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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
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Author (S):
S. GENIAUT, P. MASSIN
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V6.04 booklet: Nonlinear statics of the voluminal structures
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Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.186
SSNV186 LBB condition and contact rubbing with
X-FEM
Summary
The purpose of this test is to validate the taking into account of the contact (by the method continues [bib1]) on the lips of
fissure within the framework of method X-FEM [bib2], when the LBB condition [bib3] [bib4] is not respected.
This test brings into play a parallelepipedic block in compression. The interface the beam is represented by one
level set. The interface right, is nonpenchée and crosses completely the elements.
Code_Aster
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SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
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:
V6.04.186-A
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1
Problem of reference
Oscillations of contact pressures can appear in certain cases, in particular for
structures where the interface cuts pentahedrons, under a non-uniform loading.
That is due to the non-observance of the LBB condition [bib3] [bib4]. This phenomenon of oscillations is
comparable with that met of incompressibility [bib5]. Physically, in the case of the contact, that
amounts wanting to impose the contact in too many points of the interface (overstrained), returning the system
hyperstatic. To slacken it, it is necessary to restrict the space of the multipliers of Lagrange, like
that is done in [bib6] for the conditions of Dirichlet with X-FEM. The algorithm proposed by Moës
[bib6] to reduce the oscillations is adapted to the case 3D (algorithm version 1). This algorithm made
the object of an improvement to make it more physical and more effective (algorithm version 2). One
comparison of the two versions is carried out.
Let us note that with a mesh of hexahedrons, there are no oscillations.
1.1 Geometry
The structure is a right at square base and healthy parallelepiped. Dimensions of the block are: LX = 5 m,
LY = 20 m and LZ = 20 Mr. It does not comprise any fissure [Figure 1.1-a].
The interface is introduced by functions of levels (level sets) directly into the file of
controls using operator DEFI_FISS_XFEM [U4.82.08]. The interface is present within
the structure by the means of its representation by the level sets. The level set normal (LSN) allows
to define an interface planes not-leaning which crosses the elements completely, by the equation
following:
Z-17.5
LSN
=
éq
1.1-1
Appear 1.1-a: Geometry and positioning of the interface
1.2
Properties of material
Young modulus: E= 1000 AP.
Poisson's ratio:
= 0.
A higher Young modulus involves a very bad conditioning of the matrix of rigidity, which
results in a null pivot during factorization. For bearing this problem, a scaling of
contact pressures is in the course of implementation.
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SSNV186 LBB condition and contact rubbing with X-FEM
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1.3
Boundary conditions and loadings
The lower face is embedded.
The higher face is subjected to a parabolic pressure having for expression:
6
(Y-10) ²
100-
AP
2
10
E
pressure
=
éq
1.3-1
Displacements along axes X and are locked there for the nodes of the upper surface.
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SSNV186 LBB condition and contact rubbing with X-FEM
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2
Modeling a: hexahedrons
In this modeling, the mesh considered comprises only hexahedrons. This modeling
is used as reference for the others, because this case does not have oscillations of contact pressures.
Indeed, in the case of hexahedrons cut by an interface parallel to the faces, the number of
contact pressures (one by cut edge) is compatible with the discretization of the field of
displacement [bib1] [bib2]. The LBB condition is then respected and there are no oscillations of
contact pressures.
2.1
Characteristics of the mesh
The problem is invariant following axis OX. In order to limit the calculating time, the mesh considered here
comprise one element along this axis. The structure is then modelized by a mesh
regular composed of 1x20x20 HEXA8 to see [Figure 2.1-a].
Appear 2.1-a: Mesh of hexahedrons
This mesh 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.
2.2 Functionalities
tested
One uses the diagram of integration reduced to 4 points of Gauss per breakage of contact.
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is decontaminated.
Controls
DEFI_FISS_XFEM CONTACT
INTEGRATION=' FPG4'
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' NON'
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
S. GENIAUT, P. MASSIN
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:
V6.04.186-A
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3
Results of modeling A
3.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (0, 10, 17.5). This value is useful
of reference for other modelings.
2
9.5284410 AP
-
= -
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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4
Modeling b: pentahedrons (without algorithm)
4.1
Characteristics of the mesh
The structure is modelized by a regular mesh composed of pentahedrons.
Appear 4.1-a: Mesh of pentahedrons
4.2 Functionalities
tested
One uses the diagram of integration reduced to 4 points of Gauss per breakage of contact.
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is not activated.
Controls
DEFI_FISS_XFEM CONTACT
INTEGRATION=' FPG4'
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' NON'
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
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5
Results of modeling B
5.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (0, 10, 17.5).
Identification Reference
Aster %
difference
Not P
- 9.52844 10
- 2
- 0.186853
96.0
One carries out also a test of not-regression (compared to version 8.1.20).
Identification Reference
Aster %
difference
Not P
- 0.186853
- 0.186853
0.00
5.2 Comments
This modeling shows that without the algorithm aiming at restricting the space of the multipliers of
Lagrange of pressure, the values of contact pressures are completely false (see also
[Figure 12.2-a]).
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
S. GENIAUT, P. MASSIN
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V6.04 booklet: Nonlinear statics of the voluminal structures
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6
Modeling C: pentahedrons (algorithm version 1)
6.1
Characteristics of the mesh
The mesh is identical to that of modeling B.
6.2 Functionalities
tested
One uses the diagram of integration reduced to 4 points of Gauss per breakage of contact.
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is the n°1.
Controls
DEFI_FISS_XFEM CONTACT
INTEGRATION=' FPG4'
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' VERSION1'
7
Results of modeling C
7.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (0, 10, 17.5).
Identification Reference
Aster %
difference
Not P
- 9.52844 10
- 2
- 9.5188 10
- 2
0.10
7.2 Comments
This modeling shows that the algorithm set up makes it possible to reduce them efficiently
oscillations. However, it is observed that the algorithm version 1 tends to introduce
P0 approximations per pieces of contact pressures on the interface (see also it [Figure 12.2-a]
and it [Figure 12.2-b]).
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
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V6.04.186-A
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8
Modeling D: pentahedrons (algorithm version 2)
8.1
Characteristics of the mesh
The mesh is identical to that of modeling B.
8.2 Functionalities
tested
One uses the diagram of integration reduced to 4 points of Gauss per breakage of contact.
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is the n°2.
Controls
DEFI_FISS_XFEM CONTACT
INTEGRATION=' FPG4'
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' VERSION2'
9
Results of modeling D
9.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (0, 10, 17.5).
Identification Reference
Aster %
difference
Not P
- 9.52844 10
- 2
- 9.5188 10
- 2
0.10
9.2 Comments
This modeling shows that the algorithm version 2 makes it possible to reduce them efficiently
oscillations. It is observed that the algorithm version 2 tends to introduce P1 approximations by
pieces of contact pressures on the interface, which makes it more precise than the version1 (see too
[Figure 12.2-a] and it [Figure 12.2-b]).
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
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Page:
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10 Modeling E: tetrahedrons (algorithm version 1)
This test brings into play a free mesh made up of tetrahedrons. In order to reduce the number of elements and
thus the calculating time, the length of the structure along axis OX is LX = 1 Mr.
10.1 Characteristics of the mesh
The mesh considered is a free mesh carried out with GMSH. It consists of 3629 TETRA4.
[Figure 10.1-a] the mesh in the Oyz plan represents. The interface is traced there, only at ends
of visualization.
Appear free 10.1-a: Mesh
10.2 Boundary conditions and loadings
The lower face is embedded.
The higher face is subjected to a uniform pressure:
6
100
AP
10
E
pressure
=
éq
10.2-1
Following displacements axes X and are locked there for the nodes of the upper surface.
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
S. GENIAUT, P. MASSIN
Key
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10.3 Functionalities
tested
One uses the diagram of integration reduced to 4 points of Gauss per breakage of contact.
The contact is active as of the 1
era
iteration of active stresses but friction is not taken in
count
The algorithm aiming at restricting the space of the multipliers of Lagrange is the n°1.
Controls
DEFI_FISS_XFEM CONTACT
INTEGRATION=' FPG4'
CONTACT_INIT=' OUI'
ALGO_LAGR=' VERSION1'
11 Results of modeling E
11.1 Values
tested
One tests the value of contact pressures for all the points of the interface. The analytical solution
is quite simply:
zz
pressure
=
= -
éq
11.1-1
Identification Reference
Aster %
difference
MAX (LAGS_C)
- 0.1
0.0979 - 2.08
MIN (LAGS_C)
- 0.1
- 0.1016 1.62
To test all the points of contact in only once, the MIN and the MAX of the column are tested.
11.2 Comments
This test makes it possible to validate the robustness of the algorithm of restriction of the space of the multipliers of
Lagrange of pressure, in a free case of mesh in 3D. Even on a structure subjected to
constant pressure, the algorithm is essential bus of the oscillations of contact pressures
can appear (it is the case here if the algorithm is not activated).
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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12 Summaries of the results 3D
12.1 Summary
Within the framework of method X-FEM, one showed that, without particular processing, a structure
with a grid with pentahedrons subjected to a non-uniform loading can present the strong ones
oscillations of contact pressures (modeling B), whereas the same structure with a grid with
hexahedrons subjected to the same loading does not have such oscillations (being useful modeling A
of reference).
One proposed two algorithms allowing to reduce these oscillations significantly. The first
(modeling C) seems less precise than the second (modeling D).
Moreover even under uniform loading, of the oscillations can appear and it is essential
to use an algorithm of reduction of the space of the multipliers of Lagrange of pressure
(modeling E).
12.2 Curves of comparison
[Figure 12.2-a] gathers the curves of contact pressures along axis OY for
the first 4 modelings presented. It is noticed that the oscillations for modeling B are if
strong that in certain points the value of the contact pressure becomes positive, which would like to say
that there is separation of the interface. The two algorithms allow a visible reduction of
oscillations, and one finds the curve of reference obtained with the mesh of hexahedrons.
- 0,20
- 0,15
- 0,10
- 0,05
0,00
0
2
4
6
8
10
12
14
16
18
20
y
lambda
modeling A
modeling B
modeling C
modeling D
Appear 12.2-a: Comparison of contact pressures following modelings
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[Figure 12.2-b] compares in details the effects of the two algorithms. It is noticed that the first
often imply constant contact pressures per pieces, whereas the second tends to
to linearize the pressures. It is obvious that such differences are reduced by refining the mesh.
- 0,1
- 0,095
- 0,09
- 0,085
- 0,08
- 0,075
- 0,07
- 0,065
- 0,06
- 0,055
- 0,05
0
2
4
6
8
10
12
14
16
18
20
y
lambda
modeling C
modeling D
Appear 12.2-b: Comparison of the 2 algorithms
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SSNV186 LBB condition and contact rubbing with X-FEM
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13 Modeling F: quadrangles 2D
This modeling is the equivalent in 2D of modeling A (reference). This case does not present
oscillations of contact pressures. Just like in modeling A, the quadrangles are
half-compartments by an interface parallel to the edges, and numbers it contact pressures (one by
cut edge) is compatible with the discretization of the field of displacement [bib1] [bib2]. The LBB
condition is then observed and there are no oscillations of contact pressures.
13.1 Characteristics of the mesh
The structure is then modelized by a regular mesh composed of 20x20 QUAD4 (See
Appear 13.1-a]).
Appear 13.1-a: Mesh of quadrangles
13.2 Functionalities
tested
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is decontaminated.
Controls
DEFI_FISS_XFEM CONTACT
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' NON'
14 Results of modeling F
14.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (10, 17.5).
Identification Reference
Aster %
difference
Not P
- 9.52844 10
- 2
- 9.52844 10
- 2
2.41
10
- 4
14.2 Comments
This case test makes it possible to find the values of reference of contact pressures calculated in
modeling A, and to check that in the case of quadrangles cut their faces parallel to, these
contact pressures do not have oscillations (see [Figure 19.2-a]).
Code_Aster
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SSNV186 LBB condition and contact rubbing with X-FEM
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15 Modeling G: triangles (algorithm version 1)
15.1 Characteristics of the mesh
The structure is modelized by a regular mesh composed of triangles. The test is the equivalent in
2D of modeling B (See [Figure 15.1-a]).
Appear 15.1-a: Mesh of triangles
15.2 Functionalities
tested
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is the n°1.
Controls
DEFI_FISS_XFEM CONTACT
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' VERSION1'
16 Results of modeling G
16.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (10, 17.5).
Identification Reference
Aster %
difference
Not P
- 9.52844 10
- 2
- 9.59082 10
- 2
0.655
16.2 Comments
This modeling shows that the algorithm set up makes it possible to reduce them effectively
oscillations. (see [Figure 19.2-a]).
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SSNV186 LBB condition and contact rubbing with X-FEM
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17 Modeling H: triangles (algorithm version 2)
17.1 Characteristics of the mesh
The mesh is identical to that of modeling G.
17.2 Functionalities
tested
Friction is taken into account and the contact is active as of the 1
era
iteration of active stresses.
The algorithm aiming at restricting the space of the multipliers of Lagrange is the n°2.
Controls
DEFI_FISS_XFEM CONTACT
CONTACT_INIT=' OUI'
FROTTEMENT=' COULOMB'
ALGO_LAGR=' VERSION2'
18 Results of modeling H
18.1 Values
tested
One tests the value of the contact pressure at the point P of co-ordinates (10, 17.5).
Identification Reference
Aster %
difference
Not P
- 9.52844 10
- 2
- 9.59082 10
- 2
0.655
18.2 Comments
This modeling shows that in 2D, the algorithm version 2 has a behavior very close to that of
version 1. Indeed, put aside the first contact pressures measured on the left of
mesh, the actual values are identical, and the curves obtained with the two algorithms
cover almost completely. (see [Figure 19.2-a]).
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
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19 Summaries of the results 2D
19.1 Summary
It, initially, was checked that the behavior of the structure of reference (modeling
A) could be found in 2D (modeling F). After having observed the phenomenon of oscillations in
2D, similar to that obtained in modeling B, one tested on cases in 2D both
algorithms which make it possible to reduce these oscillations in 3D.
The two algorithms tested in modelings G and H give, in 2 dimensions, of the results
very close, and allow consequently to reduce the oscillations introduced by the mesh.
19.2 Curves of comparison
[Figure 19.2-a] represents the curves of contact pressures along axis OX for
3 modelings in 2D presented. It is noticed that the curves representative of both
algorithms overlap almost completely. The two algorithms are thus about too
efficient one that the other in 2D. It make it possible nevertheless to reduce the oscillations effectively.
Appear 19.2-a: Comparison of contact pressures following modelings 2D
It is it should be noted that a refinement of mesh increases obviously the precision of the results obtained.
Code_Aster
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Version
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Titrate:
SSNV186 LBB condition and contact rubbing with X-FEM
Date:
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Author (S):
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20 Bibliography
[1]
Massin P., Ben Dhia H., ZarrougG Mr.: Elements of contacts derived from a formulation
continuous hybrid, Manual of reference of Code_Aster, [R5.03.52]
[2]
Massin P., Geniaut S.: Method X-FEM, Manual of reference of Code_Aster, [R7.02.12]
[3]
Babuska I.: The finite element method with lagrangian multipliers, Numerische Maths 20,
179-192, 1973
[4]
Barbosa H., Hugues T.: Finite element method with lagrange multipliers one the boundary.
Circumventing the Babuska-Brezzi condition, comp. Meth. Applied Mech Engrg. 85 (1),
109-128, 1991
[5]
Vault D., Bathe K.J.: The Inf-sup test, Computers & Structures 47 (4/5), 537-545, 1993
[6]
Moës NR., Béchet E., Peaty Mr.: Imposing essential boundary conditions in the X-FEM, Int.
J. Numer. Meth. Engng, 2005, submitted.