPSILON = zero, TEMP_EXT = zero))
…
resu = THER_NON_LINE (EXCIT = chth,
SENSIBILITE
=
(
v)
…)
Note:
· The essential data of this calculation, the field derived radiation
=
T
I,
(resp.
= I), is provided by
I
=,
0
= 0
I
chth_v.
I
I
I
I
· To take into account a more sophisticated modeling of emissivity (resp. constant
of Stefan-Boltzmann)
(,
X T) = I X,
X T
I
I ()
I (
) (I)
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
source
= I in the definition of the derived loading
I
I
chth_v.
I
3.10 Derived compared to the temperature ad infinitum
3.10.1 Theoretical elements
T
T X is posed
I
I
=
() =
T I X
T
the parameter and U
required sensitivity.
I ()
()
I
I
T
T
portions being solidified, one has
= I the indicating function of the ième portion. By applying them
5 J
I
I
T
5i
formulas of the preceding paragraphs, the derivation of [éq 3-2] delivers a problem in extreme cases to us
identical to [éq 3.1.1-3] but with another voluminal source and another condition of Robin on
5
~
~
S = 0 and I = 4I
éq
3.10.1-1
I
(
T +
.
273
) 3
15
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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:
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One can thus renew the same remarks concerning the linear character of the derived problem and
theoretical and numerical simplifications that that implies. Its semi-discretization in time
conduit to seek a continuation (U
checking a system similar to [éq 3.1.1-5], [éq 3.1.1-6]
N)
V
0nN
0
whose first relation is rewritten
N 1
+
N
n+
1
U
-
one
N 1
T
T
+
- div
N 1
+
N 1
+
N 1
+
N 1
+
U T
+ U
T
T
éq
3.10.1-2
N
- (
1 -) div
unT N + N
one = 0 0 N NR - 1
T
with the new limiting condition
~n 1
+
I
= 4I
éq
3.10.1-3
I
(
T + 273.15) 3
From where a variational problem identical to [éq 3.1.1-7] comprising the same bilinear form
[éq 3.1.1-8] joined to the linear form [éq 3.1.1-9] whose only third integral is modified for
to adapt to the new “surface” source
±
L (v) =
3
3
L + 4 I
I {(
) + (+
T +
15
.
273
+ 1 -
T +
) () () - (-
15
.
273
)} vdxL
5
éq 3.10.1-4
Note:
In stationary regime, this source term is unchanged.
The space discretization and the taking into account of the homogeneous condition of Dirichlet lead to
dualized linear system [éq 2.1.1-11].
3.10.2 Establishment in Code_Aster
Compared to [§3.1.2], only the term due to the new source is to be modified
L3 K
,
= 4
+
+
+
+
15
.
273
+
5
I
K
T
K
3 NR
5
5
J (
G)
G
(I
) (
) ((
)
) J (G) éq 3.10.2-1
4
1 - I - - K
T - K
3
+
15
.
273
NR
5
5
G (
) (I
) (
) ((
)
) J (G)
What is made in the new option of calculation CHAR_SENS_RAYO_F with the fields of
standard and derived radiation.
The assembly and the resolution of the linear system are started by the analysis of the table of
correspondence associated with the significant variable. This feedback is carried out via the same one
succession of commands that with [§3.9.2] in substituent TEMP_EXT with SIGMA.
Note:
Ad infinitum to take into account a more sophisticated modeling of the temperature
T
,
X T
I
I
(
) =
T I X
,
X T
T
I ()
I (
) ()
I
in these calculations of sensitivity, it is enough to substitute for the indicating function the new term
T
source
= I in the definition of the derived loading chth_v.
I
I
I
T
We now will recapitulate all the linear systems “derived” to assemble according to
desired sensitivities.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
R4.03.02-A Page
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4
Summary of the sensitivities of the temperature
That one is in or not linear linear thermics, the sensitivity required to the moment running U+ is
solution of a dualized system of the type (with the notations of [§2.1])
With LT
LT +
U
L
~ ~ +
~
WITH U = L
B - Id
Id
=
C
B Id - Id C
where
WITH K,
WITH K,
ij (
G) = iij (
G)
I
L K,
L K,
J (
G) = ij (
G)
I
One will draw up the nomenclature of the potential elementary terms of this system and of their
option of calculation in Code_Aster. In order to be more synthetic, one agrees here on a classification
different from those practiced in the preceding paragraphs.
For the matrix, there are the seven following possible terms:
· MASS_THER (cf [§2.1.2])
A1 K, =
C K NR NR
ij (
) G
G
p (
) J (G) I (G)
T
· RIGI_THER (cf [§2.1.2])
A2 K, = K NR. NR
ij (
G)
G
() J (G) I (G)
· RIGI_THER_COEF_F/R or MTAN_THER_COEF_F/R (cf [§2.1.2], [§3.1.2])
A3 K
,
=
+
3
H
K
NR
3
NR
ij (
G)
G
(
) J (G) I (G)
In the event of exchange between walls this term is replaced by RIGI_THER_PARO_F/R or
MTAN_THER_PARO_F/R (cf [§2.1.2], [§3.1.2])
A3 K
,
3
= h+ K
NR
3
- NR
NR
ij (
G)
G
(
) (J (G) F J (G) I (G)
()
· MTAN_RIGI_MASS (cf [§3.1.2])
4
WITH K, =
T +
NR NR + T +
NR
. NR
ij (
) G
G
((G) J (G) I (G) G ((G) J (G) I (G)
T
T
· Not coded. (cf [§3.1.2])
5
WITH K, =
T +
T + NR. NR
ij (
G)
G
((G)
(G) J (G)
I (G)
T
· MTAN_THER_FLUXNL (cf [§3.1.2])
6
I
With
K
,
4
= -
T +
NR NR
ij (
G)
G
((G) J (G) I (G)
T
· MTAN_THER_RAYO_F/R (cf [§3.1.2])
A7 K
,
= 4
+
+
+ 273.15
5
K
T
5
3 NR NR
ij (
G)
G
() (
) ((G)
) J (G) I (G)
For the second member, there are sixteen possibilities:
· CHAR_SENS_EVOL (cf [§2.1.2])
L1 K, =
C K U NR + - 1 K U
. NR
J (
) G
G
p (
) (G) J (G) () G () (G)
J (G)
T
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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· CHAR_THER_TEXT_F/R (cf [§2.1.2])
L2 K
,
1
3
= - H K
U
3
NR
J (
G)
G (
) (
) (G) J (G)
In the event of exchange between walls this term is replaced by CHAR_THER_PARO_F/R (cf.
[§2.1.2])
L2 K
,
1
3
= - H K
U
3
- U F
NR
J (
G)
G (
) (
) ((G) ((G)) J (G)
· CHAR_SENS_EVOL (cf [§2.1.2])
L3 K, = -
I K T + - T -
NR
J (
)
G
G
I (
) ((G)
(G) J (G)
T
· CHAR_SENS_EVOL (cf [§2.1.2])
L4 K, =
- I K T + + 1 - T -
NR
J (
G)
G I (
) (
(G) () (G)
J (G)
· CHAR_THER_SOUR_F/R (cf [§2.3.2], [§3.3.2])
L5 K, = I K NR
J (
G)
G I (
) J (G)
· CHAR_THER_FLUN_F/R (cf [§2.5.2], [§3.5.2])
L6 K
,
=
2
I K
NR
2
J (
G)
G I (
) J (G)
· CHAR_SENS_TEXT_F (cf [§2.6.2], [§3.7.2])
T + K
3 - T
7
+ +
L
K
,
3
= I K
3
NR
J (
G)
(ext. (
)
(G)
G I (
)
-
-
(1 -) (T K
3 - T
ext. (
)
(G) J (G)
· CHAR_THER_TEXT_F (cf [§2.7.2], [§3.8.2])
L8 K
,
=
+ + 1
-
3
I K
3
H K
3
H K
NR
3
J (
G)
G I (
) {(
) (
) (
)} J (G)
· CHAR_THER_TEXT_F (cf [§2.7.2], [§3.8.2])
9
L K, =
T -
U - NR + - 1 T -
U -
. NR
J (
) G
G
((G) (G) J (G) G () ((G)
(G) J (G)
T
T
· CHAR_SENS_EVOLNI (cf [§3.1.2])
10
L K, = - 1
T -
U - T -. NR
J (
G)
G (
) ((G) (G)
(G)
I (G)
T
· CHAR_SENS_FLUNL (cf [§3.1.2])
11
I
L
K
,
4
= 1
T -
U - NR
J (
G)
G (
) ((G) (G) J (G)
T
· CHAR_SENS_RAYO_F (cf [§3.1.2])
L12 K
,
4
1
273.15
5
= -
- K
T -
5
+
3 U NR
J (
G)
G (
) () (
) ((G)
) (G) J (G)
· CHAR_SENS_FLUNL (cf [§3.6.2])
L13 K
,
=
4
I K
NR
4
J (
G)
G I (
) J (G)
· CHAR_SENS_RAYO (cf [§3.9.2])
L14 K
,
=
+
+
+
15
.
273
- +
+
15
.
273
+
5
I
K
T
K
4
T
5
5
4 NR
J (
G)
G (I
) (
) ({(
)
) ((G)
)} J (G)
(1 -) I - K
T - K
4
+
15
.
273
- T -
+
15
.
273
5
5
4 NR
G (I
) (
) ({(
)
) ((G)
)} J (G)
L15 K
,
=
+
+
+
15
.
273
- +
+
15
.
273
+
5
I
K
T
K
4
T
5
5
4 NR
J (
G)
G (I
) (
) ({(
)
) ((G)
)} J (G)
(1 -) I - K
T - K
4
+
15
.
273
- T -
+
15
.
273
5
5
4 NR
G (I
) (
) ({(
)
) ((G)
)} J (G)
· CHAR_SENS_RAYO (cf [§3.10.2])
L16 K
,
= 4
+
+
+
+
15
.
273
+
5
I
K
T
K
3 NR
5
5
J (
G)
G
(I
) (
) ((
)
) J (G)
4
1 - I - - K
T - K
3
+
15
.
273
NR
5
5
G (
) (I
) (
) ((
)
) J (G)
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 52/66
The operator B allowing to check the condition of Dirichlet is written
(+
B U) = u+ with I J
I
I
with its second associate member
C = 0
Note:
· The possible taking into account of limiting conditions of Dirichlet generalized requires
to rewrite this operator in the form
+
B U = u+ with J J
J
J
J
· As one rigorously does not take into account nonlinear thermal conductivity,
the terms A5 and L10 are not programmed yet.
The eighteen dualized systems exhumed in the preceding paragraphs can then gather
in the following table:
Type of
Variable
Stamp
Second
Key word
Characteristics
sensitivity
member
Thermics
Stamp
THER_LINEAIRE Sensibilité compared to
linear
identical to
a constant parameter
that of
by zone
problem
(possibly derived
direct
composed cf [§6.4])
Heat
T
A1+A2+A3
L1+L2+L3
THER/RHO_CP
voluminal
U =
I
Conductivity
T
Idem
L1+L2+L4
THER/LAMBDA
Sensitivity compared to
thermics
U =
a characteristic
I
orthotropic not
accessible
Source
T
Idem
L1+L2+L5
SOURCE
Three types of
U =
modeling
S
I
Temperature
T
Idem
L1+L2
TEMP_IMPO
F
imposed
U =
J
C =
=
F
K
ij
I
F
I
Three types of
modeling.
Sensitivity compared to
a coefficient
multiplier of one
condition of Dirichlet
generalized not
accessible
Normal flow
T
Idem
L1+L2+L6
FLUX_REP
Three types of
imposed
U =
modeling.
G
I
Sensitivity compared to
a component of flow
vectorial accessible.
Exchange
T
Idem
L1+L2+L7
ECHANGE/COEF_H Pas of ECHANGE_PAROI
convectif
U =
or
H
in lumpé.
I
ECHANGE_PAROI
Temperature
T
Idem
L1+L2+L8
ECHANGE/
external
U =
TEMP_EXT
I
T
ext.
Handbook of Référence
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Calculation of sensitivities in thermics
Date:
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:
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Type of
Variable
Stamp
Second
Key word
Characteristics
sensitivity
member
Thermics
Stamp Les new THER_NON_LINE Sensibilité compared to
nonlinear
near to the sources terms
a constant parameter
stamp
are similar
by zone
tangent of into linear and in
(possibly derived
problem
nonlinear.
composed cf [§6.4]).
direct.
No sensitivity by
report/ratio with the enthalpy.
Account is not held
of a conductivity not
linear
A5 and L10 not taken in
count
Heat
T
A3+A4+A5
L2+L3+L9
THER_NL/RHO_CP
voluminal
U =
+A6+A7
+L10+L11+L12
I
Conductivity
T
Idem
L2+L4+L9
THER_NL/LAMBDA linear Idem case
thermics
U =
+L10+L11+L12
I
Source
T
Idem
L2+L5+L9
SOURCE
Idem linear case
U =
S
+L10+L11+L12
I
Temperature
T
Idem
L2+ L9
TEMP_IMPO
Idem linear case
imposed
U =
F
+L10+L11+L12
I
Normal flow
T
Idem
L2+L6+L9
FLUX_REP
Idem linear case.
imposed
U =
+L10+L11+L12
linear
G
I
Normal flow
T
Idem
L2+L13+L9
FLUX_NL
Allows to model
imposed not
U =
+L10+L11+L12
sensitivity compared to
linear
I
I
a convectif exchange
nonlinear
Exchange
T
Idem
L2+L7+L9
ECHANGE/COEF_H Pas of ECHANGE_PAROI
convectif
U =
+L10+L11+L12
or
in lumpé.
linear
H
I
ECHANGE_PAROI
Temperature
T
Idem
L2+L8+L9
ECHANGE/
external
U =
TEMP_EXT
I
T
+L10+L11+L12
ext.
Emissivity
T
Idem
L2+L15+L9
RAYONNEMENT/
U =
+L10+L11+L12
EPSILON
I
Constant of
T
Idem
L2+L14+L9
RAYONNEMENT/
Stefan-
U =
+L10+L11+L12
SIGMA
Boltzmann
I
Temperature with
T
Idem
L2+L16+L9
RAYONNEMENT/
the infinite one
U =
TEMP_EXT
I
T
+L10+L11+L12
Table 4-1: Summary of the linear systems “derived”
We will now close the theoretical part this document by calculating the sensitivities of one
related size with the field of temperature: heat flow.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 54/66
5
Sensitivity of the heat flow
After having carried out a thermal calculation, one has access to the heat flow (calculation is carried out either with
nodes (field with the nodes by elements via OPTION=' FLUX_ELNO_TEMP'), is at the points of Gauss
(field at the points of Gauss by elements via OPTION=' FLUX_ELGA_TEMP')) via the operators
CALC_ELEM/CALC_NO. It is given starting from the field of temperature by using the Fourier analysis
who is written in linear thermics with isotropic materials
Q (X, T) = - (X) T (X, T)
With anisotropic materials, thermal conductivity is modelled by a diagonal matrix
(expressed in its reference mark of orthotropism). Into nonlinear (inevitably isotropic in Code_Aster) one
has on the other hand
Q (X, T) = - (X, T) T (X, T)
By taking again the notations and the step developed in the preceding paragraphs, one
synthesize the calculation of derived from the heat flow in the following table. The column entitled
“formula” expresses the relation to be set up in operators CALC_ELEM/CALC_NO for
to determine the sensitivity of the heat flow required. It depends on the field of temperature and
of its sensitivity compared to the same parameter, all the two resulting one from a thermal calculation (via
THER_LINEAIRE or THER_NON_LINE).
Note:
In practice, calculations of sensitivity of the heat flow do not take account of the possible one
non-linearity of thermal conductivity. In non-linear thermics, the first term is not
thus not programmed.
Type of
Variables
Sensitivity
Formulate/
sensitivity
of output of
sought
Options of calculation in Code_Aster
the operator
thermics
Thermics
linear
Heat
T
v (X, T) = - (X) U (X, T)
voluminal
U =
, T
= Q
v
FLUX_ELGA/NO_TEMP
I
I
Conductivity
T
v (,
X T) = - I X
,
X - X
,
X
I () T (
T)
() U (T)
thermics
U =
, T
= Q
v
FLUX_ELGA/NO_SENS
I
I
Source
T
v (X, T) = - (X) U (X, T)
U =
, T
= Q
v
S
S
FLUX_ELGA/NO_TEMP
I
I
Temperature
T
Idem
imposed
U =
, T
= Q
v
F
F
I
I
Normal flow
T
Idem
imposed
U =
, T
= Q
v
G
G
I
I
Exchange
T
Idem
convectif
U =
, T
= Q
v
H
H
I
I
Temperature
T
Idem
external
U =
, T
= Q
v
T I
I
T
ext.
ext.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
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Type of
Variables
Sensitivity
Formulate/
sensitivity
of output of
sought
Options of calculation in Code_Aster
the operator
thermics
Thermics
nonlinear
Heat
T
voluminal
U =
, T
= Q
v
v (X, T) = -
(X, T) (U T) (X, T) -
T
I
I
(X, T) U (X, T)
FLUX_ELGA/NO_TEMP
Conductivity
T
thermics
U =
, T
= Q
v
v (,
X T) = -
(, xT) U (, xt) + I, X, X
I (
T) T (T)
I
I
T
- (,
X T) U (,
X T)
FLUX_ELGA/NO_SENS
Source
T
U =
, T
= Q
v
v (X, T) = -
(X, T) (U T) (X, T) -
S
S
T
I
I
(X, T) U (X, T)
FLUX_ELGA/NO_TEMP
Temperature
T
Idem
imposed
U =
, T
= Q
v
F
F
I
I
Normal flow
T
Idem
imposed
U =
, T
= Q
v
linear
G
G
I
I
Normal flow
T
Idem
imposed not
U =
, T
= Q
v
linear
I
I
I
I
Exchange
T
Idem
convectif
U =
, T
= Q
v
linear
H
H
I
I
Temperature
T
Idem
external
U =
, T
= Q
v
T I
I
T
ext.
ext.
Emissivity
T
Idem
U =
, T
= Q
v
I
I
Constant
T
Idem
of Stefan-
U =
, T
= Q
v
Boltzmann
I
I
Temperature
T
Idem
ad infinitum
U =
, T
= Q
v
T I
I
T
Table 5-1 Sensibilités of the heat flow
Note:
While posing in the formulas of nonlinear thermics
(X, T) = 0 and (X, T) = (X)
T
one finds the formulations of linear thermics well.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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6
Implementation in Code_Aster
6.1 Difficulties
particular
The principal difficulty of these calculations of sensitivity is to detect the presence of parameters
sensitive in the standard loadings and materials, and, to associate the fields to them
suitable derivatives. To be done, a whole architecture (cf [bib6] and [§6.2], [§6.4]) has being put in
place in order to notify to the supervisor the affiliation of a derived field with such or such significant variable. Via
the commands DEFI_PARA_SENSI and MEMO_NOM_SENSI and their utilities FORTRAN associated, one
can thus make the joint within a thermal operator, between a significant parameter, the field where
it intervenes and the associated derived field.
With a particular mention for the materials which are known only in one coded form. One
detect the significant characteristic of the aforesaid material by testing the nullity of derived material (a problem
related is posed for the characteristics of the radiation).
Inter alia installations, it A was also necessary:
· To insert, within the thermal operators, a loop on the sensitivities requested,
wedged between the temporal loops and those on the loadings.
· To set up the resolution of a linear system in the non-linear process of
THER_NON_LINE.
· To organize and manage the mutualisation of the matrices between the standard problem and it (or them)
problem (S) derived (S).
· To take into account, in robust and fast manner, the insertion of possible parameters
insensitive (creation of a CHAM_NO of null components for the thermal operators and
of a CHAM_ELEM of null components for postprocessings of calculation of flow).
· To avoid an inopportune proliferation of option of calculations (and their TE000 FORTRAN
associated) while mutualisant and “bluffant” existing it (one can visualize the exact advance of
calculation (at the macroscopic level of the options) and easy ways deployed to re-use
existing the cf [§6.2]).
Beyond these fastidious developments, a large effort of validation “numérico-data processing”
was deployed on all the meshs supports, all modelings, all the loadings, all them
types of initialization of the thermal solveurs and for all the sensitivities. These hard tests on
small cases model tests (TPLL01A/H for 2D PLAN and 3D and TPNA01A for 2D AXIS) are
revealed profitable (including for standard thermal calculation in lumpé!) and essential.
Because one seldom has theoretical values allowing to validate a thermal calculation
complicated: “nothing resembles more one sensitivity… than another value of sensitivity! ”. In
crossing with converged finished differences, one thus tried to release a maximum of
confidence in all building blocs constituting the derived system.
To be exhaustive on the aspect validation, let us note that several case-tests were delivered
(SENST<00> [V1.01.15 *]) including one analytical (SENST04 cf [§6.4]).
6.2 Environments
necessary/parameter settings
C
T
C
p=10
p=v=5
2
v
1
Appear 6.2-a Désignation of thermal conductivities for a Bi-material
Handbook of Référence
R4.03 booklet: Analyze sensitivity
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Code_Aster ®
Version
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Titrate:
Calculation of sensitivities in thermics
Date:
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To calculate the sensitivity of the thermal field compared to a constant parameter by zone (here one
particular value of conductivity in one of the zones of a Bi-material during a stationary calculation),
it is first of all necessary to notify this significant parameter in the command file via one
DEFI_PARA_SENSI
v = DEFI_PARA_SENSI (VALE = 5. )
as well as the constant functions equal to zeros and the unit which will be used to define the essential one
indicating function
zero = DEFI_CONSTANTE (VALE = 0. )
one = DEFI_CONSTANTE (VALE = 1. )
Then, as for a standard problem, one definite the fields material (or the conditions limit)
associated. One does not have to modify the other standard loadings.
ma1 = DEFI_MATERIAU (THER = _F (RHO_CP = v))
MA2 = DEFI_MATERIAU (THER = _F (RHO_CP = 10. ))
affe = AFFE_MATERIAU (AFFE = (_F (GROUP_MA = < 1 >, MATER = ma1),
_F (GROUP_MA = < 2 >, MATER = MA2))
….
other insensitive data material and limiting conditions with respect to v
It is then necessary to define the “derived” fields material (or the conditions limit “derived”) and
only those concerned with this derivation. It is them which will provide information
essential with the assembly of the derived problem: the indicating function of the derived field
CP 1 on
I =
=
1.
1
v
0 out of 2
One notifies to the supervisor their affiliation with the initial field via command MEMO_NOM_SENSI (it is it
who filled a structure of data suitable sensitivity allowing to make the joint, within
thermal operators, between the significant parameter, the field where it intervenes and the derived field
associated)
MEMO_NON_SENSI (NOM =_F (NOM_SD = “my”,
PARA_SENSI = “v”,
NOM_COMPOSE = “ma_v”))
ma_v = DEFI_MATERIAU (THER = _F (RHO_CP = one))
mazero = DEFI_MATERIAU (THER = _F (RHO_CP = zero))
MEMO_NON_SENSI (NOM = _F (NOM_SD = “affe”,
PARA_SENSI = “v”,
NOM_COMPOSE = “affe_v”))
affe_v = AFFE_MATERIAU (AFFE = (_F (GROUP_MA = < 1 >, MATER = ma_v),
_F (GROUP_MA = < 2 >, MATER = mazero))
One concludes by requesting from the thermal operator the aforementioned calculation via the key word sensitivity and notifying to
supervisor the presence of a field sensitivity in the EVOL_THER resu (accessible in the field
symbolic system resu_v of the SD result)
MEMO_NON_SENSI (NOM =_F (NOM_SD = “resu”,
PARA_SENSI = “v”,
NOM_COMPOSE = “resu_v”))
resu = THER_LINEAIRE (CHAM_MATER = affe,
….
SENSIBILITE = (v))
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 58/66
One can of course make same postprocessings on the derived field as on the standard field, it
is enough to add with the suitable commands (CALC_ELEM for the calculation of the flow of this CHAM_ELEM with
nodes, CALC_NO for its transformation into a CHAM_NO, POST_RELEVE for the statement of value,
IMPR_RESU for the impression and TEST_RESU for the comparison of values) the key word
SENSIBILITE and to inform it good value of parameter of derivation. The operator
the good field sensitivity sharing will select then the same SD as the standard field. By
example, one filled here the table rlresuv with certain components of derived from T compared to v
rlresuv = POST_RELEVE_T (ACTION=_F (RESULTAT= resu,
….
SENSIBILITE = (v))
By printing the table one can then visualize the components of the sensitivity specifically
wished.
--------------------------------------------------------------------------
ASTER 6.02.24 CONCEPT RLRESUV CALCULATES THE 27/03/2002 A 14:56:43 OF TYPE
TABL_POST_RELE
ENTITLE NODE RESU NOM_CHAM PAR_SENS NUME_ORDRE INST COOR_X/Y/Z TEMP
TEMPERAT NO1 RLRESUV TEMP v 0 0. 0. 1.1 - 1.5 0. 1.69334E+01
TEMPERAT NO2 RLRESUV TEMP v 0 0. 0. 1.1 - 1.0 0. 1.88532E+01
TEMPERAT NO3 RLRESUV TEMP v 0 0. 0. 1.1 - 0.5 0. 1.19433E+01
TEMPERAT NO4 RLRESUV TEMP v 0 0. 0. 1.1 - 0.0 0. 2.15978E+01
Example 6.2-1: Layout, via IMPR_TABLE, in the file result
For further information, the reader will be able to consult [bib6], [U4.50.02] or the case-tests of
type SENST<00> [V1.01.15 *].
Note:
· By parameterizing advisedly the loadings and materials “derived” in the file from
order, one can also have access to some derivatives made up. Cf the remark of
[§6.3].
· The syntax of the specific commands of sensitivity (DEFI_PARA_SENSI,
MEMO_NOM_SENSI) can be brought to change according to the architectural choices which will be taken
(automation of the process of detection and parameter setting of the fields derived by
supervisor or not). Methodology will not be on the other hand modified and one will always be able
to pass in “manual”. The user then remaining only judge of the relevance of these calculations
“except limits”.
· During the construction of the derived fields, in AFFE_MATERIAU or AFFE_CHAR_THER, one
need does not have to specify the “nullity” of the insensitive fields. They are initialized by defect
to zero.
To be completely exhaustive on the parameter setting, let us conclude by a functionality which can
to interest of future mainteneurs/developers or the “pointed” users. Unfolding
exact of calculation (options calculated with their field IN and OUT, loadings and materials taken in
count…) as well as the easy ways deployed to re-use existing it are traced in the file
message if one notifies INFO=2 in the accused thermal solvor and in CALC_ELEM (for
options FLUX_ELGA/NO_TEMP).
*******************************************
THERMAL CALCULATION OF SECOND MEMBER: NXACMV
TYPESE/INST: 3 5.0000000000000
LINEAR CALCULATION: F
......
--> COMPLEMENTARY CALCULATION IN SENSITIVITY
--> BLUFF OF THE OPTION: T EAST REPLACES BY (DT/DS) -
Handbook of Référence
R4.03 booklet: Analyze sensitivity
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Code_Aster ®
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6.0
Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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--> AND ADDITION Of a NEW SOURCE TERM
--> OPTION:CHAR_SENS_EVOLNI
LPAIN/LCHIN:PGEOMER MALL .COORDO
LPAIN/LCHIN:PTEMPER &&OP0186VAR_____001
......
NO. LOADS: 6
LOOP ON THE LOADS OF THE TYPE NEUMANN FLAX
CHARGE:CH3
EXICHA/NOMCHS: 1????????
CHARGE:CH4
EXICHA/NOMCHS: 1????????
--> COMPLEMENTARY CALCULATION IN SENSITIVITY
--> EXCHANGE/NUMCHM: 2
--> BLUFF OF THE OPTION: CARD TEXT+/- NULL
--> K: 1
--> OPTION:CHAR_THER_TEXT_F
LPAIN/LCHIN:PT_EXTF &&VECHTH.T_EXTNUL
LPAIN/LCHIN:PGEOMER MALL .COORDO
Example 6.2-2: Layout of THER_LINAIRE or THER_NON_LINE,
via INFO=2, in the file message.
*******************************************
CALCULATION OF HEAT FLUXES
OPTION OF CALCULATION FLUX_ELGA_TEMP
MODEL MOTH
SD EVOL_THER GIVEN TH
RESULT TH
MATERIAL TAKEN INTO ACCOUNT CMAT
SEQUENCE NUMBER 1 NUMBERS
A NUMBER OF SIGNIFICANT PARAMETERS 7
*******************************************
OP0058 **********
INST/IAUX/IORDR 0. 1 0
NRPASS/TYPESE/NOPASE 1 3 PS1
CHTEMP/CHTESETH_PS1 .001.000000/TH .001.000000
--> OPTION:FLUX_ELGA_TEMP
LPAIN/LCHIN:PGEOMER MALL .COORDO
LPAIN/LCHIN:PMATERC CMAT .MATE_CODE
LPAIN/LCHIN:PTEMPER TH_PS1 .001.000000
LPAIN/LCHIN:PTEMPSR &&MECHTI.CH_INST_R
LPAIN/LCHIN:PTEREF
LPAIN/LCHIN:PMATSEN CMAT_PS1.MATE_CODE
LPAIN/LCHIN:PTEMSEN TH .001.000000
Example 6.2-3: Layout of CALC_ELEM,
via INFO=2, in the file message.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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6.3 Perimeter
of use
The perimeter of use of thermal calculations of sensitivities can be formulated in some points:
1. Thermal calculations of sensitivity is pressed on operators THER_LINEAIRE and
THER_NON_LINE. They deal with thermal, linear problems or not, isotropic or
anisotropic, stationary or transitory.
2. They were not still extended to the problems of drying and hydration which are too
treaty by THER_NON_LINE (option THER_HYDR or SECH_ * of COMP_THER_NL/RELATION).
If this type of problems were retained for a calculation of sensitivity, calculation stops in
ERREUR_FATALE after having meant the reason of it.
3. In an exhaustive way, these calculations of sensitivities relate to all the meshs supports
(TRIA3/6, QUAD4/8/9, TETRA4/10, PENTA6/13/15, PYRAM5/10 and HEXA8/20/27)
and all isoparametric modelings (PLAN, PLAN_DIAG, AXIS, AXIS_DIAG, 3D
and 3d_DIAG). If other modelings (only THER_LINEAIRE accepts other modelings
that the usual isoparametric elements) are present in grid (COQUE_ or
AXIS_FOURIER) calculation stops (option 1 front the line of the suitable options of the catalog
elements) in ERREUR-FATALE, after having meant the reason of it.
4. While being pressed on the perimeter of use of the code, one did not envisage the calculation of sensitivity
(of the temperature and its flow) in the presence of non-linear conductivity.
5. One has access to the sensitivity of the field of temperature (and its flow cf [§5]) compared to
all parameters of loadings and all the characteristics material. Those
having to depend that variables on space.
6. The only exceptions are the entropy (a constant entropy according to the temperature,
that would not have any direction!), multiplying coefficients of the conditions of Dirichlet
generalized (calculation of sensitivity not established because it has little direction with coefficients
often discrete) and conductivities anisotropic (data-processing limitations, absence of
modeling THER_ORTH_FO).
7. One can, on the other hand, calculate the sensitivity compared to one of the components of a normal flow
vectorial.
8. Calculation provides, at the same time, the field of standard temperature and the sensitivities of this same
field compared to the parameters provided to key word SENSIBILITE of the thermal operator.
On the other hand, for the calculation of flow (via CALC_ELEM and/or CALC_NO), one calculates only the part
sensitivity. To also obtain that of the field of standard temperature, it is necessary to reiterate
the operation without key word SENSIBILITE.
9. If the user, involuntarily or not, request a calculation of sensitivity compared to one
parameter known as “insensitive”, i.e. nonconcerned by thermal calculation in progress, one
message of alarm prevents it. No calculation is produced, the structure of data planned for
the greeting of this CHAMNO is initialized to zero, and calculation continues with the other sensitivities
asked.
10. The request for one or more sensitivities does nothing but enrich the structure of data
thermics (EVOL_THER) and provides also the field of temperature of which they are the derivative
(cf n°8). In term of performance, the calculation of an analytical sensitivity is much less
expensive that a standard calculation since the same factorized matrix is re-used.
11. During the calculation of the sensitivity of the heat flux via CALC_ELEM, it is necessary to specify not
only the field material used (as for the standard problem) but also them
loadings accused by the parameters of derivation.
12. To avoid any confusion (at the supervisory level, but also on the level user!), it
is not better to re-use several times a significant parameter in loadings or
different materials.
13. The calculation of the sensitivity of the heat flux was not developed, as for the problem
standard, in COQUE_PLAN and COQUE_AXIS.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
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Note:
By parameterizing advisedly the loadings and materials “derived” in the file from
order, one can also have access to some derivatives made up. Thus if one
loading or a characteristic depends, explicitly or implicitly, of the time or of
space and that one is able to exhume this dependence, one can then calculate the derivative
T (and of its flow) compared to this coefficient of dependence.
The main thing being which it also does not depend on the calculated solution (the field of temperature).
In which case, it will be necessary to develop a true made up calculation of sensitivity.
In short, the perimeter of use of this functionality gathers it or not thermal, linear,
isotropic or anisotropic, stationary or transitory, being pressed on finite elements
isoparametric lumpés or not. Within this framework there, it covers the same perimeter as that
accused thermal operators.
6.4 Example
of use
To familiarize oneself with the use of this new functionality, one can take as a starting point this
expurgée version of the case test SENST04A [V1.01.154]. In this case analytical test, it is about
to make sure of the validity of derived compared to the coefficient from exchange-wall and conductivity, in
a calculation of transitory thermal response linear of two plates separated by a play in which
a transfer of heat is carried out.
The problem is two-dimensional, but the limiting conditions make that the field of temperature reached
quickly the stationary state and depends analytically only on the X-coordinate and the data. One in
then deduced easily the analytical expressions from the sensitivities of the field of temperature and sound
flow compared to the thermal parameters which interest us.
# 1. Definition/memorizing of the constant functions
PS_UN=DEFI_CONSTANTE (VALE=1.0,);
MEMO_NOM_SENSI (NOM_UN = PS_UN);
PS_ZERO=DEFI_CONSTANTE (VALE=0.0,);
MEMO_NOM_SENSI (NOM_ZERO = PS_ZERO);
.........
# 2. Definition of the significant parameters and other parameters
PS1=DEFI_PARA_SENSI (VALE=80.0,);
PS2=DEFI_PARA_SENSI (VALE=40.0,);
.........
# 3.1. Installation of the materials (std and sensitive)
MAT=DEFI_MATERIAU (THER_FO=_F (LAMBDA=PS2, RHO_CP=A5))
MEMO_NOM_SENSI (NOM=_F (NOM_SD=' MAT',
PARA_SENSI=PS2,
NOM_COMPOSE=' MAT_PS2'));
MAT_PS2=DEFI_MATERIAU (THER_FO=_F (LAMBDA = PS_UN,
RHO_CP = PS_ZERO))
CMAT=AFFE_MATERIAU (MAILLAGE=MAIL, AFFE=_F (TOUT=' OUI', MATER=MAT))
MEMO_NOM_SENSI (NOM=_F (NOM_SD=' CMAT',
PARA_SENSI=PS2,
NOM_COMPOSE=' CMAT_PS2'));
CMAT_PS2=AFFE_MATERIAU (MAILLAGE=MAIL, AFFE=_F (TOUT=' OUI', MATER=MAT_PS2))
.........
# 3.2.2 Loadings significant (echange_paroi)
CH_1=AFFE_CHAR_THER_F (MODELE=MODTHER,
ECHANGE_PAROI=_F (GROUP_MA_1=' INTERG',
GROUP_MA_2=' INTERD', COEF_H=PS1));
MEMO_NOM_SENSI (NOM=_F (NOM_SD=' CH_1',
PARA_SENSI=PS1,
Handbook of Référence
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Titrate:
Calculation of sensitivities in thermics
Date:
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NOM_COMPOSE=' CH_1_PS1'));
CH_1_PS1=AFFE_CHAR_THER_F (MODELE=MODTHER,
ECHANGE_PAROI=_F (GROUP_MA_1=' INTERG',
GROUP_MA_2=' INTERD', COEF_H=PS_UN));
.........
# 4.1 standard Calculation + 2 calculations of sensitivity
MEMO_NOM_SENSI (NOM= (_F (NOM_SD=' RESU',
PARA_SENSI=PS1,
NOM_COMPOSE=' RESU_PS1'),
_F (NOM_SD=' RESU',
PARA_SENSI=PS2,
NOM_COMPOSE=' RESU_PS2')));
RESU=THER_LINEAIRE (MODELE=MODTHER, CHAM_MATER=CMAT,
EXCIT= (_F (CHARGE=CH_0), _F (CHARGE=CH_1)),
TEMP_INIT=_F (CHAM_NO=TEMPINIT),
SENSIBILITE= (PS1, PS2),
INCREMENT=_F (LIST_INST=LINST))
# 4.2 Calculation of flow to the nodes
RESU=CALC_ELEM (reuse=RESU, MODELE=MODTHER, CHAM_MATER=CMAT,
RESULTAT=RESU, OPTION=' FLUX_ELNO_TEMP',
EXCIT= (_F (CHARGE=CH_0), _F (CHARGE=CH_1)),
SENSIBILITE= (PS1, PS2))
RESU=CALC_NO (reuse=RESU,
RESULTAT=RESU, OPTION=' FLUX_NOEU_TEMP',
SENSIBILITE= (PS1, PS2))
Example 6.4-1: Installation of a calculation of sensitivity in linear thermics
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
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Author (S):
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:
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7 Outline Conclusion/
During digital simulations obtaining a rough result is not sufficient any more. The user is of
more in petitioning of calculation of sensitivity compared to the data input of the problem. That
he makes it possible to estimate the uncertainty which the field result according to the law of variation answers
data. This derivative is also the basic substrate of problems opposite (retiming of
parameters…) and of problems of optimization.
This sensitivity can be obtained “manually”, but the experiment shows that these studies
parametric are often expensive, little mutualisables and less reliable than an analytical calculation
established in the computation software.
In this note, one places oneself in the perimeter of use of the thermal operators
standards of Code_Aster and one is interested in this analytical sensitivity of the field of
temperature and of its flow compared to the characteristics material and the loadings. One y
described the process allowing to exhume the linear system which this derivative checks. In order to
to minimize the overcost calculation, a particular effort was brought to bind its resolution to that of
initial problem.
One details theoretical, numerical work and data processing which governed the establishment
of these calculations of sensitivity in the code. One specifies their properties and their limitations all in
connecting these considerations to a precise parameter setting of the accused operators and to the choices of
modeling of the code. One tried constantly to bind different the items approached while detailing,
has minimum, the a little technical demonstrations.
Required environment, the parameter setting and the perimeter of use of this news
functionality are described. An example extracted from an official case-test is clarified.
Thereafter, the prospects for this work are several commands:
· From a functional point of view, one could extend truly the perimeter of use
thermal calculations of sensitivity (and also that of the standard problem on which they
lean) with non-linear and/or anisotropic conductivities and the conditions of exchange
wall in lumpé. With few expenses, one could also deal with the problems of hydration, of
drying and of convection-diffusion.
· In addition, of the developments still remain to establish to be able to treat
suitably a broad spectrum of derivative made up and, in particular, to carry out
chainings thermomechanical (which are the genuine target of the current developments).
One will be able to then obtain the mechanical sensitivities of variables (displacements, deformations
and constraints) compared to loadings or characteristics material of the problem
thermics.
· From a theoretical point of view, it remains to carry out a study “numérico-functional calculus”
similar to that of this document, to exhume the same thermal sensitivities in
modeling COQUE and FOURIER. That will open another field of investigation then:
sensitivities compared to the geometrical characteristics of the structural elements.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 64/66
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]
H. BREZIS. Analyze functional, theory and applications. ED. Masson, 1983.
[4]
V. VENTURINI and probabilistic Al Etude of the tank by a coupling mechanic-reliability engineer. Assessment of
P1-97-04 project: PROMETE. Note HP-26/99/012, Nov. 1999.
[5]
G. NICOLAS. Feasibility study of the I7-01-02 project. Note HI-72/00/020, 2000.
[6]
G. NICOLAS & J. PELLET. Structure for the sensitivity. Note HI-72/01/009.
[7]
G. CORLISS and Al AD off algorithms: from simulation to optimization. ED. Springer Verlag,
2002.
[8]
S. CAMBIER. Vibratory sensitivities of sizes. Theory and algorithms for one
establishment in Code_Aster. Note HT-62/01/011.
[9]
C. DUVAL and Al Applicabilité of the automatic differentiation to a system of governing EDP
thermohydraulic phenomena in a heating tube. Note 96NJ00017, 01/02/96.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Code_Aster ®
Version
6.0
Titrate:
Calculation of sensitivities in thermics
Date:
01/07/03
Author (S):
O. BOITEAU Key
:
R4.03.02-A Page
: 65/66
Appendix 1 Notion of derived “within the meaning of the distributions”
Let us place within the framework of a Bi-material and a derivation compared to the voluminal heat of one of these
materials, in linear thermics. The same reasoning can be led into non-linear and compared to
do not matter that it another characteristic material, loading or limiting condition.
One considers the modeling of following total voluminal heat (I the indicating function of ième
I
part)
I
C
éq
A1.1
p (X):=
I X + I X
,
1 1 ()
2 2 ()
(1
)
2
2
1
2
1
2
Appear A1-a: Désignation of voluminal heats for a Bi-material
That is to say the family of distributions parameterized by
(
*
Of ×,
0
+
T)
(] [)
éq A1.2
T
such as
T is the solution of the problem in extreme cases
(
T
I I
T
S
1 (1 +
) 1 + 2 2)
- div () =
×],
0 [
T
T
= F
1 ×],
0 [
T
(P
)
= G
éq
A1.3
2 ×],
0 [
N
T
+ HT
= hText
3 ×],
0 [
N
T (X)
0
,
= 0
T (X)
One of course recognizes the solutions of as many slightly shifted problems of linear thermics (cf [éq 2
T
1] in order to be able to approximate gradually, when the small parameter tends towards zero,
.
1
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R4.03 booklet: Analyze sensitivity
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Code_Aster ®
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Titrate:
Calculation of sensitivities in thermics
Date:
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By writing that T0 and T check, respectively, (P0) and (P), and by withdrawing members from members both
EDP one obtains
(T - T
T
0)
- div ((T - T
I
0) = -
1
1
×],
0 [
T
T
T
- T0 = 0
1 ×],
0 [
(T - T0)
(P P0
= 0
éq
A1.4
2 ×],
0 [
)
N
(T - T0)
+ H (T
- T0) = 0
3 ×],
0 [
(N
T - T X
0) (
)
0
, = 0
It any more but does not remain to divide by
1 this new EDP and to make tend towards zero the parameter. The problem
in extreme cases then becomes a simple problem interfered similar homogeneous Cauchy-Dirichlet-Neumann-Robin type
with the initial problem and comprising however a different source term
U
T
- div (U) = -
0
I
1
×],
0 [
T
T
U = 0
1 ×],
0 [
1
U
lim
(P - P = 0
éq
A1.5
2 ×],
0 [
0)
0
1
N
U
+ hu = 0
3 ×],
0 [
N
U (X)
0
, = 0
1
which one notes U:= lim
(T - T the solution. This one exists and is single because it inherits all them
0)
0
1
good properties of the initial problem. By definition, one then indicates it under the term of “distribution
derived compared to characteristic 1 from voluminal heat “:= T
U
.
1
Taking into account the process of obtaining of this type of derived (translation, passage in extreme cases then definition
of a distribution derived as soon as one is ensured of his existence and his unicity), it appears that that returns
“formally” to directly derive the terms from the initial problem in extreme cases [éq A1-2], [éq A1-3].
It is in this way that one proceeded for all the derivative exhumed in this document. Moreover, for
each one of them, one checked that this “formal” derivation produced the same result as it
process describes previously.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HI-23/03/001/A
Outline document