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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R5.02 booklet: Thermics
Document: R5.02.01
Algorithm of linear thermics transitory
Summary:
One presents the algorithm of transitory thermics linear established within the control
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].
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
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 Main 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
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
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:
-
+
=
div ((,)) S (,)
(,)
Q R T
R T
C
T R T
T
p
with:
Q
vector heat transfer rate (directed according to the decreasing temperatures),
S
heat per unit of volume dissipated by the internal sources,
C
p
voluminal heat with constant pressure,
T
temperature,
R
variable of space,
T
variable time.
This equation translates the phenomenon of change of the temperature (only through
phenomenon of dissemination, 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 transfer rate 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:
-
-
+
=
div (
(,)) S (,)
(,)
T R T
R T
C
T R T
T
p
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
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
1
on
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
I I
I
N
T R T
T
(,)
()
=
=
1
·
between couples of nodes: with an equation of the form
1
1
2
1
12
1
21
2
I I
I
N
I I
I
N
T
R T
T
R T
T
/
/
(,)
(,)
()
=
=
+
=
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 interval.
2.3
Imposed normal flow
It is of the conditions of the Neumann type, defining flow entering the field.
-
=
Q
N
(,).
(,)
R T
F R T
on
2
where
F R T
(,)
is a function of the variable of space and/or time and
N
indicate the normal with
border
2
.
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
2.4 Exchange
It is of the conditions of the Neumann type modelizing the convectifs transfers on the edges of
field.
-
=
-
Q
N
(,).
(,) (
(,)
(,))
R T
H R T
T
R T
T R T
ext.
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 the 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 modelizes a thermal resistance of interface.
T
N
H R T
T R T
T R T
1
1
2
1
12
=
-
(,) ((,)
(,))
on
N
1
normal external with
12
T
N
H R T
T R T
T R T
2
2
1
2
21
=
-
(,) ((,)
(,))
on
N
2
normal external with
21
(
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
T R
(,)
()
0
0
=
where
T R
0
()
is a function of the variable of space.
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
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
, of border
=
1
2
3
.
The weak formulation of the equation of heat is:
C
T
T v D
T
v D
T
N v D
S v D
p
+
-
=
.
.
.
.
where
v
is a sufficiently regular function cancelling itself uniformly on
1
. With the conditions with
following limits:
T T R T
T
N
Q R T
T
N
H R T T
R T
T
ext.
=
=
=
-
1
1
2
3
(,)
(,)
(,) (
(,)
)
on
on
on
The variational formulation of the problem is:
C
T
T v D
T
v D
H T v D
S v D
Q v D
H T
v D
p
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
, of border
=
1
12
21
.
The boundary conditions are in this case:
T T R T
T
N
H R T
T R T
T R T
T
N
H R T
T R T
T R T
=
=
-
=
-
1
1
1
1
2
1
12
2
2
1
2
21
(,)
(,) ((,)
(,))
(,) ((,)
(,))
on
on
on
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
In substituent in the weak formulation of the equation of heat, one obtains:
C
T
T v D
T
v D
H T
T
v D
H T
T
v D
p
+
+
-
+
-
=
.
.
(
).
(
).
/
/
/
/
12
21
12
21
12
21
12
21
0
where
v
cancel yourself 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 conventional way to discretize a first order differential equation consists in using one
- method. Let us consider the following differential equation:
T y T
T y T
y
y
()
(, ())
()
=
=
0
0
- method consists in discretizing the equation by a diagram with the finished differences
1
1
1
1
1
T y
y
T
y
T y
N
N
N
N
N
N
(
)
(
,
) (
) (,
)
+
+
+
-
=
+ -
where
y
N
+
1
is an approximation of
y T
N
(
)
+
1
,
y
N
being supposed known
and
is the parameter of the method,
[]
0 1
,
.
Note:
if
=
0
the diagram is known as explicit,
if
0
the diagram is known as implicit.
5.1.1 Precision of the method
Let us suppose
y
sufficient regular (at least 3 times differentiable), by a development of Taylor
at the point
T
N
one obtains:
y T
y T
T y T
T y T O T
N
N
N
N
(
)
()
'()
'' ()
(
)
+
-
=
+
+
1
2
2
2
and
(
, (
)) (
) (, ())
'(
) (
) '()
'(
)
('(
)
'())
'()
'' ()
(
)
T
y T
T y T
y T
y T
y T
y T
y T
y T
T y T
O T
N
N
N
N
N
N
N
N
N
N
N
+
+
+
+
+
+ -
=
+ -
=
+
-
=
+
+
1
1
1
1
1
2
1
1
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:
8/12
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
The solution thus checks roughly:
(
)
1
1
1
2
1
1
1
2
T y T
y T
T
y T
T y T
T y T
O T
N
N
N
N
N
N
N
(
)
()
(
, (
)) (
) (, ()) (
)
'' ()
(
)
+
+
+
-
=
+ -
+
-
+
The diagram is of command 1 in time if
1
2
, and of command 2 if
=
1
2
(diagram of Crank-Nicolson).
5.1.2 Stability of the method
Let us consider the following differential equation:
y
y T
R
y
y
'
()
= -
=
0
0
0
While using
- method in this differential equation one obtains:
y
T
T
y
N NR
N
N
+
= - -
+
-
1
1 1
1
0
1
(
)
That is to say still:
y
R
T y
R X
X
X
N
N
+
=
= - -
+
1
0
1 1
1
(
)
()
(
)
with
The approximate solution
y
N
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
()
, it is noted easily that:
·
if
12
the condition is checked whatever
T
, the diagram is unconditionally
stable;
·
if
<
12
the condition is checked that if
T
-
2
1 2
(
)
, the diagram is conditionally
stable.
In the control
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 at the Department MN
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.
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
5.1.3 Application to the equation of heat
Let us use
- 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
S
S R T
T
S
S R T
T
T R T
T
T
T R T
ext.
ext.
ext.
ext.
+
-
+
-
+
-
+
-
+
-
+
-
=
+
=
=
+
=
=
+
=
=
+
=
=
+
=
=
+
=
(,
)
(,)
(,
)
(,)
(,
)
(,)
(,
)
(,)
(,
)
(,)
(,
)
(,)
1
1
1
1
Let us introduce following spaces of functions:
{
}
{
}
{
}
V
v H
v
T R T
V
v H
v
T R T
V
v H
v
T
T
+
-
=
=
=
=
=
=
+
-
1
1
1
1
0
1
1
1
1
0
()
(,)
()
(,)
()
/
/
/
The field
T
V
T
-
-
being supposed known, one seeks
T
V
T
+
+
:
C T
T
T
v D
T
v
T
v D
F
F
v D
H T
H T
H T
H T
v D
S
S
v D
v V
p
ext.
ext.
+
-
+
-
+
-
+
+
-
-
+
+
-
-
+
-
-
+
+ -
-
+ -
-
+ -
-
- -
=
+ -
(
.
(
)
.
)
(
(
)
)
(
(
)
(
)
)
(
(
)
)
1
1
1
1
1
2
3
0
2
3
While posing:
(
)
(
)
(
)
HT
H T
H T
F
F
F
ext.
ext.
ext.
=
+ -
=
+ -
+ +
- -
+
-
1
1
one obtains finally:
C
T T v D
T
v D
H T v D
C
T T v D
T
v D
F v D
HT
H T
v D
S
S
v D
v V
p
p
ext.
+
+
+
+
-
-
- -
+
-
+
+
=
-
-
+
+
- -
+
+ -
.
(
)
.
((
)
(
)
)
(
(
)
)
3
2
3
0
3
2
3
1
1
1
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
6 Discretization
space
That is to say
P
H
a space division
, let us indicate by
NR
the number of nodes of the mesh,
p
I
function of form associated with the node
I
. One indicates by
J
the whole of the nodes belonging to
border
1
.
Are:
{
}
{
}
{
}
V
v
v p X
v
T X T
J J
V
v
v p X
v
T X T
J J
V
v
v p X
v
J J
T
H
I
I
I
NR
J
J
T
H
I
I
I
NR
J
J
H
I
I
I
NR
J
+
-
=
=
=
=
=
=
=
=
=
=
+
=
-
=
()
;
(,)
()
;
(,)
()
;
,
,
,
1
1
1
1
0
1
0
Let us pose:
K T
C
T T p p D
T p
p D
H T p D
L
C
T T p D
T
p D
F p D
HT
H T
p D
S
S
p D
ij I
p
I I J
H
I
I
J
H
I
I
H
J
p
J
H
J
H
J
H
ext.
J
H
J
H
H
H
H
H
H
H
=
+
+
=
-
-
+
+
- -
+
+ -
+
-
-
- -
+
-
.
(
)
.
((
)
(
)
)
(
(
)
)
3
2
3
3
2
3
1
1
1
H
H
By dualisant the boundary conditions in imposed temperature ([R3.03.01]), one reveals
the operator
B
defined by:
(
)
Bv
if
J J
v
if
J J
J
J
=
0
One obtains finally the following system:
()
()
K T
B
L
J
B T
T X T
J J
ij I
I
NR
T
J
J
J
J
=
+
=
=
1
1
(,)
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
7
Implementation in Code_Aster
7.1 Equations
introduced
The control
THER_LINEAIRE
[U4.33.01] the equation in the transitory case such makes it possible to treat
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
T
N
Q R T
T
N
H R T T
R T
T
S
S
ext.
S
=
=
=
-
1
1
2
3
(,)
(,)
(,) (
(,)
)
on
on
on
T
S
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 the control
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.
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 the control
DEFI_LIST_REEL
[U4.21.04].
A thermal transient can be calculated by carrying out several calls to the control
THER_LINEAIRE
[U4.33.01] by enriching the same concept of the type
evol_ther
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 mesh 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 mesh. The concept result of the type
evol_ther
will contain fields then with
nodes being based on different classifications. The operators of Code_Aster interpolate
then only with the nodes of the mesh when classification differs.
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
Manual of Reference
R5.02 booklet: Thermics
HI-75/95/020/A
7.2
Main thermal options calculated in Code_Aster
7.2.1 Boundary conditions and loadings
TEMP_IMPO
DDLI_R
DDLI_F
T
D
v D
+
+
*
1
1
1
1
DDLI_R
DDLI_F
*
T D
1
1
1
FLUX_REP
CHAR_THER_FLUN_R
CHAR_THER_FLUN_F
Q v D
2
2
EXCHANGE
CHAR_THER_COEF_R
CHAR_THER_COEF_F
H T v D
+
+
3
3
CHAR_THER_TEXT_R
CHAR_THER_TEXT_F
((
)
(
)
)
HT
H T
v D
ext.
- -
-
-
1
3
3
ECHANGE_PAROI
RIGI_THER_PARO_R
RIGI_THER_PARO_F
H T
T
v D
+
+
+
-
(
)
/
/
12
21
12
1
12
CHAR_THER_PARO_R
CHAR_THER_PARO_F
(
)
(
)
/
/
1
21
12
12
1
12
-
-
-
-
-
H T
T
v D
SOURCE
CHAR_THER_SOUR_R
CHAR_THER_SOUR_F
(
(
)
)
S
S
v D
+
-
+ -
1
7.2.2 Calculation of the elementary matrices and transitory term
RIGI_THER
+
T
v D
.
MASS_THER
C
T T v D
p
+
CHAR_THER_EVOL
C
T T v D
T
v D
p
-
-
-
-
(
)
.
1