Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
1/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
Organization (S):
EDF/AMA, EDF/UTO/LOCATED, Delta CAD
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
V4.22.302 document
TTNL302 - Infinite wall subjected to a constant flow
with variable properties
Summary:
This test results from the validation independent of version 3 in nonlinear transitory thermics.
It is about a linear problem 1D represented by five modelings, one planes, four the other voluminal ones.
The functionalities tested are as follows:
·
element of plane thermics,
·
voluminal element of thermics,
·
algorithm of transitory thermics non-linear,
·
variable properties,
·
limiting condition: imposed flow.
The interest of the test lies in the taking into account of properties variable (thermal conductivity and heat
voluminal).
The results are compared with an analytical solution.
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
2/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
L
C
D
B
With
y
X
With
E
Q
T (S)
Q (W/m ²)
1
T (°C)
,
C
1
1,5
1
L = 2 m
AE=EB= 1m
1.2
Properties of material
= 1.0 + 0.5 T W/m.°C
Thermal conductivity
C = 1.0 + 0.5 T J/m
3
.°C Heat
voluminal
1.3
Boundary conditions and loadings
·
dimensioned [AD]: flow imposed Q = 1 W/m ² for T > 0,
·
dimensioned [AB], [BC], [CD]
= O.
1.4 Conditions
initial
T (X, 0) = 0°C for any X
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
3/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Semi-analytical solution utilizing functions of error:
T X T
T
X
T
X erfc
X
T
(,)
(/) exp
.
=
+
-
+
-
-
2
1 2
4
2
1
2
with
()
erfc X
E dt
X
T
=
-
2
2
where X = X-coordinate
T = time
This formula is valid only for
(T) =
C (T) = 1. + 0.5T
2.2
Results of reference
Temperature at points A (X = 0) and E (X = 1) at the moment T following: T = 0.1, 0.3, 0.5, 0.7 and 1s
2.3
Uncertainty on the solution
Unknown factor, due to the evaluation of the functions of error.
2.4 References
bibliographical
[1]
Segal, NR. Praagman, “A fast implementation off explicit time stepping algorithms with the finite
element method for has class off nonlinear evolution problems ", Int. J. num. Meth. Engng, flight 23,
p 155-168, 1986.
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
4/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
PLAN (QUAD9)
m10
m20
Limiting conditions:
- dimensioned AB, BC, CD
= 0
- dimensioned AD
Q
= 1 W/m ² °C
y
X
0.1 m
With
2.m
1. m
E
D
m11
C
B
m1
F
Not
X
y
Node
With
0.0
0.0
N1
D
0.0
0.1
N3
E
1.0
0.0
N61
F
1.0
0.1
N63
3.2
Characteristics of the mesh
A number of nodes:
123
A number of meshs and types: 20 QUAD9
3.3 Functionalities
tested
Controls
AFFE_MODELE
THERMICS
PLAN
ALL
DEFI_MATERIAU
THER_NL
AFFE_CHAR_THER
FLUX_REP
THER_NON_LINE
TEMP_INIT
VALE
INCREMENT
LIST_INST
RECU_CHAMP
INST
3.4 Remarks
The discretization in pitch of time is as follows:
10 pitches for [0. , 5.D-2] is
T = 5.D-3
19 pitch for [5.D-2, 1.D0] is
T = 5.D-2
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
5/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
tolerance
Temperature (°C)
Node N1 T = 0.1s
0.330
0.329
-
0.204%
1%
““T = 0.3s
0.544
0.544
-
0.048%
1%
““T = 0.5s
0.682
0.681
-
0.075%
1%
““T = 0.7s
0.789
0.789
-
0.036%
1%
““T = 1.0s
0.918
0.920
0.254%
1%
Node N3 T = 0.1s
0.330
0.329
-
0.204%
1%
““T = 0.3s
0.544
0.544
-
0.048%
1%
““T = 0.5s
0.682
0.681
-
0.075%
1%
““T = 0.7s
0.789
0.789
-
0.036%
1%
““T = 1.0s
0.918
0.920
0.254%
1%
Node N61 T = 0.1s
0.004
0.004
1.161%
1%
““T = 0.3s
0.071
0.071
-
0.377%
1%
““T = 0.5s
0.160
0.161
0.573%
1%
““T = 0.7s
0.247
0.251
1.616%
1%
““T = 1.0s
0.366
0.380
3.951%
1%
Node N63 T = 0.1s
0.004
0.004
1.161%
1%
““T = 0.3s
0.071
0.071
-
0.377%
1%
““T = 0.5s
0.160
0.161
0.573%
1%
““T = 0.7s
0.247
0.251
1.616%
1%
““T = 1.0s
0.366
0.380
3.951%
1%
4.2 Remarks
The relative error is to the maximum of 3.9%.
4.3 Parameters
of execution
Version: 5.03
Machine: SGI - ORIGIN 2000 - R12000
Overall dimension memory:
8 megawords
Time CPU To use: 19.44 seconds
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
6/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
5 Modeling
B
5.1
Characteristics of modeling
3D (HEXA20)
D
E
B
With
Z
X
Limiting conditions:
- faces ABCD, ABFE
= 0
- faces EFGH, DCGH
= 0
- face BFGC
= 0
- face AEHD
Q
= 1 W/m ²
0.1 m
C
y
H
F
G
0.1 m
1 m
Nodes
N1
N8
Nodes
N121
N126
Nodes
X
y
Z
N1
0.0
0.0
0.05
N8
0.0
0.1
- 0.05
N121
1.0
0.0
- 0.05
N126
1.0
0.1
0.05
5.2
Characteristics of the mesh
A number of nodes:
248
A number of meshs and types: 20 HEXA20
5.3 Functionalities
tested
Controls
AFFE_MODELE
THERMICS
3D
ALL
DEFI_MATERIAU
THER_NL
AFFE_CHAR_THER
FLUX_REP
THER_NON_LINE
TEMP_INIT
VALE
INCREMENT
LIST_INST
RECU_CHAMP
INST
5.4 Remarks
The discretization in pitch of time is as follows:
10 pitches for [0. , 5.D-2] is
T = 5.D-3
19 pitch for [5.D-2, 1.D0] is
T = 5.D-2
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
7/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
6
Results of modeling B
6.1 Values
tested
Identification Reference
Aster %
difference
tolerance
Temperature (°C)
Node N1 T = 0.1s
0.330
0.330
-
0.129
1%
““T = 0.3s
0.544
0.543
-
0.149
1%
““T = 0.5s
0.682
0.681
-
0.154
1%
““T = 0.7s
0.789
0.788
-
0.092
1%
““T = 1.0s
0.918
0.920
0.222
1%
Node N8 T = 0.1s
0.330
0.330
-
0.129
1%
““T = 0.3s
0.544
0.543
-
0.149
1%
““T = 0.5s
0.682
0.681
-
0.154
1%
““T = 0.7s
0.789
0.788
-
0.092
1%
““T = 1.0s
0.918
0.920
0.222
1%
Node N121 T = 0.1s
0.004
0.004
10.931
1%
““T = 0.3s
0.071
0.071
-
0.242
1%
““T = 0.5s
0.160
0.161
0.587
1%
““T = 0.7s
0.247
0.251
1.619
1%
““T = 1.0s
0.366
0.380
3.95
1%
Node N126 T = 0.1s
0.004
0.004
10.931
1%
““T = 0.3s
0.071
0.071
-
0.242
1%
““T = 0.5s
0.160
0.161
0.587
1%
““T = 0.7s
0.247
0.251
1.619
1%
““T = 1.0s
0.366
0.380
3.95
1%
6.2 Remarks
The relative error is to the maximum of 3.95%, except for X = 1 at the moment T = 0.1s the error is 11%.
This error was obtained for the smallest value of the temperature (T = 0.004°C). This variation
be explained by the fact that the function of error in this point is 0.025347 and that uncertainty on
calculation of the function of error is unknown.
6.3 Parameters
of execution
Version: 5.03
Machine: SGI - ORIGIN 2000 - R12000
Overall dimension memory:
8 megawords
Time CPU To use: 8.30 seconds
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
8/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
7 Modeling
C
7.1
Characteristics of modeling
3D (TETRA4)
Longitudinal axis X following the trisecting one (X=Y=Z)
Limiting conditions:
- faces ABCD, ABFE
= 0
- faces EFGH, DCGH
= 0
- face BFGC
= 0
- face AEHD
Q = 1 W/m ²
Y
X
X
Z
With
B
E
0.2 m
0.2 m
C
D
G
H
N104
N107
N103
N106
N1
N6
N8
N2
N5
N7
N3
N4
N9
N105
N108
With
E
D
H
Z
X
Y
Z
X
Y
Section X = 0.
Section X = 1.
X = 1.
7.2
Characteristics of the mesh
A number of nodes:
224
A number of meshs and types: 692 TETRA4 (and 8 TRIA3)
7.3 Functionalities
tested
Controls
AFFE_MODELE
AFFE
THERMICS
3D
DEFI_MATERIAU
THER_NL
AFFE_CHAR_THER
FLUX_REP
THER_NON_LINE
TEMP_INIT
VALE
INCREMENT
LIST_INST
RECU_CHAMP
INST
7.4 Remarks
The discretization in pitch of time is as follows:
10 pitches for [0. , 5.D-2] is
T = 5.D-3
19 pitch for [5.D-2, 1.D0] is
T = 5.D-2
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
9/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
8
Results of modeling C
8.1 Values
tested
Identification Reference
Aster
Relative variation (%)
Absolute deviation (°C)
difference
tolerance
difference
tolerance
Temperatures in °C:
Face
X = 0.m
N7 node
T = 0.1s
0.330
0.3295
-
0.162%
1.%
-
0.000536
0.005
““
T = 0.3s
0.544
0.5425
-
0.273%
1.%
-
0.00149
0.005
““
T = 0.5s
0.682
0.6796
-
0.351%
1.%
-
0.00239
0.005
““
T = 0.7s
0.789
0.7861
-
0.362%
1.%
-
0.00285
0.005
““
T = 1.0s
0.918
0.9165
-
0.159%
1.%
-
0.00146
0.005
N5 node
T = 0.1s
0.330
0.3279
-
0.627%
1.%
-
0.00207
0.005
““
T = 0.3s
0.544
0.5418
-
0.406%
1.%
-
0.00221
0.005
““
T = 0.5s
0.682
0.6791
-
0.422%
1.%
-
0.00288
0.005
““
T = 0.7s
0.789
0.7858
-
0.409%
1.%
-
0.00323
0.005
““
T = 1.0s
0.918
0.9162
-
0.192%
1.%
-
0.00176
0.005
Section
X = 1.m
Node N107 T = 0.1s
0.00394
0.004140
5.085%
1.% 0.000200 0.005
““
T = 0.3s
0.0706
0.07013
-
0.665%
1.%
-
0.000470
0.005
““
T = 0.5s
0.160
0.1596
-
0.228%
1.%
-
0.000364
0.005
““
T = 0.7s
0.247
0.2488
0.730%
1.%
0.00180
0.005
““
T = 1.0s
0.366
0.3766
2.889%
1.%
0.0106
0.005
Node N108 T = 0.1s
0.00394
0.004002
1.577%
1.%
-
0.0000621
0.005
““
T = 0.3s
0.0706
0.06937
-
1.742%
1.%
-
0.00123
0.005
““
T = 0.5s
0.160
0.1586
-
0.895%
1.%
-
0.00143
0.005
““
T = 0.7s
0.247
0.2476
0.238%
1.%
0.000587
0.005
““
T = 1.0s
0.366
0.3753
2.534%
1.%
0.00928
0.005
8.2 Remarks
The selected nodes correspond to the extreme results on the same section.
8.3 Parameters
of execution
Version: 5.03
Machine: SGI - ORIGIN 2000 - R12000
Overall dimension memory:
8 megawords
Time CPU To use: 13.08 seconds
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
10/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
9 Modeling
D
9.1
Characteristics of modeling
3D (TETRA10)
Y
X
X
Z
With
B
E
C
D
G
H
0.2 m
0.2 m
Longitudinal axis X following the trisecting one (X=Y=Z)
Limiting conditions:
- faces ABCD, ABFE
= 0
- faces EFGH, DCGH
= 0
- face BFGC
= 0
- face AEHD
Q = 1 W/m ²
With
E
D
H
Z
X
Y
Section X = 0.
N4
N7
N12 N16 N25
N1
N9
N15 N17 N23
N2
N10 N13 N20 N22
N3
N6
N14 N18 N21
N5
N8
N11 N19 N24
Z
X
Y
Section X = 1
N602
N601
N605
N611
N599
N606
N610
N600
N604
N609
N603
N607
N608
X = 1.
9.2
Characteristics of the mesh
A number of nodes:
1310
A number of meshs and types: 697 TETRA10 (and 8 TRIA6)
9.3 Functionalities
tested
Controls
AFFE_MODELE
AFFE
THERMICS
3D
DEFI_MATERIAU
THER_NL
AFFE_CHAR_THER
FLUX_REP
THER_NON_LINE
TEMP_INIT
VALE
INCREMENT
LIST_INST
RECU_CHAMP
INST
9.4 Remarks
The discretization in pitch of time is as follows:
10 pitches for [0. , 5.D-2] is
T = 5.D-3
19 pitch for [5.D-2, 1.D0] is
T = 5.D-2
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
11/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
10 Results of modeling D
10.1 Values
tested
Identification Reference
Aster
Relative variation (%)
Absolute deviation (°C)
difference
tolerance
difference
tolerance
Temperatures in °C:
Face
X = 0.m
N4 node
T = 0.1s
0.330
0.3291
-
0.281%
1.%
-
0.000926
0.005
““
T = 0.3s
0.544
0.5423
-
0.318%
1.%
-
0.00173
0.005
““
T = 0.5s
0.682
0.6794
-
0.383%
1.%
-
0.00261
0.005
““
T = 0.7s
0.789
0.7860
-
0.384%
1.%
-
0.00303
0.005
““
T = 1.0s
0.918
0.9164
-
0.180%
1.%
-
0.00165
0.005
N25 node
T = 0.1s
0.330
0.3292
-
0.255%
1.%
-
0.000843
0.005
““
T = 0.3s
0.544
0.5423
-
0.314%
1.%
-
0.00171
0.005
““
T = 0.5s
0.682
0.6794
-
0.382%
1.%
-
0.00261
0.005
““
T = 0.7s
0.789
0.7860
-
0.383%
1.%
-
0.00303
0.005
““
T = 1.0s
0.918
0.9163
-
0.180%
1.%
-
0.00165
0.005
Section
X = 1.m
Node N606 T = 0.1s
0.00394
0.004331
9.913%
1.% 0.000391 0.005
““
T = 0.3s
0.0706
0.07021
-
0.551%
1.%
-
0.000389
0.005
““
T = 0.5s
0.160
0.1596
-
0.251%
1.%
-
0.000402
0.005
““
T = 0.7s
0.247
0.2488
0.710%
1.%
0.00175
0.005
““
T = 1.0s
0.366
0.3764
2.855%
1.%
0.0104
0.005
Node N611 T = 0.1s
0.00394
0.004332
9.944%
1.% 0.000392 0.005
““
T = 0.3s
0.0706
0.07021
-
0.550%
1.%
-
0.000388
0.005
““
T = 0.5s
0.160
0.1596
-
0.251%
1.%
-
0.000402
0.005
““
T = 0.7s
0.247
0.2488
0.710%
1.%
0.00175
0.005
““
T = 1.0s
0.366
0.3764
2.855%
1.%
0.0104
0.005
10.2 Remarks
The calculated results are almost identical on the nodes of the same section.
10.3 Parameters
of execution
Version: 5.03
Machine: SGI - ORIGIN 2000 - R12000
Overall dimension memory:
8 megawords
Time CPU To use: 25.88 seconds
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
12/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
11 Modeling
E
11.1 Characteristics of modeling
3D (HEXA27)
Longitudinal axis X following the trisecting one (X=Y=Z)
Limiting conditions:
- faces ABCD, ABFE
= 0
- faces EFGH, DCGH
= 0
- face BFGC
= 0
- face AEHD
Q = 1 W/m ²
N3
N85
N2
N4
N87
N1
N86 N249 N88
With
E
D
H
Z
X
Y
Z
X
Y
Section X = 0.
Section X = 1.
B
C
G
0.1 m
0.1 m
Y
X
X
Z
With
E
D
H
N44 N167 N41
N43 N165 N42
N166 N308 N168
X = 1.
11.2 Characteristics of the mesh
A number of nodes:
369
A number of meshs and types: 20 HEXA27 (and 1 QUAD9)
11.3 Functionalities
tested
Controls
AFFE_MODELE
AFFE
THERMICS
3D
DEFI_MATERIAU
THER_NL
AFFE_CHAR_THER
FLUX_REP
THER_NON_LINE
TEMP_INIT
VALE
INCREMENT
LIST_INST
RECU_CHAMP
INST
11.4 Remarks
The discretization in pitch of time is as follows:
10 pitches for [0. , 5.D-2] is
T = 5.D-3
19 pitch for [5.D-2, 1.D0] is
T = 5.D-2
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
13/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
12 Results of modeling E
12.1 Values
tested
Identification Reference
Aster
Relative variation (%)
Absolute deviation (°C)
difference
tolerance
difference
tolerance
Temperatures in °C:
Face
X = 0.m
Node N249 T = 0.1s
0.330
0.3291
-
0.283%
1.%
-
0.000933
0.005
““
T = 0.3s
0.544
0.5423
-
0.317%
1.%
-
0.00173
0.005
““
T = 0.5s
0.682
0.6794
-
0.376%
1.%
-
0.00256
0.005
““
T = 0.7s
0.789
0.7860
-
0.378%
1.%
-
0.00298
0.005
““
T = 1.0s
0.918
0.9165
-
0.168%
1.%
-
0.00154
0.005
Section
X = 1.m
Node N308 T = 0.1s
0.00394
0.004331
9.926%
1.% 0.000391
0.005
““
T = 0.3s
0.0706
0.07021
-
0.554%
1.%
-
0.000391
0.005
““
T = 0.5s
0.160
0.1596
-
0.227%
1.%
-
0.000363
0.005
““
T = 0.7s
0.247
0.2488
0.726%
1.%
0.00179
0.005
““
T = 1.0s
0.366
0.3766
2.886%
1.%
0.0106
0.005
12.2 Remarks
The calculated results are identical (to 10
7
near) on the nodes of the same section.
12.3 Parameters
of execution
Version: 5.03
Machine: SGI - ORIGIN 2000 - R12000
Overall dimension memory:
8 megawords
Time CPU To use: 9.68 seconds
Code_Aster
®
Version
5.0
Titrate:
TTNL302 infinite Wall subjected to a constant flow with variable properties
Date:
20/09/02
Author (S):
C. DURAND,
E. SCREWS, F. LEBOUVIER
Key
:
V4.22.302-A
Page:
14/14
Manual of Validation
V4.22 booklet: Non-linear transitory thermics of the linear structures
HT-66/02/001/A
13 Summary of the results
Five modelings carried out, have same cutting in the direction of propagation of
temperature, they are different only by their type of meshs.
Five modelings give results whose certain values exceed the fixed tolerance
initially (1%). The maximum change is 9.9%. It appears for the smallest value of reference
located in the middle of the wall and at the beginning of the transient.
A mesh finer associate with a finer temporal discretization should improve quality of
results.
Moreover, the reference solution utilizes a function of error whose precision is unknown.
The results are regarded as acceptable taking into account the points evoked above.
This test made it possible to test hexahedral and tetrahedral meshs in non-linear thermics
transient as well as the main following thermal controls:
·
DEFI_MATERIAU
associated the key word
THER_NL
,
allowing to define the characteristics of one
material whose characteristics vary according to the temperature (conductivity and
enthalpy),
·
THER_NON_LINE
order allowing the resolution of a thermal nonlinear problem
stationary or not.