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Model of thermics for the thin hulls
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Organization (S):
EDF/IMA/MN
Manual of Reference
R3.11 booklet: Thermal elements on average surface
Document: R3.11.01
Formulation of a model of thermics
for the thin hulls
Summary:
The model presented here results from the asymptotic analysis of the equations of thermics when the thickness from
the structure tends towards zero.
The temperature is described by 3 fields defined on the average surface of the hull.
One shows on some examples, the capacities of the model by reference to 3D solutions.
The applications concerned are thermomechanical calculations of hulls, the thermal restitution of wall for
thermohydraulics of pipings, problems of identification.
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Titrate:
Model of thermics for the thin hulls
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Contents
1 Introduction ............................................................................................................................................ 3
2 Presentation of the model ......................................................................................................................... 5
2.1 Position of the thermal problem in the hulls .......................................................................... 5
2.1.1 Description of the geometry ................................................................................................... 5
2.1.2 Equation of heat ........................................................................................................... 6
2.1.3 Thermics for a mean structure ..................................................................................... 7
2.2 Recall of the results resulting from the asymptotic development ......................................................... 8
2.2.1 The limiting model obtained ......................................................................................................... 8
2.2.2 An application ...................................................................................................................... 9
2.3 Formulation of the stationary model of thermics of hull ........................................................ 11
2.3.1 Equations of the model ........................................................................................................... 11
2.3.2 Case of a homogeneous plate ............................................................................................... 13
2.3.3 Link with the asymptotic model ....................................................................................... 15
2.3.4 Generalization with the problems of thermal evolution .......................................................... 17
2.3.5 Equations of the model with usual variables .............................................................. 19
2.3.5.1 Case of a homogeneous plate ................................................................................... 21
2.3.5.2 Relation between the variables of the two representations ............................................ 22
2.3.6 Synthesis .............................................................................................................................. 22
3 Validation of the model on some examples ..................................................................................... 24
3.1 The infinite cylinder subjected to a uniform interior flow .................................................................... 24
3.2 The infinite plate under a couple of antisymmetric flows ............................................................. 26
3.3 The infinite plate under a couple of symmetrical flows ................................................................... 28
3.4 The infinite cylinder subjected to a horizontal stratification ............................................................... 32
4 Remarks on the numerical discretization ........................................................................................ 35
4.1 Resolution by finite elements ........................................................................................................ 35
4.2 Numerical blocking of a finite element of thermal hull ........................................................... 36
5 ........................................................................................................................................... Conclusion 39
6 References .......................................................................................................................................... 40
Appendix 1
Infinite plate under a couple of symmetrical flows ................................................. 41
Appendix 2
Mixed formulation of the stationary problem for the plate ................................... 43
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Titrate:
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1 Introduction
The mechanical models of mean structures (hulls and plates) arrived at a stage of
extremely at least advanced development for the homogeneous elastic structures in
the thickness. The problem is known since strong a long time and various theories were born,
generally dedicated to specific problems (thick hulls, buckling etc…). However one
basic model, that of LOVE-KIRCHHOFF, achieve the unanimity in the most current applications.
The difficulties lie rather in the numerical calculation of the aforementioned of the fact, on the one hand, of the need
to approach correctly the surface of the hull (in particular its curvature), and in addition of the command
raised partial derivative equations which should be solved (4
ème
command).
In thermics on the other hand, the situation is much less clear and a great number of approaches
coexist. It is indeed only recently that the problem arose with the possibilities (and
need) for thermomechanical calculations. The first models neglect conduction in parallel
on average surface to retain only the thermal transfers in the thickness of the hull, this
step is completely paradoxical among that of the mean structures where, on the contrary, the low thickness
structure leads to simplifying assumptions on the variation in the thickness of the fields
physical sizes.
The most recent modelings take as a starting point the the mechanical ideas of thin hulls being attached
with the second approach, one can classify them according to a completely similar command.
1) Models utilizing a more or less thorough polynomial development of the temperature
in the thickness [bib2], [bib9], [bib10]. It is primarily about formulation of finite elements.
2) Models associated with the theories with surfaces to Directors (Surfaces of COSSERAT) [bib5], [bib8].
The Director is here the gradient of the temperature in the thickness. The problem of these approaches
reside in the law of behavior to introduce. Coherence with the three-dimensional law led
with choices which are interpreted like an assumption of linear distribution of the temperature in
the thickness. This formalism thus joined practically the preceding models (the introduction of
several Directors being identified with various commands of development of the polynomials).
3) Models of degenerated finite elements [bib11]: on the basis of a three-dimensional finite element, the introduction
stresses between the degrees of freedom located on the same normal at average surface
allows by condensation to deduce an element from “thermal hull”. Practically, there still,
the basic element using a parabolic interpolation according to the thickness, the element of hull
corresponds to a linear distribution in the thickness.
Parallel to these approaches numerical (1) and (3) or based on assumptions a priori (2),
results on the shape of the field of temperature of a thin section and problem of which it is
solution were obtained by asymptotic methods [bib3], [bib1].
As for the mechanical model, those make it possible to justify the assumptions made a priori in
mean theories of hulls, to even obtain the equations of the problem of hull. Results of
[bib 1] are recalled low and will be used as a basis for the model suggested. Let us note simply here that the idea
subjacent with any step of the asymptotic type is to introduce a small parameter
(here the report/ratio
thickness of the plate on a dimension characteristic of the aforementioned), then having obtained the problem
limit when
tends towards zero starting from the three-dimensional problem, to approach in the applications
(where
obviously a nonnull value) the solution by its limit takes.
From a practical point of view, the limit obtained for the equations of stationary thermics seems
to be too “poor” to be of a real interest, (one will give of it an illustration in [ß2.2.2]). More
precisely values of
to reach to identify the solution with its limit are very small in
real situations met.
This is why one proposes in this note to keep
the form of the limiting solution
(parabolic distribution in the thickness)
but to lay out about it like assumption a priori on
the three-dimensional solution
allowing to bring back itself to a problem arising on average surface.
Code_Aster
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Titrate:
Model of thermics for the thin hulls
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One has one thus
approximate model of mean structure converging towards the model limits
three-dimensional equations
. In this direction, it is “optimal” since a linear assumption of distribution
in the thickness leads to a model
not converging towards the limiting solution
and that a model
based on a richer development in the thickness sees its terms of a nature higher than two
to converge towards zero
when the hull is thin.
The plan of the note is as follows:
·
one starts by pointing out the equations of the stationary thermal problem for the solid
three-dimensional and their expressions in a frame of reference adapted to the cases where the solid
is a “thin hull”,
·
then, having pointed out the results of an asymptotic study of these equations carried out in the case
of a plate, one gives the complete description of the model suggested,
·
one then applies the model to a certain number of geometries and thermal loadings and
a comparison is made compared to analytical solutions or numerical calculations
three-dimensional,
·
finally, one gives some indications on the numerical aspects of the use of the model in one
calculation by surface and linear finite elements.
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Model of thermics for the thin hulls
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2
Presentation of the model
2.1
Position of the thermal problem in the hulls
In this paragraph, we first of all will point out the description of the geometry of the hulls, sights
like thin three-dimensional solids. One will pose then the thermal problem of conduction.
2.1.1 Description of the geometry
A hull is defined as being a solid, thin perpendicular to an average surface
.
One notes 2h the thickness of the hull; one chooses a frame of reference (X
1,
, X
2
) on surface
.
One notes G the associated metric tensor,
N
the normal vector, C the tensor curvature of
.
2h
X
1
N
X
2
I =] - H, H
-
+
Appear: 2.1.1-a
The hull is described by the frame of reference (X
, X
3
), X
3
according to
N
: =
X] - H, H [
(Greek indices
, ß,
are dedicated to the surface co-ordinates on
).
This description is appropriate of course for a hull thickness 2h lower than the smallest radius of
curvature of
.
In an unspecified point (X
, X
3
) of the hull, the metric tensor G is expressed according to
fundamental tensors G and C of average surface
by:
G
X
, X
3
= G
X
-
2 X
3
C
X
G
3
= 0, G
33
= 1
éq 2.1.1-1
and
det G = det G 1
-
X
3
tr C
= det G 1 + X
3
1
R
1
+ 1
R
2
where R
1
, R
2
are the main radii of curvature of
at item X
considered.
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Note:
It is known indeed that the trace
(tr)
of a tensor is an invariant (by basic change). One has
the practice however to write the sizes in orthonormée physical base i.e. As follows:
G
phy
X
, X
3
=
- 2x
3
C
phy
X
.
And if the base is main of curvature:
C
phy
= -
1
R
(without summation).
One will note:
1
R
1
+ 1
R
2
=
H
1
and
1
R
1
-
1
R
2
= H
2
.
One limited oneself here under the first command in X
3
tr C; as it subsequently will be done. In
practical, indeed, the thinness of the hull allows such a simplification. There will be advantage also with
to place in a main reference mark of curvature, orthonormé. The tensor G is then the identity, C is
diagonal. It is what one will do henceforth.
2.1.2 Equation of heat
The equations of three-dimensional thermal conduction are written (for a rigid conductor):
-
div
K
grad T +
C
T
T =
éq 2.1.2-1
where
K
indicate the tensor of conductivity,
C heat-storage capacity and R possible sources.
There is advantage to write the expression of the differential operator according to metric G's generated by
surface average
. Tensors of conductivity indeed will be considered
K
isotropic transverses
according to these axes of co-ordinates (multi-layer cf materials).
K
I
J
=
K
0
0
K
,
K
and K which can vary with
X
1
,
X
2
,
X
3
.
The expression of the operator:
-
div
K
grad T =
-
det G.
-
1/2
.
I
det G.
1/2
K
I
J
G
I J
J
T
is written then with the first command in X
3
/tr C, for an orthotropic conductivity according to directions' of
curvature main:
-
1
-
X
3
H
1
.
1
1
-
X
3
H
2
K
11
1
T +
2
1 + X
3
H
2
K
22
2
T + H
1
K
3
T
-
3
K
3
T
éq 2.1.2-2
If the curvatures are constant, this becomes:
-
1
-
2x
3
/R
1
.
1
K
11
1
T
-
1
-
2x
3
/R
2
.
2
K
22
2
T
-
H
1
1
-
X
3
H
1
K
3
T
-
3
K
3
T
The effect of the curvature is thus in the same way standard than a modified distribution of conductivity in
the thickness.
Code_Aster
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Titrate:
Model of thermics for the thin hulls
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2.1.3 Thermics for a mean structure
The equations of stationary thermics on the hull can be written in the form of one
problem of minimization.
One supposes in particular that the boundary conditions on the ends
X I of the hull are of
even standard on all thickness I. One partitionne
X I in:
T
X I (area at imposed temperature),
and
X I (area in condition of exchange or imposed flow).
To find the field of temperature T:
T =
Arg Min
V
J (
), with J (
) =
1
2
With (
,
)
-
F (
), with:
WITH (T,
) =
K
.
T.
D
+
+
-
T.
D
±
+
X I
T.
dS
F (
) =
+
-
.
D
±
+
X I
.
dS.
éq 2.1.3-1
One notes:
·
V =
H
1
(
X I),
= 0 on
T
X I
.
·
Boundary conditions on
+
-
(
X I) are of the type exchanges or imposed flow
:
. N =
T
-
=
-
(
K
T).
N
being a coefficient of exchange.
The term of conductivity in A (T,
) is written:
K
T.
D
=
I
K
1 - X
3
1
R
+ 1
R
T.
+ K
3
T.
3
1 + X
3
H
1
dx
1
dx
2
dx
3
,
this, in a orthonormé main reference mark of curvature of
(K
and K are then the components
physiques of the tensor of conduction
K
).
Terms of exchange on surfaces
+
and
-
are:
+
-
T.
D
±
=
±
T
±
±
. 1
±
h. H
1
dx
1
. dx
2
The object of a thermal model of hull is thus to bring back from three to two variables of space
dependence of the field of temperature T in the expression of the differential operator corresponding to
[eq 2.1.2-2] or [eq 2.1.3-1], with the help of the choice and the justification of suitable assumptions.
The model suggested in [§2.3] rests on the results of the asymptotic development of the equations
thermics presented in [§2.2] hereafter.
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2.2
Recall of the results resulting from the asymptotic development
2.2.1 The limiting model obtained
One summarizes here the main results obtained in [bib1] by a technique of development
asymptotic. The case of a plate is considered:
X I
, thickness 2
h. The temperature is fixed at 0
on the edge
X I
, and of flows
+
,
-
on the faces
+
and
-
.
One seeks to study the dependence of the solution T
thermal problem [éq 2.1.3-1] with respect to
the thickness of plate 2
h. One uses for that a technique of change of open which brings back it
problem with a fixed field
X I, with I =]
-
H, + H [. The parameter
appears then explicitly in
equations of the problem transported (P
), of
X I
with
X I.
On
X I
, the initial problem is written in variational form:
To find:
T
V
=
H
1
X
I
,
= 0 on
X
I
such as:
X
I
K
.
T,
.
,
+ K
33
T,
3
.
,
3
=
+
+
+
-
-
,
V
.
éq 2.2.1-1
The results of the asymptotic development [bib 1] consist of the checked following properties
by T (
), the solution of the problem transported (P
), posed on
X I:
(I) 1
T (
) tends towards T
1
X
; X
3
= T
1
X
in H
1
(
X I).
T
1
X
COM m E a tem pérature m oyenne appears on thickness I, at item X
.
(II)
T,
3
(
), which is the derivative of
T (
) according to the variable thickness X
3
I, tends towards
derived according to X
3
cham p
X
; X
3
in L
2
(
) X H
m
1
(I), where H
m
1
(I) the space indicates of
functions of H
1
(I) with m oyenne null.
éq 2.2.1-2
In conclusion, the solution T
initial problem on
X I
can represent itself by the two first
terms of its development:
T
X
, X
3
= 1
T
1
X
+
X
, X
3
= X
3
/
+…
éq 2.2.1-3
However the gradient of T
does not represent itself by the gradient of the representation of T
. This
situation is generic problems of singular disturbances encountered in the study of
mean structures (plates, beams…) :
T
X
, X
3
= 1
T
1
X
,
. E
+
X
, X
3
= X
3
/
,
3
E
3
.
éq 2.2.1-4
The field of the “gradient of T
“is thus not a field of gradient!
Code_Aster
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Titrate:
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The fields T
1
and
are calculated on average surface
. If conductivity is
homogeneous in the thickness, one a:
T
1
H
0
1
(
), solution of:
H K
T
1,
.
,
=
1
2
+
+
-
,
H
0
1
(
)
éq 2.2.1-5
X
, X
3
=
+
X
+
-
X
4 K X
. X
3
2
H
-
H
3
+
+
X
-
-
X
2 K X
. X
3
.
éq 2.2.1-6
It is noted that T
1
is the solution of a problem arising on
, whereas
is obtained explicitly in
function of imposed flows. These two equations constitute the “limiting” model obtained by
asymptotic development.
Note:
·
In a language more coloured and more blur, the preceding results are interpreted while saying
that for a thin section, the average temperature is governed by received average flow and
conduction in the plan of the plate. The distribution in the thickness is not a function, in one
not given, that the flows imposed in this point on the faces higher and lower, it
is not affected by the presence of the close points.
·
The distribution of temperature in the thickness is “parabolic” according to the representation
[éq 2.2.1-6].
2.2.2 One
application
One can illustrate the results of the asymptotic development for a simple example, which shows too
limitations of the model obtained by using the representation of the temperature [éq 2.2.1-3] using
fields T
1
and
, [éq 2.2.1-5] and [éq 2.2.1-6].
One considers an infinite plate subjected on his half X
2
< 0 with a couple of constant flows (
+
=
,
-
=
-
) balanced, and insulated on other half X
2
> 0.
= 0
-
= 0
X
2
T (0, 0) = 0
I
X
1
3
X
+
=
-
=
Appear: 2.2.2-a
The problem [eq 2.2.1-5] of determination of the average temperature T
1
is an equation here
differential in X
2
:
X
2
2
2
T
1
(X
2
) = 0
since flow average J
+
+ J
-
is null. The solution is then T
1
= 0
everywhere.
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The field
(X
2
, X
3
) is calculated easily by [éq 2.2.1-6]:`
X
2
, X
3
=
K
. X
3
for X
2
< 0,
= 0
for X
2
> 0.
The discontinuity of the boundary condition of NEUMANN on
±
thus refers directly on
field of temperature: opposite the higher temperature T is as follows:
T
0
X
2
/K
+
H
Appear: 2.2.2-b
This discontinuity appears of more independent the thickness H in this limiting model, once it
flow
brought standardized per h.
This limitation of the limiting model obtained by asymptotic development is inherent in
purely local determination of the parabolic complementary term
(X
, X
3
). Discontinuities
induced will be awkward for the applications, in particular in thermomechanics.
One is thus brought to differently formulate the model of thermics of hull, while keeping them
results of this asymptotic development.
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2.3
Formulation of the stationary model of thermics of hull
One saw on the results of the asymptotic study of the three-dimensional equations on the solid =
X
I, that the limiting model obtained comprised an average temperature solution of a problem of the 2
ème
command posed on
, and that the additional parabolic term was given only locally (not
by point on
). This thus had the disadvantage of providing discontinuous solutions when them
thermal “loadings” are it.
One thus presents in this paragraph a representation of the temperature, always parabolic
in the thickness, but avoiding the preceding pitfall. One describes the equations obtained, and their properties.
2.3.1 Equations of the model
Following the results of the asymptotic development, one
chooses
the representation in the thickness
following on =
X I:
T X
,
X
3
=
T
1
X
+
T
2
X
.
W
2
X
3
+
T
3
X
.
W
3
X
3
éq 2.3.1-1
with (W
1
= 1, W
2
, W
3
) a given base of the polynomials of degree 2.
One thus replaces the determination of the field T to three variables of space by that of three fields
scalars T
1
, T
2
, T
3
with 2 surface variables on
. This decomposition [éq 2.3.1-1] is practical for
to show its link with the asymptotic model. But one will use another representation for
digital model: to see [§ 2.3.5].
One will inject this representation of the temperature T (X
, X
3
) directly in the thermal problem
[éq 2.1.3-1] posed on =
X I.
From the definition of space
v
in [éq 2.1.3 - 1], one adopts for the fields T
I
:
W = V =
1
,
2
,
3
H
1
(
)
3
,
I
= 0 on
T
.
While posing
T
= (T
1
, T
2
, T
3
) the formulation of the thermal problem on becomes:
To find
T
W,
T =
J
(
), with
J
(
) =
1
2
With
(
,
)
-
F
(
), and
Arg Min
V
W
With
(
T,
) =
T
T
.
With
.
+
T.
B
.
T
D
F
(
) =
C
.
D
+
T
D
.
ds
T
éq 2.3.1-2
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Model of thermics for the thin hulls
Date:
03/11/93
Author (S):
F. VOLDOIRE, S. ANDRIEUX
Key:
R3.11.01-A
Page:
12/44
Manual of Reference
R3.11 booklet: Thermal elements on average surface
HI-75/93/098/A
Indeed, from [éq 2.3.1-1] one deduces the expressions:
(
T)
=
T
.
1
W
2
W
3
3
T =
T.
0
w'
2
w'
3
The tensor
With
of command 4 corresponds to surface average conductivities:
With
I J
X
=
I
K
W
I
.
W
J
.
1 - X
3
. 1
R
+ 1
R
1 + X
3
H
1
dx
3
éq 2.3.1-3
(by using the metric sight in [éq 2.1.1-1]).
Dependence of
With
according to (X
) comes from that of K
ß
and of that of the average curvature H
1
surface
.
The tensor
B
of command 2 described transverse conduction as well as the exchanges on the faces
+
and
-
:
B
ij
X
=
I
K W
I
'
.
W
J
'
1 + X
3
H
1
dx
3
+
+
W
I
(H).
W
J
(H) 1 + H
H
1
+
-
W
I
(
-
H).
W
J
(
-
H) 1 - H H
1
éq 2.3.1-4
With regard to the second member
F
, the vector
C
is:
C
X
=
+
1
W
2
(H)
W
3
(H)
1 + H
H
1
+
-
1
W
2
(
-
H)
W
3
(
-
H)
1
-
H
H
1
éq 2.3.1-5
(One supposes the absence of heat sources in the thickness to simplify.)
Finally:
D
X
=
I
1
W
2
X
3
W
3
X
3
.
1 + X
3
H
1
dx
3
, for
X
éq 2.3.1-6
With the examination of the formulation [éq 2.3.1-2] obtained for the thermics of hull, one notes that
the differential operator remains of command 2, contrary to mechanics where the aforementioned passes to 4. In
thermics the curvature of average surface intervenes only in one amendment of metric, and
not directly in the operators, as an inhomogeneousness of conductivities would do it in
the thickness.
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Model of thermics for the thin hulls
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2.3.2 Case of a homogeneous plate
If a plate is considered, or if one neglects the variation of metric in the thickness of
the hull (1
>>
hH
1
) and by supposing homogeneous material in the thickness to simplify, one can
to propose the choice of a base (1, W
2
, W
3
) of the polynomials of degree 2 (polynomials of Legendre), of kind
that tensors of conduction
With
and
B
diagonalisent themselves on indices I, J (out of U
I
, V
J
):
W
2
X
3
=
X
3
/H;
W
3
X
3
=
3
2
X
3
2
H
2
-
1
3
éq 2.3.2-1
that is to say:
W
I
(H) = 1,
I;
W
2
(
-
H) =
-
1 =
- W
3
(
-
H)
and:
I
W
2
= 0 =
I
W
3
=
I
W
2
.
W
3
=
I
W
2
'
.
W
3
'
I
W
2
2
=
2h
3
;
I
W
3
2
=
2h
5
;
I
W
2
'2
=
2
H
;
I
W
3
'2
=
6
H
Thus T
1
will be the average temperature, T
2
will be associated the gradient in the thickness.
One finds then:
With
1 1
= 2 KH
;
With
2 2
=
2
3
KH
;
With
3 3
=
2
5
KH
;
With
I J
= 0 if I
J
Moreover:
B
=
2K
H
0 0 0
0 1 0
0 0 3
+
+
+
-
1 0 1
0 1 0
1 0 1
+
+
-
-
0 1 0
1 0 1
0 1 0
C
=
+
+
-
1
0
1
+
+
-
-
0
1
0
D
=
I
I
.
X
3
/H
I
3
2
X
3
2
/H
2
-
1/3
,
on
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Titrate:
Model of thermics for the thin hulls
Date:
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By writing the variational formulation of the problem [éq 2.3.1-2]:
To find U =
T
1
T
2
T
3
H
1
(
)
3
such as,
V
H
1
(
)
3
:
U.
With
.
V +
T
U.
B
. V
T
dx
1
dx
2
=
C
T
. V
dx
1
dx
2
+
D
T
. V
one establishes the local equations to solve in
:
-
2 KH
T
1
+
+
+
-
T
1
+ T
3
+
+
-
-
T
2
=
+
+
-
-
2
3
KH
T
2
+ 2
K
H
T
2
+
+
+
-
.
T
2
+
+
-
-
T
1
+ T
3
=
+
-
-
-
2
5
KH
T
3
+ 6K
H
T
3
+
+
+
-
T
1
+ T
3
+
+
-
-
T
2
=
+
+
-
éq 2.3.2-2
with the boundary conditions following:
T
1
, T
2
, T
3
given on
T
T
1,
= 1
4k
2
H
2
I
T
2,
= 9
4k
2
H
2
I
X
3
/H
T
3,
= 25
4k
2
H
2
I
.
3
2
.
X
3
2
/H
2
-
1/3
on
The equations [éq 2.3.2-2] are thus valid for the thin plates and hulls which one neglects
terms of curvature in the metric one (1
>>
hH
1
), and for a homogeneous material in the thickness.
General solutions [T
I
] of [éq 2.3.2-2] comprise the exponential ones of type
E
X
has
-
/!
with
lengths of damping
!
has
depending on the values on
K
K
and
±
H
K
. For example, in the absence of
conditions of the type exchanges on the walls
+
-
(
±
= 0), one obtain for the fields T
2
and T
3
respective lengths of damping:
!
2
has
= H
K
3K
!
3
has
= H
K
15K
.
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Model of thermics for the thin hulls
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It in practice frequently arrives that to neglect the terms of curvature (hH
1
<<
1) in the operator
deteriorate that little the solution; on the other hand, one often may find it beneficial to keep the expression supplements in
second member. Indeed this makes it possible to calculate the true quantity of heat brought by flows
applied to the faces
±
(cf example in [§3.1]). In this case, it is necessary to take
C
in [éq 2.3.1-2] and
[éq 2.3.2-2].
C
=
+
+
-
1
H H
1
1
+
+
-
-
H H
1
1
H H
1
éq 2.3.2-3
2.3.3 Link with the asymptotic model
One can check easily that the model suggested here has well as a limit when the thickness
H tends towards 0
results of the asymptotic development presented in [éq 2.2.1-5] and [éq 2.2.1-6].
Indeed the thickness H intervenes here explicitly in the coefficients of the differential operator in
local equations [éq 2.3.2-2], which are solved on average surface
.
In the case without heat exchange (
+
=
-
= 0) considered in the asymptotic study, these equations
[éq 2.3.2-2] have the form:
-
2 K
T
1
= 1
H
+
+
-
-
2
3
KH
2
2
.
T
2
+ 2 K T
2
=
H
+
-
-
-
2
5
KH.
2
2
T
3
+ 6 K T
3
=
H
+
+
-
After a formal asymptotic development of the solution (T
I
) according to the thickness
in these
equations, it is checked well that:
·
T
1
is solution of the problem [éq 2.2.1-5] giving the main term of the development
asymptotic (cf [§2.2]).
·
1
T
2
and
1
T
3
are:
1
T
2
= H (
+
-
-
)
2 K
; 1
T
3
= H (
+
+
-
)
6 K
, which corresponds well to the definition
[éq 2.2.1-6] of the complementary field
.
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The model [éq 2.3.1-1] to three scalar fields T
1
, T
2
, T
3
, parabolic in the thickness, appears in
some kind the optimal model with respect to the asymptotic behavior of the equations of
stationary thermics in the mean structures. The following diagram indicates the overlap of
various possible models, with their behavior when the thickness tends towards zero (arrows
):
Model with
2 fields (refines)
Limiting model
asymptotic
Model with 3 fields
(parabolic)
Models more
rich person
2
… +…
+
+
+
+
+
+
+
:
T
1
X
T
1
T
2
X
3
H
1
T
1
X
T
1
X
T
2
X
W
2
X
3
+
T
3
X
W
3
X
3
T
2
X
W
2
X
3
+
T
3
X
W
3
X
3
X
X
X
,
X
3
T
I
X
W
I
X
3
One saw the interest of the additional term
to describe the evolutions of temperature in the thickness
X
3
(whereas
T
1
(X
)
is constant on the thickness).
However the preceding result proves that the term
T
2
model with 2 fields does not converge towards
: it is necessary
at least a representation with 3 fields for that. However, knowing that mechanical models of
hulls consider thermal deformations closely connected in the thickness, one could have believed sufficient
a thermal model with 2 fields. One will see in [§3.3] an example illustrating (for a thickness
data) the effect of the parabolic term T
3
on the average temperature T
1
between the various models.
Other authors propose richer models of thermics (cf for example [bib9], [bib10],
[bib2], probably interesting for thick hulls, but of which terms higher than the command
2 become useless for the mean structures.
Indeed, as shows it the preceding diagram, the terms of a higher nature only come
to correct (when
0) of the expressions of which the main parts are given per T
1
on the one hand, T
2
and T
3
in addition. Qualitatively, they thus do not bring anything (contrary to T
2
and T
3
), quantitatively
their contribution quickly becomes negligible in general in front of the main parts.
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2.3.4 Generalization with the problems of thermal evolution
The model of thermics in the hulls presented previously was justified starting from the results of
asymptotic development of the three-dimensional equations of stationary thermics. One
do not have however results on the problem of evolution, except the convergence of the term in
average temperature <T> (cf [bib3]) (see also the remark made hereafter in [éq 2.3.4-5]).
One can however give some indications on the resolution of the problem of evolution, in particular
within the framework of a modal approach (contrary to a direct integration in time).
The three-dimensional equations are:
-
K
T +
C
T
T = R on
with:
T =
T
D
on
T
,
-
K
N
T =
on
T (X, T = 0) =
T
0
(X) on
éq 2.3.4-1
One notes:
(
µ
Q
, T
Q
)
eigenvalues and clean vectors of the following problem:
K
T +
µ
CT = 0 on
;
T = 0 on
T
,
N
T = 0 on
éq 2.3.4-2
The solution (three-dimensional) of [eq 2.3.4 - 1] is then given by:
T (X, T) =
Q = 1
T
0
.
T
Q
E
-
µ
Q
T
+
0
T
R (S).
T
Q
+
(S).
T
Q
E
-
µ
Q
(T - S)
ds.
T
Q
(X)
éq 2.3.4-3
Μ
Q
, opposite of relaxation time, are characteristic of the space modes of the problem
[éq 2.3.4-2]. To solve the equations [éq 2.3.4-1] on a thin hull, one can adopt as in
stationary the representation [éq 2.3.1-1] for the field of temperature in the hull:
T X
; X
3
, T =
I = 1
3
T
I
X
. F
I
(T). W
I
X
3
.
One then obtains the problem of clean modes, posed on average surface
, in form
variational:
To find
µ
Q
, T
Q
R
+
3
X
H
1
(
)
3
such as,
H
1
(
)
3
:
T
T
Q
.
With
.
+ T
Q
T
.
B
- 2h
C.
µ
Q
1
0 0
0
µ
Q
2
0
0 0
µ
Q
3
.
D
= 0
éq 2.3.4-4
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Note: the operator
T
(.).
With
.
(.) + (.).
B
. (.
is quite elliptic; it is reminded the meeting that
B
described
transverse conduction (coefficient K) as well as the exchanges on the two walls of the hull, then
that
With
corresponds to surface conduction (coefficient K). It was supposed here that
C was homogeneous
in the thickness.
For example, if one neglects the effect of curvature in the thickness, in the absence of condition of exchange on
walls
+
,
-
, and with a homogeneous material, one obtains the partial derivative equations
following, to solve on
(cf [éq 2.3.2-2]):
T
1
+
µ
1
C
K
T
1
= 0
T
2
+ 3
C
K
-
K
H
2
C
+
µ
2
T
2
= 0
T
3
+ 5
C
K
-
3 K
H
2
C
+
µ
3
T
3
= 0 with
µ
I
> 0
éq 2.3.4-5
It is noted here that the thickness H does not affect the modes of average temperature T
1
. On the other hand, one
relative increase in transverse conductivity K/K
or a reduction thickness H cause of
to decrease times characteristic for the modes of “temperatures” T
2
and T
3
.
The complete solution according to this representation thus appears in the form:
T X
; X
3
, T =
2h T
1
0
. T
1q
E
-
µ
Q
1
T
+
0
T
+
(S) +
-
(S)
.
T
1q
E
-
µ
Q
1
(T
-
S)
ds. T
1q
X
+
2h
3 T
2
0
. T
2q
E
-
µ
Q
2
T
+
0
T
+
(S)
-
-
(S)
.
T
2q
E
-
µ
Q
2
(T
-
S)
ds. T
2q
X
X
3
H
+
2h
5 T
3
0
. T
3q
E
-
µ
Q
3
T
+
0
T
+
(S) +
-
(S)
.
T
3q
E
-
µ
Q
3
(T
-
S)
ds. T
3q
X
32 X
3
2
H
2
-
13
Q
=
1
Q
=
1
Q
=
1
éq 2.3.4-6
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where an initial temperature was considered:
T
0
(X
; X
3
) = T
0i
(X
). W
I
(X
3
)
and where one supposed the absence of heat sources in the thickness.
By comparing the 3D solution [éq 2.3.4-3] and the model of hull [éq 2.3.4-6], one notes that in it
the last transverse modes
T
Q
according to X
3
are represented only by the functions W
I
(X
3
) data;
what amounts truncating the series
T
Q
. But another limitation appears in the product of convolution
for the relaxation times
1
µ
Q
characteristics of the transverse modes in [éq 2.3.4-3] which
disappear in the model [éq 2.3.4-6] beyond from a parabolic “mode”.
In a diffusion of purely transverse heat (described by
(T
i0
,
µ
i0
)
in the model
[éq 2.3.4-6]), the eigenvalue lowest being K/H
2
C one can hope for a correct solution with
model of hull if relaxation times T
C
loadings applied are such as:
T
C
>
C
K
H
2
éq 2.3.4-7
This inequality can be used as practical limit of application of the model.
2.3.5 Equations of the model with usual variables
The choice of the variables T
1
, T
2
, T
3
representation [éq 2.3.1-1] corresponded to the development of
the temperature according to the thickness.
For the applications, it is however more convenient to replace them by the variables: T
m
, T
S
, T
I
:
T
m
indicate the temperature on the average surface of the hull,
T
S
the temperature on “external” surface (X
3
= + H),
T
I
the temperature on “interior” surface (X
3
= - H).
The representation in the thickness uses the polynomials of LAGRANGE then: P
1
, P
2
, P
3
:
T X
;
X
3
=
T
m
X
.
P
1
X
3
+ T
S
X
.
P
2
X
3
+ T
I
X
.
P
3
X
3
with:
P
1
X
3
= 1 -
X
3
/H
2
P
2
X
3
= X
3
2h
1 + X
3
/H
P
3
X
3
= -
X
3
2h
1 - X
3
/H
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The formulation of the thermal problem on
is similar to [éq 2.3.1-2], but where one considers:
T
= T
m
,
T
S
,
T
I
T
=
T
.
P
1
P
2
P
3
;
3
T =
T
.
P
1
'
P
2
'
P
3
'
The tensor
With
of command 4 is written then:
With
ij
X
=
K
P
I
P
J
1 + X
3
.H
1
1 - X
3
1
R
+ 1
R
dx
3
I
The tensor
B
of command 2 is:
B
ij
X
=
K P
I
'
P
J
'
1 + X
3
H
1
dx
3
+
+
P
I
H P
J
H 1 + hH
1
+
-
P
I
- H P
J
- H 1 - hH
1
I
For the second member,
C
becomes:
C
X
=
+
0
1
0
1 + hH
1
+
-
0
0
1
1 - hH
1
And
D
:
D
X
=
I
P
1
X
3
P
2
X
3
P
3
X
3
.
1 + X
3
H
1
dx
3
, for
X
.
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2.3.5.1 Case of a homogeneous plate
Various integrals on I =] - H, H [necessary to the calculation of
With
and
B
are gathered hereafter:
P
1
2
dx
3
=
16h
15
;
I
P
2
2
dx
3
=
I
P
3
2
dx
3
= 4h
15
I
P
1
.P
2
dx
3
=
I
P
1
.P
3
dx
3
= 2h
15
;
I
P
2
.P
3
dx
3
= - H
15
I
P
1
'2
dx
3
= 8
3h
;
I
P
2
'2
=
I
P
3
'2
= 7
6h
I
P
1
'
.P
2
'
dx
3
=
I
P
1
'
.P
3
'
dx
3
= - 4
3h
;
I
P
2
'
.P
3
'
dx
3
= 1
6h
I
One finds then (by neglecting the correction of curvature):
With
11
=
16hk
15
;
With
22
=
With
33
=
4hk
15
With
12
=
With
21
=
With
13
=
With
31
= 2hk
15
;
With
23
=
With
32
= -
HK
15
Then:
B
= K
6h
16
- 8
- 8
- 8
7
1
- 8
1
7
+
0 0 0
0
+
0
0 0
-
C
=
0
+
-
D
=
1 - X
3
/H
2
.dx
3
I
.
X
3
2h
.
1 + X
3
/H dx
3
I
.
- X
3
2h
1 - X
3
/H dx
3
I
on
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2.3.5.2 Relation between the variables of the two representations
T
m
X
=
T
1
X
-
1
2
T
3
X
T
S
X
=
T
1
X
-
T
2
X
+ T
3
X
T
I
X
=
T
1
X
+ T
2
X
+ T
3
X
and:
T
1
X
=
1
6
4 T
m
X
+ T
S
X
+ T
I
X
T
2
X
=
1
2
T
S
X
+ T
I
X
T
3
X
=
1
3
- 2 T
m
X
+ T
S
X
+ T
I
X
2.3.6 Synthesis
The problem to be solved on the hull
, thickness 2h is written:
To find
T = T
m
,
T
S
,
T
I
W =
=
m
,
S
,
I
H
1
(
)
3
,
m
=
S
=
I
= 0 on
T
such as:
T
T
.
With
.
+
T
T
.
B
.
. D
=
C
T
.
D
+
D
T
.
ds,
W
with:
With
ij
X
=
-
H
H
K
P
I
.
P
J
1 + X
3
H
1
1 - X
3
1
R
+ 1
R
dx
3
B
ij
X
=
-
H
H
K P
I
'
.
P
J
'
1 + X
3
H
1
dx
3
+
±
.
P
I
(
±
H).
P
J
(
±
H) 1
±
H H
1
C
I
X
=
±
.
P
I
(
±
H) 1
±
H H
1
+
- H
H
r.
P
I
1 + X
3
H
1
dx
3
D
I
X
=
-
H
H
.
P
I
1 + X
3
H
1
dx
3
, for
X
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and:
T X
;
X
3
=
T
m
X
.
P
1
X
3
+ T
S
X
.
P
2
X
3
+ T
I
X
.
P
3
X
3
P
I
(X
3
):
three polynomials of LAGRANGE in the thickness [
-
H, H]:
P
1
X
3
= 1 -
X
3
/H
2
;
P
2
X
3
= X
3
2h
1 + X
3
/H;
P
3
X
3
= -
X
3
2h
1 - X
3
/H
H
1
:
average curvature:
H
1
= 1
R
1
+ 1
R
2
;
(X
1
, X
2
):
frame of reference orthonormées according to main curvatures' of
; D
= dx
1
. dx
2
;
K
ß
:
surface components of the tensor
K
of conductivity;
K:
transverse component of the tensor
K
of conductivity;
±
:
coefficients of exchange on the faces
+
and
-
;
±
:
flows applied to the faces
+
and
-
;
R:
sources distributed in the thickness;
:
flow imposed on the end
hull.
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3
Validation of the model on some examples
One presents here applications on cylinders and plates. First draft in fact of a case
unidimensional in the thickness and allows to evaluate the effect of the terms of curvature, in particular in
the second member of the equations. The others make it possible to judge the capacity of the model to treat the case
discontinuous thermal loadings, by reference to 3D solutions.
3.1
The infinite cylinder subjected to a uniform interior flow
One considers an infinite cylinder (radius R, thickness 2h), subjected to a uniform flow inside:
I
, and with
a condition of exchange in external skin
+
(T - T
ext.
) =
+
T -
E
.
One notes K the coefficient of transverse conductivity.
The analytical solution of this axisymmetric problem 1D is:
T X
3
=
T
1
+ T
0
ln
1 + X
3
/R
R
X
2
X
1
2h
Z
R
-
H
R + H
X
3
R
- KT
, 3
=
I
> 0
+
T -
E
= - KT
, 3
with:
T
0
= -
R
I
K
1 -
H
R
T
1
=
R
I
K
1 -
H
R
ln
1 +
H
R
+
I
1 - H/R
1 + H/R
+
E
.
1
+
A development limited to the 2
ème
command in X
3/
R is:
T (X
3
)
I
+
E
+
-
2
I
+
. H
R
1
-
H
R
-
+
R
2 K
1
-
3 H
2 R
-
I
H
K
1
-
H
R
X
3
H
-
X
3
2
2 H
2
. H
R
+…
Now let us use the model with 3 fields
T
= (T
1
, T
2
, T
3
) defined in [§ 2.3.2]. Because of
independence in X
1
and X
2
solution, one is reduced to the resolution of:
B T
=
C
.
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For the representation
T (X
3
) = T
1
+ T
2
X
3
H
+ 3
2
. T
3
X
3
2
H
2
-
1
3
one a:
B
=
2K
H
0
0
0
0
1
H/R
0
H/R
3
+
+
1 + H
R
1 1 1
1 1 1
1 1 1
C
=
I
1
-
H
R
1
-
1
1
+
E
1 + H
R
1
1
1
for the second member.
If one neglects the intervention of the curvature in the metric one, one removes the terms in
H
R
in the preceding expressions.
The solution is, if the curvature completely is neglected
1
>>
H
R
:
T X
3
=
I
+
E
+
+
I
. H
K
1
-
X
3
H
i.e. the solution of the plane “wall”.
If one takes account of the curvature in the second member like in the terms of exchange
+
(true surfaces of application of flows):
T X
3
=
I
+
E
+
-
2
I
+
.
H
R
1 - H
R
-
+
R
2K
1
-
H
R
-
I
H
K
1
-
H
R
.
X
3
H
+ 0
One finds the analytical solution developed to the 1
Er
command in X
3
/R. the taking into account of the curvature
in the terms of conductivity in
B
would intervene on the level of the terms in (X
3
/R)
2
.
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3.2
The infinite plate under a couple of antisymmetric flows
Let us take again the case of the infinite plate subjected on its half X
2
< 0 with a couple of constant flows (
+
=
,
-
= -
) balanced, and adiabat on other half X
2
> 0.
The antisymetry of the loading imposes that: T (X
1
, X
2
, 0) = 0. One can also show that T is linear
in the thickness in X
2
= -, 0, +.
The equations [éq 2.3.2-2] are reduced here to:
-
2kh
1
0
0
0
1/3
0
0
0
1/5
T
1
T
2
T
3
“
+ 2K
H
0 0 0
0 1 0
0 0 3
T
1
T
2
T
3
=
0
2
0
= 0
= 0
X
2
T (0, 0) = 0
I
X
1
3
X
The derivative T
I, 2
cancelling itself ad infinitum, T
1
and T
3
are identically null everywhere. It remains to determine T
2
(depending only on X
2
) such as:
-
+
=
T
K
KH T
KH
2
2
2
3
3
“
.
whose solution is form:
T
2
X
2
= E has
3K/K. X
2
/H
+
H
K if X
2
< 0,
T
2
X
2
= B E
-
3K/K
.
X
2
/H
if X
2
> 0.
The continuity of T
2
and
T
2 '
in 0 gives:
T
2
X
2
=
H
2K 2
-
E
3K/K. X
2
/H
if X
2
0,
T
2
X
2
=
H
2K. E
-
3K/K
.
X
2
/H
if X
2
0.
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After change for the variables T
m
, T
S
, T
I
, one finds:
T
m
X
2
= 0;
T
S
X
2
=
T
2
X
2
;
T
I
X
2
= -
T
2
X
2
The temperature of the plate, calculated within the framework of this model is thus linear in the thickness and
express yourself with T
2
(X
2
), or T
S
(X
2
) and T
I
(X
2
) by:
T X
1
,
X
2
,
X
3
=
T
2
X
2
.
X
3
H
=
T
S
X
2
.x
3
2h
1 + X
3
/H -
T
I
X
2
.x
3
2h
1 - X
3
/H
[Figure 3.2-a] allows to compare the temperatures in higher skin (X
3
= + H) of the plate under
a standardized external flow (
= K/H, with K = K = 1, H = 1), obtained by a numerical calculation 3D
(
Code Aster
), the model hull, and the asymptotic limiting model (with the discontinuity observed in
[§2.2]).
One notes the good capacity of the model to describe the boundary layer appearing in the vicinity of one
discontinuity of external flow.
Temperature in higher skin of the plate pb1
Homogeneous plate
H = 1; K = K = 1
Unit flows
Opposed on X < 0
Null flows on X > 0
Caption:
0 = model 3D
= model of Hull
X = Asymptot model.
Appear 3.2-a: Temperature compared in higher skin of the plate
subjected to antisymmetric flows.
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3.3
The infinite plate under a couple of symmetrical flows
In the preceding example, the antisymetry of the loading involved the nullity of the even terms in X
3
(T
1
= T
3
= 0). One treats now another case of loading, symmetrical, (compared to X
3
= 0)
allowing to judge the effect of the term T
3
, in particular on T
1
, which requires to take conditions
in extreme cases of the type exchanges, for dédiagonaliser
B
[§ 2.3.2].
In X
3
= + H, one has like condition:
-
KT
=
T
+
if X
2
< 0
=
T
-
if X
2
> 0
In X
3
=
-
H, one a:
-
KT
=
T +
if X
2
< 0
=
T
-
if X
2
> 0
T -
X
2
T (0, 0) = 0
I
T +
T +
T -
X
3
The conditions of symmetry and antisymetry force the solution to check:
-
T (X
1
,
-
X
2
, X
3
) = T (X
1
, X
2
, X
3
) = T (X
1
, X
2
, -
X
3
)
from where: T (X
1
, 0, X
3
) = 0, and
3
T (X
1
, X
2
, 0) = 0.
The equations (18) are written in our case (T
I
depends only on X
2
):
-
+
+
=
<
-
>
-
+
+
=
-
+
+
+
=
<
-
>
KH T
T
T
X
X
KH
T
K
H
T
KH
T
T
K
H
T
X
X
.
/.
/.
“
“
“
1
1
3
2
2
2
2
3
1
3
2
2
0
0
3
0
5
3
0
0
for
or
for
for
or
for
T
2
is thus identically null (what is coherent with the conditions of symmetry).
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The solutions T
1
and T
3
are given by (cf [Year 1]):
T
1
X
1
,
X
2
=
-
1
-
/KH
-
S
2
2
S
1
2
-
S
2
2
.
E
-
S
1
X
2
+
/KH
-
S
1
2
S
1
2
-
S
2
2
.
E
-
S
2
X
2
sgn
X
2
T
3
X
1
,
X
2
=
-
.
KH
.
/KH
-
S
1
2
/KH
-
S
2
2
S
1
2
-
S
2
2
E
-
S
1
X
2
-
E
-
S
2
X
2
sgnx
2
S
1
and S
2
being positive roots of the characteristic polynomial.
After change for the variables T
m
, T
S
, T
I
, one finds:
T
m
X
1
,
X
2
=
T
1
X
1
,
X
2
T
S
X
1
,
X
2
=
T
1
X
1
,
X
2
+ T
3
X
1
,
X
2
T
I
X
1
,
X
2
=
T
S
X
1
,
X
2
If one adopts to solve the thermal problem a model with 2 fields
(T
1
,
T
2
)
, with one
representation closely connected in the thickness, one obtains as solution:
T
1
X
1
, X
2
=
/
1
-
E
/KH. X
2
if X
2
< 0
= -
/
1
-
E
-
/KH. X
2
if X
2
> 0
T
2
X
1
, X
2
= 0
In such a model the temperature appears constant in the thickness. The asymptotic limiting model
product the same solution.
One compares the numerical solution 3D and that of a model with 2 fields
(T
1
)
. The latter
comparison makes it possible to judge effect of the parabolic term on the distribution of the temperature
average. Indeed it is the latter which, in mechanical theory of the hulls, generates one
membrane deformation.
These comparisons are made for unit values of K, K,
, H,
in units If. They are visualized
isovaleurs 3D in the thickness on [Figure 3.3-a].
Average temperatures T
1
and
T
1
are represented [Figure 3.3-b]. Finally it [Figure 3.3-c] shows
change of the temperature in higher skin (X
3
= + H) of the plate, for the three solutions
considered, like by that of the asymptotic limiting model; [Figure 3.3-d] the same one presents
comparison for the average layer of the plate.
One notes on these results the good adequacy between the solution supplements 3D (items 0) and that
obtained with the model of hulls with 3 fields (points
), whereas the model with 2 fields (points +)
appears insufficient.
These observations remain valid for other choices of K, K,
, H,
, since the problem is linear
in
, and that the variable of space X
2
normalisable appears by
KH
in the equations.
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Appear: 3.3-a: Isovaleurs of temperature by numerical calculation 3D
model with 2 fields
model of hull
Appear: 3.3-b: Comparison of the average temperatures: effect of the parabolic term
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model with 2 fields
model 3D
model of hull
asymptotic model
Appear: 3.3-c: Temperature compared in higher skin of the plate
subjected to symmetrical exchanges
m odèle of hull
m odèle 3D
m odèle with 2 cham PS
m odèle asym ptotic
Appear: 3.3-d: Temperature compared on the average layer
plate subjected to symmetrical exchanges
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3.4
The infinite cylinder subjected to a horizontal stratification
One is interested in this paragraph in a situation more industrial than the preceding cases. It is about one
thermal problem of stratification in a horizontal pipe [bib12]. Under certain conditions
thermohydraulics, the temperature of the fluid can vary very quickly with dimension Z (cf appears
below). One can practically consider that there are two areas at constant temperatures of
leaves and other of a horizontal interface.
0
y
Z
0
·
Geometrical characteristics:
R = 1.0m
H = 0.075m
0
= - 30 °
·
Physical characteristics:
Conductivity
K = 17 W/m/°C
Exchanges:
outside (air)
=
E
= 12 W/m
2
/°C
interior (hot water)
=
C
= 1000 W/m
2
/°C
interior (cold water)
=
H
= 1000 W/m
2
/°C
Temperatures: external: 25 °C
interior: heat 250 °C
cold 50 °C
The determination of the temperature in the pipe is of two interests, first is to be able
to lead to the distribution of stress in the vicinity of the stratification, second is to estimate them
heat transfers enters the “cold” water area and the “hot” water area via conduction
tube.
The problem being independent of variable X, it becomes unidimensional within the framework of the model of
hull. To solve it, one first of all seeks the general solutions of the equation without second
member on each segment
]
-
2
,
0
[and]
0
,
2
[
:
-
With
T
1
T
2
T
3
+
B
T
1
T
2
T
3
= 0
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Solving for that numerically a cubic equation (characteristic polynomial in S),
one writes then the conditions of continuity of the fields T
I
and their derivative tangential with
the interface, by expressing those by the combination of the general solutions and the solutions
particular in each field. The linear system to solve (12 X 12) is brought back to one
system of dimension 6 X 6 by considerations of symmetries, then solved numerically.
One has an semi-analytical solution thus [bib4] (numerical resolution of the equation of the third
degree and of the linear system) although the situation is complex. The comparison with a calculation 2D
by finite elements is given on [Figure 3.4-a] and [Figure 3.4-b]: the difference between the two
solutions is indistinguishable.
Extreme values
47.3717
247.6520
Appear 3.4-a: laminated Piping: analytical solution by the thermal model of hull
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extreme values
47.1831
247.2360
Appear 3.4-b: laminated Piping: solution finite elements thermal 2D Aster
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4
Remarks on the numerical discretization
In this paragraph one limits oneself to some observations as for the numerical resolution of
equations of the thermal model of hull: first of all on the use of a method finite elements
and then on numerical blocking appearing when the thickness 2h is low. This last comes
intervention of H to powers different in the coefficients from the equations.
4.1
Resolution by finite elements
The model of thermics of hull describes in [§ 2.3] shows the following characteristics:
·
it leads to an operator of a nature 2 acting out of the three scalar fields
T
= (T
m
, T
S
, T
I
);
·
these three fields are defined on a field
surface, plunged in IR
3
;
·
curvature of surface
intervenes, possibly, only in the expression of the coefficients
With
,
B
,
C
,
D
.
In the general case of a hull of an unspecified form plunged in IR
3
, one can discretize
geometry of its average surface
by a mesh in plane triangular elements (this method
present certainly the defect not to be able to explicitly take into account the curvature of
).
The thermal problem (see [§ 2.3.6]) being scalar, with 3 fields, of the second command, one proposes them
usual finite elements: the plane triangles P1 (with 3 nodes) or P2 (with 6 nodes).
Their formulation is the same one, as
that is to say plane or curved: one neglects the corrections thus of
metric in the operators of rigidity
With
and
B
, (one saw in the cases of validation that that had little
of effect in practice). On the other hand the user, if he knows the expression of the curvature, will have interest with in
to hold account in the values of the coefficients
± and of flows
±, as in the expressions
[éq 2.3.1-4] and [éq 2.3.1-5].
In the case of materials composite (just as if one wanted to take account of the curvature), one has
to envisage a preprocessing providing the coefficients
With
,
B
,
C
,
D
, as well as a postprocessing
allowing to reconstitute the temperature and flows in any point thickness.
There are situations where the problem does not depend any more but on one variable of space: they are the hulls
of axisymmetric revolution loading, or “sections of hulls”, axis
E
3
.
The geometry is then represented by a meridian line: (see [Figure 4.1-a]).
Y
y
X
B
Z
X
3
S
R
X
T
T
With
T
N
The average curve is then:
- case revolution:
- case “slices”, or arc:
where R indicates the radius of curvature of the line
meridian A B.
H
R
X
1
1
=
+
cos
H
R
1
1
=
Appear: 4.1-a
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For these types of problems, one proposes also a finite element P2 with 3 nodes, using the same one
formulation, where one neglects the correction of metric in the thickness, for the coefficients
With
and
B
. One
use a formula of quadrature at 4 points of GAUSS.
This element is associated exactly that proposed in mechanics for studies
thermomechanical chained [R3.07.02].
4.2
Numerical blocking of a finite element of thermal hull
Blocking is a phenomenon appearing in the numerical resolution by finite elements of
certain problems such as that of the thin hulls or the arcs when the element is curved (blocking
of membrane), that of the hulls or beams with taking into account of shearing (blocking of
shearing), or that of plasticity (plastic blocking of incompressibility [bib7]). It was
met initially in mechanics of the incompressible fluids and it is within this framework that its study
theoretical began [bib6].
This phenomenon of blocking appears by a very great loss of precision and oscillations
important on certain calculated quantities when a physical parameter of the model becomes “small”.
The illustration of these nuisances is given in note HI-71/7131, (§4.2). The origin of these
problems lies in the difference in order of magnitude which appears between certain components of
the bilinear form of “rigidity” when the physical or geometrical parameter tends towards zero (thickness
hull for the blocking of membrane, reverses tangent module of compressibility for
plastic blocking for example). Here, it is the thickness of the hull which will play the part of small parameter.
Let us take again the equations of the stationary thermal problem posed on a plate in form
variational; let us note 2
H its thickness (
reality without dimension):
To find
T = T
1
,
T
2
,
T
3
W
= H
0
1
(
)
3
such as
With (
T,
) + 1
B (
T,
) = F (
)
= (
1
,
2
,
3
)
W
éq 4.2-1
with:
With (
T,
) = 2kh
T.
1 0 0
0 1
3
0
0 0 1
5
.
D
. indicating the usual scalar product,
B (
T,
) =
2K
H
T.
0 0 0
0 1 0
0 0 3
.
D
=
1
,
2
,
3
(gradients surfaciqu
An equivalent mixed formulation of this problem is obtained with the variables Q
2
and Q
3
, flow of
heat in the thickness (cf [Year 2]): one notes
= (L
2
(
))
2
.
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To find (
T, p)
W X Q such as
WITH (T,
)
-
M (
, p) = F (
)
W
-
B (p, Q)
-
M (
, Q) = 0
Q
Q
éq 4.2-2
where:
Q = (
Q
2
,
Q
3
)
M (
, Q) =
2
Q
2
+
3
Q
3
D
B (p, Q) =
H
2K
p
1 0
0 1
3
Q
D
On this formulation, numerical blocking appears clearly (at least formally). Indeed,
discretization of
W
X Q being carried out, (one notes it
W
D
X Q
D
), the problem tends formally when
tends towards zero towards the following problem:
With (
T
D
,
)
-
M (
,
p
D
) = F (
)
W
D
M (
T
D
, Q) = 0
Q
Q
D
What amounts solving on the core of M:
With (
T
D
,
) = F (
)
W
D
éq 4.2-3
Blocking appears when the discretized core of M is too small or reduced to zero: the resolution of
[éq 4.2-3] is done on a very small space even reduced to zero. Even if the mesh is fine, the solution is
then of very bad quality.
The core of M in
W
D
being by definition, space:
Ker M
D
=
W
D
|
M (
, Q) = 0
Q
Q
D
It is seen that the choice of the discretization of Q is not innocent and strongly conditions it
behavior of the solution when
tends towards zero. There is a condition of convergence carrying
on spaces
W
D
and Q
D
who ensures the good numerical behavior of the solution with
small, it is
condition known as LBB discrete, version adapted to the discrete case of continuous condition LBB
(LADYJENSKAIA - BREZZI - BABUCHKA). We return to [bib 7] for a case study (plasticity)
and a bibliography on this subject.
Parallel to the theoretical studies quickly mentioned above, a remedy practices with
blocking, appearing once the discretization in
W
D
chosen if this choice were unhappy, consists with
under-to integrate the term “locking” in the construction of rigidity, i.e. the term B here. Some
choices of under-integration, in the primal formulation [éq 4.2-1] are interpreted like choices
of interpolation of
W
D
and Q
D
in the mixed formulation and can thus, via the checking (sometimes
hard) of discrete condition LBB, being justified on the theoretical level.
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Let us consider indeed, a triangular finite element with 3 nodes and P1 interpolation to solve it
problem [éq 4.2-2]. Then let us choose for discretization of Q a discontinuous P0 interpolation, it is
with-statement a representation of [Q] constant by element.
The second equation of [éq 4.2-2] is then a local equation, i.e to be solved on each element
separately since p is unspecified on each element E.
-
H
2K
E
p
2
Q
2
+ 1
3
p
3
Q
3
-
E
T
2
Q
2
+ T
3
Q
3
= 0
(Q
2
,
Q
3
)
from where the immediate solution if |E| is the surface of E.
p
2
=
-
2K
H
1
E
E
T
2
p
3
=
-
6K
H
1
E
E
T
3
By deferring these results in the form M, one has on the element E:
M (
, p) =
2K
H
1
E
E
T
2
E
2
+ 3
E
E
T
3
E
3
for all
W
D
having thus eliminated p, one is brought back to a primal formulation on T only:
M (
, p) =
2K
H
1
E
E
T
2
E
2
+ 3
E
E
T
3
E
3
for all
W
D
who corresponds very exactly to the formulation [éq 4.2-1] in which the elementary term:
B
E
(
T,
) =
2K
H
E
T
2
2
+ 3
T
3
3
under-is integrated by a diagram into a point of GAUSS:
E
F. G
1
E
.
E
F.
E
G
The examination of discrete condition LBB remains to be made for this discretization in order to conclude with its
convergence (cf [bib7]).
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5 Conclusion
An asymptotic analysis of the equations of thermics in a mean structure when the thickness
tends towards zero leads to a limiting model characterized by an average temperature, solution of one
problem in extreme cases, and a parabolic complementary term in the thickness, definite locally.
One deduced the formulation from it from a model with 3 scalar fields definite on average surface from
hull, giving a parabolic representation of the temperature in the thickness. The operator
differential obtained is of command 2; the thickness of the hull appears in its coefficients.
This model seems “optimal” for the mean structures:
·
its limit when the thickness tends towards zero is identical to the asymptotic limiting model;
·
possible additional terms would tend towards zero with the thickness.
In a standard version the curvature of the average surface of the hull does not intervene directly.
Test examples show a good adequacy of the temperature obtained with solutions
three-dimensional complete.
This model thus appears completely entitled with:
·
to be used in a finite elements formulation to calculate the temperature in a hull
thin of an unspecified form; the solution obtained being able to be easily injected into one
thermomechanical calculation of the hull; one proposes surface and linear elements thus
for the cases where a variable of space does not intervene;
·
to be introduced directly (or by coupling) into a method of resolution of the equations
governing the thermohydraulic state of a piping for example, in order to take account of
thermal restitution of the wall on the fluid;
·
to be used as model integrated in the resolution of problems of identification (problem
opposite) starting from experimental measurements (for example for laminated conduits);
·
to seek analytical solutions in cases with simple geometry.
The model describes here can also be used in the problems of thermal evolution, provided that
the thermal loadings do not vary too quickly.
Lastly, it remains to study the numerical methods to use to avoid the blocking which could appear
in a calculation by finite elements, when the thickness becomes low.
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6 References
[1]
ANDRIEUX S., MARIGO. J.J. : Application of the asymptotic methods to the problem of
thermal conduction in the thin sections. Note EDF-DER-MMN: HI-71/5963, 1987.
[2]
BLANCHARD J.L, CARNOY E.: Finite element of thermal hull for the analysis
thermomechanics of thin hulls. In “c.f. and Structural analyzes A.O”.
[3]
FRANKFURT G.: Asymptotic transient thermoelastic behavior in thermo mechanical
couplings in solids. IUTAM. Bui and Guyen Eds. Elsevier 1987.
[4]
FAZOUANNE A.: Study of an opposite thermal problem. Scientific report/ratio of training course,
ENPC. 1990.
[5]
GREEN A.E., NAGHDI TOKEN ENTRY: One thermal effects in the theory off shells. Proc. Roy. Plowshare
London. A365, A367. 1979.
[6]
HUGHES T.J.R., MALKUS D.S.: “Mixed Finite Element Methods Reduced and selective
integration techniques. Meth computer. Appl. Mech. Eng. 15-1, pp. 63-81, 1978.
[7]
MIALON P., THOMAS B.: Incompressibility in plasticity: under-integration and others
digital techniques. Note EDF-DER-MMN: HI-72/6404, of January 19, 1990. See also:
Bull DER Series C, n°3, 1991.
[8]
RUBIN Mr. B.: Heat conduction in punts and shells with emphasis one has conical Shell, in Int.
J. off Solids and Structures, vol. 22, N° 5, pp. 527-551. 1986.
[9]
SURANA K., ABUSALEH G.: Curved Shell elements for heat conduction with p-approximation
in the Shell thickness direction. Computers and Structures, vol. 34, N° 6, 1990.
[10]
SURANA K.S., ORTH N.J. : Axisymmetric Shell elements for heat conduction with
p-approximation in the thickness direction, Computers and Structures, vol. 33, N° 3, pp.
689-705. 1989.
[11]
SURANA K., PHILIPS R.: Three dimensional curved Shell finite elements for heat
conduction. Computers and Structures, vol. 25, N° 5. 1987.
[12]
SABATON Mr., BIMONT G., PURPLE P.L., VOLDOIRE F., MASSON J.C., GRATTIER J.:
Stratifications in pipings of the pressurized water reactors. Note EDF-DER:
HP/109/88/01. Feb. 1988.
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Appendix 1 Plates infinite under a couple of symmetrical flows
In X
3
= + H, the boundary conditions are:
-
K
3
T
=
T +
if X
2
< 0
=
T
-
if X
2
> 0
In X
3
= - H, one a:
-
K
3
T
=
T +
if X
2
< 0
=
T
-
if X
2
> 0
T -
X
2
T (0, 0) = 0
I
T +
T +
T -
X
3
The conditions of symmetry and antisymetry force the solution to check:
T (X
1
,
-
X
2
, X
3
) = T (X
1
, X
2
, X
3
) = T (X
1
, X
2
,
-
X
3
)
and thus: T (X
1
, 0, X
3
) = 0,
3
T (X
1
, X
2
, 0) = 0.
The equations [éq 2.3.2-2] are written in our case:
-
KH T
1
“+
T
1
+
T
3
=
for X
2
< 0 or -
for X
2
> 0
-
KH/3. T
2
“+ K
H
+
T
2
= 0
-
KH/5. T
3
“+
T
1
+ 3 K
H
+
T
3
=
for X
2
< 0 or -
for X
2
> 0
T
2
is thus identically null (what is coherent with the conditions of symmetry). The system
precedent admits as particular solution:
(
)
(
)
T
X X
X
X
T
X X
R
p
p
1
1
2
2
2
3
1
2
2
0
0
0
,
,
=
<
-
>
=
if
and
if
on
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The polynomial characteristic in S of the homogeneous system is:
K
2
H
2
5
S
4
-
K 6 H
5
+ 3 K S
2
+ 3
K
H
= 0
, including the 4 roots S
I
are:
S
I
=
±
1
H
.
3
K
.
H + 5
2
K
±
2
H
2
+ 2 5 K
2
4
+ 1 0 K
H
3
, S
1
> S
2
> 0 > S
3
> S
4
The solutions T
1
(X
1
, X
2
) and T
3
(X
1
, X
2
), finished in
X
2
=, thus express themselves:
T
1
X
1
, X
2
=
+
E
S
1
X
2
+
E
S
2
X
2
=
-
-
E
-
S
1
X
2
-
E
-
S
2
X
2
T
3
X
1
, X
2
=
E
S
1
X
2
+
E
S
2
X
2
=
-
E
-
S
1
X
2
-
E
-
S
2
X
2
for X
2
> 0
for X
2
> 0
for X
2
> 0
for X
2
> 0
Conditions of connection in X
2
= 0 are naturally expressed by the conditions of antisymetry of
T, already used above. Four constants
, ß,
,
are determined by:
+
=
-
/
+
= 0
nullity of T in X
2
= 0
-
KH S
1
2
+
= 0
-
KH S
2
2
+
= 0
m odes T
1
- T
3
associated S
1
, S
2
From where:
=
-
.
/KH
-
S
2
2
S
1
2
-
S
2
2
=
.
/KH
-
S
1
2
S
1
2
-
S
2
2
=
. KH
2
.
/KH
-
S
1
2
.
/KH
-
S
2
2
S
1
2
-
S
2
2
=
. KH
2
.
/KH
-
S
1
2
.
/KH
-
S
2
2
S
1
2
-
S
2
2
The solutions T
1
and T
3
are thus written:
T
1
X
1
, X
2
=
-
. 1
-
/KH
-
S
2
2
S
1
2
-
S
2
2
E
-
S
1
X
2
+
/KH
-
S
1
2
S
1
2
-
S
2
2
E
-
S
2
X
2
sgn X
2
T
3
X
1
, X
2
=
-
KH
2
.
/KH
-
S
1
2
/KH
-
S
2
2
S
1
2
-
S
2
2
. E
- S
1
X
2
-
E
-
S
2
X
2
sgn X
2
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Mixed appendix 2 Formulation of the stationary problem for
plate
In the case of the plate, the variational problem [éq 4.2-1] is equivalent to a problem of
minimization (while revealing explicitly the thickness
H in B) of the functional calculus:
J (
) =
1
2
With (
,
) + 1
2
B (
,
)
-
F (
).
To obtain the mixed formulation, let us notice that:
Proposal:
1
2
B (
,
) =
-
Q
2
2
+ Q
3
3
-
4
H
K
Q
1
2
+ 1
3
Q
3
2
Sup
Q
2
, Q
3
L
2
(
)
2
Demonstration:
Let us write the condition of extremality of the functional calculus between hooks (its opposite is strictly
convex, coercive and semi-continuous in a lower position) and let us note p the couple where the sup is reached:
Q
2
2
+ Q
3
3
+
H
2 K
Q
2
p
2
+ 1
3
Q
3
p
3
= 0
Q
from where:
p
2
=
-
2 K
H
2
p
3
=
-
6 K
H
3
The value of the functional calculus in this point is thus:
+ 2 K
H
2
2
+ 6 K
H
3
2
-
H
4 K
4 K
2
2
H
2
2
2
+ 3 6 K
2
3
2
H
2
3
2
= 1
2
2 K
H
2
2
+ 3 K
H
3
2
that is to say indeed the announced result.
There is thus a formulation equivalent to the minimization of J on
W
:
Min
W
Max
Q
Q
1
2
With (
,
) -
2
B (Q, Q) - M (
, Q)
-
F (
).
while noting:
M (
, Q) =
Q
2
2
+ Q
3
3
,
B (Q, Q) =
H
2K
p
2
Q
2
+
1
3
p
3
Q
3
The condition of point-saddle of this Lagrangian led to the formulation [éq 4.2 - 2]:
WITH (T,
)
-
M (
, p) = F (
)
W,
-
B (p, Q)
-
M (
T, Q) = 0
Q
Q
Code_Aster
®
Version
2.6
Titrate:
Model of thermics for the thin hulls
Date:
03/11/93
Author (S):
F. VOLDOIRE, S. ANDRIEUX
Key:
R3.11.01-A
Page:
44/44
Manual of Reference
R3.11 booklet: Thermal elements on average surface
HI-75/93/098/A
Intentionally white left page.