Code_Aster
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Titrate:
Diagonalisation of the thermal matrix of mass
Date:
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Author (S):
J.M. PROIX
Key:
R3.06.07-A
Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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Organization (S):
EDF/MTI/MN
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
R3.06.07 document
Diagonalisation of the thermal matrix of mass
Summary:
To improve the regularity of the solution in the problems of transitory thermics, one of the solutions
consist with “lumper” (i.e.: to condense on the diagonal) the thermal matrix of mass (matrix of
capacity).
This possibility is accessible by modelings
PLAN_DIAG
,
AXIS_DIAG
and
3d_DIAG
for the phenomenon
THERMICS
. It is activated at the time of the call to the controls of thermal calculation
THER_LINEAIRE
and
THER_NON_LINE
.
When these modelings are used, only the linear finite elements (2D and 3D) have a matrix of mass
lumpée. Indeed, the direct diagonalisation does not give satisfactory results for the finite elements
quadratic. Consequently, for the quadratic finite elements 2D, one carries out a cutting in
linear elements, which are lumpés. On the other hand, for the quadratic finite elements 3D, one does not make
diagonalisation of the matrix of mass.
The theoretical results are illustrated by the thermomechanical calculation of a cylinder subjected to a thermal shock.
Code_Aster
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Titrate:
Diagonalisation of the thermal matrix of mass
Date:
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Key:
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Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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Contents
1 Introduction ............................................................................................................................................ 3
2 Principle of the maximum ............................................................................................................................ 3
2.1 Statement of the principle for the continuous case ........................................................................................... 3
2.2 Respect of the principle of the maximum at the discrete level ....................................................................... 4
2.3 Conditions sufficient for the respect of the principle of the maximum at the discrete level ........................ 5
3 Method of diagonalisation selected and types of elements ...................................................................... 7
3.1 Elements such as the extra-diagonal terms of
K
............................................ 7 are negative
3.1.1 Linear elements .................................................................................................................. 8
3.1.2 Quadratic elements ........................................................................................................... 8
3.1.3 Conclusion on the elements: properties of the matrices
K
.................................................... 8
3.2 Method of diagonalisation: Integration with the nodes of the elements ............................................... 9
4 Implementation in Code_Aster ..................................................................................................... 10
4.1 Modelings 2D ........................................................................................................................... 10
4.2 Modeling 3D ............................................................................................................................. 11
5 thermal Calculation of a cylinder subjected to a cold shock ........................................................................ 12
5.1 Data ........................................................................................................................................ 12
5.2 ........................................................................................................................................ Results 13
6 Conclusion ........................................................................................................................................... 17
7 Bibliography ........................................................................................................................................ 18
Code_Aster
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Version
5.0
Titrate:
Diagonalisation of the thermal matrix of mass
Date:
29/08/00
Author (S):
J.M. PROIX
Key:
R3.06.07-A
Page:
3/18
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/00/006/A
1 Introduction
One is interested in transitory thermal calculations where intervene of abrupt variations of
loadings - for example, thermal shocks. In certain cases, one notes that the temperature
oscillate spatially and temporally. Moreover, if one observes a profile of temperature at one moment
given transient, the temperature can in certain nodes exceed the terminals min. and max.
imposed by the initial conditions and the boundary conditions. This result physically
unacceptable violate what is called the “principle of the maximum”.
The diagonalisation of the matrix of mass can solve these problems of going beyond of
maximum. This is detailed in the note [bib1]. One is satisfied here to recall the main results of them.
One points out the principle of the maximum in the continuous case, then sufficient conditions are expressed
who allow to check it for the discrete equations. It is shown in particular that the diagonalisation
thermal matrix of mass is one of these sufficient conditions and one presents different
methods for diagonaliser
M
.
Another sufficient condition depends on the thermal matrix of rigidity (conduction). One studies more
particularly from this point of view finite elements of thermics used in Code_Aster.
It results from it that in the case of the linear elements, all the sufficient conditions to check it
principle of the maximum are gathered. In particular, the diagonalisation of the mass allows
indeed to obtain a regular solution. On the other hand, for the quadratic elements, one cannot
to prevent the oscillations.
One thus describes the solution suggested in Code_Aster: the modelings developed in 2D
(
AXIS_DIAG
,
PLAN_DIAG
) function with linear elements (if the mesh is of command 2, one
cut out in linear elements for thermal calculation). In 3D, one treats only the linear elements.
A numerical study of a thermal shock on a cylinder makes it possible to illustrate these results.
2
Principle of the maximum
2.1
Statement of the principle for the continuous case
One gives here one of the statements possible of the principle of the maximum for the operator of heat (in
the absence of terms of source, and in isotropic homogeneous linear thermics) [bib2].
That is to say
open limited
IR
N
of border
, whose adherence is noted
.
That is to say
()
U X, T
such as:
U
T
U
T
T
-
×
>
0
0
0
on
], [, (
)
of class
compared to
and
of class
compared to on
]
[
2
1
C
C
X
U
T
, T
×
0
Then
[]
Max U max U
T
P
×
=
0,
, where
{}
[]
P
T
=
×
×
(
) (
,)
0
0
is the border of the cylinder
] [
×
0, T
.
This result thus ensures that the maximum of
U
is inevitably reached either at the time of the initial conditions
maybe on an edge of the field during the transient.
Code_Aster
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Titrate:
Diagonalisation of the thermal matrix of mass
Date:
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Author (S):
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Key:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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2.2
Respect of the principle of the maximum at the discrete level
The equation of heat (thermal conduction) is considered:
div (.
)
(,)
+
=
T
S X T
C
T
T
p
+ Conditions limit + initial condition
()
()
T T, X
T X
0
0
=
with
T
temperature
S
heat per unit of volume (sources intern)
T
variable of time
X
variable of space
thermal coefficient of conductivity
C
p
voluminal heat with constant pressure
Limiting types of conditions:
·
Imposed temperature: condition limits of Dirichlet
()
()
T X T
T
X T
imp
imp
,
,
=
on
·
Imposed normal flow: condition of Neumann defining flow entering the field
()
()
-
=
Q
N
X T
F X T
flow
.
,
on
·
Exchange: condition limits of Fourier modelizing the convectifs exchanges on the edges of
field
()
()
() ()
(
)
-
=
-
Q
N
X T
H X T T
X T
T X T
ext.
exchange
.
,
,
,
on
The variational formulation of the problem is as follows: [bib3]
C
T
T
D
T
D
HT D
S D
F
D
HT
D
p
ext.
exchange
flow
exchange
+
+
=
+
+
.
.
.
.
.
.
v
checking
()
v T
X T
imp
imp
=
,
on
After discretization in space of this equation, one obtains the system:
()
()
M
T
K T
F
T T
T
T
+
=
()
.
with
()
T T
: vector of the nodal temperatures
M
: stamp of thermal mass
M
NR
=
C
FD
E
E
T
K
: stamp thermal rigidity
K
NR NR
NR NR
=
+
.
.
T
T
E
E
echangee
FD
H
D
F
: vector of the second member
F
NR
NR
NR
=
+
+
S D
F
D
H T
D
E
flux
echangee
ext.
E
NR
: (functions of form)
Code_Aster
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Titrate:
Diagonalisation of the thermal matrix of mass
Date:
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Author (S):
J.M. PROIX
Key:
R3.06.07-A
Page:
5/18
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/00/006/A
For the discretization in time, one applies one
- method
[]
(
)
0 1
,
, which leads to:
(
)
(
)
(
)
M
K
T
M
K
T
F
F
+
=
+
-
+
+ -
+
+
T
T
N
N
N
N
1
1
1
1
(
)
where
T T
N
N
,
+
1
are the vectors of the nodal temperatures at the moments
T, T
N
n+1
.
2.3 Conditions sufficient for the respect of the principle of the maximum with
discrete level
One of the characteristics of nonrespect of the principle of the maximum is the appearance of oscillations
(temporal or space): if one observes the variation in the temperature in a node during
time, it is noted that the solution oscillates and exceeds the values minimal and maximum determined by
initial conditions and limiting conditions. Or, at a given moment, one observes oscillations
space.
One thus seeks sufficient conditions on
T
,
K
and
M
so that the solution does not oscillate with
run from time ([bib1], [bib4], [bib5]). Indeed, one cannot obtain conditions necessary and
sufficient. One thus seeks conditions of nonoscillation of the solution in the course of time. If
those are checked, it will be checked that the space oscillations also disappeared, and then it
respect of the principle of the maximum is assured.
Assumptions:
To be able to express these sufficient conditions of nonoscillation, two should be added
assumptions:
·
one places oneself at the elementary level. The respect of the properties at the elementary level is enough for
that the conditions of nonoscillation are checked for the assembled matrices.
·
it is considered that the matrix of rigidity
K
is formed only of the voluminal term
K
NR NR
V
=
.
T
E
FD
This assumption is valid whenever there is no condition of exchange or when it
coefficient of exchange H is sufficiently large: one can then approach the condition of exchange
by a condition in imposed temperature.
The sufficient conditions of nonoscillation amount expressing conditions on the pitch of time
and on the diagonal and extra-diagonal terms of
M
and
K
so that certain properties of these
matrices are checked (based on the monotony of the matrices) [bib1]:
M
tK
I
J
ij
ij
+
.
0
éq 2.3-1
M
tK
I
J
ij
ij
+ -
(
)
1
0
éq 2.3-2
M
tK
I
II
II
+
-
(
)
1
0
éq 2.3-3
In the general case, the extra-diagonal terms can be of unspecified sign. A fast study
allows to determine the conditions on
T according to their signs so that equations
the preceding ones are checked:
Code_Aster
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Diagonalisation of the thermal matrix of mass
Date:
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K
ij
0
K
ij
0
M
ij
0
M
tK
I
J
ij
ij
+
.
0
[éq 2.3-1] unconditionally distorts
except
M
K
ij
ij
=
=
0
max
min
I J
ij
ij
I
II
II
M
K
T
M
K
()
1
M
ij
0
(
)
M
tK
I
J
ij
ij
+ -
1
0
[éq 2.3-2] unconditionally distorts
except
M
K
ij
ij
=
=
0
max
min
I J
ij
ij
I
II
II
M
K
T
M
K
-
-
(
)
(
)
1
1
Some is
T
and forms it
M
,
there is risk of oscillations.
Interval to be respected on
T
.
The diagonalisation of
M
allows
to remove the lower limit.
The sufficient conditions to avoid the oscillations are then:
K
I
J
ij
0
T
T
T
min
max
with:
·
T
M
K
I
II
II
max
()
=
min 1
and
·
T
= max
min
I J
ij
ij
ij
ij
M
K
M
K
-
-
(
)
,
1
,
Consequently, it is necessary initially that the elementary matrices check
K
ij
0
(it is the case of
linear finite elements studied further).
With regard to the interval on the pitch of time:
If the oscillations are due to a pitch of too large time
(
)
T
T
>
max
, one can advise:
·
that is to say to choose a diagram of integration in time of the implicit type
(
)
=
1
, to eliminate
the upper limit of the interval.
·
that is to say to decrease
T
. (In practice, it is difficult to know an order of magnitude of
T
max
).
Code_Aster
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Date:
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Enough often, the problem of the oscillations arises for pitches of small times
(
)
T
T
<
min
; in
effect, to take into account the variations of the solution (for example, at the time of a thermal shock), one
is brought to choose a fine discretization in time. In this case, to avoid the oscillations, one can
to suggest:
·
that is to say to increase the step value of time. In practice, this is not always possible
because
T
can be imposed by the nature of the problem (fast variation of the loading). Moreover,
it is difficult to have an order of magnitude of
T
min
.
·
that is to say to decrease the size of the meshs and thus to increase the number of elements. Indeed,
value of
T
min
depends directly on the space discretization:
The forms of the elementary matrices are indeed:
M
C NR NR FD
ij
I
J
E
=
K
NR
NR FD
ij
I
J
E
=
For the elements 2D, the terms of
M
are thus of the form
C
×
Surface whereas
those of
K
are related only to
. This solution remains the best if one is not
not limited by the cost calculation, because the thermal solution and especially mechanics will be of as much
more precise.
·
Maybe of diagonaliser the matrix
M
, which removes the lower limit of the interval.
It is the solution suggested here.
In the continuation of the study, one is interested only in the problem of the oscillations which appear for
pitches of too small times:
T
T
<
min
. One presents the method more precisely of
diagonalisation of the matrix
M
chosen, and various types of elements to which it applies.
3
Method of diagonalisation selected and types of elements
3.1
Elements such as the extra-diagonal terms of
K
are negative
It was seen that the diagonalisation of
M
is effective only when the extra-diagonal terms of
stamp rigidity
K
are negative. In the contrary case, one of the sufficient conditions of
not-oscillation is unconditionally false, whatever the form of
M
.
For each finite element used in thermics in Code_Aster, one checks if the matrix
elementary of rigidity of the element has negative extra-diagonal terms, while resting
mainly on [bib11], which gives the analytical expressions of the elementary matrices for
conventional finite elements. One summarizes here the observations made in [bib11].
Code_Aster
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3.1.1 Elements
linear
3.1.1.1 Elements TRIA3, TETRA4, PENTA6
The elementary matrix
K
is a function of the cotangents of the angles. If one of the angles is blunt,
certain extra-diagonal terms of
K
are positive. If all the angles are acute, the property is
checked.
One has the same type of result in 3D for the tetrahedron with 4 nodes and the pentahedron with 6 nodes.
3.1.1.2 Elements QUAD4 and HEXA8
Certain extra-diagonal terms of
K
can be positive if the element is lengthened too much in one
direction. If not, the property is checked.
One has the same type of result in 3D for element HEXA8.
3.1.1.3 Element 3D pyramid with 5 nodes
For this element, the functions of form are not any more of the polynomials but rational fractions in
X, y, Z. For this type of element, one does not have the expression, even approximate of
K
.
3.1.2 Elements
quadratic
3.1.2.1 Element
TRIA6
In
K
, certain extra-diagonal terms are necessarily positive.
3.1.2.2 Element
QUAD9
In the same way, on the analytical expression of the terms of
K
, it is noted that some of the terms
extra-diagonal are necessarily positive.
3.1.2.3 Element
QUAD8
For this element, there is not the expression supplements
K
for the real element. But for the element of
reference, one notes that certain extra-diagonal terms are positive.
3.1.3 Conclusion on the elements: properties of the matrices
K
For the linear elements, if the real element is not too irregular, extra-diagonal terms of
K
are quite negative. For the quadratic elements (in 2D), certain extra-diagonal terms of
K
are positive. Even while diagonalisant
M
, one cannot ensure that the solution will not oscillate.
In Code_Aster, to eliminate the problems from oscillation and going beyond of the maximum, one
diagonalise only matrices of mass for the thermal calculations carried out on elements
linear. For the quadratic elements, one saw that one could not diagonaliser directly
stamp of mass. One thus cuts out these elements in linear elements which themselves are lumpés.
This is applied to the quadratic elements 2D in Code_Aster, but not to the elements
quadratic 3D, not for reasons of method but because automatic cutting is
difficult to implement in 3D.
Code_Aster
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Diagonalisation of the thermal matrix of mass
Date:
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3.2
Method of diagonalisation: Integration with the nodes of the elements
If the elementary matrix of mass is calculated by numerical integration, its terms are written under
the form [bib8]
M
C NR NR FD
W
C NR NR
ij
I
J
Q
Q
NR
I
J Q
C
=
=
1
(
)
where
NR
C NR
I
J
is evaluated with Q
ième
not integration
and
W
Q
is the weight of integration associated with this point.
Classically, the points of integration are the points of Gauss; the position of the NR points and their weight
are defined so that the diagram integrates the polynomials of 2N-1 degree exactly. If one
chooses the points of integration to the nodes of the element, one obtains:
M
ij
= 0 for
I
J
. This
method of integration is also called method of Newton-Dimensions.
Notice 1: Axisymmetric problems:
If the points of integration are with the nodes, one will have, for any type of element, of the null masses
on the axis of symmetry.
Indeed,
M
C NR NR
R Dr. dz
ij
I
J
=
2
M
C W Jac R X
ij
ij
I
I
2
()
If the point of integration
I
is a node of the axis,
()
R X
I
=
0
and the corresponding mass is null.
For the axisymmetric elements, the method of integration to the nodes is thus not
adapted close to the axis. In this case, it is necessary to integrate into the points of Gauss the elements which touch
the axis of revolution, by using usual modeling (AXIS).
Notice 2: other possible methods of diagonalisation:
Other methods are studied in [bib1], in particular to test diagonaliser them
quadratic elements. In practice, it are not retained at present in
Code_Aster.
·
Scaling of the diagonal terms ([bib9], [bib10]): Hinton suggests the scaling of
diagonal terms of the matrix
M
consistent, so that total mass of
the element is preserved. It is noted that the lumpées masses are always positive, even
for the elements quadrangles to 8 and 9 nodes.
·
Summation by line ([bib10]): One summons the values of Mij per line and one concentrates it
result on the diagonal. Unfortunately, this process can lead to masses
negative, in particular for the quadrangle with 8 nodes.
Notice 3:
For the quadratic elements, one notes in [bib1] that, even while diagonalisant with
method of scaling of the diagonal terms, one obtains oscillations. One thus cannot
not to use these elements within the framework of the diagonalisation (i.e. for a mesh
relatively coarse with respect to the speed of the thermal transient).
One can of course use the quadratic elements in thermics, with the proviso of adapting the smoothness
mesh with the stiffness of the thermal shock.
Code_Aster
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Titrate:
Diagonalisation of the thermal matrix of mass
Date:
29/08/00
Author (S):
J.M. PROIX
Key:
R3.06.07-A
Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/00/006/A
4
Implementation in Code_Aster
In order to eliminate the oscillations from the temperature in space and time, modelings
AXIS_DIAG
,
PLAN_DIAG
and
3d_DIAG
carry out the diagonalisation of the matrices of mass during calculation thermal
linear (
THER_LINEAIRE
). The diagonalisation is not operational for
THER_NON_LINE
currently. To guarantee the effectiveness of it, it was seen that it should be carried out on linear elements.
If the mesh is linear, one carries out simply a diagonalisation of the matrices of
mass by integration with the nodes.
In the case of a quadratic mesh, in 2D, one carries out a thermal calculation ISO-P2: calculation on one
QUAD9 is brought back to a calculation on 4 QUAD4; in the same way, one passes from a TRIA6 to 4 TRIA3.
This makes it possible not to lose the smoothness of the discretization of the mesh, as well for the solution of
thermal problem that for that of the mechanical problem. Indeed, one shows in [bib1] that
this solution is preferable with that which consists in carrying out thermal calculation on linear meshs
who are based on the nodes nodes of the quadratic meshs (what is normal since
discretization is finer).
Modelings available are thus:
4.1 Modelings
2D
Modeling
PLAN_DIAG
AXIS_DIAG
Net
Element
Element
TRIA3
THPLTL3
THAXTL3
QUAD4
THPLQL4
THAXQL4
SEG2
THPLSL2
THAXSL2
TRIA6
THPLTL6
THAXTL6
QUAD9
THPLQL9
THAXQL9
SEG3
THPLSL3
THAXSL3
Comments on elementary calculations 2D:
For the linear elements: terms of mass (matrix to the first member and vector with the second
member) are lumpés by integration with the nodes. The new elements have options of calculations
elementary identical to the conventional elements. The only modified elementary options are thus
MASS_THER
and
CHAR_THER_EVOL
.
For the quadratic elements: calculation is ISO-P2. Calculation on an element QUAD9 (resp. TRIA6)
is brought back has a calculation on 4 linear elements QUAD4 (resp. 4 TRIA3) whose terms of mass are
lumpés by the preceding method. The matrices and vectors of each of the 4 linear elements are
assembled on the level of the elementary routine of calculation. By homogeneity, on the elements of edges,
one calculates the elementary terms on 2 SEG2, then one assembles.
Elements THPLTL6, THAXTL6, THPLQL9, THAXQL9 have the functions of form of the elements
linear in which they are cut out.
Caution:
There is no element associated with mesh QUAD8. Consequently, if the mesh is composed of
quadratic meshs, it is initially necessary to change the QUAD8 into QUAD9 using
order
CREA-MAILLAGE
:
CREA_MAILLAGE (MODI_MAILLE: (OPTION: “QUAD8_9”)).
Code_Aster
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Version
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Titrate:
Diagonalisation of the thermal matrix of mass
Date:
29/08/00
Author (S):
J.M. PROIX
Key:
R3.06.07-A
Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/00/006/A
Into axisymmetric: if elements of the mesh touch the axis, one should not integrate into the nodes which
find on the axis. It is thus necessary to isolate this layer from elements and to affect modeling AXIS to him.
4.2 Modeling
3D
Modeling
3d_DIAG
Net
Element
HEXA8
THER_HEXA8_D
PENTA6
THER_PENTA6_D
TETRA4
THER_TETRA4_D
PYRAM5
THER_PYRAM5_D
QUAD4
THER_FACE4_D
TRIA3
THER_FACE3_D
Comments on elementary calculations 3D:
For the linear elements: as in 2D, terms of mass (matrix to the 1st member and vector
with the 2nd member) are lumpés by integration with the nodes (3rd family of points of Gauss).
For the quadratic elements, it would be necessary to cut out those in linear elements. This cutting is
delicate to implement, because it results in creating a new element (PENTA18) with nodes with
medium of each quadrilateral face (and it would also be necessary to create a new element PYRAM14).
One diagonalise thus currently only the linear elements 3D.
With regard to the pyramids with 5 nodes, integration with the nodes was tested but
do not function well. Cf [§3.1.1.3] (it is not known if all the extra-diagonal terms are negative).
modeling “3d_DIAG” exists for the pyramids with 5 nodes but it is identical to modeling
“3D”. In any case these element are minority in a mesh 3D: it are generated only
by the voluminal free maillor of GIBI, which creates some with the need of them to supplement the mesh
hexahedral.
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Diagonalisation of the thermal matrix of mass
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/00/006/A
5
Thermal calculation of a cylinder subjected to a cold shock
One illustrates on a numerical example what was shown previously; namely that
diagonalisation is effective to check the principle of the maximum.
One takes as a starting point the the industrial example of the cooling of a moulded elbow: a shock is applied
cold thermics (289°C with 20°C) on a fissured elbow. During the transient of cooling,
temperature calculated in certain nodes reached 310° without diagonalisation of the matrices of mass. For
the example treated here, one restricts with a hollow roll of the same dimension than the elbow on which one
apply a cold thermal shock.
5.1 Data
A presumedly infinite hollow roll is studied. Like there is no dependence compared to Z (cylinder
infinite), one limits the study to a plane calculation. By reason of symmetry, one nets only one portion of
structure.
Z
Rint
Rext
Rint = 417 mm
Rext = 496 mm
With
B
C
D
M1
M2
45°
Co-ordinates of the points:
X (mm)
y (mm)
Z (mm)
M
1
436.75
0.
0.
M
2
436.75 cos 45°
436.75 sin 45°
0.
Calculations are carried out on a linear mesh (meshs TRIA3-QUAD4):
M
1
M
2
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Diagonalisation of the thermal matrix of mass
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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Characteristics of the mesh:
A number of nodes: 90
Numbers and type of meshs: 64 TRIA3, 32 QUAD4
Characteristics of material:
= 19.97 W/m °C
CP = 4.89488 106 J/m3 °C
Limiting conditions and loading:
To ensure invariance by rotation, one forces conditions of null heat transfer rate on faces AB
and CD. The external wall is supposed perfectly insulated. On internal skin AD, the transfer
thermics between the cylinder and the fluid is modelized by a coefficient of high convectif exchange:
H = 40.000 W/m2 °C.
The cold thermal shock applied to the moulded elbows is represented by a linear variation of
temperature of the fluid circulating in the pipe: 289°
20° in 12 S. In order to accentuate the problem of
going beyond of the maximum and thus to better highlight the influence of the diagonalisation, one
adopt a more brutal shock: 289°
20° in 1 S.
T
Tfluide
289°
20°
10 S 11 S
The following discretization in time is adopted:
T = 0 S
with T = 10 S,
1 pitch of time
T = 10 S
with T = 11 S,
2 pitches of time
T = 11 S
with T = 25 S,
7 pitches of time
T = 25 S
with T = 60 S,
10 pitches of time
Numerically, the value retained for the parameter of the discretization in time is
= 0,57.
5.2 Results
The following figures show the profiles of temperature in the thickness of the cylinder at the moment t=15s
(moment when the goings beyond of the maximum are largest) without diagonalisation of the matrices of
mass.
One gives also the temporal evolution T (T) to the nodes M
1
and M
2
located at a quarter thickness of
internal skin.
Without diagonalisation, one notes that the temperature oscillates in time and in space exceed the value
maximum of 289° at the beginning of the transient.
With diagonalisation on the linear elements, one observes a regular solution without going beyond
maximum.
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A similar study was undertaken on linear elements 3D (tetrahedron to 4 nodes, pentahedron with 6
nodes, hexahedron with 8 nodes). The results lead to the same conclusions: with diagonalisation,
the oscillations of the temperature disappear for calculation on the linear elements 3D.
Complementary remark concerning thermomechanical calculation:
Another study was carried out in [bib1] estimating the consequences of the diagonalisation
thermics on the mechanical results. It is noted that calculation ISO-P2 (elements quadratic
divided into linear elements, whose matrices of mass are lumpées) provides results
satisfactory. One eliminates the space oscillations from the temperature. But in the studied case, with one
relatively coarse mesh, the mechanical solution remains not very precise. Although the thermal solution
that is to say correct, to improve the solution in stresses, the mesh should in any case be refined.
For meshs TRIA3, the diagonalisation leads to a regular solution without going beyond of
maximum:
MOMENT T (S)
T
E
M
PER
AT
U
R
E
WITH
NR
O
HAVE
D
M
1
EDF
Electricity
from France
Mechanical department and Digital Models
CHANGE OF the TEMPERATURE TO the QUARTER THICKNESS - MESH TRIA3
agraf 13/11/98 (c) EDF/DER 1992-1997
WITHOUT LUMPING
LUMPING
Thermal calculation
of a subjected cylinder
has a cold shock
DT 289 --> 20 in 1s
260
270
280
290
300
310
320
10
12
14
16
18
20
22
24
26
28
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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SKIN INTERNS CURVILINEAR X-COORDINATE (m) EXTERNAL SKIN
T
E
M
P
E
R
With
T
RUE
T
EDF
Electricity
from France
Mechanical department and Digital Models
PROFILE OF TEMPERATURE IN the THICKNESS - LINEAR MESH - ELEMENTS TRIA3
agraf 13/11/98 (c) EDF/DER 1992-1997
35.6814
294.365
WITHOUT LUMPING
LUMPING
Thermal calculation
of a subjected cylinder
has a cold shock
DT 289 --> 20 in 1s
Moment t=15 S
50
100
150
200
250
0
10
20
30
40
50
60
70
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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For meshs QUAD4, the diagonalisation leads to a regular solution without going beyond of
maximum:
MOMENT T (S)
T
E
M
PER
AT
U
R
E
WITH
NR
O
HAVE
D
M
2
EDF
Electricity
from France
Mechanical department and Digital Models
CHANGE OF the TEMPERATURE TO the QUARTER THICKNESS - MESH QUAD4
agraf 13/11/98 (c) EDF/DER 1992-1997
WITHOUT LUMPING
LUMPING
Thermal calculation
of a subjected cylinder
has a cold shock
DT 289 --> 20 in 1s
260
270
280
290
300
310
320
10
12
14
16
18
20
22
24
26
28
SKIN INTERNS CURVILINEAR X-COORDINATE (m) EXTERNAL SKIN
T
E
M
PER
AT
U
R
E
T
EDF
Electricity
from France
Mechanical department and Digital Models
PROFILE OF TEMPERATURE IN the THICKNESS - LINEAR MESH - ELEMENTS QUAD4
agraf 13/11/98 (c) EDF/DER 1992-1997
34.9119
298.594
WITHOUT LUMPING
LUMPING
Thermal calculation
of a subjected cylinder
has a cold shock
DT 289 --> 20 in 1s
Moment t=15 S
50
100
150
200
250
0
10
20
30
40
50
60
70
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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6 Conclusion
Modelings
AXIS_DIAG
,
PLAN_DIAG
and
3d_DIAG
are proposed in order to solve them
problems of going beyond of the maximum with oscillation of the solution in space and time which
appear during certain transitory thermal calculations with abrupt variation of the loading.
At the discrete level, the analysis leads to a sufficient condition of not-oscillation on the pitch of
discretization in time which must belong with an interval:
T
T
T
min
max
where values of
T
min
and
T
max
depend on the coefficients of matrices of mass and rigidity
thermics as well as parameter
discretization in time.
In practice, if the oscillations come from a pitch of too large time
(
)
T
T
>
max
, it is suggested
choice of an implicit scheme in time
(
)
=
1
. If the pitches of time are too small, the diagonalisation
matrix of mass can make it possible to remove the oscillations.
For the linear elements, one shows that the diagonalisation makes it possible indeed to avoid them
oscillations of the solution. For the quadratic elements, a direct diagonalisation is not enough with
to avoid the oscillations. For this type of element, one cuts out them in linear elements, and one carries out one
diagonalisation of the linear elements resulting by integration with the nodes (this is carried out only
in 2D for the moment).
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R3.06 booklet: Machine elements and thermal for the continuous mediums
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7 Bibliography
[1]
Mr. A. REDON, J.M. PROIX: Diagonalisation of the thermal matrix of mass. Note
EDF/DER/HI-75/97/008/0
[2]
H. BREZIS: Analyze functional, Théorie and applications. Masson (1983).
[3]
Manual of reference. Document [R3.02.01]: Linear algorithm of thermics.
[4]
H. OUYANG, D. XIAO: Criteria for eliminating oscillation in analysis off heat-conduction
equation by finite-element method. Communications in numerical methods in engineering.
Vol.10, 453-460 (1994).
[5]
B. NITROSSO: Study of the principle of the discrete maximum - September 1996. Communication
intern EDF.
[6]
O.C. ZIENKIEWICZ: Finite element method (3rd edition). Mc Graw-Hill (1977).
[7]
I. FRIED, D.S. MELKUS: Finite element farmhouse matrix diagonalisation by numerical integration
with No convergence misses loss. Int. Newspaper off solids and structures. Vol.11, 461-466 (1975).
[8]
G. COHEN, A. ELMKIES: Triangular finite elements P2 with condensation of mass for
the equation of the waves. Research report INRIA n°2418 - September 1994.
[9]
E. HINTON, A. ROCK, O.C. ZIENKIEWICZ: With note one farmhouse diagonalisation in related
processes in the finite element method. Int. J. Earthquake Structural Engineering and
Dynamics, Vol.4, 245-249 (1976).
[10]
T. HUGHES: The finite element method. Linear static and dynamic finite element analysis.
Prentice Hall (1987).
[11]
J.P. GREGOIRE: Collection of elementary matrices of mass and the Laplacian for
main elements of simple form. Note EDF HI-76/96/012 - June 1996.