Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
Document: R3.06.02
Linear modeling of the elements of medium
continuous in thermics
Summary:
One describes the expression of the elementary terms intervening in the linear modeling of the equation of
heat and in postprocessings. One gives the mathematical expression of the integral to be evaluated, and for
each element one provides the number of points of integration used.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
2/8
Contents
1 options of linear modeling ...................................................................................................... 3
2 Expression of the elementary terms for the various options of calculation ......................................... 4
2.1 General notations ......................................................................................................................... 4
2.2 Elementary terms contributing a share ............................................................................ 4
2.2.1 Thermal rigidity .................................................................................................................. 4
2.2.2 Mass thermal .................................................................................................................... 5
2.2.3 Rigidity due to the boundary conditions of exchange ................................................................. 5
2.2.4 Rigidity due to the conditions of exchange between walls ............................................................... 6
2.3 Elementary terms contributing a share to the second member ............................................ 6
2.3.1 Discretization in time .......................................................................................................... 6
2.3.2 Voluminal term of source ................................................................................................... 7
2.3.3 Term of convectif exchange .................................................................................................... 7
2.3.4 Term of normal flow imposed ................................................................................................ 7
2.3.5 Term of exchange between walls ............................................................................................... 7
3 Bibliography .......................................................................................................................................... 8
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
3/8
1
Options of linear modeling
In this document, one considers only the linear modeling of the physical phenomenon of evolution
temperature in a continuous medium. All coefficients intervening in the equation of
heat will be constants or many functions being able to depend on time or space.
boundary conditions could be only linear functions of the temperature.
By defect the material is supposed to be isotropic, the Fourier analysis connecting the heat flow to the gradient of
temperature utilizes a scalar coefficient thermal conductivity:
Q = - T
In the general case, unspecified medium, this relation is expressed with a tensor of conductivity
thermics. The definite associated matrix being positive, it is always possible to be brought back to one
stamp diagonal in the reference mark associated with the clean directions. Processing of the anisotropy
thermics (see [bib 1] is thus carried out in Code_Aster by providing the values of conductivity
thermics for each principal direction and the clean reference mark. The evaluation of the elementary terms
be carried out then by recovering the various coefficients and while changing reference mark. Two types
of anisotropy are treated in Code_Aster, it acts:
· Cartesian anisotropy where the privileged directions remain fixed in a reference mark
Cartesian, the data of the three nautical angles, and makes it possible to pass from the reference mark
total with the principal reference mark of anisotropy,
· cylindrical anisotropy where the privileged directions remain fixed in a reference mark
cylindrical, the data of the two nautical angles and defining the direction of the axis and
of the three punctual coordinates of this axis allows to pass from the total reference mark to the reference mark
the main thing of anisotropy.
The variational formulation of the linear equation of heat (cf [R5.02.01]) led to the evaluation
of a certain number of expressions in the form of integrals which constitute finally a system
matric. The matrix and the second member are built starting from various bricks: options of
calculation which gathers one or more integrals. The options described here are common to the unit
isoparametric finite elements. Their evaluation depends on the type of element: degree of the functions
of form, numbers and family of points of integration used.
One will be able to refer to the documents [U1.23.01], [U1.23.02] and [U1.24.02] concerning the different ones
modelings (type of mesh supporting the finite elements).
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
4/8
2
Expression of the elementary terms for the different ones
options of calculation
2.1 Notations
general
We indicate by:
open of ¤3 or ¤2 of border,
T
the variable representing time,
T
the step of time used,
R
the variable of space,
T
the temperature (unknown of the problem),
Tn
the temperature at the previous moment (known),
T *
the function test,
density,
C
specific heat with constant pressure,
p
C = C
heat-storage capacity with constant pressure per unit of volume,
p
p
the parameter of - method for the transitory thermal analysis.
2.2
Elementary terms contributing a share
2.2.1 Rigidity
thermics
Term utilizing variations in temperatures and the coefficient of conduction in the case
isotropic mediums (denomination used by analogy at the end of rigidity intervening in
the equation of the modeling of the mechanical phenomenon of elasticity). The coefficient can depend
time.
· mathematical expression:
().
*
T
T T
D
,
when the medium is anisotropic, the evaluation of flow (T) T is carried out in the reference mark
the main thing of anisotropy after a first change of reference mark (the tensor of conductivity
thermics is diagonal there) then by a change locates opposite, one returns in the reference mark
total,
· denomination of the option in the catalogs: RIGI_THER,
· a number of points of integration used: (first family of points of integration
cf [R3.01.01]).
net support
a number of nodes
a number of points
triangle
3
1
6
3
quadrangle
4
4
8 or 9
9
tetrahedron
4
4
10
15
pentahedron
6
6
15
21
hexahedron
8
8
20
27
27
27
Table 2.2.1-1
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
5/8
2.2.2 Mass
thermics
Term utilizing the coefficient of heat-storage capacity to constant pressure C = C
p
p
(denomination used by analogy at the end of mass intervening in the equation of modeling
equations of dynamics). The coefficient CP can depend on time.
·
1
mathematical expression: (). *
T
C T T T
D
p
· denomination of the option in the catalogs: MASS_THER
· a number of points of integration used: (second family of points of integration)
net support
a number of nodes
a number of points
triangle
3
3
6
6
quadrangle
4
4
8 or 9
9
tetrahedron
4
4
10
15
pentahedron
6
6
15
21
hexahedron
8
8
20
27
27
27
Table 2.2.2-1
2.2.3 Rigidity due to the boundary conditions of exchange
Term utilizing the coefficient of exchange H having for origin a boundary condition modelling
the convectifs exchanges with the edge of the field. The coefficient H can depend on time and space.
· mathematical expression: H (R, T) T.T * D
· denomination of the option in the catalogs: RIGI_THER_COEF_R or RIGI_THER_COEF_F
· a number of points of integration used:
net support
a number of nodes
a number of points
segment
2
4
3
4
triangle
3
3
6
4
quadrangle
4
4
8 or 9
9
Table 2.2.3-1
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
6/8
2.2.4 Rigidity due to the conditions of exchange between walls
Term due to the boundary condition of the Neumann type bringing into play two pennies left the border
in opposite and utilizing a single coefficient of exchange h. This type of boundary condition
create new relations between the degrees of freedom of the border.
In this case, one uses a particular finite element whose mesh support is obtained by associating two
meshs of identical edge or face, the functions of form used and the points of integration are
those of the starting mesh.
Into three-dimensional, the meshs support of the elements of face are of type TRIA3-TRIA3, QUAD4-
QUAD4, TRIA6-TRIA6, QUAD8-QUAD8 or QUAD9-QUAD9.
Into two-dimensional, they are of type SEG2-SEG2 or SEG3-SEG3.
One will be able to refer to [U4.25.02 § 3.1.3] for the description of the algorithm of search for meshs
in opposite.
· mathematical expression:
((, +) (2 -)). *
H R T
T
T
T
T D
1
and
((, +) (1 -)). *
H R T
T
T
T
T D
2
1
2
1
2
where 1 and 2 is two pennies left the border in opposite.
· denomination of the option in the catalogs: RIGI_THER_PARO_R or RIGI_THER_PARO_F
· a number of points of integration used: cf [Tableau 2.2.3-1].
2.3
Elementary terms contributing a share to the second member
2.3.1 Discretization in time
Term due:
· with the discretization of derived in time utilizing part of the term of mass with
the coefficient of heat-storage capacity CP,
· with
- method utilizing part of rigidity in the second member with
coefficient of conduction,
· mathematical expression in the case of isotropic mediums:
1
N
. *
-
1
N
*
(-).
T
C T T
D
T
T
D
p
when the medium is anisotropic, the evaluation of flow (T) T is carried out in the reference mark
the main thing of anisotropy after a first change of reference mark (the tensor of conductivity
thermics is diagonal there) then by a change locates opposite, one returns in the reference mark
total,
· denomination of the option in the catalogs: CHAR_THER_EVOL,
· a number of points of integration used: cf [Tableau 2.2.2-1].
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
7/8
2.3.2 Voluminal term of source
Term due to the voluminal source of heat.
· mathematical expression:
(S (R, T + T) + (1 -) S (R, T)). *
T D
,
· denomination of the option in the catalogs: CHAR_THER_SOUR_R or
CHAR_THER_SOUR_F,
· a number of points of integration used: cf [Tableau 2.2.1-1].
2.3.3 Term of convectif exchange
Term due to the boundary condition of convectif exchange utilizing the coefficient of exchange H and
the temperature of the “external” medium Tex.
· mathematical expression:
(H (R, T
T) T (R, T
T
) (1
) H (R, T) (T (R, T) T N
+
+
+ -
-
)). *
T D
ex
ex
· denomination of the option in the catalogs: CHAR_THER_R or CHAR_THER_F,
· a number of points of integration used: cf [Tableau 2.2.3-1].
2.3.4 Term of imposed normal flow
Term due to the boundary condition of flow imposed according to the normal on the border, utilizing one
function being able to depend on the variables R and T.
· mathematical expression:
(F (R, T + T) + (1 -) F (R, T)). *
T D
,
· denomination of the option in the catalogs: CHAR_THER_FLUN_R or
CHAR_THER_FLUN_F,
· a number of points of integration used: cf [Tableau 2.2.3-1].
2.3.5 Term of exchange between walls
Term due to the boundary condition of the Neumann type bringing into play two pennies left the border
in opposite and utilizing a single coefficient of exchange h.
· mathematical expression:
((,) (-) (N
N
1
-
)). *
H R T
T
T
T D
N
N
2
1
1
and
((,) (1 -) (
-
)). *
H R T
T
T
T D
1
2
2
1
2
where 1 and 2 is two pennies left the border in opposite
· denomination of the option in the catalogs: CHAR_THER_PARO_R or
CHAR_THER_PARO_F,
· a number of points of integration used: cf [Tableau 2.2.3-1].
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
Code_Aster ®
Version
3
Titrate:
Linear modeling of the elements of continuous medium in thermics
Date:
30/08/95
Author (S):
J.P. LEFEBVRE, X. DESROCHES
Key:
R3.06.02-A
Page:
8/8
3 Bibliography
[1]
N.RICHARD: Development of the thermal anisotropy in the Aster software. Note
EDF/DER HM-18/94/0011 of the 05/07/1994.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/95/002/A
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