Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
V7.03.101 document
HPLV101 - Homogénéisation 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.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
2/8
1
Problem of reference
1.1 Geometry
Z
N4
N5
N6
N3
X y Z
N7
N4
0 0 16.410
N2
y
N8
N8 1. 1. 16.410
N3 0.5 0 16.410
0
N1
C
16.410
1
X
With
B
1
1.2
Material properties
E = 1.0 MPa
= 0.3
K = 1.0 W/(m.°C)
CP = 0 J/(°C.m3)
1.3
Boundary conditions and loadings
· Mechanics 3D:
Plan Z = 0:
dz = 0
for the membrane loading;
dx = 0, Dy = 0
for the loading of inflection
Plans y = 0, y = 1:
Dy = 0
Plans X = 0, X = 1:
dx = 0
Node: O
dz = 0
(for the only loading of inflection)
1 0
0
membrane deformation:
E = 0
0
0
Loading:
0 0
0
Z 0
0
imposed uniform inflection:
E = 0 0
0
0 0 0
· Mechanics 2D, constraints plane:
Center: X = 0
dx = 0
(these conditions do not correspond to the application of
Node: O
Dy = 0
method of homogenization).
1 0
0
Loading: deformation E = 0
0
0 uniform imposed
0 0 0
· 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.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
3/8
2
Reference solution
2.1
Method of calculation used for the reference solution
· In thermics: the stationary thermal problem is solved:
1
T.K. =
.
G K., V, with G
=
0
0
Note:
The boundary conditions chosen here are not those necessary to the method
of homogenization: one would find T indeed = 0 everywhere.
The solution is then (checking the conditions defined in [§1.3]): T (X, y, Z) = X
1
1
The potential energy is then with balance: W HT =
T
.K. T
=
here
2
2
· In mechanics: one solves the problem of elastostatic:
(U) .A. (v) = E.A. (v), v
W
,
for the cases:
loading 3D
loading 3D
loading 2D
membrane
of inflection
plane constraints
1 0
0
Z 0
0
1 0
0
E = 0
0
0
E = 0 0
0
E = 0
0
0
0 0 0
0 0 0
0 0 0
The solutions are:
Z
· in 3D, membrane loading: U (X, y, Z) =,
0,
0
(
;
1 )
the potential energy with balance is:
1
2
W pot =
(U) .A. (U) =
.
(2µ)
2
1
+
2
+ z2
· in 3D, loading of inflection: U (X, y, Z) =,
0,
0
;
2 (1 )
2
h3
W pot =
.
(2µ)
.
1
+
2
3
· in 2D, plane loading: U (X, y) = ( X,)
0;
W pot = 2 (1 2)
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
4/8
3 Modeling
With
3.1
Characteristics of modeling
y
N6
C
B
GRNM14
O
With
X
Boundary conditions and loading:
Thermics:
GROUP_NO: GRNM14: TEMP: 0.0
GRAD_TEMP_INIT: FLUX_X: 1.0
Mechanics:
GROUP_NO: GRNM14: DX: 0.0
(plane constraints)
NOEUD: O DY: 0.0
EPSI_INIT: EPXX: 1.0
3.2
Characteristics of the grid
A number of nodes: 8
A number of meshs and types: 1 QUAD8
3.3 Functionalities
tested
Commands
Keys
AFFE_CHAR_THER
GRAD_TEMP_INIT
TOUT
“OUI”
[U4.25.02]
FLUX_X
AFFE_CHAR_MECA
EPSI_INIT
TOUT
“OUI”
[U4.25.01]
EPXX
POST_ELEM
ENER_POT
TOUT
“OUI”
[U4.61.04]
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
5/8
4
Results of modeling A
4.1 Values
tested
Not
Size
Reference
Aster
% difference
Tolerance %
CMP
With
TEMP
1.0000
1.00000
0.000
106
With
DX
1.0000
1.00000
0.000
106
N6
DX
0.5000
0.50000
0.000
106
Net
Energy
Reference
Aster
% difference
potential
with balance
M1
Thermics
0.500000000
0.500000
108
M1
Mechanics
0.549450550
+0.549451
108
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
Obstruction memory:
16 megawords
Time CPU To use:
3.7 seconds
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
6/8
5 Modeling
B
5.1
Characteristics of modeling
Z
N4
N6
N3
N2
N8
O
C
y
With
B
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:
ZERO: DEFI_CONSTANTE (VALE: 0.0);
FCT1:DEFI_FONCTION (Nom_para:“Z”, VALE: (0.0 0.0.1.0.1.0));
Thermics:
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 inflection:
GROUP_NO: ZEGALO: DX = ZERO, DY = ZERO
NOEUD: 0 DZ = ZERO
EPSI_INIT: EPXX: FCT1
5.2
Characteristics of the grid
A number of nodes: 20
A number of meshs and types: 1 HEXA20
5.3 Functionalities
tested
Commands
Keys
AFFE_CHAR_THER
GRAD_TEMP_INIT
TOUT
[U4.25.02]
FLUX_X
AFFE_CHAR_MECA
EPSI_INIT
EPXX
[U4.25.01]
AFFE_CHAR_MECA_F
EPSI_INIT
EPXX
[U4.25.01]
POST_ELEM
ENER_POT
TOUT
“OUI”
[U4.61.04]
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
7/8
6
Results of modeling B
6.1 Values
tested
Case
Size
Not
Reference
Aster
% difference
Thermics
temp
N8
1.000000
1.000000
1010
temp
N3
0.500000
0.5000000
1010
Mechanics
dz
N4
7.03285714
7.03285714
1010
membrane
dz
N8
7.03285714
7.03285714
1010
Mechanics
dz
N4
57.70459285
57.70459285
1010
inflection
dz
N8
57.70459285
57.70459285
1010
Net
Energy
Reference
Aster
% difference
potential
with balance
M1
Thermics
8.20500
8.20500
107
M1
Mechanics
Membrane
2.0287088
2.02871
107
Inflection
1.8210238 102
1.82102 102
107
6.2 Parameters
of execution
Version: 3.05.30
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
16 megawords
Time CPU To use:
16.51 seconds
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Code_Aster ®
Version
3
Titrate:
HPLV101 - Homogénéisation of a homogeneous material
Date:
21/05/96
Author (S):
I. EYMARD, F. VOLDOIRE
Key:
V7.03.101-A Page:
8/8
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.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HI-75/96/032/A
Outline document