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