Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
1/16
Organization (S): EDF/MTI/MN
Handbook of Référence
R5.02 booklet: Thermics
Document: R5.02.02
Nonlinear thermics
Summary
Operator THER_NON_LINE [U4.54.02] allows to solve the problems of transitory thermics in
solids in the presence of non-linearities of the properties of the materials (heat-storage capacity and conductivity), or of
boundary conditions (heat exchange of radiation type). One presents here the formulation and the algorithm
employee, this last being close to that related to operator STAT_NON_LINE [R5.03.01]. Various options
of calculation necessary were presented in the plane, axisymmetric elements of structure and
three-dimensional [U3.22.01], [U3.23.01] and [U3.24.01].
Handbook of Référence
R5.02 booklet: Thermics
HI-75/99/013 - Ind A

Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
2/16
Contents
1 Expression of the equation of heat in nonlinear thermics ......................................................... 3
1.1 Equation of heat for a motionless solid .............................................................................. 3
1.2 Fourier analysis .................................................................................................................................. 4
1.3 Equation of heat in the case of the model of transitory thermics non-linear ..................... 4
1.4 Numerical advantage of the formulation in enthalpy for the problems with change of
phase ............................................................................................................................................... 5
2 Boundary conditions, loading and initial condition ........................................................................ 5
2.1 Non-linear normal flow .................................................................................................................. 5
2.2 Non-linear normal flow - condition of the type radiation ad infinitum ................................................. 6
3 variational Formulation of the problem ................................................................................................. 6
4 Discretization in time of the differential equation ................................................................................ 8
4.1 Introduction of - method ........................................................................................................... 8
4.2 Application to the equation of heat .............................................................................................. 8
5 space Discretization and adaptation of the algorithm of Newton to the problem ..................................... 10
5.1 Space discretization .................................................................................................................... 10
5.2 Stationary calculation ......................................................................................................................... 11
5.3 Transitory calculation ............................................................................................................................ 12
5.4 Convergence ................................................................................................................................. 13
6 Principal options of non-linear thermics calculated in Code_Aster .................................... 14
6.1 Boundary conditions and loadings .......................................................................................... 14
6.2 Calculation of the elementary matrices and transitory term .................................................................. 14
6.3 Calculation of the residue ............................................................................................................................. 14
7 Bibliography ........................................................................................................................................ 15
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
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Key:
R5.02.02-A
Page:
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1
Expression of the equation of heat in thermics not
linear
1.1
Equation of heat for a motionless solid
In all this document, one treats only it thermal of the solid bodies, even if the change of
liquid/solid phase is taken into account. There is thus no heat transfer by convection but
only by conduction.
The first principle of thermodynamics connects the temporal variation of total energy dEtotale of one
system included/understood in a volume of control to the work of the external efforts W and heat
Q received by this same system:
of
= D (E
+ E
) = W
+ Q
total
intern
kinetics

éq 1.1-1
By injecting the theorem of the kinetic energy in this equation, one reveals thus
power of the interior efforts, function of the field speed [bib1]:
!Einterne =!Q - P (U)
I
éq 1.1-2
For the resolution of the problem of thermics, the system is supposed without movement. Power
interior efforts pi (U) is thus null. Indeed, in the majority of the applications concerned, them
thermal phenomena and mechanics are uncoupled; density power density dissipated by
plastic deformations, pi = C.!plastic, is neglected in front of the heat exchanged on the surface or
other voluminal heat sources.
The equation [éq 1.1-2] which expresses the variation of heat in volume is written then:
D
S

E D = Q! = (rvol - div Q D
éq 1.1-3
dt

)
S
S
where one noted:
E
internal energy,

density,
rvol
the voluminal rate of contribution external of heat,
Q
the vector heat flow.
Moreover, since the solid is motionless, for any volume of control (
T) =, one obtains then
the local equation of conservation of heat:
of

= R - div Q
éq 1.1-4
dt
flight
If all the system is actuated by a rigid movement of body, an additional term
appears in the member of left, utilizing the speed of the solid and the gradient of energy.
This situation is treated by command THER_NON_LINE_MO [R5.02.04].
Handbook of Référence
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Code_Aster ®
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Titrate:
Nonlinear thermics
Date:
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Author (S):
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Key:
R5.02.02-A
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In the case of a reversible transformation, the equation [éq 1.1-4] becomes, with the assistance of the second
principle of the thermodynamics which makes it possible to write in our case of
= TdS:
intern
T S! = R - divq
éq 1.1-5
flight
and finally the equation of heat in its traditional form:
C T! = R - div Q
éq 1.1-6
P
flight
S
with the heat-storage capacity with constant pressure defined by: C = T
P
T
P
As it is explained in chapter 1.4, it can be advantageous to write the term of left of the equation
[éq 1.1-6] with the enthalpy which does not depend whereas temperature:
!
= R - div Q
flight
éq 1.1-7
T
where (T) =
C dT
P
T0
1.2
Fourier analysis
In thermal conduction, the Fourier analysis provides an equation connecting the heat flow to the gradient
temperature (normal vector on the isothermal surface). This law reveals, in its form
more general, a tensor of conductivity. In the case of an isotropic material, this tensor is reduced to one
coefficient (being able to depend on the temperature), thermal conductivity:
Q (X, T) = - (T) T
(X, T)
éq 1.2-1
1.3
Equation of heat in the case of the model of thermics
non-linear transient
By combining the equations [éq 1.1-5] and [éq 1.2-1], one obtains:
D
R
div ((T) T
flight -
-
) =
éq 1.3-1
dt
or, if the heat-storage capacity does not depend on the temperature:
dT
R
- div (- (T) T
) = C
flight
p
éq 1.3-2
dt
Handbook of Référence
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Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
5/16
1.4
Numerical advantage of the formulation in enthalpy for
problems with phase shift.
The relation between enthalpy and heat-storage capacity is:

T
(T) =
C (U) of

p
T0
When this function enthalpy presents abrupt variations, it is more precise to handle (T)
that its derivative. Indeed, paces characteristic of these functions in the vicinity of the melting point
are as follows:
Enthalpy
Liquid
C
H (T)
C
H
2
p
has
L
have
R
late
nte
C (T)
p
1
C (T)
p
2
Solid
Solid
H (T)
Liquid
1
Temperature
Temperature
During an iteration, either because the thermal transient is violent, or because the range of
phase shift very small (pure substance), two the is reiterated successive ones of the temperature can
to be located discontinuity on both sides. The evaluation of the slope of the function enthalpy with
vicinity of the melting point will be very false if one considers C (T
p
1), C (T
p
2) or an average
balanced of both. On the other hand, the slope of the straight line in dotted lines is always an approximation
correct of D dT at the melting point.
2
Boundary conditions, loading and initial condition
One will refer to [R5.02.01] for the boundary conditions thermal and the loadings leading
with linear equations in temperature like for the initial condition.
2.1
Non-linear normal flow
It is of the conditions of the Neumann type, defining flow entering the field.
- Q (X, T) .n = G (X, T) on the border
éq 2.1-1
where G (X, T) is a function of the temperature and possibly of the variable of space X and/or of
time T and N indicates the normal external with the border, Q is the vector heat flow (directed
according to the decreasing temperatures).
This expression makes it possible to introduce for example conditions of the type exchanges with a coefficient
of convectif exchange depend on the temperature:
- Q (X, T) .n = G (X, T) = H (X, T) (T (X, T)
ext.
- T)
éq 2.1-2
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Code_Aster ®
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Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
6/16
2.2
Non-linear normal flow - condition of the radiation type ad infinitum
A particular case of the boundary conditions preceding is the radiation ad infinitum of gray body which
results in a particular case of function G (X, T):
- Q (X, T) .n = [(T (X) +.
273)
15 4 - (
4

T +
.
273)
15]
éq 2.2-1
The characteristics to be defined at the time of the definition of this loading are emissivity, the constant of
Stefan-Boltzmann =5,73.10­8 usi and the temperature ad infinitum.
T (R) and T are then expressed in degrees Celsius. ­ 273.15°C is the temperature of the absolute zero.
3
Variational formulation of the problem
We will restrict ourselves here to present the problem with only the boundary conditions of
imposed temperature [R5.02.01 §2.1], of imposed normal flow [R5.02.01 §2.3], of exchange
[R5.02.01 §2.4], of nonlinear flow [§2.1] and radiation [§2.2].
That is to say open of R3, border = 1 2 3 4 5.
One must solve the equation [éq 1.1-4] in T on ×] 0, T [with the boundary conditions:
T
= Td (R, T)
on

1
T
(T)
= F (R, T)
on

N
2
T
(T)
= H (R, T) (T (R, T) - T

)
on
N
ext.
3
éq 3-1

T
(T)
= G (R, T)
on
N
4

T
(T)
=
4
4

[(T +.
273)
15 - (T +
.
273)
15] on

N
5
and with, possibly, of the initial conditions T (T =)
0. If these last are not specified, one
solves as a preliminary the stationary problem, i.e. the equation [éq 1.3-1] without the term of evolution
temporal.
That is to say a v sufficiently regular function cancelling itself on 1, while noticing:
D

(T) .v D
!(T) .v D

=
dt


T

éq 3-2
(T) T
.v D = - div ((T) T
) .v D + (T)
.v D





N



Handbook of Référence
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Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
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the weak formulation of the equation of heat can then be written:
D

T

(T) .v D
(T) T
. v
D - (T)
.v D = R
.v D

+


éq 3-3
dt
N
flight




One deduces the variational formulation from it from the problem:
D (T) .v D + (T) T .v d+ hT.vd



3 =
dt


3
r. v D + F .v D
éq 3-4
2 +
H T .v D



3 +
flight
ext.

2
3
g.v D
4
4
27315
27315
4 +


[.(T +. ) - (T +. )] .vd5
4
5
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Code_Aster ®
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Titrate:
Nonlinear thermics
Date:
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Author (S):
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Key:
R5.02.02-A
Page:
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4
Discretization in time of the differential equation
4.1
Introduction of - method
A traditional way to discretize a first order differential equation is it - method.
Let us consider the following differential equation:
!y (T) = (T, y (T))

éq 4.1-1
y ()

0 = y0
- Method consists in discretizing the equation [éq 4.1-1] by a diagram with the finished differences
1 (y - y) = (T, y) + 1
N 1
+
N
N 1
+
N 1
(-) (T, y)

éq 4.1-2
T
+
N
N
where yn+1 is an approximation of y (tn)
+1, yn being supposed known and is the parameter of
method, [0]
1
.
Note:
if = the 0 diagram is known as explicit,
if the 0 diagram is known as implicit.
4.2
Application to the equation of heat
Let us use it - method in the variational formulation of the equation of heat, where one posed:
T + = T (R, T + T
)
T - = T (R, T)
h+ = H (R, T + T
)
H = H (R, T)
F + = F (R, T + T
)
F - = F (R, T)
T + = T (R, T + T
) T - = T (R, T)

ext.
ext.
ext.
ext.
R + = R (R, T + T
) R - = R (R, T) T d+ = T D (R, T + T
) T D = T D (R, T
flight
flight
flight
flight
)
G + = G (R, T +)
G - = G (R, T -)
where T D (R, T) represents the temperature imposed on the border of the field, according to time and
space.
Let us introduce following spaces of functions:
V
1
D
=

()
/=
+
t+
{v H
v
T
1
}
V
1
D
=

()
/=
-
T
{v H
v
T
1
}
V
1
0 = {v H () v/= 0
1
}
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Code_Aster ®
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Titrate:
Nonlinear thermics
Date:
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Author (S):
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Key:
R5.02.02-A
Page:
9/16
The field T -
V

+
v V
T being supposed known, one seeks T
Vt+ such as 0:
(T +) - (T -) v D
+
((T +) T +

. v
+ (1 -) (T -) T -

. v
) D



T



-
(F + + (1 -) F -) v d2
-
(h+ T + + (
-
-
+
+
-
-


ext.
1 -) H Text - H T - (1 -) H T) v d3
2
3
-
(G + + (1 -) G -) v D
-
4
=
(R + + (

flight 1) rvol) v D
4

éq 4.2-1
Not to weigh down the writing excessively and insofar as the process is identical to the different one
terms, one did not make appear the term of radiation in these equations (integral on 5).
While posing:
(HT) = h+T + + (1 -) HT -
ext.
ext.
ext.
F = F + + (1 -) F -
R = R + + (1 -) R +
flight
flight
,
one obtains finally:
(T +) v D + (T+) T+

. v
D + h+ T + v D




T

3


3
- G (T +) v d4 = L (v, T -)

1
éq 4.2-2
4
v V
0
where one posed:
(T -)
L (v, T -) =
v D - 1
(-) (T -) T -

. v D + F v D
1




T
2



2

éq 4.2-3
+ ((HT) - 1
(-) HT -) v D + R v D + 1
(-) G (T -) v D
ext.
3


4

3


4

At one moment of calculation given, this term is known. Indeed, only the temperature at the previous moment,
T -, as well as the known values at the moment running of function of time, intervene.
If the distribution of temperature in the system at the initial moment is not provided, one
solves the stationary problem. The terms of evolution disappear, = 1; the field of temperature
to the initial moment is given by:
(T t=0) T t=

0. v
D + ht=0 T t=0 v D
0
3
-
G (T t=) v D



4


3

4

=
F t=0 v D
0
0
0
2
+ ht= T t= v D 3
+ rt= v D


ext.


éq 4.2-4
2

3


v V
0
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Code_Aster ®
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Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
10/16
The problem is written finally in the condensed form:
Oneself
T - - known tr
ouver
,

+
T
V
T
V
+ such as



T
T

éq 4.2-5
v V

has (v, T +) = L (v, T


-
0
1
)
5
Space discretization and adaptation of the algorithm of
Newton with the problem
The principle of the method of Newton is very detailed in [R5.03.01], one will expose only them here
adaptations specific to the nonlinear algorithm of thermics.
5.1 Discretization
space
That is to say pH a division space, let us indicate by NR the number of nodes of the grid, pi
function of form associated with node I. One indicates by J the whole of the nodes belonging to
border 1.
Are:
V H
D
=
=
()
;
=
+
(,)

t+
{v
v p X
v
T
X T
J
J
I
I
J
J
}
i=,
1 NR
V H
D
=
=
()
;
=
-
(,)

éq 5.1-1
T
{v
v p X
v
T
X T
J
J
I
I
J
J
}
i=,
1 NR
V H
0 = {v = v p (X)
;
v = 0
J J
I
I
J
}
i=,
1 NR
The problem [éq 4.2-5] can be replaced by the problem discretized with a finished number of unknown factors
according to:
Oneself
T -
H
- known tr
ouver
,

+
T
V
T
V H
+ such as



T
T

éq 5.1-2
v
V H

has (v, T +
0
) = L (v, T


-
H
H
1
H
)
that one can as write, with the same formalism as STAT_NON_LINE [R5.03.01], in form
vectorial:

vT R (T+, t+) = vT L (T, t+) v
such as B
v = 0

éq 5.1-3
B T+


= Td (T +)
where the operator B expresses the boundary condition of imposed temperature T +
V H
t+. It is defined by:
0 if J J
(Bv) J = v

éq 5.1-4
J
if
J J

The case where the application R is linear is treated by command THER_LINEAIRE [R5.02.01].
Handbook of Référence
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Titrate:
Nonlinear thermics
Date:
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Author (S):
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Key:
R5.02.02-A
Page:
11/16
The dualisation of the boundary conditions, detailed in [R3.03.01], led to the nonlinear problem
in T+:
R

(T+, t+) + BT +
= L (T, t+)

éq 5.1-5
B T+
Td

+
=


(T)
The unknown factors are the couple (T+, +
), where + represents the “multipliers of Lagrange” of
boundary conditions of Dirichlet.
To solve the system [éq 5.1-5] amounts cancelling in (T+, +
)
+
+
I
I
the vector F (T,), called residue,
defined by:
L (T, +
T) - R (T+, +
T) - BT +

F (T+, +
) =


Td
éq 5.1-6
(+
T) - B T+

The method of Newton consists in building a vector series {xn} converging towards the solution
N
of F (X) = 0 using the tangent linear application of F.
5.2 Calculation
stationary
The variational problem is that of the equation [éq 4.2-4]. To note: in stationary calculation, enthalpy
does not intervene in application R.
One introduces the matrix of the tangent linear application of the function R (Tn):
R
K N = T Tn
That of the function F (Tn, N
) is then:
K N

BT


B
0



In the case of stationary calculation, one must reiterate starting from a uniform value of initialization of the field
of temperature; in fact T0 = 0 in any node. The first iteration of calculation, known as iteration
of prediction, consists in solving the following system:
K
(T) BT T
- H T - R (T) - BT
0
1
0
0
0


=

éq 5.2-1
B
0
D
1 - 0


T

- BT



0


As one can see it in the equation of the stationary problem [éq 4.2-4], the temperature does not appear
not with the second member: are written L and not L (T)
0.
If the problem is linear, R (T) = K (T) T = K .T
0
0
0
0. All the terms in T0 disappear by
simplification. The solution is obtained in an iteration by inversion of a system identical to that
described in [R5.02.01 §6].
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
12/16
The following iterations are iterations of Newton, with reactualization or not of the matrix
tangent K.
K
(T) BT T
+ - T
T
I 1
I
L - R (T) - B
(I)
I
I


=

éq 5.2-2
B
0 I
1
+ - I

0





For the iteration of prediction, the writing of the lower subsystem of the equation [éq 5.2-1], afterwards
simplification, ensures us that B T
T
1 =
D. The reiterated first and all the following thus check them
conditions of Dirichlet.
Brackets around the index of iteration in the expression K (T)
(I) mean that one can
to reactualize or not the tangent matrix with the wire of the iterations.
Note:
The temperature of T0 initialization has of influence only for one nonlinear stationary calculation. In
being of about size of the awaited temperatures, it would make it possible “to leave” less
far from the solution that a null field everywhere; and thus the iteration count would decrease. Today,
it is not possible to enter a value of T0. The vector temperature is initialized, into hard, to zero.
5.3 Calculation
transient
For the first iteration of the step of time, known as iteration of prediction, one “makes like if” it
problem describes in [éq 5.1-5] was linear. This formulation must make it possible to obtain directly
solution to a linear problem of thermics. But here, the situation is a little different from calculation
stationary because of the formulation in enthalpy. The linearization of [éq 5.1-5] gives:
R

(T, t+) + K (T, t+) (T+ - T) + BT +
= L (T, t+)

éq 5.3-1
B T+
Td +


=
(T)
What amounts solving, for the problem presented in matric form:
K
(T) BT T+
1
L
(T, +
T) + K (T) T - R (T)

=
D

B
+
0

éq 5.3-2
T (+
T)

1

The function enthalpy is known with a constant of integration close which appears in the relation flexible
R (T)
-
-
-
+
with K (T) T. This same constant is found in the expression of L (T, T). One can
then to eliminate it while leading to the system from equations according to:
K
-
T
+
~
(T) B T1
L
(T, +
T)

= D

B
+
0

éq 5.3-3
T (+
T)

1

~
where L (T, T +) is the second member calculated with the heat-storage capacity and not the enthalpy (option
CHAR_THER_EVOLNI [§6.2]).
Lastly, as for the stationary case seen in the preceding chapter, the following iterations are
iterations of Newton:
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Titrate:
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Date:
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Author (S):
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K (T, T +) BT T+
1 - T+ L (T, t+) - R (T, T +

) - BT
(I)
i+
I
I
I


=

éq 5.3-4

B
+
+

0 i+1 - I
0


This time, on the other hand, L (T, T +) is calculated with the enthalpy and not the heat-storage capacity to be
coherent with R (T+)
I
.
5.4 Convergence
Since time intervenes in the form of the tangent matrix, and also the step of time, one
prefer systematically to bring up to date this one at the beginning of each step not not to degrade too much
speed of convergence. On the other hand, freedom is left with the user control his frequency of
calculation during a step of time.
With each iteration, one can carry out the search for an optimum step of progression towards the solution by
some iterations (2 or 3) of linear search. This method is described in detail in [R5.03.01].
Calculation famous is converged when the vector residue is null [éq 5.1-6]:
L (T, T +) - R (T+ +
T +

I, T
) - B
I
F (T+ + +



I,
I, T
) =
éq 5.4-1
Td (T +)

- B T+

I

The lower part of the vector is always null (conditions of Dirichlet). One thus checks:
L (T, +
T) - R (T+
T
I, +
T) - B
+
I 2
éq 5.4-2
L (T, +
T) - BT +
I 2
The user also has the possibility of stopping the iterations on an absolute criterion:
L (T, +
T) - R (T+
T
I, +
T) - B
+i


éq 5.4-3

Handbook of Référence
R5.02 booklet: Thermics
HI-75/99/013 - Ind A

Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
14/16
6
Principal calculated options of non-linear thermics
in Code_Aster
6.1
Boundary conditions and loadings
One will refer to [R5.02.01] for the boundary conditions and the loadings linear.
(1 -) (-)
Flow not
G T v D
CHAR_THER_FLUNL
4
linear
4


[(T +.
273)
15 4 - (
-
4

1 -) (T +
.
273)
15]
Radiation
v D
CHAR_THER_RAYO_R
4
CHAR_THER_RAYO_F
4

6.2
Calculation of the elementary matrices and transitory term
Inertia
CP
thermics,
MTAN_RIGI_MASS
v.v D +
(T) v. v
D


+
conductivity
T



4 +
Radiation
(T + 27315 3
. ) v.v D


MTAN_THER_RAYO_R
4
MTAN_THER_RAYO_F
4

hv.v D
Coefficient

MTAN_THER_COEF_R
4
of exchange
MTAN_THER_COEF_F
4
dg
Flow not
MTAN_THER_FLUXNL
-
+
(T) v.v d4
linear
dT
4
Term
(T -)
transient
CHAR_THER_EVOLNI
v D - (1 -) (T -) T -. v D



T



CpT - v D - (1) (T) T
. v
D



T



6.3
Calculation of the residue
1
I
I
I
RESI_RIGI_MASS
(T) v D + ((T)) T. v
D


T


I
Radiation
(T +
. ) v D

27315 4

RESI_THER_RAYO_R
4
RESI_THER_RAYO_F
4

(h+ Ti) v D
Coefficient


RESI_THER_COEF_R
3
of exchange
RESI_THER_COEF_F
3
Flow not
RESI_THER_FLUXNL
-
I
(
)

linear
G T v D 3
3
Handbook of Référence
R5.02 booklet: Thermics
HI-75/99/013 - Ind A

Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
15/16
7 Bibliography
[1]
SALENCON. Mechanics of the continuous mediums. Ellipses. 1988.
[2]
RUUP, PENIGUEL. SYRTHES - Conduction and radiation, theoretical Manuel of the version
3.1. HE-41/98/048/A
Handbook of Référence
R5.02 booklet: Thermics
HI-75/99/013 - Ind A

Code_Aster ®
Version
5.0
Titrate:
Nonlinear thermics
Date:
22/06/00
Author (S):
C. DURAND
Key:
R5.02.02-A
Page:
16/16
Intentionally white left page.
Handbook of Référence
R5.02 booklet: Thermics
HI-75/99/013 - Ind A

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