Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
1/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Organization (S):
EDF/IMA/MN
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
V7.03.101 document
HPLV101 - Homogenization of a material
homogeneous
Summary:
This test tests, in a commonplace situation where the material is homogeneous, the thermal resolution of the problems
and mechanics stationary, with loadings corresponding to a variation in temperature and to one
imposed deformation, close to those corresponding to the elementary problems of the method
of periodic homogenization.
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
2/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
1
Problem of reference
1.1 Geometry
1
Z
N6
With
B
C
X
y
N5
N4
0
N7
N1
N2
N3
N8
1
16.410
X y Z
N4
0 0 16.410
N8 1. 1. 16.410
N3 0.5 0 16.410
1.2
Material properties
E = 1.0 MPa
= 0.3
K = 1.0 W/(m.°C)
C
p
= 0 J/(°C.m
3
)
1.3
Boundary conditions and loadings
·
Mechanics 3D:
Plan Z = 0:
dz = 0
dx = 0, Dy = 0
for the membrane loading;
for the loading of bending
Plans y = 0, y = 1:
Dy = 0
Plans X = 0, X = 1:
dx = 0
Node: O
dz = 0
(for the only loading of bending)
Loading:
membrane deformation:
imposed uniform bending:
E
E
=
=
1 0 0
0
0 0
0
0 0
0 0
0 0 0
0 0 0
Z
·
Mechanics 2D, stresses plane:
Center: X = 0
Node: O
dx = 0
Dy = 0
(these conditions do not correspond to the application of
method of homogenization).
Loading: deformation
E
=
1 0 0
0
0 0
0
0 0
uniform imposed
·
Thermics 3D and 2D:
Plan X = 0
temp = 0
(this condition does not correspond to the application of
method of homogenization).
Loading: gradient
(
)
G
=
,
1 0 0
imposed uniform.
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
3/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
2
Reference solution
2.1
Method of calculation used for the reference solution
·
In thermics: the stationary thermal problem is solved:
=
=
T.
.
,
,
K
G K
G
V
with
1
0
0
Note:
The boundary conditions chosen here are not those necessary to the method
of homogenization: one would find indeed
T
=
0
everywhere.
The solution is then (checking the conditions defined in [§1.3]):
(
)
T X y Z
X
,
=
The potential energy is then with balance:
W
HT
T
T
=
=
.
1
2
1
2
K
here
·
In mechanics: one solves the problem of elastostatic:
()
()
()
U With
v
With
v
v
=
.
.
,
,
E
W
for the cases:
loading 3D
membrane
loading 3D
of bending
loading 2D
plane stresses
E
=
1 0 0
0
0 0
0
0 0
E
=
Z 0 0
0 0 0
0 0 0
E
=
1 0 0
0
0 0
0
0 0
The solutions are:
·
in 3D, membrane loading:
(
)
(
)
U X y Z
Z
,
,
=
0 0
1
;
the potential energy with balance is:
()
()
(
)
W
pot
=
=
+
.
.
1
2
1
2
2
2
µ
U With
U
·
in 3D, loading of bending:
(
)
(
)
U X y Z
Z
,
,
=
+
0 0 2 1
2
;
(
)
W
pot
H
=
+
.
.
µ
1
2
2
3
2
3
·
in 2D, plane loading:
()
(
)
U X y
X
,
,
=
0
;
(
)
W
pot
=
2 1
2
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
4/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
3 Modeling
With
3.1
Characteristics of modeling
y
N6
B
X
With
O
C
GRNM14
Boundary conditions and loading:
Thermics:
GROUP_NO: GRNM14: TEMP: 0.0
GRAD_TEMP_INIT: FLUX_X: 1.0
Mechanics:
(plane stresses)
GROUP_NO: GRNM14: DX: 0.0
NODE: O DY: 0.0
EPSI_INIT: EPXX: 1.0
3.2
Characteristics of the mesh
A number of nodes: 8
A number of meshs and types: 1 QUAD8
3.3 Functionalities
tested
Controls
Keys
AFFE_CHAR_THER
GRAD_TEMP_INIT
ALL
FLUX_X
“YES”
[U4.25.02]
AFFE_CHAR_MECA
EPSI_INIT
ALL
EPXX
“YES”
[U4.25.01]
POST_ELEM
ENER_POT
ALL
“YES”
[U4.61.04]
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
5/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
4
Results of modeling A
4.1 Values
tested
Not
Size
CMP
Reference
Aster
% difference
Tolerance %
With
TEMP
1.0000
1.00000
0.000
10
6
With
DX
1.0000
1.00000
0.000
10
6
N6
DX
0.5000
0.50000
0.000
10
6
Net
Energy
potential
with balance
Reference
Aster
% difference
M1
Thermics
0.500000000
0.500000
10
8
M1
Mechanics
0.549450550
+0.549451
10
8
4.2 Remarks
Code_Aster provides the value of the deformation energy, equal contrary to the potential energy to
balance (elastic case).
4.3 Parameters
of execution
Version: 3.02.18
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
16 megawords
Time CPU To use:
3.7 seconds
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
6/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
5 Modeling
B
5.1
Characteristics of modeling
Z
N2
N6
y
C
B
With
O
N8
N3
N4
X
Name of the meshs of the faces:
ZEGAL0
YEGAL0
YEGAL1
XEGAL0
XEGAL1
Summits:
B C O WITH
O WITH N2 N4 B C N6 N8
C O N4 N6 A B N8 N2
Boundary conditions:
Thermics:
ZERO: DEFI_CONSTANTE (VALE: 0.0);
FCT1:DEFI_FONCTION (Nom_para:“Z”, VALE: (0.0 0.0.1.0.1.0));
GROUP_NO: XEGAL0: TEMP: 0.0
GRAD_TEMP_INIT: FLUX_X: - 1.0
Mechanics:
GROUP_NO: YEGALO: DY = 0.0
XEGAL1: DX = 0.0
YEGAL1: DY = 0.0
XEGALO: DZ = 0.0
Membrane case:
GROUP_NO: ZEGALO: DZ = 0.0
EPSI_INIT: EPXX: - 1.0
Case bending:
GROUP_NO: ZEGALO: DX = ZERO, DY = ZERO
NODE: 0 DZ = ZERO
EPSI_INIT: EPXX: FCT1
5.2
Characteristics of the mesh
A number of nodes: 20
A number of meshs and types: 1 HEXA20
5.3 Functionalities
tested
Controls
Keys
AFFE_CHAR_THER
GRAD_TEMP_INIT
ALL
FLUX_X
[U4.25.02]
AFFE_CHAR_MECA
EPSI_INIT
EPXX
[U4.25.01]
AFFE_CHAR_MECA_F
EPSI_INIT
EPXX
[U4.25.01]
POST_ELEM
ENER_POT
ALL
“YES”
[U4.61.04]
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
7/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
6
Results of modeling B
6.1 Values
tested
Case
Size
Not
Reference
Aster
% difference
Thermics
temp
N8
1.000000
1.000000
10
10
temp
N3
0.500000
0.5000000
10
10
Mechanics
dz
N4
7.03285714
7.03285714
10
10
membrane
dz
N8
7.03285714
7.03285714
10
10
Mechanics
dz
N4
57.70459285
57.70459285
10
10
bending
dz
N8
57.70459285
57.70459285
10
10
Net
Energy
potential
with balance
Reference
Aster
% difference
M1
Thermics
8.20500
8.20500
10
7
M1
Mechanics
Membrane
Bending
2.0287088
1.8210238 10
2
2.02871
1.82102 10
2
10
7
10
7
6.2 Parameters
of execution
Version: 3.05.30
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
16 megawords
Time CPU To use:
16.51 seconds
Code_Aster
®
Version
3
Titrate:
HPLV101 - Homogenization of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A
Page:
8/8
Manual of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
7
Summary of the results
The results are exact with round-off errors close, since the sought solutions belong to
the space of the finite elements selected for modeling.