Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
1/12
Manual of Reference
R5.02 booklet: Thermics
HI-74/98/007/A
Organization (S):
EDF/IMA/MN, ENPC
Manual of Reference
R5.02 booklet: Thermics
Document: R5.02.04
Nonlinear thermics in pointer
Summary
One presents the formulation and the algorithm of the problem of convection-dissemination in nonlinear thermics
stationary introduced within the control
THER_NON_LINE_MO
[U4.33.04].
The goal is to solve the equation of heat in a mobile reference frame related to a loading and moving
in a given direction and at a speed.
Nonthe linearities of the problems come as well from the characteristics of material which depend on
temperature, that boundary conditions of the radiation type.
The problems of this type can be dealt with with models using of the finite elements of structure plans,
axisymmetric and three-dimensional.
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
2/12
Manual of Reference
R5.02 booklet: Thermics
HI-74/98/007/A
Contents
1 Presentation of the problem ...................................................................................................................... 3
2 Boundary conditions. Problem of reference to solve ................................................................... 5
3 variational Formulation of the problem ................................................................................................. 6
4 Processing of nonthe linearities ............................................................................................................... 7
4.1 Processing of nonthe linearity related to the enthalpy ............................................................................... 7
4.2 Processing of nonthe linearities related on the nonlinear condition of Fourier and conductivity
thermics ........................................................................................................................................ 8
5 Algorithm established in Code_Aster ............................................................................................... 9
6 Main options of calculation in Code_Aster ............................................................................... 10
7 Bibliography ........................................................................................................................................ 11
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
3/12
Manual of Reference
R5.02 booklet: Thermics
HI-74/98/007/A
1
Presentation of the problem
The equation of heat has strong not linearities under certain conditions. It is the case
when the material undergoes phase shifts: those are accompanied by abrupt
variation of the characteristic sizes (heat-storage capacity, enthalpy). This nonlinearity is
all the more accentuated when the problem of convection-dissemination is dealt with, where the term appears of
transport depend on the function enthalpy. The goal of this modeling is to treat this last
problem in steady state (stationary case).
In all the cases, one supposes that the field speed is known a priori. The case of a mobile solid
is rather frequent in practice. It relates to in particular the applications of welding or the processing
of surface which brings into play a heat source moving in a direction and at a speed
data. The problem of thermics is then studied in a reference frame related to the source.
The problem with the derivative partial results from the equation of the total heat balance on any field
who is written:
D
dt
D
Qd
D
=
-
q.n
accumulation
creation + input-output
éq 1-1
In this equation,
represent a related, interior field with the studied system, which one follows
in its movement,
represent the specific enthalpy of material and
indicate its mass
voluminal.
Q
is a voluminal heat source,
Q
is the heat transfer rate through the border
(
N
being the external normal), and
D dt
/
is the particulate derivative.
The first term of [éq 1-1] is written (see for example [bib1]):
()
(
)
D
dt
D
T
div
D
=
+
V
éq 1-2
or, taking into account the conservation of mass
()
T
div
+
=
V
0
[bib1]:
D
dt
D
T
D
=
+
V grad
.
éq 1-3
where
V
is the Flight Path Vector of displacement of the field
.
V
is well informed under the single-ended spanner word
CONVECTION
controls
AFFE_CHAR_THER
and
AFFE_CHAR_THER_F
.
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
4/12
Manual of Reference
R5.02 booklet: Thermics
HI-74/98/007/A
The second term of the second member of [éq 1-1] is written, taking into account the theorem of the divergence
and of the Fourier analysis
()
(
)
Q
grad
= -
K T
T
:
()
(
)
q.n
Q
grad
D
div D
div K T
T D
=
= -
éq 1-4
where
T
is the temperature and
()
K T
is the thermal conductivity of material, function of the temperature.
The equation [éq 1-1] having to be satisfied for any field
, it comes then:
()
(
)
T
div K T
T
Q
+
-
=
V.grad
grad
,
,
in
éq 1-5
Note:
Let us note that the conventional case with,
()
K T
K
=
(constant) and
V
=
0
, and where the specific enthalpy is
a linear function of the temperature,
()
T
CT
=
give again the conventional equation good known:
C Tt K T Q
-
=
,
,
in
where
is the Laplacian and
C
(constant) the specific heat represents.
The problem with the derivative partial treaty by the control
THER_NON_LINE_MO
[U4.33.04], consists
to solve the equation [éq 1-5] in the stationary case (directly at the permanent state) with
boundary conditions on the border
.
This problem is formally written in the following form:
()
()
(
)
V.grad
grad
U T
div K T
T
Q
-
=
+
,
,
in
boundary conditions
on
éq 1-6
where we adopted the notation, valid for all the continuation,
()
()
U T
T
=
where
is constant,
defining the voluminal enthalpy.
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
5/12
Manual of Reference
R5.02 booklet: Thermics
HI-74/98/007/A
2
Boundary conditions. Problem of reference to be solved
One will refer, for example, with [R5.02.01] for more information on the boundary conditions
thermics of the type Dirichlet, Neumann and linear Fourier, and with [R5.02.02] for the conditions with
limits of the nonlinear normal flow type (nonlinear Fourier).
Of enthalpic formulation, the stationary problem of thermics thus consists in solving in one
field
of border on
.
()
()
(
)
V.grad
grad
U T
div K T
T
Q
-
=
,
,
in
éq 2-1
()
(
)
with
on
K T
T
T
T
ext.
N
=
-
1
,
éq 2-2
()
K T
T
Q
N
=
0
2
on
,
éq 2-3
()
()
K T
T
T
N
=
on
3
,
éq 2-4
T
T
=
0
4
on
,
éq 2-5
where:
·
T
0
: is the temperature imposed on
4
;
·
Q
0
: is the normal flow imposed on
2
;
·
: is the coefficient of heat exchange;
·
T
ext.
: is the outside air temperature;
·
()
T
: is the normal flow of nonlinear Fourier type (radiation).
The equations [éq 2-2], [éq 2-5] the boundary conditions of the types represent, respectively:
Linear Fourier, Neumann, nonlinear Fourier and Dirichlet.
The problem of reference [éq 2-1], [éq 2-5] is strongly nonlinear because of nonthe linearities on
() ()
K T
U T
,
(phase shift) and
()
T
(radiation).
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
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Key:
R5.02.04-A
Page:
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R5.02 booklet: Thermics
HI-74/98/007/A
3
Variational formulation of the problem
That is to say
open of
R
3
, of border
=
1
2
3
4
such as,
for I
J and I, J = 1,…, 4, one a:
I
J
=
.
That is to say still
a sufficiently regular function which is cancelled on
4
:
{
}
=
=
V
regular and
4
0
.
Let us multiply by
the two members of the equation [éq 2-1], then let us integrate on
. An integration by
parts gives then:
()
()
(
)
()
()
()
Q D
U T
D
div K T
T
D
U T
D
K T
T
D
K T
T
D
=
-
=
+
-
-
V.grad
grad
V.grad
grad grad
N
.
4
éq 3-1
since
is null on
4
.
From where, by taking account of the boundary conditions [éq 2-2], [éq 2-3] and [éq 2-4], the formulation
variational of the problem of reference which is given by the following equation:
V
()
()
()
K T
T
D
U T
D
T
D
T
D
Q
D
T
D
Q
D
ext.
grad grad
V grad
.
.
,
+
+
-
=
+
+
3
1
2
1
0
éq 3-2
Code_Aster
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Version
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Titrate:
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Key:
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Page:
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R5.02 booklet: Thermics
HI-74/98/007/A
4
Processing of nonthe linearities
In the sight of the numerical resolution of the nonlinear problem that we consider, it is necessary of
to treat all nonlinearities.
In our case, let us quote the strong not linearity related to the function enthalpy
()
U T
who takes into account it
solid-liquid phase shift, as well as nonthe linearity related to the possible presence of one
boundary condition of nonlinear normal flow (radiation).
Let us recall that in the conventional case of the problems of transitory thermics nonlinear without
convection, i.e.
V
=
0
, several methods of resolution are proposed in the literature. There exists
as well methods using of the enthalpic formulations as methods using of
formulations in temperature, all having for goal as well as possible to treat nonthe linearity related to the enthalpy
(phase shift).
We return the reader to the reference [bib5] for a summary of the main methods met
in the literature. However, let us note that because of the difficulty related to the presence of the term of
transport
()
''
''
V.grad U T
in the problem, none of these methods will be employed in
continuation.
As in any iterative process, the goal of the numerical diagram in sight is to find a field of
temperature
T
N
+
1
with the iteration
N
+
1
, starting from the field of temperature
T
N
, solution of the iteration
the preceding one.
4.1
Processing of nonthe linearity related to the enthalpy
In order to treat this nonlinearity, the strategy employed in this study was inspired by a technique
of resolution of the free problems of borders [bib3], which, in the beginning was proposed in [bib4].
Let us consider the function enthalpy
()
U T
as being given in a reciprocal form:
Temperature function of the enthalpy (opposite of the function
()
U T
). In other terms one will have to treat
relation following Temperature-enthalpy:
()
T
U
=
éq 4.1-1
The reason of this choice will be clearer in what follows. Indeed we will have to deal with problem with
two fields: a field of temperature and a field enthalpy. Discretization of the function
opposite [éq 4.1-1] allows to increment the field enthalpy according to the current field of
temperature (and not the reverse) as follows:
The development with the first command of the function
()
U
is as follows,
() () (
)
T
U
U
U
U
N
N
N
N
N
+
+
=
+
-
1
1
'
,
éq 4.1-2
where
'is the derivative of the function defined by [éq 4.1-1] compared to its argument.
Code_Aster
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Version
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Titrate:
Nonlinear thermics in pointer
Date:
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Key:
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Page:
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R5.02 booklet: Thermics
HI-74/98/007/A
In order to take into account this nonlinearity, and from [éq 4.1-2], one replaces
U
N
+
1
by one
approximation according to the unknown field of temperature
T
N
+
1
in the following way:
()
(
)
U
U
T
U
N
N
N
N
+
+
-
=
-
1
1
,
éq 4.1-3
where
is a parameter of relieving, constant on all the field and during all the iterative process,
representing the term
()
1
'U
N
.
Because of the nonconvexity of the function
()
U
, this parameter of relieving necessarily must
to check the following condition [bib2], [bib3]:
()
1
max '
N
N
U
éq 4.1-4
In practice one takes
()
=
1
max '
N
N
U
.
By taking of account the approximation [éq 4.1-3], discretization of the second term of the equation
[éq 3-2] is expressed in the following way:
()
V.grad
V.grad
V.grad
V.grad
U
D
U
D
T
D
U
D
N
N
N
N
+
+
=
+
-
1
1
,
éq 4.1-5
4.2 Processing of nonthe linearities related to the condition of Fourier not
linear and with thermal conductivity
Nonthe linearity related to the condition of normal flow nonlinear is treated by considering it
development with the first command of the function (supposed sufficiently regular)
()
T
who is given
by:
() () () (
)
T
T
T
T
T
N
N
N
N
N
+
+
=
+
-
1
1
'
,
éq 4.2-1
where (.)'the derivative of the function indicates (.) compared to its argument.
It appeared necessary to decide on a strategy of discretization of the term
()
“
“
K T
T
grad
in
the equation [éq 3-2] in order to be able to treat this nonlinearity for the stationary problem that us
let us consider. For that, we adopted the following approximation:
()
()
() ()
[
]
K T
T
K T
T
K T
K T
T
N
N
N
N
N
N
N
+
+
+
-
=
-
-
1
1
1
1
grad
grad
grad
éq 4.2-2
Code_Aster
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Version
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Titrate:
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Date:
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Key:
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Page:
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R5.02 booklet: Thermics
HI-74/98/007/A
This discretization is in fact a simplification of the development to the first command of the term
()
K T
T
grad
. It is effective being in particular because of the low not linearity of the function
()
K T
in practice.
Note:
Also let us note that the following purely explicit approximation:
()
()
K T
T
K T
T
N
N
N
N
+
+
+
1
1
1
grad
grad
,
also give satisfactory results. This observation was checked from several
numerical experiments.
5
Algorithm established in Code_Aster
The numerical diagram employed for the resolution of the problem of reference [éq 2-1], [éq 2-5] is
deduced from the variational formulation [éq 3-2] and from the processing of the various not linearities, [éq 4.1-5],
[éq 4.2-1], [éq 4.2-2], discussed in the preceding section.
The algorithm of resolution is consisted the sequence of two successive operations with each
iteration of calculation.
Knowing the fields solutions with the iteration
N
:
T
N
with the nodes and
U
N
at the points of Gauss, one
seek the solutions
T
U
N
N
+
+
1
1
and
with the iteration
N
+
1
as follows:
V,
()
()
() ()
(
)
() ()
[
]
()
K T
T
D
T
D
T
D
T T
D
Q
D
T
D
Q
D
T
T T
D
K T
K T
T
D
U
D
U
D
N
N
N
N
N
N
ext.
N
N
N
N
N
N
N
N
grad
grad
V.grad
grad
grad
V.grad
V.grad
+
+
+
+
-
+
+
-
=
+
+
+
-
+
-
+
-
1
1
1
1
0
1
3
1
2
1
3
.
'
'
.
,
éq 5-1
()
(
)
U
U
T
U
N
N
N
N
+
+
=
+
-
1
1
éq 5-2
With each iteration, a linear problem of convection-dissemination is solved to obtain the field with
nodes
T
N
+
1
[éq 5-1], and then a simple on-the-spot correction is carried out to obtain the field with
points of Gauss
U
N
+
1
[éq 5-2].
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
25/03/98
Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
10/12
Manual of Reference
R5.02 booklet: Thermics
HI-74/98/007/A
The criterion of stop adopted in Code_Aster utilizes at the same time the two fields solutions:
the field of temperature, and the field enthalpy.
The algorithm continues the iterations as long as at least one of the relative variations of reiterated is
higher than the corresponding tolerance given by the user:
(
)
()
(
)
()
T
T
T
U
T
U
in
in
I
nddl
in
I
nddl
in
in
I
npg
in
I
npg
+
=
+
=
+
=
+
=
-
>
-
>
1
2
1
1 2
1 2
1
1 2
1
2
1
1 2
1 2
1
1 2
,…,
/
,…,
/
,…,
/
,…,
/
sheet 1
sheet 2
where nddl is the total number of the degrees of freedom to the nodes, and npg is the total number of the points of
Gauss.
sheet 1 is indicated under the key word
crit_temp_rela
key word factor convergence of the operator
ther_non_line_mo
.
sheet 2 is indicated under the key word
crit_enth_rela
key word factor convergence of the operator
ther_non_line_mo
.
6
Main options of calculation in Code_Aster
One presents below the main options of Code_Aster specific to the unfolding of
the algorithm [éq 5-1], [éq 5-2] above. On the other hand, we will not mention the options not
specific of Code_Aster and which is used in calculation:
·
Boundary conditions:
Linear Fourier
RIGI_THER_COET_R
RIGI_THER_COET_F
T
D
N
+
1
1
Nonlinear Fourier
RIGI_THER_FLUTNL
()
'T
T
D
N
N
+
1
3
CHAR_THER_FLUTNL
() ()
(
)
T
T T
D
N
N
N
-
'
3
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
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Author (S):
F. WAECKEL, B. NEDJAR
Key:
R5.02.04-A
Page:
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R5.02 booklet: Thermics
HI-74/98/007/A
·
Elementary matrices and second member:
RIGI_THER_TRANS
()
K T
T
D
N
N
grad
grad
+
1
.
RIGI_THER_CONV_T
V. gradT
D
N
+
1
CHAR_THER_TNL
() ()
[
]
()
K T
K T
T
D
U
D
U
D
N
N
N
N
N
-
+
-
-
1
grad
grad
V.grad
V.grad
.
7 Bibliography
[1]
Duvaut, G., Mechanics of the continuous mediums, Masson, 1990.
[2]
Nochetto, R.H., Numerical methods for free boundary problems, Free Boundary Problems:
Theory years Applications, éds. K.H. Hoffmann and J. Spreckels, (Pitman, Boston, 1988).
[3]
Paolini, Mr., Sacchi, G. and Made green, C., Finite Element approximations off singular parabolic
problems, Int. J. Numer., Meth. Engng., 26, pp. 1989-2007, 1988.
[4]
Friedman, A., Variational Principles and Free Boundary Problems, Pure and Applied
Mathematics (Wiley-Interscience, New York, 1982).
[5]
Tamma, K.K. and Namburu, R.R., Recent Advences, Trends and New Prospects via
enthalpy-based finite element formulations for applications to solidification problems, Int.
J. Numer., Meth. Engng., 30, pp. 803-820, 1990.
Code_Aster
®
Version
4.0
Titrate:
Nonlinear thermics in pointer
Date:
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Author (S):
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Key:
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Page:
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