Code_Aster ®
Version
8.2
Titrate:
SSNV182 ­ Bloc in contact with X-FEM


Date:
25/11/05
Author (S):
S. GENIAUT, Key P. MASSIN
:
V6.04.182-B Page:
1/10

Organization (S): EDF-R & D/AMA

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.182

SSNV182 ­ Bloc with interface in rubbing contact
with X-FEM

Summary

The purpose of this test is to validate the taking into account of the contact on the lips of the fissure, while being limited if
the fissure crosses the structure completely. The contact is taken into account by the method continues [bib1]
adapted to the framework of method X-FEM [bib2].

This test brings into play a parallelepipedic block in compression. The interface the beam is represented by one
levet set within the framework of X-FEM. One takes into account several angular positions of the interface: = 0°
(the interface follows the faces of the elements) and = 22.5° (the interface cuts the elements). By taking a coefficient
of friction of sufficiently high Coulomb so that there is adherence, one finds the solution of same
problem without interface.

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

Code_Aster ®
Version
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Titrate:
SSNV182 ­ Bloc in contact with X-FEM


Date:
25/11/05
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S. GENIAUT, Key P. MASSIN
:
V6.04.182-B Page:
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1
Problem of reference

1.1 Geometry

The structure is a right at square base and healthy parallelepiped. Dimensions of the block
(see [Figure 1.1-a]) are: LX = 5 m, LY = 20 m and LZ = 20 Mr. It does not comprise any fissure.

The interface is introduced by functions of levels (level sets) directly into the file
order using operator DEFI_FISS_XFEM [U4.82.08]. The interface is present in the middle of
the structure by the means of its representation by the level sets. The level set normal (LSN) allows
to define a plane interface forming an angle with the Oxy plan by the following equation:

LSN= Z - (aY + b)







éq 1.1-1

LZ
LY
where A is the slope of the interface, that is to say has = - tan ()
and B =
- has
.
2
2

Appear 1.1-a: Géométrie of the bar and positioning of the interface

1.2
Properties of material

Young modulus: E= 100 MPa
Poisson's ratio: = 0.

1.3
Boundary conditions and loadings

The nodes of the lower face of the bar are embedded and a displacement UZ = - 10-6 m is
imposed on those of the higher face which corresponds to a loading in pressure along axis Z.
Displacements along axes X and are blocked there for the nodes of the upper surface.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

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SSNV182 ­ Bloc in contact with X-FEM


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1.4
Characteristics of the grid

The structure is modelled by a regular grid composed of 5x20x20 HEXA8 [Figure 1.4-a].


Appear 1.4-a: Grid

This grid is composed of linear finite elements. However, within the framework of the continuous method
[bib1] with X-FEM [bib2], it is necessary to pass to a little special linear elements. These
elements have linear functions of form and a quadratic mesh support. On these elements, them
nodes node carry the unknown factors of displacement, and the nodes medium carry the dependant unknown factors
with the contact. Moreover, when the interface follows the edge of an element, its nodes node carry too
unknown factors of contact.

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

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SSNV182 ­ Bloc in contact with X-FEM


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:
V6.04.182-B Page:
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2
Modeling a: interfaces right

In this modeling, one represents a right interface, the angle is worth 0 then. The interface coincides
with the faces of certain finite elements.

2.1 Resolution
analytical

The interface being right, and the state of uniaxial pressing and normal to the interface, it does not have there
possible slip. The solution of the problem is that of the same problem without interface.
The constraint in the structure is:

UZ
= E









éq 2.1-1
zz
LZ

and the value of the contact pressure on the interface is:

=









éq 2.1-2
zz

With the numerical values previously introduced, = - 5.0 Pa.

2.2 Functionalities
tested

Commands



DEFI_FISS_XFEM CONTACT




This case does not require the activation of friction. Under key word CONTACT of the operator
DEFI_FISS_XFEM, one stipulates FROTTEMENT=' SANS' then.

Moreover, as of the first iteration of the active constraints, one makes the assumption that the points of contact
have a contacting statute. This is possible by specifying CONTACT_INIT=' OUI'.
If not, at the end of the first iteration, the contact not being activated, the higher block re-enters in
lower block but the two blocks did not become deformed. Their state of stresses is thus null, and it
is then necessary to choose a total criterion (RESI_GLOB_MAXI) for the convergence of the algorithm of
Newton-Raphson [bib3], criterion which is likely to be unsuited in the continuation of calculations when the contact
will be activated.
To avoid that, and to have a relative criterion, one needs a state of stresses not no one as of the first
iteration, and thus to activate the contact as of the beginning.

The algorithm of the active constraints thus converges in an iteration.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

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3
Results of modeling A

3.1 Values
tested

One tests the values of the normal pressure of contact after convergence of the iterations of
operator STAT_NON_LINE and of the loop on the active constraints. One tests all the points of
contact, which corresponds to the nodes of the grid on the interface. It is checked that one finds them well
values determined with [§2.1].

Identification Reference
Aster %
difference
LAGS_C for all the nodes of
- 5.00
- 5.00
0.00
the interface

To test all the nodes in only once, the MIN and the MAX of the column are tested.

3.2 Comments

This modeling shows the possibilities of the continuous method of contact applied to the framework
X-FEM. The advantage is that the procedure of pairing is intrinsic with method X-FEM since
here, there are not really surface Master and slave considering whom one has only one surface.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
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SSNV182 ­ Bloc in contact with X-FEM


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4
Modeling b: interfaces leaning

In this modeling, one represents a leaning interface. The angle is worth 22.5°, that is to say a slope has
being worth - 1/2. The interface does not coincide any more with the faces of the finite elements, and cuts them now
elements. The normal with the interface is noted N and the tangent vector is noted:

0
0




N = 1 5, = 2 5






éq 4-1








2 5
- 1 5

4.1 Resolution
analytical

The interface being leaning, it is likely y to have slip. To avoid that, one forces adherence in
choosing a coefficient of friction of sufficiently high Coulomb. Theoretically, it is enough to
to take:
µ > tan ()







éq 4.1-1

Thus, the solution of the problem remains identical to that of the same problem without interface.
The constraint in the structure is always that of [éq 2.1-1], and the value of the contact pressure
on the interface is a function of normal N to the interface:

= N N = N N
éq
4.1-2
Z
zz Z
where nz is the component according to Z of N.

The semi-multiplier of friction is defined by:

R =
µ









éq 4.1-3

With the density of tangential stress being written as follows:

R =









éq 4.1-4
(N)

From where:
1 N
1
=
=
Z



éq
4.1-5
µ N N





µ nz
Μ = 1 is taken.
With the numerical values previously introduced, = - 0
.
4 Pa and = - 5
.
0.

4.2 Functionalities
tested

Commands



DEFI_FISS_XFEM CONTACT




This case requires the activation of friction. Under key word CONTACT of the operator
DEFI_FISS_XFEM, one stipulates FROTTEMENT=' COULOMB' then.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

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SSNV182 ­ Bloc in contact with X-FEM


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5
Results of modeling B

5.1 Values
tested

One tests the values of the normal pressure of contact and the semi-multiplier of friction afterwards
convergence of the iterations of operator STAT_NON_LINE, the loop on the active constraints and
loop on the thresholds of friction. All the points of contact are tested. It is pointed out that these
points of contact can be of two kinds: either of the nodes node, or of the nodes medium,
according to whether the interface cuts or not the elements.
It is checked that one finds well the values determined with [§4.1]. LAGS_F1 corresponds to
semi-multiplier of friction in direction OX (it is thus null), whereas LAGS_F2 corresponds
with the semi-multiplier of following friction.

Identification Reference
Aster %
difference
LAGS_C for all the points of contact
- 4.00
- 4.00
0.00
LAGS_F1 for all the points of contact
0.00
2.10-14
0.00
LAGS_F2 for all the points of contact
- 0.50
- 0.50
0.00

To test all the points of contact in only once, the MIN and the MAX of the column are tested.

5.2 Comments

Let us specify that in this study, the key word CONTACT_INIT = “OUI” makes it possible to begin the loop
on the active constraints with an assumption of statute contacting for all the points of contact.
That authorizes to take a relative criterion (“RESI_RELA_MAX”) for the convergence of the iterations of
Newton. Indeed, if one selected CONTACT_INIT = “NON”, at the time of the phase of prediction of Newton, it
contact not being activated, the higher structure moves without becoming deformed, and that lower
remain motionless. The constraints are then null and a relative criterion is not usable, only a criterion
total is, whose value is left with the choice of the user. The problem is that this value can
to reveal calculation (active contact thereafter….) inadequate with the loadings and the constraints then in
play. Thus, it is to better provide to take a single relative criterion as of the beginning.

Moreover, the initial value of the threshold of friction was taken with - 1011 in order to be sure that one has
adherence as of the 1ère iteration on the thresholds of friction.

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
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6
Modeling C: interface right and under-integration

This modeling is exactly the same one as modeling A, except that the diagram of integration
numerical the terms of contact changed.
In modeling A, one uses a diagram of Gauss at 12 points by triangular facets of
contact. In modeling C, one only uses a diagram reduced to 4 points.
Indeed, the diagram must allow the exact integration of a constant field of pressure. The intégrande
on the facet is then a students'rag procession in
xi yi
with I + J 3.

According to [bib4], a diagram at 4 points of Gauss is enough.

6.1 Functionalities
tested

Commands


DEFI_FISS_XFEM CONTACT
INTEGRATION=' FPG4'


7
Results of modeling C

7.1 Values
tested

One tests the same values as for modeling A.

Identification Reference
Aster %
difference
LAGS_C for all the points of contact
- 5.00
- 5.00
0.00
LAGS_F1 for all the points of contact
0.00
0.00
0.00
LAGS_F2 for all the points of contact
0.00
0.00
0.00

To test all the nodes of the interface in only once, one tests the values min and max.

7.2 Comments

This modeling shows that a diagram of integration reduced to 4 points makes it possible to pass the patch test
where the solution in pressure is constant.

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

Code_Aster ®
Version
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Titrate:
SSNV182 ­ Bloc in contact with X-FEM


Date:
25/11/05
Author (S):
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:
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8
Summaries of the results

The objectives of this test are achieved:

·
It is a question of showing the feasibility of the taking into account of the contact rubbing on the lips of
fissure with the method continues adapted to framework X-FEM. Only the case of a fissure
crossing the structure completely was considered (interface).
·
The cases where the interface follows the border of the elements (= 0°) and where the interface cut them
elements (= 22.5°) were validated.

9 Bibliography

[1]
MASSIN P., BEN DHIA H., ZARROUG Mr.: Elements of contacts derived from a formulation
continuous hybrid, Manuel of reference of Code_Aster, [R5.03.52]
[2]
MASSIN P., GENIAUT S.: Method X-FEM, Manuel of reference of Code_Aster, [R7.02.12]
[3]
TARDIEU NR., VAUTIER I., LORENTZ E.: Quasi-static nonlinear algorithm, Manuel of
Reference of Code_Aster, [R5.03.01]
[4]
DHATT G., TOUZOT G.: A presentation of the finite element method, Maloine ED.,
PARIS

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/05/005/A

Code_Aster ®
Version
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Titrate:
SSNV182 ­ Bloc in contact with X-FEM


Date:
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Author (S):
S. GENIAUT, Key P. MASSIN
:
V6.04.182-B Page:
10/10

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