Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
1/12
Organization (S): EDF/IMA/MN
Handbook of Référence
R5.02 booklet: Thermics
Document: R5.02.01
Algorithm of linear thermics transitory
Summary:
One presents the algorithm of transitory thermics linear established within command THER_LINEAIRE
[U4.33.01]. The various options of calculation necessary were presented in the elements of structure
plans, axisymmetric and three-dimensional [U1.22.01], [U1.23.01] and [U1.24.01].
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
2/12
Contents
1 Expression of the equation of heat in linear thermics ................................................................ 3
1.1 Equation of heat ..................................................................................................................... 3
1.2 Fourier analysis .............................................................................................................................. 3
1.3 Equation of heat in the case of the linear model of thermics ............................................. 3
2 Boundary conditions, loading and initial condition ........................................................................ 4
2.1 Imposed temperatures ................................................................................................................. 4
2.2 Linear relations ........................................................................................................................... 4
2.3 Normal flow imposed ......................................................................................................................... 4
2.4 Exchange .......................................................................................................................................... 5
2.5 Exchange wall ................................................................................................................................. 5
2.6 Voluminal source ............................................................................................................................ 5
2.7 Initial condition .............................................................................................................................. 5
3 variational Formulation of the problem ................................................................................................. 6
4 variational Formulation of the problem with condition of exchange between two walls .......................... 6
5 Discretization in time of the differential equation ................................................................................ 7
5.1.1 Precision of the method ......................................................................................................... 7
5.1.2 Stability of the method ........................................................................................................... 8
5.1.3 Application to the equation of heat ..................................................................................... 9
6 space Discretization .......................................................................................................................... 10
7 Implementation in Code_Aster .................................................................................................... 11
7.1 Introduced equations ..................................................................................................................... 11
7.2 Principal thermal options calculated in Code_Aster ..................................................... 12
7.2.1 Boundary conditions and loadings ................................................................................ 12
7.2.2 Calculation of the elementary matrices and transitory term ......................................................... 12
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
3/12
1
Expression of the equation of heat in linear thermics
1.1
Equation of heat
One places oneself in open of ¤3 of regular border.
In any point of, the equation of heat can be written:
T (R, T)
- div ((
Q R, T)) + S (R, T) = C
p T
with:
Q
vector heat flow (directed according to the decreasing temperatures),
S
heat per unit of volume dissipated by the internal sources,
C
voluminal heat with constant pressure,
p
T
temperature,
R
variable of space,
T
variable time.
This equation translates the phenomenon of change of the temperature (only through
phenomenon of diffusion, convection having been neglected) in any point of opened and at any moment. It
in theory an infinity of solutions admits, but the data of the initial conditions and variation of
boundary conditions in the course of time determines the evolution of the phenomenon perfectly.
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
simple coefficient, the thermal coefficient of conductivity.
Q (R, T) = - T (R, T)
For the elements of anisotropic thermics one will refer to Implantation of the elements 2D and 2D
Axisymmetric in mechanics and thermics [R3.06.02].
1.3
Equation of heat in the case of the linear model of thermics
By combining the two equations above, one obtains:
T (R, T)
- div (- T (R, T)) + S (R, T) = CP T
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
4/12
2
Boundary conditions, loading and initial condition
One describes here only the boundary conditions thermal leading to linear equations in
temperature, which excludes the conditions of the radiation type.
2.1 Temperatures
imposed
The conditions of the Dirichlet type, are usually treated by dualisation in Code_Aster
(cf [R3.03.01]), but they can also be eliminated in certain cases (loads kinematics).
T (R, T) = T (R, T)
1
on
1

where T R T
1 (,) is a function of the variable of space and/or time.
2.2 Relations
linear
It is of the conditions of the Dirichlet type, making it possible to define a linear relation between the values of
the temperature:
· between two or several nodes: with an equation of the form
N
I iT (R, T) = (T)
i=1
· between couples of nodes: with an equation of the form
1
N
2
N
1i iT (R, T) + 2i iT (R, T) = (T)
/
/
I 1
12
=
I 1
21
=
where 12 and 21 are two under-parts of the border which one binds two to two the values of
temperature. This type of boundary condition makes it possible to define conditions of periodicity.
2.3
Imposed normal flow
It is of the conditions of the Neumann type, defining flow entering the field.
- Q (R, T) .n = F (R, T)
on 2
where F (R, T) is a function of the variable of space and/or time and N the normal indicates with
border 2.
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
5/12
2.4 Exchange
It is of the conditions of the Neumann type modelling the convectifs transfers on the edges of
field.
- Q (R, T) .n = H (R, T) (T (R, T)
ext.
- T (R, T))
on 3
where T
R T
ext. (,) is a function of the variable of space and/or time representing the temperature of
external medium, and H (R, T) is a function of the variable of space and/or time representing it
coefficient of convectif exchange on border 3.
2.5 Exchange
wall
It is of the conditions of the Neumann type bringing into play two pennies left the border in opposite.
This type of boundary condition models a thermal resistance of interface.
T

1

= H (R, T) (T (R, T) - T (R, T)) on
N
N
2
1
12
1 normal external to 12
1
T

2

= H (R, T) (T (R, T) - T (R, T)) on
N
N
1
2
21
2 normal external to 21
2
(N = - N
1
2 in general)
2.6 Source
voluminal
It is the term S (R, T) function of the variable of space and/or time.
2.7 Condition
initial
It is the expression of the field of temperature at the initial moment T = 0:
T (R,)
0 = T (R)
0
where T R
0 () are a function of the variable of space.
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
6/12
3
Variational formulation of the problem
We will restrict ourselves here to present the problem with only the boundary conditions of
imposed temperature [§2.1], of imposed normal flow [§2.3] or of exchange [§2.4]. Conditions with
limits of exchange wall [§2.5] are treated with [the §4] and those with linear relations [§2.2] are brought back
without difficulties with that of [§2.1].
That is to say open of ¤3, border = 1 2 3.
The weak formulation of the equation of heat is:
T
T
C
.v D + T
. v
D -
.
=
.



v D
S v D
p



T
N




where v are a sufficiently regular function cancelling itself uniformly on 1. With the conditions with
following limits:
T
= T (R, T)
on

1
1
T

= Q (R, T)
on
N
2
T

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



on
N
ext.
3
The variational formulation of the problem is:
T
C
.v D +

T
. v
D + hT.v D = S. v D + q.v D + HT .v D
p







T
ext.


3

2
3
4
Variational formulation of the problem with condition
of exchange between two walls
One considers the “simplified” problem where does not appear any more source term and where boundary conditions
are only of imposed the temperature type and exchanges wall.
That is to say open of ¤3, border = 1 12 21.
The boundary conditions are in this case:

T
= T (R, T

1
)
on 1

T

1

= H (R, T) (T (R, T
2
) - T (R, T))
on
N
1
12

1
T

2

= H (R, T) (T (R, T
1
) - T (R, T

))
on
N

2
21
2
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
7/12
In substituent in the weak formulation of the equation of heat, one obtains:
T
C
.v D
p

+

T
. v
D

T


+
H (T/
- T) .v D
/
12 + H (T/- T) .v D





/
0
12
21
21

21 =
12
12
21
where v are cancelled uniformly on 1.
This type of boundary conditions reveals new terms connecting degrees
of freedom located on the two borders in relation.
5
Discretization in time of the differential equation
A traditional way to discretize a first order differential equation consists in using one
- method. Let us consider the following differential equation:


y (T) =

(T, y (T))
T y () = y


0
0
- Method consists in discretizing the equation by a diagram with the finished differences
1 (y - y) = (T, y) + 1
N 1
+
N
N 1
+
N 1
(-) (T, y)
T
+
N
N
where yn+1 is an approximation of y (tn)
+1, yn being supposed known
and is the parameter of the method, [
0]
1
.
Note:
if = the 0 diagram is known as explicit,
if the 0 diagram is known as implicit.
5.1.1 Precision of the method
Let us suppose there sufficiently regular (at least 3 times differentiable), by a development of Taylor
at the point tn one obtains:
t2

y (T
) y (T)
T
y' (T)
y '' (T) O (t2
N 1
N
N
N
)
+ -
=
+
+
2
and
(T, y (T))
1
1
+ (
n+
n+
1 -) (T, y (T))
N
N
= y' (T)
1 + (
n+
1) y' (T)
N
=
y' (T
)
n+1 + (y' (T
)
1 - y' (T))
n+
N
=
y' (T)
2
N +
T
y '' (T) + O (T
)
N
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
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Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
8/12
The solution thus checks roughly:
1 (
2
y (tn 1) - y (T
+
N))
1
= (T, y (T)) + 1
N 1
+
N 1
(-) (T, y (T
+
N
N)) + (
-) T y '' (T) + O
N
(T)
T
2
1
1
The diagram is of command 1 in time if, and of command 2 if = (diagram of Crank-Nicolson).
2
2
5.1.2 Stability of the method
Let us consider the following differential equation:
y' = - y

T 0 R
y () 0 = y

0
By using it - method in this differential equation one obtains:
1 - 1
(-) T

y
y
1
0 N NR
n+ =
- 1
1+ T
N

That is to say still:
1 - 1
(-) X
y
RN (T
) y
with R (X
n+1 =
0
) =
1+ X

The approximate solution yn must be limited (the exact solution of the initial problem being it), which imposes
the following condition:
R (T
) 1
By studying the variations of the function R (X), one notes easily that:
· if
12 the condition is checked whatever the T, the diagram is unconditionally
stable;
2
· if
< 12 the condition is checked that if T
, the diagram is conditionally
1
(- 2)
stable.
In command THER_LINEAIRE [U4.33.01], the parameter is a data being able to be provided
by the user, the default value is fixed at 0.57. This value with the reputation in Département MMN
to be preferable with the value of Crank-Nicolson (0,5) and “optimal” for the quadratic interpolations,
but we did not find trace of the justifications.
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
9/12
5.1.3 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.
)
s+ = S (R, T + T
)
S = S (R, T)
T +
1 = T (R, T
1
+ T
)
T -
1 = T (R, T)
1
Let us introduce following spaces of functions:
V =
1 ()
/=
+
(,)
t+
{v H
v
T R T
1
1
}
V =
1 ()
/=
-
(,)
T
{v H
v
T R T
1
1
}
V
1
0 = {v H () v/= 0
1
}
The field T -
V

+
T being supposed known, one seeks T
Vt+:
T + - T -
C
v D
+
-
p

+
(T

. v + (1) T

. v
) D


T



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


ext. + (1 -) H Text - H T
- (1) H T) v d3
2
3
=
(s+ + (1) S) v D



v V
0
While posing:
(HT) = h+T + + (1) HT -
ext.
ext.
ext.
F = F + + (1) F -
one obtains finally:
CP T+ v D + T+

. v
D + h+ T+ v D



T

3


3
CP
=
T - v D - (1)
-

T. v D + F v D



T

2


2

+ ((HT
- -
+
-
ext.) - (1) H T) v d3 + (S + (1) S) v D

3

v V
0
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
10/12
6 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 =
=
()
;
=
+
(,)
1

t+
{v
v p X
v
T X T
J J
I
I
J
J
}
i=,
1 NR
V H =
=
()
;
=
-
(,)
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
Let us pose:
C

K T
p
=
T p p D
+ T p. p

D + h+ T p D
ij I




T I I J
H
I
I
J
H
I
I
H

3
H
H
H
3
C

L
p
=
T - p D - (1) T
-. p

D +
F p D
J




T
J
H
J
H
J
H

2
H
H
H
2
+ ((HT
) - (1) HT -) p D 3 +

(s+ + (1) S) p D
ext.
J
H

J
H
H
3
H
By dualisant the boundary conditions in imposed temperature ([R3.03.01]), one reveals
the operator B defined by:
0 if J J
(Bv) J = v

if
J J
J

One obtains finally the following system:
NR

K T
T
ij I
+ (B) = L
J

J
J
i=1

(B T) = T (X, T)
J
1
J
J
J

Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
11/12
7
Implementation in Code_Aster
7.1 Equations
introduced
Command THER_LINEAIRE [U4.33.01] makes it possible to treat the equation in the transitory case such
that it is described above, but it also makes it possible to solve the stationary problem which is reduced
with the following equation:
- div (T) = S
in
and boundary conditions following:
T
= T (R, T
1
S)
on

1
T

= Q (R, T)
on
N
S
2
T

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



on
N
ext.
S
3
ts being the moment taken to evaluate the boundary conditions of the equation.
In the transitory case, it is necessary to provide an initial state, this initial state (field of temperature)
can be selected among the following:
· a field which can uniform or unspecified be created by command AFFE_CHAM_NO,
· a field result of a stationary problem describes by the equations above, the moment of
calculation is taken at the first moment defined in the list of realities describing the discretization
temporal defined by the user,
· a field extracted the result of a transitory problem.
The discretization in time (value of T) must be provided in the shape of one or more lists
moments. These lists are created by the user by command DEFI_LIST_REEL [U4.21.04].
A thermal transient can be calculated by carrying out several calls to the command
THER_LINEAIRE [U4.33.01] by enriching the same concept of the evol_ther type while providing to
to leave the second call the initial moment of resumption of calculation (to obtain T -) and possibly
the final moment.
The fields of temperatures resulting from a calculation contain at the same time the value with the nodes of the grid and
with the nodes of Lagrange. During a resumption of calculation, it is possible to vary the type of
boundary conditions, the field used to initiate new in-house calculation is then tiny room to
only nodes of the grid. The concept result of the evol_ther type will contain fields then with
nodes being based on different classifications. The operators of Code_Aster interpolate
then only with the nodes of the grid when classification differs.
Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

Code_Aster ®
Version
3
Titrate:
Algorithm of linear thermics transitory
Date:
04/05/95
Author (S):
J.P. LEFEBVRE
Key:
R5.02.01-A
Page:
12/12
7.2
Principal thermal options calculated in Code_Aster
7.2.1 Boundary conditions and loadings
TEMP_IMPO
DDLI_R
T + * D

1 +
v D



DDLI_F
1
1
1
DDLI_R
* T D
1
DDLI_F
1
1
FLUX_REP
CHAR_THER_FLUN_R
qv D

CHAR_THER_FLUN_F
2
2
ECHANGE
CHAR_THER_COEF_R
h+ T+ v D


CHAR_THER_COEF_F
3
3
CHAR_THER_TEXT_R
((HT) - (-) H T -) v D

1

CHAR_THER_TEXT_F
ext.
3
3

ECHANGE_PAROI
RIGI_THER_PARO_R
h+ T+
T +
/-
v D
(
) 1
RIGI_THER_PARO_F
/
12
21
12
12
CHAR_THER_PARO_R
(1 -) - (- - -)
H T
T
v D
CHAR_THER_PARO_F
/
/
21
12

1
12
12

SOURCE
CHAR_THER_SOUR_R
(s+ (
) S
+ -
) v D

1

CHAR_THER_SOUR_F

7.2.2 Calculation of the elementary matrices and transitory term
RIGI_THER
+.

T
v D

MASS_THER
CP T v D

+
T


CHAR_THER_EVOL
CP T v D - (1) T
. v
D


T



Handbook of Référence
R5.02 booklet: Thermics
HI-75/95/020/A

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