Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
1/14

Organization (S): EDF/IMA/MN

Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
Document: V4.04.100

TPLV100 - Cylindre subjected to conditions
with the nonaxisymmetric limits

Summary:

It is about a test in stationary thermics with modeling of Fourier.

This test validates all the elements of Fourier in thermics (5 different modelings) with various types of
boundary conditions: imposed temperature, exchange, imposed flow, heat source.

The interest of the test, in addition to the validation of Fourier thermics, lies in the following points:

· comparison between the results and an analytical solution on various harmonics of Fourier (1, 2 and
3),
· homogeneity of the elements between them.

Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
2/14

1
Problem of reference

1.1 Geometry

Z
Z
E
G
D
C
R
R
With
B
F


Radius of the cylinder R = 1 Mr.

1.2 Properties
of
materials

= 1 W/m °C

1.3
Boundary conditions and loadings

[EA]: imposed temperature
T = To = 0.°C
[BC]: imposed flow
= O = 2. W/m2 °C
[CD]: exchange
h= 2. W/m2 °C

Text = 2. °C

Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
3/14

2
Reference solution

2.1
Method of calculation used for the reference solution

T (R, Z,) = r2 cos L

with L number of the harmonic of Fourier

- T = (l2 - 4) cos L = S
- 2r cos L
R

= -


T = 0.+ (lrsinl)
[
on AB] and [ED]
R R
0
= N = 0.

[
on BC]
0
= 2R = 2.
[
on CD]
2
NR = 2R =
(2R2 - R2) = (hText - T)
R
2
from where H

=
= 2.
R
T
2
ext.
= 2R
= 2.

Only the source term varies according to harmonic (SSL (R, Z) = l2 - 4).

In following modelings, one will solve the problem on harmonics 1, 2 and 3.

2.2
Results of reference

Temperatures and flow at the points B, C, D, F, G.

Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
4/14

3 Modeling
With

3.1
Characteristics of modeling

AXIS-FOURIER (TRIA6)

y (Z)
R Z node
To 0. 0. N1
G
B 1. 0. N7
E
D
C 1. 0.5
N8
D 1. 1. N9
C
E 0. 1. N3
F 0.5
0. N4
X (R)
With
G 0.5
1. N6
F
B


The axes of description of the grid are X (R) and y (Z).

Mode - Fourier: 1 T (A) = 0.

S = - 3.
on all the field
[BC]:
= 2.
[CD]:
H = 2. Text = 2.

3.2 Characteristics

grid

A number of nodes: 25.
A number of meshs and types: 8 TRIA6

3.3 Functionalities
tested

Commands

Keys
THERMAL AFFE_MODELE
“AXIS_FOURIER”
TOUT
[U4.22.01]
AFFE_CHAR_THER TEMP_IMPO
NOEUD

[U4.25.02]
FLUX_REP
GROUP_MA


ECHANGE
GROUP_MA


SOURCE
TOUT


CALC_MATR_ELEM “RIGI_THER”
MODE_FOURIER
[U4.41.01]
CALC_VECT_ELEM “CHAR_THER”


[U4.41.02]
ASSE_MATRICE


[U4.42.02]
ASSE_VECTEUR


[U4.42.03]
FACT_LDLT

[U4.51.01]
RESO_LDLT

[U4.51.02]
CALC_CHAM_ELEM “FLUX_ELNO_TEMP”
MODE_FOURIER
[U4.61.01]
COMB_CHAM_NO COMB_FOURIER

[U4.53.02]
COMB_CHAM_ELEM COMB_FOURIER


[U4.53.03]
POST_RELEVE CHAM_GD
“EXTRACTION”

[U4.74.03]

3.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM.

The use of commands COMB_CHAM_NO and COMB_CHAM_ELEM key word COMB_R is not one
recombination of Fourier but a simple validation of this key word.
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
5/14

4
Results of modeling A

4.1 Values
tested

Identification
Reference
Aster %
difference
= 0
T (B)
1.
0.9981
­ 0.19
T
(F)
0.25
0.2484
­ 0.66

R (B)
­ 2 ­ 1.993
­ 0.36

R (F)
­ 1. ­ 0.9924
­ 0.76

(B)
1. 0.9996
­ 0.04

(F)
0.5 0.4982
­ 0.37

Z (B)
0. ­ 5.
10­3 -

Z (F)
0. 7.
10­4 -





= 45
T (B)
0.7071
0.7057
­ 0.192
T
(F)
0.177
0.1756
­ 0.65

R (B)
­ 1.414 ­ 1.4018
­ 0.87

R (F)
­ 0.7071 ­ 0.6848 3.15

(B)
­ 0.707 ­ 0.7069 0.027

(F)
­ 0.3535 ­ 0.3512 ­ 0.65

Z (B)
0. 0.36
10­3 -

Z (F)
0. 0.12
10­2 -





= 135
T (B)
­ 0.707
­ 0.7057
0.19
T
(F)
­ 0.177
­ 0.1756
0.65

R (B)
1.414 1.4018
­ 0.87

R (F)
0.707 0.685
­ 3.15

(B)
­ 0.707 ­ 0.7069 0.027

(F)
­ 0.3535 ­ 0.3533 0.06

Z (B)
0. ­ 0.36
10­3 -

Z (F)
0. ­ 0.12
10­2 -

4.2 Remarks

The values of flows to the nodes are realized on the elements containing this node.

It is noticed that the exact solution is not found. This is with the fact that numerical integration
thermal matrix of rigidity is approximate (formula at 3 points of GAUSS). If one were used
formulate at 6 points, one would find the solution exactly.

4.3 Parameters
of execution

Version: 4.00.02



Machine: CRAY C90

System UNICOS:
8.04
Obstruction memory:
8 megawords
Time CPU To use:
7.0 seconds
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
6/14

5 Modeling
B

5.1
Characteristics of modeling

AXIS-FOURIER (QUAD8)

y (Z)
R Z node
To 0. 0. N1
G
B 1. 0. N4
E
D
C 1. 0.5
N5
D 1. 1. N6
C
E 0. 1. N3
F 0.5
0. N7
X (R)
With
G 0.5
1. N12
F
B


The axes of description of the grid are X (R) and y (Z).

Mode - Fourier: 2 T (A) = 0.
No the term source bus S L (R, Z)

=.
0 for L = 2

[BC]: = 2.
[CD]: H = 2. Text = 2.

5.2 Characteristics

grid

A number of nodes: 13.
A number of meshs and types: 2 QUAD8

5.3 Functionalities
tested

Commands

Keys
“THERMAL” AFFE_MODELE “AXIS_FOURIER”
TOUT
[U4.22.01]
AFFE_CHAR_THER TEMP_IMPO
NOEUD

[U4.25.02]
FLUX_REP
GROUP_MA


ECHANGE
GROUP_MA


CALC_MATR_ELEM “RIGI_THER”
MODE_FOURIER
[U4.41.01]
CALC_VECT_ELEM “CHAR_THER”


[U4.41.02]
ASSE_MATRICE


[U4.42.02]
ASSE_VECTEUR


[U4.42.03]
FACT_LDLT

[U4.51.01]
RESO_LDLT

[U4.51.02]
CALC_CHAM_ELEM “FLUX_ELNO_TEMP”
MODE_FOURIER
[U4.61.01]

5.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM.
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
7/14

6
Results of modeling B

6.1 Values
tested

Identification Reference
Aster %
difference
T (B) 1. 1.
0.
T (C) 1. 1.
0.
T (D) 1. 1.
0.
T (F) 0.25 0.25
0.
T (G) 0.25 0.25
0.
R (B)
­ 2. ­ 2.
0.
R (C)
­ 2. ­ 2.
0.
R (D)
­ 2. ­ 2.
0.
R (F)
­ 1. ­ 1.
0.
R (G)
­ 1. ­ 1.
0.
(B)
2. 2.
0.
(C)
2. 2.
0.
(D)
2. 2.
0.
(F)
1. 1.
0.
(G)
1. 1.
0.
Z (B)
0.
2.10­15
0.
Z (C)
0.
­ 1.2 10­14
0.
Z (D)
0.
­ 1.2 10­13
0.
Z (F)
0.
­ 1.4 10­14
0.
Z (G)
0.
­ 3.7 10­15
0.

6.2 Remarks

The analytical solution is found exactly.

6.3 Parameters
of execution

Version: 4.00.02



Machine: CRAY C90

System UNICOS:
8.04
Obstruction memory:
8 megawords
Time CPU To use:
3.8 seconds

Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
8/14

7 Modeling
C

7.1
Characteristics of modeling

AXIS-FOURIER (QUAD9)

y (Z)
R Z node
To 0. 0. N1
G
B 1. 0. N4
E
D
C 1. 0.5
N5
D 1. 1. N6
C
E 0. 1. N3
F 0.5
0. N7
X (R)
With
G 0.5
1. N12
F
B


The axes of description of the grid are X (R) and y (Z).

Mode - Fourier: 3 T (A) = 0.

S = 5.
on all the field
[BC]:
= 2.
[CD]:
H = 2. Text = 2.

7.2 Characteristics

grid

A number of nodes: 15.
A number of meshs and types: 2 QUAD9

7.3 Functionalities
tested

Commands

Keys
“THERMAL” AFFE_MODELE “AXIS_FOURIER”
TOUT
[U4.22.01]
AFFE_CHAR_THER TEMP_IMPO
NOEUD

[U4.25.02]
FLUX_REP
GROUP_MA


ECHANGE
GROUP_MA


SOURCE
TOUT


CALC_MATR_ELEM “RIGI_THER”
MODE_FOURIER
[U4.41.01]
CALC_VECT_ELEM “CHAR_THER”


[U4.41.02]
ASSE_MATRICE


[U4.42.02]
ASSE_VECTEUR


[U4.42.03]
FACT_LDLT

[U4.51.01]
RESO_LDLT

[U4.51.02]
CALC_CHAM_ELEM “FLUX_ELNO_TEMP”
MODE_FOURIER
[U4.61.01]

7.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM.
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
9/14

8
Results of modeling C

8.1 Values
tested

Identification Reference
Aster %
difference
T (B) 1. 1.
0.
T (C) 1. 1.
0.
T (D) 1. 1.
0.
T (F) 0.25 0.25 0.
T (G) 0.25 0.25
0.
R (B)
­ 2. ­ 2.
0.
R (C)
­ 2. ­ 2.
0.
R (D)
­ 2. ­ 2.
0.
R (F)
­ 1. ­ 1.
0.
R (G)
­ 1. ­ 1.
0.
(B)
3. 3.
0.
(C)
3. 3.
0.
(D)
3. 3.
0.
(F)
1.5 1.5
0.
(G)
1.5 1.5
0.
Z (B)
0.
1.2 10­14
0.
Z (C)
0.
5.5 10­14
0.
Z (D)
0.
4.6 10­15
0.
Z (F)
0.
­ 1.1 10­15
0.
Z (G)
0.
1.8 10­14
0.

8.2 Remarks

The analytical solution is found exactly.

8.3 Parameters
of execution

Version: 4.00.02



Machine: CRAY C90

System UNICOS:
8.04
Obstruction memory:
8 megawords
Time CPU To use:
3.9 seconds
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
10/14

9 Modeling
D

9.1
Characteristics of modeling

AXIS-FOURIER (QUAD4)

y (Z)
50 subdivisions in R
R Z node
To 0. 0. N1
G
D
B 1. 0. N151
E
C 1. 0.5
N152
D 1. 1. N153
C
E 0. 1. N3
F 0.5
0. N76
X (R)
G 0.5
1. N78
With
F
B


The axes of description of the grid are X (R) and y (Z).

Mode - Fourier: 2 T (A) = 0.

S = 0.
on all the field
[BC]:
= 2.
[CD]:
H = 2. Text = 2.

9.2 Characteristics

grid

A number of nodes: 153
A number of meshs and types: 100 QUAD4

9.3 Functionalities
tested

Commands

Keys
“THERMAL” AFFE_MODELE “AXIS_FOURIER”
TOUT
[U4.22.01]
AFFE_CHAR_THER TEMP_IMPO
NOEUD

[U4.25.02]
FLUX_REP
GROUP_MA


ECHANGE
GROUP_MA


SOURCE
TOUT


CALC_MATR_ELEM “RIGI_THER”
MODE_FOURIER
[U4.41.01]
CALC_VECT_ELEM “CHAR_THER”


[U4.41.02]
ASSE_MATRICE


[U4.42.02]
ASSE_VECTEUR


[U4.42.03]
FACT_LDLT

[U4.51.01]
RESO_LDLT

[U4.51.02]
CALC_CHAM_ELEM “FLUX_ELNO_TEMP”
MODE_FOURIER
[U4.61.01]

9.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM.
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
11/14

10 Results of modeling D

10.1 Values
tested

Identification Reference
Aster %
difference
T (B) 1. 0.9998
­ 2.10­3
T (C) 1. 0.9998
­ 2.10­3
T (D) 1. 0.9998
­ 2.10­3
T (F) 0.25
0.2498
­ 0.02
T (G) 0.25
0.2498
­ 0.02
R (B)
­ 2. ­ 1.9800 1.
R (C)
­ 2. ­ 1.9800 1.
R (D)
­ 2. ­ 1.9801 1.
R (F)
­ 1. ­ 1.0000 4. 10­3
R (G)
­ 1. ­ 1.0000 4. 10­3
(B)
2. 2.0000 4. 10­3
(C)
2. 2.0000 4. 10­3
(D)
2. 2.0001 5. 10­3
(F)
1. 1.0000 4. 10­3
(G)
1. 1.0000 4. 10­3
Z (B)
0.
­ 2.10­5
-
Z (C)
0.
­ 2.10­5
-
Z (D)
0.
­ 2.10­5
-
Z (F)
0.
­ 2.10­8
-
Z (G)
0.
­ 2.10­8
-

10.2 Remarks

The bad precision recorded on R (B), R (C), R (D) is explained by the fact why B, C and D are
nodes of the edge, therefore flows are not realized on adjacent elements in the direction
variation in temperature (direction R).

This phenomenon is not found on, because is balanced by 1/R.

10.3 Parameters
of execution

Version: 4.00.02



Machine: CRAY C90

System UNICOS:
8.04
Obstruction memory:
8 megawords
Time CPU To use:
5.8 seconds
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
12/14

11 Modeling
E

11.1 Characteristics of modeling

AXIS-FOURIER (TRIA3)

y (Z)
50 subdivisions in R
R Z node
To 0. 0. N1
G
D
B 1. 0. N151
E
C 1. 0.5
N152
D 1. 1. N153
C
E 0. 1. N3
F 0.5
0. N76
X (R)
G 0.5
1. N78
With
F
B


The axes of description of the grid are X (R) and y (Z).

Mode - Fourier: 2 T (A) = 0.

S = 0.
on all the field
[BC]:
= 2.
[CD]:
H = 2. Text = 2.

11.2 Characteristics

grid

A number of nodes: 153
A number of meshs and types: 200 TRIA3

11.3 Functionalities
tested

Commands

Keys
“THERMAL” AFFE_MODELE “AXIS_FOURIER”
TOUT
[U4.22.01]
AFFE_CHAR_THER TEMP_IMPO
NOEUD

[U4.25.02]
FLUX_REP
GROUP_MA


ECHANGE
GROUP_MA


SOURCE
TOUT


CALC_MATR_ELEM “RIGI_THER”
MODE_FOURIER
[U4.41.01]
CALC_VECT_ELEM “CHAR_THER”


[U4.41.02]
ASSE_MATRICE


[U4.42.02]
ASSE_VECTEUR


[U4.42.03]
FACT_LDLT

[U4.51.01]
RESO_LDLT

[U4.51.02]
CALC_CHAM_ELEM “FLUX_ELNO_TEMP”
MODE_FOURIER
[U4.61.01]

11.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM.
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
13/14

12 Results of modeling E

12.1 Values
tested

Identification Reference
Aster %
difference
T (B) 1. 0.9995 ­ 0.049
T (C) 1. 0.9999 ­ 7.10­3
T (D) 1. 1.0003 0.033
T (F) 0.25
0.2500 9.10­3
T (G) 0.25
0.2498 ­ 0.077
R (B)
­ 2. ­ 1.977
­ 1.14
R (C)
­ 2. ­ 1.9819
0.90
R (D)
­ 2. ­ 1.9856
0.72
R (F)
­ 1. ­ 0.993
0.68
R (G)
­ 1. ­ 1.007
0.68
(B)
2. 1.9992 ­ 0.04
(C)
2. 2.0000 -
(D)
2. 2.0008 0.04
(F)
1. 1.0004 0.04
(G)
1. 0.9995 ­ 0.05
Z (B)
0.
­ 4.10­3

Z (C)
0.
­ 4.10­3

Z (D)
0.
­ 4.10­3

Z (F)
0.
1.10­3

Z (G)
0.
1.10­3


12.2 Remarks

The bad precision recorded on R (B), R (C), R (D) is explained by the fact why B, C and D are
nodes of the edge, therefore flows are not realized on adjacent elements in the direction
variation in temperature (direction R).

This phenomenon is not found on, because is balanced by 1/R.

12.3 Parameters
of execution

Version: 4.00.02



Machine: CRAY C90

System UNICOS:
8.04
Obstruction memory:
8 megawords
Time CPU To use:
6.1 seconds
Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
TPLV100 Cylindre subjected to boundary conditions

Date:
22/12/98
Author (S):
X. DESROCHES Key
:
V4.04.100-C Page:
14/14

13 Summary of the results

This problem is correctly solved:

· whatever the number of harmonic of Fourier,
· by the various types of elements (degree 1 or 2).

Handbook of Validation
V4.04 booklet: Stationary thermics of the voluminal structures
HI-75/01/010/A

Outline document