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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
1/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
Organization (S):
EDF-R & D/AMA















Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.185



SSNV185 ­ Fissure emerging in a plate
3D of width finished with X-FEM



Summary

The purpose of this test is to validate method X-FEM [bib1] on an academic case 3D, within the framework of
linear elastic breaking process.

This test brings into play a plate 3D comprising an emerging fissure plane at right bottom. Complete calculation
as well as the extraction of the stress intensity factors is realized within the framework of method X-FEM.
mesh is healthy, the fissure being represented virtually with level sets.

Several configurations of mesh are tested and compared with the analytical solution. The same problem
treated in a conventional way (with a fissured mesh) is used as reference in order to compare the precise details of
two methods.
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
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HT-66/05/005/A
1
Problem of reference
1.1 Geometry
The structure is a plate 3D of dimensions LX = 1 m, LY = 10 m and LZ = 30 m, comprising one
emerging plane fissure length has = 5 m, being located at middle height (see [Figure 1.1-a]).
If the problem is dealt with by conventional method, the fissure is with a grid. On the other hand,
if method X-FEM is employed, the fissure is not with a grid, and the geometry is in fact
a healthy plate without fissure. The fissure will then be introduced by functions of levels (level sets)
directly in the file orders using operator DEFI_FISS_XFEM [U4.82.08]. Level
set normal (LSN = distance to the plan of cracking) makes it possible to define the plan of fissure and the level set
tangent (LST = distance to the bottom of fissure) makes it possible to define the position of the bottom of fissure.
Appear 1.1-a: Geometry of the fissured plate
One defines the points A (1, 0, 15), B (0, 0, 15) and C (1, 3, 15) which will be used to lock the rigid modes.
1.2
Properties of material
Young modulus: E= 205.000 MPa (except contrary mention)
Poisson's ratio:
= 0.
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
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Author (S):
P. MASSIN, S. GENIAUT
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2
Modeling a: fissures with a grid (conventional case)
In this modeling, it fissure is with a grid, and one uses the standard method of the finite elements
to carry out calculation. This modeling will be used as reference and will allow the comparison with
method X-FEM.
2.1
Characteristics of the mesh
The structure is modelized by a regular mesh composed of 5x30x50 HEXA8, respectively
according to axes X, y, Z (see [Figure 2.1-a]). Two superimposed surfaces are the lips of
fissure.
Appear 2.1-a: fissured Mesh
2.2
Boundary conditions and loadings
Two types of loading will be studied: a loading of traction on the faces lower and
higher of the structure, then a loading which consists in imposing a field of displacement in
any node, identical to the asymptotic field of displacement in mode I (solution of Westergaard
for an infinite medium [bib2]).
2.2.1 Loading of traction
A pressure distributed is forced on the faces lower and higher of the structure (see
[Figure 2.1-a]). The pressure is
6
10
-
=
p
AP
(
)
p
zz
-
=
, which makes it possible to request the fissure in
mode of opening I pure.
The rigid modes are locked in the following way:

· Point A is locked according to the 3 directions:
N4265
N4265
N4265
0
0
0
DX
DY
DZ
=
=

=
· The point B is locked along axis OZ:
N3751
0
DZ
=
· The point C is locked along axes OX and OZ:
N4256
N4256
0
0
DX
DZ
=

=

fissure
higher face
lower face
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
4/18
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HT-66/05/005/A
2.2.2 Loading with the asymptotic field in mode I
The asymptotic field in pure mode I, solution of a problem of elastic rupture linear is known
in an analytical way [bib2]. In the defined reference mark, this field takes the following form:
0
=
X
U
éq 2.2.2-1
(
)
cos
4
3
2
cos
2
1
-
-
+
-
=
R
E
U
y
éq
2.2.2-2
(
)
cos
4
3
2
sin
2
1
-
-
+
=
R
E
U
Z
éq
2.2.2-3
This field is imposed on all the nodes of the structure by the means of formulas in the operator
AFFE_CHAR_MECA_F [U4.44.01]. These formulas utilize the polar co-ordinates
()
,
R
in
base local at the bottom of fissure:
(
)




-
-
=
-
+
-
=
y
Z
Z
y
R
5
15
arctan
,
) ²
15
(
²
5
éq 2.2.2-4
However, it is advisable to treat separately the nodes belonging to the lips of the fissure. Indeed, for
nodes of the lower lip, the formula being used to calculate the angle
(it is not valid
would give
whereas theoretically,
is worth ­
). For the nodes of the lower lip, the value of
the angle is thus not calculated by the equation [éq 2.2.2
- 4] but is directly put at ­. For
nodes of the upper lip, the formula is nevertheless valid.

2.3
Solutions of the problem
2.3.1 Loading of traction
The stress intensity factor in mode I is given [bib3] by:


=
LY
has
F
has
K
zz
I
éq 2.3.1-1
where
B
has
B
has
B
has
B
has
has
B
B
has
F
2
cos
02
.
2
2
sin
1
37
.
0
752
.
0
2
tan
2
3
2
1
+


-
+


=


éq 2.3.1-2
The precision of this formula reaches 0.5% whatever the report/ratio
B
has
.
2.3.2 Loading with the asymptotic field in mode I
In the presence of such a loading, the theoretical value is
1
=
I
K
éq 2.3.2-1
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
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V6.04.185-A
Page:
5/18
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HT-66/05/005/A
2.4 Functionalities
tested
Option CALC_K_G of operator CALC_G_LOCAL_T [U4.82.04] allows the calculation of the factors
of intensity of stresses by the energy method “G-theta”. This functionality is tested with
the loading n°1. This case of loading is used as a basis of comparison for method X-FEM.
When one of the loads is a function or a formula (coming from AFFE_CHAR_MECA_F
[U4.44.01]), the option becomes CALC_K_G_F. This functionality is tested with the loading n°2.
Controls
CALC_G_LOCAL_T CALC_K_G
CALC_K_G_F

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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
6/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
3
Results of modeling A
3.1 Values
tested
The values of K are tested
I
along the bottom of fissure, for various crowns of fields theta.
The values of the radii inf and sup of the core are as follows:
Crown 1 Crown 2 Couronne 3 Crown 4 Couronne 5
Crown 6
Rinf 2 0.666 1 1 1 2.1
Rsup 4 1.666 2 3 4 3.9
Table 3.1-1
To test all the nodes of the bottom of fissure in only once, one tests the min and max values of K
I
on all the nodes of the bottom of fissure.
3.1.1 Loading of traction
Identification
Aster
Reference %
difference
Crown 1: MAX (K
I
)
1.051 10
7
1.120
10
7
- 6.1
Crown 1: MIN (K
I
)
1.051 10
7
1.120
10
7
- 6.1
Crown 2: MAX (K
I
)
1.049 10
7
1.120
10
7
- 6.3
Crown 2: MIN (K
I
)
1.048 10
7
1.120
10
7
- 6.3
Crown 3: MAX (K
I
)
1.051 10
7
1.120
10
7
- 6.2
Crown 3: MIN (K
I
) 1.051
10
7
1.120
10
7
- 6.2
Crown 4: MAX (K
I
)
1.051 10
7
1.120
10
7
- 6.2
Crown 4: MIN (K
I
)
1.051 10
7
1.120
10
7
- 6.2
Crown 5: MAX (K
I
)
1.051 10
7
1.120
10
7
- 6.2
Crown 5: MIN (K
I
)
1.051 10
7
1.120
10
7
- 6.2
Crown 6: MAX (K
I
)
1.051 10
7
1.120
10
7
- 6.1
Crown 6: MIN (K
I
) 1.051
10
7
1.120
10
7
- 6.1
3.1.2 Loading with the asymptotic field in mode I
Identification
Aster Reference
%
difference
Crown 1: MAX (K
I
)
0.999987 1.0
- 0.001
Crown 1: MIN (K
I
)
0.999987 1.0
- 0.001
Crown 2: MAX (K
I
)
0.998279 1.0
- 0.172
Crown 2: MIN (K
I
)
0.998279 1.0
- 0.172
Crown 3: MAX (K
I
)
1.000162 1.0
0.016
Crown 3: MIN (K
I
) 1.000162
1.0
0.016
Crown 4: MAX (K
I
)
1.000058 1.0
0.006
Crown 4: MIN (K
I
)
1.000058 1.0
0.006
Crown 5: MAX (K
I
)
1.000045 1.0
0.005
Crown 5: MIN (K
I
)
1.000045 1.0
0.005
Crown 6: MAX (K
I
)
0.999981 1.0
- 0.002
Crown 6: MIN (K
I
) 0.999981
1.0
- 0.002
3.2 Comments
The 1st loading of this modeling is used as a basis of comparison for method X-FEM.
2nd case of loading makes it possible to validate option CALC_K_G_F for the elements 3D.
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
4
Modeling b: fissures nonwith a grid (case X-FEM)
In this modeling, the fissure is not with a grid any more, but it is represented by level sets:
15
-
= Z
LSN
éq 4
- 1
y
has
LY
LST
-
-
=
éq 4
- 2
4.1
Characteristics of the mesh
The structure is modelized by a healthy, regular mesh composed of 5x30x50 HEXA8, respectively
according to axes X, y, Z in order to have the same number of elements as for the mesh of
modeling A (see [Figure 4.1-a]). Thus, the plan of fissure is in correspondence with faces
HEXA8 and bottom of fissure with edges of HEXA8.
Appear 4.1-a: healthy Mesh
4.2
Boundary conditions and loadings
Only one type of loading is studied here: it is about a pressure distributed imposed on the faces
lower and higher of the structure (identical to the 1
Er
case of loading of modeling A).
The rigid modes are locked in the following way:

· Point A is locked according to the 3 directions:
N3751
N3751
N3751
0
0
0
DX
DY
DZ
=
=

=
· The point B is locked along axis OZ:
N9276
0
DZ
=
· The point C is locked along axes OX and OZ:
N3760
N3760
0
0
DX
DZ
=

=

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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
8/18
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
4.3 Functionalities
tested
Controls
DEFI_FISS_XFEM
CALC_G_LOCAL_T CALC_K_G


5
Results of modeling B
5.1 Values
tested
The values of K are tested
I
along the bottom of fissure, for various crowns of fields theta.
The values of the radii inf and sup of the core are as follows:
Crown 1 Crown 2 Couronne 3 Crown 4 Couronne 5 Crown 6
Rinf 2 0.666 1 1 1 2.1
Rsup 4 1.666 2
3
4 3.9
Table 5.1-1
To test all the nodes of the bottom of fissure in only once, one tests the min and max values of K
I
on all the nodes of the bottom of fissure.
Identification
Aster
Reference %
difference
Crown 1: MAX (K
I
)
1.111 10
7
1.120
10
7
- 0.8
Crown 1: MIN (K
I
)
1.110 10
7
1.120
10
7
- 0.9
Crown 2: MAX (K
I
)
1.112 10
7
1.120
10
7
- 0.8
Crown 2: MIN (K
I
)
1.111 10
7
1.120
10
7
- 0.9
Crown 3: MAX (K
I
)
1.111 10
7
1.120
10
7
- 0.8
Crown 3: MIN (K
I
) 1.110
10
7
1.120
10
7
- 0.9
Crown 4: MAX (K
I
)
1.111 10
7
1.120
10
7
- 0.8
Crown 4: MIN (K
I
)
1.110 10
7
1.120
10
7
- 0.9
Crown 5: MAX (K
I
)
1.111 10
7
1.120
10
7
- 0.8
Crown 5: MIN (K
I
)
1.110 10
7
1.120
10
7
- 0.9
Crown 6: MAX (K
I
)
1.111 10
7
1.120
10
7
- 0.8
Crown 6: MIN (K
I
) 1.110
10
7
1.120
10
7
- 0.9

5.2 Comments
The results are stable for any selected crown.
With same number of elements, the precision of the results obtained with X-FEM is much better than
that obtained in the conventional case (less than 1% for X-FEM against 6% for a method
conventional).
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
9/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
6
Modeling C: fissure nonwith a grid (case X-FEM)
In this modeling, the fissure is not with a grid, but it is represented by level sets:
15
-
= Z
LSN
éq 6
- 1
y
has
LY
LST
-
-
=
éq 6
- 2

6.1
Characteristics of the mesh
The structure is modelized by a healthy, regular mesh composed of 5x31x51 HEXA8, respectively
according to axes X, y, Z. In this manner, the bottom of fissure is in the center of elements and the plan
of fissure does not correspond any more to faces of elements. [Figure 6.1-a] represents out of Oyz cut
enrichment in an area near bottom of fissure.
Appear 6.1-a: Various enrichments in bottom of fissure

6.2
Boundary conditions and loadings
Just as previously, a pressure distributed is imposed on the faces lower and higher
structure (identical to the 1
Er
case of loading of modeling A).
In order to reproduce the preceding cases, it is necessary to lock the same points A, B and C.
however here, there are no nodes in the median plane. To lock the rigid modes, it is necessary then
to impose relations between the ddls nodes just above and below the median plane
[Figure 6.2-a]:

· point A is locked according to the 3 directions:


=
+
=
+
=
+
0
0
0
3876
4031
3876
4031
3876
4031
NR
NR
NR
NR
NR
NR
DZ
DZ
DY
DY
DX
DX
·
· the point B is locked along axis OZ:
0
4041
3886
=
+
NR
NR
DZ
DZ
· the point C is locked along axes OX and OZ:



=
+
=
+
0
0
9767
9768
9767
9768
NR
NR
NR
NR
DZ
DZ
DX
DX
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Code_Aster
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
10/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
Appear 6.2-a: Conditions of Dirichlet around the median plane
[Figure 6.2-a] is a diagrammatic sight of the Oyz plan, on which the number of finite elements is not
not respected. It is simply used to include/understand the linear relations forced in order to lock them
displacements of points A and C. For the point B, one acts in the same way.

6.3 Functionalities
tested
Controls
DEFI_FISS_XFEM
CALC_G_LOCAL_T CALC_K_G
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
11/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
7
Results of modeling C
7.1 Values
tested
The values of K are tested
I
along the bottom of fissure, for various crowns of fields theta.
The values of the radii inf and sup of the core are as follows:
Crown 1 Crown 2 Couronne 3 Crown 4 Couronne 5 Crown 6
Rinf 2 0.666 1 1 1 2.1
Rsup 4 1.666 2
3
4 3.9
Table 7.1-1
To test all the nodes of the bottom of fissure in only once, one tests the min and max values of K
I
on all the nodes of the bottom of fissure.
Identification
Aster
Reference %
difference
Crown 1: MAX (K
I
)
1.084 10
7
1.120
10
7
- 3.2
Crown 1: MIN (K
I
)
1.079 10
7
1.120
10
7
- 3.7
Crown 2: MAX (K
I
)
1.087 10
7
1.120
10
7
- 3.0
Crown 2: MIN (K
I
)
1.083 10
7
1.120
10
7
- 3.4
Crown 3: MAX (K
I
)
1.083 10
7
1.120
10
7
- 3.3
Crown 3: MIN (K
I
) 1.079
10
7
1.120
10
7
- 3.7
Crown 4: MAX (K
I
)
1.084 10
7
1.120
10
7
- 3.3
Crown 4: MIN (K
I
)
1.079 10
7
1.120
10
7
- 3.7
Crown 5: MAX (K
I
)
1.083 10
7
1.120
10
7
- 3.3
Crown 5: MIN (K
I
)
1.079 10
7
1.120
10
7
- 3.7
Crown 6: MAX (K
I
)
1.084 10
7
1.120
10
7
- 3.2
Crown 6: MIN (K
I
) 1.079
10
7
1.120
10
7
- 3.7

7.2 Comments
The results are stable for any selected crown.
The precision of the results obtained is worse than for modeling B. That can be explained
by the fact that the area of enrichment is less wide here.
However, the results remain better than in the conventional case.
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
25/11/05
Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
12/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
8
Modeling D: fissure nonwith a grid (case X-FEM)
This modeling is exactly the same one as modeling B, except that the length of the fissure
is: has = 4.8333, so that the bottom of fissure does not coincide with edges of
elements.
Appear 8-a: Enrichment in an area close to the bottom of fissure
9
Results of modeling D
9.1 Values
tested
The values of K are tested
I
along the bottom of fissure, for various crowns of fields theta.
The values of the radii inf and sup of the core are as follows:
Crown 1 Crown 2 Couronne 3 Crown 4 Couronne 5 Crown 6
Rinf 2 0.666 1 1 1 2.1
Rsup 4 1.666 2
3
4 3.9
To test all the nodes of the bottom of fissure in only once, one tests the min and max values of K
I
on all the nodes of the bottom of fissure.
Identification
Aster
Reference %
difference
Crown 1: MAX (K
I
)
1.020 10
7
1.045
10
7
- 2.4
Crown 1: MIN (K
I
)
1.033 10
7
1.045
10
7
- 1.1
Crown 2: MAX (K
I
)
1.021 10
7
1.045
10
7
- 2.3
Crown 2: MIN (K
I
)
1.034 10
7
1.045
10
7
- 1.1
Crown 3: MAX (K
I
)
1.021 10
7
1.045
10
7
- 2.3
Crown 3: MIN (K
I
) 1.033
10
7
1.045
10
7
- 1.1
Crown 4: MAX (K
I
)
1.021 10
7
1.045
10
7
- 2.3
Crown 4: MIN (K
I
)
1.033 10
7
1.045
10
7
- 1.1
Crown 5: MAX (K
I
)
1.021 10
7
1.045
10
7
- 2.3
Crown 5: MIN (K
I
)
1.033 10
7
1.045
10
7
- 1.1
Crown 6: MAX (K
I
)
1.020 10
7
1.045
10
7
- 2.3
Crown 6: MIN (K
I
) 1.033
10
7
1.045
10
7
- 1.1
9.2 Comments
These results confirm that the size of the area of enrichment influences the precision of the results.
Here, the area of enrichment is intermediate between the of the same case B and case C, and precision.
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Code_Aster
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
25/11/05
Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
13/18
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
10 Modeling E: fissure nonwith a grid (case X-FEM)
In this modeling, the fissure is not with a grid, but it is represented by level sets:
15
-
= Z
LSN
éq 10
- 1
y
has
LY
LST
-
-
=
éq 10
- 2
In this modeling, the Young modulus is equal to 100 MPa.
10.1 Characteristics of the mesh
The structure is modelized by a healthy, regular mesh composed of 3x11x31 HEXA8, respectively
according to axes X, y, Z (see [Figure 10.1-a]). Such a discretization leads to a configuration
of enrichment similar to that of modeling C.
Appear 10.1-a: Mesh
10.2 Boundary conditions and loadings
One wishes to apply the same loading as the loading n°2 of modeling A, i.e.
to impose on all the nodes of the mesh the asymptotic field of displacement in mode I pure.
For all the conventional nodes (not nouveau riches), one then imposes the fields previously definite.
For the nodes nouveau riches in bottom of fissure, one seeks to impose each ddl enriched.
With this intention, one rewrites the analytical expressions of the fields of displacements to be imposed on
nodes in the base of the functions of enrichment:
0
=
X
U
éq 10.2-1
(
)


+
-
+
-
=
sin
2
sin
4
2
2
cos
2
1
1
R
R
E
U
y
éq
10.2-2
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Code_Aster
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
25/11/05
Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
14/18
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
(
)


-
-
+
=
sin
2
cos
4
4
2
sin
2
1
1
R
R
E
U
Z
éq
10.2-3
It is reminded the meeting that the base of the functions of enrichment is as follows:




sin
2
cos
,
sin
2
sin
,
2
cos
,
2
sin
R
R
R
R
éq
10.2-4
For the nodes nouveau riches by the Heaviside function, it is also necessary to carry out a calculation as a preliminary.
Let us consider a couple of nodes A and B, and note C
+
and C
-
points located on the upper lips
and lower of the fissure, the aforementioned cutting a symmetrical element of way. One is in
following configuration:
Appear 10.2-a: Heaviside Enrichment
The nodes represented by rounds [Figure 10.2-a] carry conventional ddls has and of the ddls
Heaviside
H
.
According to approximation X-FEM, the fissure passing in the middle of the elements, displacements
are written:




-
+
-
=
+
+
+
=
-
=
+
=
-
+
2
2
)
(
2
2
)
(
)
(
)
(
B
B
With
With
B
B
With
With
B
B
With
With
H
has
H
has
C
U
H
has
H
has
C
U
H
has
B
U
H
has
With
U
éq
10.2-5
By reversing this linear system, one obtains the expressions of the nodal unknown factors according to
analytically known displacements:
() ()
()
() ()
()
() ()
()
() ()
()






+
-
-
=
+
-
=
-
+
=
+
-
=
-
+
-
-
C
U
With
U
B
U
H
C
U
With
U
B
U
has
C
U
B
U
With
U
H
C
U
B
U
With
U
has
B
B
With
With
2
2
2
2
éq
10.2-6
10.3 Functionalities
tested
Controls
CALC_G_LOCAL_T CALC_K_G_F
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Code_Aster
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
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15/18
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
11 Results of modeling E
11.1 Values
tested
The values of K are tested
I
along the bottom of fissure, for various crowns of fields theta.
The values of the radii inf and sup of the core are as follows:
Crown 1 Crown 2 Couronne 3 Crown 4 Couronne 5 Crown 6
Rinf 2 0.666 1 1 1 2.1
Rsup 4 1.666 2
3
4 3.9
Table 11.1-1
To test all the nodes of the bottom of fissure in only once, one tests the min and max values of K
I
on all the nodes of the bottom of fissure.
Identification
Aster
Reference %
difference
Crown 1: MAX (K
I
)
0.9996 1.0
- 0.04
Crown 1: MIN (K
I
)
0.9996 1.0
- 0.04
Crown 2: MAX (K
I
)
1.0272 1.0
2.7
Crown 2: MIN (K
I
)
1.0272 1.0
2.7
Crown 3: MAX (K
I
)
1.0053 1.0
0.53
Crown 3: MIN (K
I
) 1.0053
1.0
0.53
Crown 4: MAX (K
I
)
1.0023 1.0
0.23
Crown 4: MIN (K
I
)
1.0023 1.0
0.23
Crown 5: MAX (K
I
)
1.0015 1.0
0.15
Crown 5: MIN (K
I
)
1.0015 1.0
0.15
Crown 6: MAX (K
I
)
9.9969 1.0
- 0.03
Crown 6: MIN (K
I
) 9.9969
1.0
- 0.03

11.2 Comments
The results are stable for any selected crown.
They make it possible to validate option CALC_K_G_F for the elements 3D X-FEM.
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Code_Aster
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
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Author (S):
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Key
:
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
12 Modeling F: fissure nonwith a grid (case X-FEM)
This modeling is exactly the same one as modeling C. the only difference is as the area
of enrichment in bottom of fissure has now a size fixed by the user, it is not thus more
limited to only one elements layer in bottom of fissure.

12.1 Enrichment in bottom of fissure
The nodes being at a distance from the bottom of fissure equal or lower than a certain criterion are
nouveau riches by the singular functions. This criterion is selected as in [bib4], equal to a tenth of
cut structure. Here, it is worth 1 m since LY are worth 10 Mr.

12.2 Functionalities
tested
Controls
DEFI_FISS_XFEM RAYON_ENRI
CALC_G_LOCAL_T CALC_K_G

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Code_Aster
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Version
8.2
Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
25/11/05
Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
17/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
13 Results of modeling F
13.1 Values
tested
The values of K are tested
I
along the bottom of fissure, for various crowns of fields theta.
The values of the radii inf and sup of the core are as follows:
Crown 1 Crown 2 Couronne 3 Crown 4 Couronne 5 Crown 6
Rinf 2 0.666 1 1 1 2.1
Rsup 4 1.666 2
3
4 3.9
Table 13.1-1
To test all the nodes of the bottom of fissure in only once, one tests the min and max values of K
I
on all the nodes of the bottom of fissure.
Identification
Aster
Reference %
difference
Crown 1: MAX (K
I
)
1.109 10
7
1.120
10
7
- 1.0
Crown 1: MIN (K
I
)
1.104 10
7
1.120
10
7
- 1.5
Crown 2: MAX (K
I
)
1.111 10
7
1.120
10
7
- 0.8
Crown 2: MIN (K
I
)
1.106 10
7
1.120
10
7
- 1.2
Crown 3: MAX (K
I
)
1.110 10
7
1.120
10
7
- 0.9
Crown 3: MIN (K
I
) 1.105
10
7
1.120
10
7
- 1.4
Crown 4: MAX (K
I
)
1.110 10
7
1.120
10
7
- 1.0
Crown 4: MIN (K
I
)
1.105 10
7
1.120
10
7
- 1.4
Crown 5: MAX (K
I
)
1.110 10
7
1.120
10
7
- 1.0
Crown 5: MIN (K
I
)
1.104 10
7
1.120
10
7
- 1.4
Crown 6: MAX (K
I
)
1.109 10
7
1.120
10
7
- 1.0
Crown 6: MIN (K
I
) 1.104
10
7
1.120
10
7
- 1.5

13.2 Comments
The results are stable for any selected crown.
The precision of the results obtained is better than for modeling C. That proves the influence
beneficial of the increase size of the area of enrichment, with identical mesh.
However, if it one compares with the precision of modeling B (lower than 1%) one could
to be astonished not to find better results with the fixed area. The explanation is in [bib4].
Indeed, on modeling B, the approximation of displacement is exactly in
R
on one
lay down element around the bottom. On the other hand, in modeling F, the approximation is
elt nouveau riches
R
on the area of enrichment. In this case (relatively coarse mesh), the approximation by one
summon square roots on a wide area is worse than the approximation by only one
square root on a more restricted area.

However, when one refines sufficiently the mesh, precision obtained with an enrichment
on a fixed area becomes better than that obtained with an enrichment on only one layer
elements.
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Code_Aster
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Titrate:
SSNV185 ­ Fissure emerging in 3D with X-FEM
Date:
25/11/05
Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.185-A
Page:
18/18
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A
14 Summaries of the results
The objectives of this test are achieved:
·
To validate on a simple case the taking into account of singular enrichment in bottom of fissure
with method X-FEM.
·
To validate the calculation of the stress intensity factors (here only mode I) for
elements X-FEM, whatever the load (fixed or function).
It will be retained that the use of method X-FEM makes it possible to appreciably improve the precision of
calculation of K
I
, and that the aforementioned increases when the area of enrichment is not restricted with one
only layer of elements in bottom of fissure.


15 Bibliography
[1]
MASSIN P., GENIAUT S.: Method X-FEM, Manual of reference of Code_Aster, [R7.02.12]
[2]
SCREW E.: Calculation of the coefficients of intensity of stresses, Manual of reference of
Code_Aster, [R7.02.05]
[3]
BARTHELEMY B.: Concepts practice breaking process, Eyrolles, 1980.
[4]
LABORDE P., APPLE TREE J., FOX Y., SALÜN Mr.: “High-order extended finite element
method for cracked domains ", International Newspaper for Numerical Methods in Engineering, 64
(3), 354-381, 2005.