Code_Aster
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Titrate:
Unilateral contact by conditions kinematics
Date:
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Author (S):
NR. TARDIEU, I. VAUTIER
Key:
R5.03.50-B
Page:
1/24
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Organization (S):
EDF/MTI/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
R5.03.50 document
Unilateral contact by conditions kinematics
Summary:
One describes in this document the numerical method used by defect to deal with the problems of contact
unilateral in great displacements in the operator
STAT_NON_LINE
. One uses conditions kinematics of
not interpenetration which is dualisées. The formulation used is of main type slave (node-breakage or
nodal) with reactualization of the geometry during iterations, and the resolution of the problem of contact is
carried out by a method of active stresses within each iteration of the total method of Newton
of the operator
STAT_NON_LINE
.
Code_Aster
®
Version
5.0
Titrate:
Unilateral contact by conditions kinematics
Date:
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Author (S):
NR. TARDIEU, I. VAUTIER
Key:
R5.03.50-B
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1 Introduction
The key word
CONTACT
control
AFFE_CHAR_MECA
allows to define conditions of contact
unilateral which is treated (in the control
STAT_NON_LINE
) in initial or reactualized geometry,
in a nodal formulation or node-breakage, adapted better to the incompatible mesh and to
slip of surfaces one compared to the other. It will replace the key word in the long term
LIAISON_UNIL_NO
who is valid only for compatible mesh undergoing of small
slips.
One presents here an algorithm based on the method of the active stresses [bib2]. It is that which is
used by defect and which corresponds to
METHOD:“FORCED”
key word
CONTACT
. Another
algorithm is available under
METHOD:“LAGRANGIAN”
. It is similar to the precedent except for the detail
that the connections are not activated there one by one (as we will see it), but by package. For
more precise details, one will be able to refer to the document [R5.03.51].
1.1 General
Two solids are known as in contact when they “are touched” by part of their borders. To treat it
unilateral contact consists in preventing that one of the solids “does not cross” the other: it is the principle of
not interpenetration of the matter, which results in relations of inequality between the variables
kinematics (displacements). These relations are written in a discretized form: it is thus
necessary to identify the entities between which one writes them (it is what is called pairing).
In Code_Aster, the use of the key word
CONTACT
allows to pair a node with another node or
with a mesh: there is then a potential couple of contact, i.e. a couple for which one will write
relations of nonpenetration. If the contact takes place really (the two nodes find with
even position, or the node is found on the mesh), one will say that the two entities are associated
center of an effective couple of contact.
Note:
The expression “to make a calculation with contact” wants to say that one writes such relations of not
penetration, but does not imply that there is effective contact for the loading considered.
There are four products in an algorithm of processing of the unilateral contact:
·
the identification: definition of potential surfaces of contact (cf [§1.2]),
·
pairing: determination of the potential couples of contact (cf [§2]),
·
the relation of nonpenetration: direction of writing and coefficients (cf [§3]); the relation is written
between the main node slave and one or more nodes, according to the formulation used,
·
the resolution: one uses a method of active stresses here (cf [§4]); it is an algorithm
iterative which determines, step by step, the list of the couples indeed in contact while examining
geometrical conditions of contact and the sign of the associated multipliers of Lagrange,
by duality, in these conditions.
1.2
Areas and surfaces of contact
One considers the 3 solids of the figure [Figure 1.2-a], represented in 2D. 3 possible areas were defined
of interpenetration enters the solids: an area enters the solid A and the solid B, and two areas between
solid B and the solid C. the user, who defines these areas in the command file, supposes here
that apart from these areas, there is no risk of interpenetration, taking into account the loading.
Code_Aster
®
Version
5.0
Titrate:
Unilateral contact by conditions kinematics
Date:
17/04/01
Author (S):
NR. TARDIEU, I. VAUTIER
Key:
R5.03.50-B
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area 1
area 2
surface 1
surface 2
surface 2
surface 1
area 3
surface 1
surface 2
Solid A
Solid B
Solid C
Appear 1.2-Error! Argument of unknown switch. : Definition of 3 areas of contact
Each area of contact is defined in the operator
AFFE_CHAR_MECA
by an occurrence of the key word
CONTACT
. An area is composed by definition of two surfaces which one seeks to prevent
interpenetration: first is defined under the key word
GROUP_MA_1
(or
MAILLE_1
), the second
under the key word
GROUP_MA_2
(or
MAILLE_2
), i.e. by the data of the meshs of edge which them
constitute. These meshs are SEG2 or SEG3 for a mesh 2D, TRIA3, TRIA6,
QUAD4, QUAD8 or QUAD9 for a mesh 3D.
Note:
The meshs of edge necessary to the contact will not be created by the code starting from the elements
voluminal and must thus already exist in the file of mesh.
It is imperative that the meshs of contact are defined so that the normal is outgoing:
connectivity of the segments must be defined in order AB, that of the triangles in the order ABC, and
that of the quadrangles in order ABCD, as indicated on the figure [Figure 1.2-b]. For one
better reading of the drawing, one a little drew aside the mesh of edge being used here in contact with the “face” of
the voluminal element 2D or 3D on which it is based.
Particular case: contact for a cable or a beam in 3D
It is possible in 3D to treat the contact between a mesh SEG2 or SEG3 (modelized cables some or
beam) and a surface. In this case, it is imperatively necessary to use the method of pairing
“MAIT_ESCL”
and to give the segments under the key word
GROUP_MA_2
(meshs slaves).
section of the beam can then be taken into account by the use of the key word
DIST_2
(cf [§3.3]).
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
17/04/01
Author (S):
NR. TARDIEU, I. VAUTIER
Key:
R5.03.50-B
Page:
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HI-75/01/001/A
2D
3D
With
B
With
B
C
C
N
N
With
C
B
N
D
With
B
C
D
N
N
T
With
B
With
B
T
N
Appear 1.2-Error! Argument of unknown switch. : Classification of the meshs of contact to have
an outgoing normal
Note:
One advises to use disjoined areas of contact, i.e. not having no node in
commun run.
Chapter 2 details the method of pairing for the formulations node-breakage and nodal:
the establishment of the potential couples of contact is made area by area. In chapter 3, one gives
form relations of nonpenetration (inequations). The imposition of these conditions of nonpenetration
is realized by an iterative method, called method of the active stresses, described in
chapter 4: the resolution of the problem obtained is total, i.e. it takes into account them
couples of all the areas simultaneously.
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
17/04/01
Author (S):
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Key:
R5.03.50-B
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2
Establishment of the couples of contact
2.1 Principle
Potential surfaces of contact were defined in the operator
AFFE_CHAR_MECA
like
specified in [§1.2]. The effective processing of the contact is done, him, in the operator
STAT_NON_LINE
.
The resolution of a nonlinear problem in the operator
STAT_NON_LINE
is described in detail in
document [R5.03.01]. We recall here briefly the main phases of them, for a calculation
comprising two pitches of time:
1st pitch of load
(1/has)
prediction
(1/b1)
iteration of Newton n°1
(1/b2)
iteration of Newton n°2
.............................................................
(1/bm)
iteration of Newton n°m
(1/c)
storage of the results with convergence
2nd not of load
(2/has)
prediction
(2/b1)
iteration of Newton n°1
(2/b2)
iteration of Newton n°2
.............................................................
(2/LP)
iteration of Newton n°p
(2/c)
storage of the results with convergence
The unilateral contact is treated after the phases (1/has), (1/b1), (1/b2),…, (1/bm), (2/has), (2/b1), (2/b2),…,
(2/LP) i.e after the phase of prediction and each iteration of Newton of
STAT_NON_LINE
. It is
there the essential difference between this algorithm and the algorithm of contact friction (see
documentation [R5.03.51]) where the processing of the contact is effective only at the end of the pitch of load and not
during iterations.
One calls “master key of contact” each occurrence of processing of the contact.
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
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2.1.1 Formulation
node-breakage
This formulation, chosen by the key word
PAIRING: “MAIT_ESCL”
, a role does not grant
equivalent on the two surfaces: the surface described under
GROUP_MA_1
or
MAILLE_1
(S1) is called
surface main and the other (S2) surface slave. The conditions of noninterpenetration express that
the nodes of surface slave (of stars on the figure [2.1.1-a]) do not penetrate in
meshs of surface Master: one can see that, on the other hand, it is possible that the main nodes (
rounds) penetrate in surface slave.
surface main
*
*
*
*
surface slave
S
1
2
S
Appear 2.1.1-a: Surfaces main and surfaces slave
The relation of noninterpenetration will be written between a node and a mesh: one seeks initially it
main node of surface nearest to the node slave (cf [§2.2]), then one examines (cf [§2.3])
all the meshs Masters containing this node (the distance obtained by projection of the node slave on
each mesh Master makes it possible to choose the mesh nearest). One uses the normal with the mesh Master
to write the relation of nonpenetration.
Note:
The nodes slaves are by defect all the nodes belonging to the meshs of contact
defining surface slave. Key words
SANS_NO
and
SANS_GROUP_NO
allow to give,
area by area, the list of the nodes which must be removed list of the nodes slaves (but they
could be used as main nodes). That makes it possible to remove the nodes subjected to
boundary conditions of Dirichlet incompatible with the contact.
To symmetrize the role of two surfaces, it would be interesting to use a functionality of the type
PAIRING: “MAIT_ESCL_SYME”
who would exchange the roles of Master and slave with each
pass from processing of the contact. It is a development under consideration in version 6.
2.1.2 Formulation
nodal
The nodal formulation (
PAIRING: “NODAL”
) imposes that relative displacement enters a node
slave and the main node which is paired to him, projected on the direction of the normal to the node slave,
that is to say lower than the initial play in this direction. The use of this formulation is disadvised because it
require to have compatible mesh (nodes “opposite”) which remains compatible during
deformation (assumption of small slips), and for which the normals Master and slave are with
little close colinéaires. Without these assumptions, the made approximation becomes hazardous just like (
the use of
LIAISON_UNIL_NO
) and it is preferable to use the node-breakage formulation.
One chooses to take as surface slave that which comprises less nodes (with an equal number,
it is that described under
GROUP_MA_2
or
MAILLE_2
), in order to maximize the chances to have one
injective pairing (a main node is paired only with one node slave). The main node
paired with each node slave is determined by a calculation moreover nearer close explained in
[§2.2]. One uses the normal with the main node to write the relation of noninterpenetration (cf [§3]).
Note:
Even in the case of nodal pairing, surfaces of contact are defined in terms of
meshs (cf [§1.2]). The nodes slaves and Masters are then the nodes of the meshs thus defined.
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
17/04/01
Author (S):
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Key:
R5.03.50-B
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2.2
Seek nearer close to a node
The method used to seek the main node nearest to a node slave is very simple:
it is enough to calculate the distance (in current geometry, cf [§2.4]) between the node slave and the nodes
Masters candidates. The only alternative used consists in being able to restrict the whole of the nodes
Masters a priori candidates.
The key word
SEEK: “NOEUD_BOUCLE”
start the examination of all the main nodes
belonging to the same area of contact as the node slave.
One stores with each master key of contact the main node which was closest to each node slave
(it is called the former “neighbor”). If the relative slip of two surfaces is small (a mesh or two),
one can choose to examine only the main nodes connected to this old node by meshs of
contact. This method is activated by
SEEK: “NOEUD_VOISIN”
. One chooses among these nodes
candidates nearest like “new neighbor”, and one will examine (cf [§2.3]) the meshs containing it
new neighbor. Thus, meshs potentially likely to be paired with the node slave (round
black) in the new configuration are those having scratches on the figure [Figure 2.2-a].
***
*
*
*
*
*
old main node nearest (old “vo
meshs containing the former “neighbor”
*
nodes candidates to be the new “neighbor
meshs likely to be paired with the node
Appear 2.2-a: Territory covered by
SEEK: “NOEUD_VOISIN”
Note:
If the discretization in time is sufficiently fine (what is the case in general out the problems
of elasticity), it is reasonable to think that the slip will be small of a pitch of time to the other.
One thus can has minimum to use the option
SEEK: “NOEUD_VOISIN”
.
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2.3
Search of the mesh nearest (node-breakage formulation)
Knowing the main node nearest to the node slave, one examines successively the meshs
Masters containing this node. The method of selection is simple: one determines projection M of
node slave P on the mesh Master (according to the normal with the mesh Master), one brings back it on the edge of
the mesh if it is outwards, and one calculates the scalar product between vector PM and the normal with
the mesh. The mesh carrying out the smallest value of this scalar product (before correction while bringing back
on the edge) is selected to be paired with the node slave.
2.3.1 Projection on a segment (contact in 2D)
One considers the situation described on the figure [Figure 2.3.1-a]. Surface Master is “below”
surface slave, therefore the direction of main course of surface must be of
With
towards
B
(the mesh
of main edge is defined as being
AB
, and not
BA
): thus normal N with the mesh is
outgoing, i.e. point towards surface slave (cf [§1.2]). On the other hand, from a point of view
algorithmic, one uses
NR
=-n, the vector opposed to the outgoing normal of the mesh.
The parametric co-ordinate is sought
point
M
, projection of the node slave
P
according to
entering normal
NR
with the mesh
AB
, defined by:
AM
AB
MP
NR
AB NR
=
=
=
(
,)
0
where
(,)
indicate the scalar product.
P
B
NR
slave
Master
P'
Me
M
NR
With
Be reproduced 2.3.1-a: Projection on a segment
One a:
=
(
,
)
(
,
)
AP AB
AB AB
.
The point
M
belongs to the mesh
AB
if
0; 1
[]
. If
>
1
(case of
P'
projected in
Me
), one brings back
projection in
With
while posing
=
1
; if
<
0
, one brings back projection in
B
while posing
=
0
. One
evaluate then the scalar product of
PM
with the normal
NR
entering to the mesh (i.e opposed to
outgoing normal of the mesh Master), whose components are:
(
) (
)
NR
= -
-
-
=
-
+
-
y
y
L
X
X
L
L
AB L
X
X
y
y
B
With
B
With
B
With
B
With
,
:
.
with length
of
2
2
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
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Key:
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Page:
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HI-75/01/001/A
The value of
(
)
PM NR
,
the play between the node slave and the mesh Master is called. Direction
NR
is
reserve like direction of writing of the relations of nonpenetration (cf [§3]).
Note:
One could have defined the direction of writing of the relations of nonpenetration by the vector
PM
, and
thus play by the standard of
PM
. However, the vector
PM
tends towards the null vector (direction
unspecified) when the points approach (when one tends towards the effective contact) and becomes
very sensitive to the round-off errors: to the extreme, when
P
M
=
, one can find
PM
= (10
15
; 0)
(for a mesh Master horizontal), which is a horizontal direction, perfectly erroneous for
the writing of the relations of nonpenetration here. For this reason one chooses to use
normal Master which, it, does not vary because of the only bringing together of the solids.
The fact of privileging a surface compared to another can generate errors of
modeling (loss of symmetry) which one can minimize by refining the mesh. Another solution
would consist in using the average between the normals Master and slave. This approach is with
the study in version 6.
2.3.2 Projection on a triangle (contact in 3D)
P
With
B
C
*
M
NR
slave
Master
Be reproduced 2.3.2-a: Projection on a triangle
The parametric co-ordinates are sought
1
and
2
point
M
, projection of the node slave
P
according to the normal
NR
entering to the triangular mesh
ABC
(one uses the normal with the mesh, but in
the direction slave towards Master), defined by:
AM
AB
AC
PM NR
=
+
=
1
2
0
that is to say:
1
2
= -
=
((
),
)
(
)
((
),
)
(
)
AP NR AC
AB AC
AP NR AB
AB AC
with
NR
AB AC
AB AC
= -
(entering unit normal).
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
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Author (S):
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If one poses
3
=
1
-
1
-
2
, values of the three parametric co-ordinates
1
,
2
,
3
allow
to find if the point
M
belongs or not to the triangle
ABC
, like the figure illustrates it [Figure 2.3.2-b].
With
B
C
2
<
0
1
<
0
3
<
0
1
<
0
3
<
0
2
<
0
3
<
0
1
<
0
2
<
0
4
6
2
5
3
1
0
1
>
0
2
>
0
3
>
0
Appear 2.3.2-b: possible Areas for the point
M
, prolongation
node
P
according to the direction of the normal to the mesh
If the point
M
is in sectors 1, 2, or 3, one brings back it on the corresponding edge
(
)
AC AB
BC
,
or
. If it is in sectors 4, 5 or 6, one brings back it on the corresponding point
(not
B WITH
C
,
or
). That amounts cancelling the parametric co-ordinates which are negative.
Let us take the example of sector 1 where
1
<
0
. The point is brought back
M
at the point
Me
, defined by:
AM
AC
AM
AB
AC
AM MM
'
(
',
')
'
=
=
+
=
2
1
2
0
One finds:
2
2
1
'
(
,
)
(
,
)
=
+
AB AC
AC AC
.
In sector 2, an identical reasoning gives:
1
1
2
'
(
,
)
(
,
)
=
+
AB AC
AB AB
and
AM
AB
'
'
=
1
.
In sector 3, one has
AM
AB
AC
'
(
)
'
'
=
+ -
1
1
1
with
1
1
2
1
'
(
,
) (
) (
,
)
(
,
)
= -
+ -
AB BC
AC BC
BC BC
.
The play is calculated like the scalar product between the vector
PM
and the normal
NR
entering to the mesh
Master.
Code_Aster
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Titrate:
Unilateral contact by conditions kinematics
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2.3.3 Projection on a quadrangle (contact in 3D)
For the quadrangles, the determination of the parametric co-ordinates in the current element
would require the return to the element of reference, and thus the resolution of a nonlinear problem, which is
expensive.
An approached solution was initially chosen, which is to cut (virtually) it
quadrangle in two triangles, according to the two possible manners (cf [Figure 2.3.3-a]), to calculate
outdistance node slave with each of the four triangles thus defined (cf [§2.3.2]), and to choose it
triangle carrying out the smallest distance. The relation of noninterpenetration is then written between
node slave and 3 main nodes of the selected triangle. If the quadrangle remains plane,
projection on the selected triangle is equivalent to projection on the quadrangle; in the case more
General where the quadrangle is left, this operation is a means of taking into account, of one
certain way, curvature.
Appear 2.3.3-a: Cutting of a linear quadrangle
2.3.4 Case of the quadratic elements
Projection on the quadratic elements is made for the moment while being reduced to the linear case
(triangle with three nodes and quadrangle with four nodes). On the other hand, the writing of the relation of not
interpenetration utilizes all the nodes of the main elements with the functions of form
associated (cf [§3.2]). It is thus considered that the contribution of the nodes mediums to the result must be
taking into account even if their contribution to the geometrical deformation of the element is neglected.
Note:
For the quadrangles, one uses only the functions of form relating to the three nodes of
triangle chosen, and that even for the QUAD8 and QUAD9.
Important warning:
The contact in 3D for quadratic elements gives, for exposed theoretical reasons
in CR MN 97/023, results which can surprise the user. Us
let us not recommend the use of such elements; if such is the case, however, we advise
to refine “sufficiently” the mesh on the edges of the structures in contact.
Code_Aster
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Version
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Titrate:
Unilateral contact by conditions kinematics
Date:
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2.4 Reactualization
geometrical
In the framework of the modeling of the contact in great displacements, the evolution of the geometry of
surfaces plays a fundamental part. Indeed, it is it which conditions the calculation of the normals to the faces
potentially in contact and thus which conditions pairing carried out.
The geometrical reactualization is defined by the key word
REAC_GEOM_INTE
key word factor
CONTACT
. Its operation is as follows:
REAC_GEOM_INTE=0
There is no geometrical reactualization. All calculation is carried out on
initial configuration with initial pairing.
REAC_GEOM_INTE=1
A geometrical reactualization is carried out with convergence of
each pitch of load i.e right before the phases (1/c), (2/c),…
presented to [§2.1]. This reactualization is accompanied again
pairing.
REAC_GEOM_INTE=2
One places oneself at a pitch of load given. With convergence of this last,
a geometrical reactualization then a new pairing are
carried out. One does not pass to the pitch of load according to but one begins again
the same pitch of load until convergence. A reactualization
geometrical then a new pairing are carried out and one passes to
no the load according to.
REAC_GEOM_INTE=n
(n>2)
It is a generalization of the preceding case. Within the same pitch of
charge, one carries out N time the cycle iteration until convergence,
geometrical reactualization, pairing.
One can first of all notice that pairing is subjected to the phase of reactualization
geometrical. Moreover, the fact of carrying out several times within the same pitch of load the cycle
iteration until convergence, geometrical reactualization, pairing makes it possible to follow the evolution of
geometry of the structure. It should indeed be stressed that this geometrical evolution is one of
nonlinear components of a calculation of contact in great displacements.
In practice, one can advise the following values for the key word
REAC_GEOM_INTE
:
·
for a calculation in small displacements, the natural value is 0. One works on
initial configuration,
·
for calculation in great displacements, the selected value depends of course on the importance on
the geometrical evolution of surfaces but values 1 or 2 is generally with advising.
Value 2 is the default value besides of this key word.
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3
Relation of noninterpenetration
3.1 Condition
kinematics
One carries out a idealized modeling of the phenomenon of contact, in the sense that it supposes them
borders of the bodies perfectly defined by a line or a surface: one writes a condition then of
not discrete and linearized interpenetration [bib3].
That is to say
P
a node slave,
M
its projection on the mesh Master which was given at the time of
pairing. In 2D, this mesh Master has 2 nodes (SEG2) or 3 nodes (SEG3). In 3D, it can in
to have 3, 4, 6, 8 or 9 (TRIA3, QUAD4, TRIA6, QUAD8, QUAD9). The displacement of the point
M
is one
linear combination of displacements of the nodes of the finite element, with for coefficients values
functions of form
in
M
. We place if the mesh Master is a SEG2 for
to simplify the talk. One has then:
U
U
U
M
With
With
B
B
M
M
=
+
()
(
)
The relation of nonlinearized penetration consists in saying that relative displacement enters
P
and
M
according to
a given direction cannot exceed the initial play in this direction. One chose to take
like direction
NR
the entering normal of the mesh Master (cf [3.1-a]).
With
B
NR
P
M
surface m
aître
surface slave
Be reproduced 3.1-a: Projection of a node slave on a mesh SEG2
The relation of nonpenetration is written then like a scalar sign of product (noted by one.) :
PM NR
.
0
, that is to say
P MR. NR
U
U
NR
-
-
+
-
.
(
).
M
P
0
,
if
U
is the increment of displacement since the preceding configuration where displacement was noted
U
-
.
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One must thus check
(
).
.
U
U
NR P MR. NR
P
M
-
-
-
. By noticing that
P MR. NR
-
-
.
is the play
D
-
in the preceding configuration, the relation of nonpenetration is also written:
(
).
U
U
NR
P
M
D
-
-
,
maybe, by using the relation
U
U
U
M
With
With
B
B
M
M
=
+
()
(
)
:
[
]
U
U
U
NR
P
With
With
B
B
M
M
D
-
+
-
(
()
()
).
Extension of the formula for a mesh Master comprising N
maît
noted nodes
B
J
, is immediate:
U
U
NR
P
B
B
J
N
J
J
maît
M
D
-
=
-
()
.
1
If one writes such a relation for all the couples of contact, one obtains the geometrical conditions
of nonpenetration in matric form:
With
U
D
-
Note:
Effective play in the configuration
U
-
+
U
is
D
0
-
With (U
-
+
U)
, that is to say
D
-
-
With
U
. The condition
of nonpenetration thus expresses that the effective play remains positive or null in any configuration.
The matrix
With
, called matrix of contact, contains 1 line by couple of contact, and as much of
columns that physical degrees of freedom of the problem.
Let us suppose that one has 2 meshs of contact of the type SEG2, according to the diagram of the figure [Figure 3.1-b]:
L/2
L/4
M
1
B
D
C
P
*
M
2
*
NR
NR
Q
D
1
2
D
Be reproduced 3.1-b: Writing of the matrix of contact A on an example
If one notes for example
U
B
X
the increment of displacement of the node
B
according to the direction
X
, and
D
D
1
2
and
the current plays for the two couples, the two relations of nonpenetration are written
matriciellement:
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NR
NR
NR
NR
NR
NR
U
U
U
U
U
U
U
U
U
U
X
y
X
y
X
y
X
y
X
y
X
y
P
P
Q
Q
B
B
C
C
D
D
.
.
.
.
-.
-.
-.
-.
D
D
X
y
X
y
X
y
X
y
X
y
0
0
0 5
0 5
0 5
0 5
0
0
0
0
0
0
0 75
0 75
0 25
0 25
1
2
-
-
-
-
NR
NR
NR
NR
NR
NR
Note:
One considered here only the degrees of freedom of the nodes implied in the contact; in
reality, the matrix
With
is hollower.
3.2
Coefficients of the relation of nonpenetration
The relation of nonpenetration is written:
U
U
NR
P
B
B
J
N
J
J
maît
M
D
-
=
-
()
.
1
One gives below the values of the functions of form
B
J
M
()
main nodes at the point
M
for the various treated meshs of contact.
3.2.1 Meshs
SEG2
The parametric co-ordinate of the projection of the node slave on the SEG2 is noted
. Values
functions of form at the parametric point of co-ordinate
are as follows:
With
B
M
M
()
()
= -
=
1
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3.2.2 Meshs
SEG3
One projected the node slave by supposing the rectilinear mesh: the point
M
thus does not belong
inevitably rigorously with the SEG3 if the aforementioned is curved. Nevertheless, one calculates the values of
functions of form associated to the SEG3 (nodes
With
and
B
, node medium
C
) starting from the co-ordinate
parametric
paid to the SEG2:
With
B
C
M
M
M
()
(
) (
)
()
(
)
()
(
)
=
-
-
=
-
=
-
2 1
1
2
2
1
2
4
1
3.2.3 Meshs
TRIA3
One explained in [§2.3.2] how one finds the co-ordinates parametric
1
and
2
projection
M
node slave in the triangle. The values of the functions of form are in fact those
parametric co-ordinates:
With
B
C
M
M
M
()
()
()
= - -
=
=
1
1
2
1
2
3.2.4 Meshs
TRIA6
As in the case of the segments, one carried out projection by taking account only of the 3
nodes of the triangle: on the other hand, one uses the parametric co-ordinates thus obtained (while posing
With
B
C
= - -
=
=
1
1
2
1
2
,
,
) to deduce the values from them from the functions of form to the 6
nodes (
WITH B C
,
nodes,
D E F
,
nodes mediums respectively on the sides
AB BC
CA
,
and
):
With
With
With
B
B
B
C
C
C
D
WITH B
E
B C
F
A.C.
M
M
M
M
M
M
()
(
)
()
(
)
()
(
)
(
)
()
()
=
-
=
-
=
-
=
=
=
2
1
2
1
2
1
4
4
4
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3.3
Introduction of a fictitious play
One can want to modelize the contact between structures whose mesh did not take account of
certain characteristics (“hole” or “bump” nonwith a grid (cf [Figure 3.3-a])).
N
real structure with a hole
mesh
NB:
N
=
NR
real structure with a bump
mesh
Appear 3.3-Error! Argument of unknown switch. : Holes and bumps
In this case, it is necessary to correct the value of the play intervening with the second member of the inequation of not
penetration, according to the following model (
NR
is the normal entering to the mesh Master):
(
)
(
)
U
U
NR
P
M
D
D
D
-
-
+
-
.
1
2
where
D
D
1
2
and
are given by the user respectively under the key words
DIST_1
and
DIST_2
for
each area of contact. These “distances” have a sign: they represent the translation to be applied to
node of the mesh in the direction of the normal
N
outgoing to obtain the point of the real structure
(cf [Figure 3.3-b]).
N
D
1
0
=
D
2
0
<
N
D
1
0
=
D
2
0
>
Appear 3.3-Error! Argument of unknown switch. : Definition of
D
D
1
2
and
These key words make it possible to also give an account of the contact between hulls of which only them
average surfaces are with a grid:
D
D
1
2
and
are worth then the half-thickness of the hulls (values
positive) (cf [Figure 3.3-c]).
half-épa
issor E
2
half-thickness E
1
surface average
real edge of the hull
real edge of the hull
surface average
D
1
+ D
2
= E
1
+ E
2
> 0
Appear 3.3-Error! Argument of unknown switch. : Contact between hulls
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Note:
If one uses
DIST_1
and
DIST_2
, it is necessary to take guard with the visual interpretation of the results. If
D
D
1
2
0
+
>
, the code will be able to announce contact whereas visualization shows one
spacing of the two mesh. If
D
D
1
2
0
+
<
, the code will be able to announce contact whereas
visualization will show two interpenetrated mesh.
Help memory:
To remember the signs, to think of:
D
or D
1
2
0
0
>
>
: “matter addition” compared to the mesh,
D
or D
1
2
0
0
<
<
: “ablation” of matter compared to the mesh.
3.4
Dualisation of the conditions of nonpenetration
To simplify the writing, we in this chapter in linear elasticity place (matrix
C
, loading
F
), by forgetting the boundary conditions of Dirichlet. If one dualise conditions of nonpenetration
(cf [bib3]), one must solve the following system, including/understanding equations and inequations:
Cu
+
With
T
µ
µ
µ
µ =
F
With
D
-
µ
µ
µ
µ
0
J
,
,
,
, ((((
With
-
-
-
-
D
-
-
-
-
)
)
)
)
J
µ
µ
µ
µ
J
=
0
The first line expresses the equilibrium equations: the vector
With
T
µ
µ
µ
µ
can be interpreted like
nodal forces due to the contact. The second line represents the geometrical conditions of not
interpenetration: the inequality is understood component by component (each line relates to one
potential contact couples). The third line expresses the absence of opposition to separation (them
surfaces of contact can know only compressions). The last line is the condition of
compatibility: when for the connection J the multiplier of Lagrange
µ
µ
µ
µ
J
is nonnull, there is contact and
thus play
(D
-
-
With)
J
is null; when the play is nonnull (two surfaces are not in contact),
the associated multiplier must be null (not compression).
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4
Resolution of the problem of contact
4.1
Position of the problem
One treats the unilateral contact after the phase of prediction and each iteration of Newton. Thus it
field of displacements passes by the following states:
beginning of the pitch of time I:
U
I
-
1
prediction
~
U
I
0
processing of the contact
U
I
0
update:
U
U
U
I
I
I
0
1
0
=
=
=
=
+
+
+
+
-
iteration of Newton number 1
~
U
I
1
processing of the contact
U
I
1
update:
U
U
U
I
I
I
1
0
1
=
=
=
=
+
+
+
+
…
iteration of Newton number N
~
U
I
N
processing of the contact
U
I
N
update:
U
U
U
I
N
I
N
I
N
=
=
=
=
+
+
+
+
-
1
…
When the contact is not treated, the systems to be solved are as follows (with the notations of
[R5.03.01]):
()
K
B
B
U
L
L
U
K
B
B
U
L
R U
B
0
0
0
1
1
1
0
0
0
T
I
I
I
méca
I
ther
I
D
I
N
T
I
N
I
N
I
méca
I
N
T
I
N
=
+
=
-
-
-
-
-
~
~
~
~
with the phase of prediction
with N
iteration of Newton
ičme
One can write the generic form of the system to be solved when the contact is not treated:
C U
F
=
,
where
U
gather the degrees of freedom of displacement
U
and multipliers of Lagrange
associated
in the boundary conditions of Dirichlet (it
~
indicate that the contact is not taken into account),
C
is
stamp tangent supplements, and
F
the second member.
The relation of noninterpenetration is written:
With
D
0
(
D
0
is the initial play, measured on the mesh),
or:
With
U
D
-
=
D
0
-
With
-
if
U
=
U
-
+
U
(cf [§3.2]).
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In the presence of conditions of unilateral contact, systems to solve (dualisation of the conditions of
not interpenetration) are thus written:
C U WITH
F
WITH U D
+
=
-
T
µ
µ
µ
µ
where
With
is the complete matrix of the conditions of contact,
µ
µ
µ
µ
the vector of the multipliers of Lagrange
associated the contact (they must be positive or null),
With
T
µ
µ
µ
µ
the vector of the nodal forces of contact, and
D
-
the vector (plunged in the whole of, the multiplier degrees of freedom of Lagrange included/understood)
containing the play running (for
U
-
):
D
D
D
With
-
-
-
= = -
0
0
0
4.2
Method of the active stresses
One will be able to find a description complete of the method with the theoretical justifications
necessary in [bib2] and [bib3]. The principle is as follows: a whole of stresses is postulated
known as active, which corresponds to a null play (the relation inequality becomes an equality); it is solved
system of equations obtained in this subspace, and it is looked at if the starting postulate were justified. If
the selected unit was too small (active connections had been forgotten), one adds with the unit the connection
violating more the condition of noninterpenetration; if the selected unit were too large (connections
presumedly active are not it in fact not), one removes from the unit the most improbable connection i.e that
of which the multiplier of Lagrange violating condition 3 of the system of [§3.4] to the greatest value
absolute. The fact of removing or of adding only one connection with each iteration of the method guarantees
convergence in a finished number of iterations [bib2].
One notes
U
-
the field of displacements obtained before treating the contact: it is about
U
I
-
1
when one
draft the contact at the end of the phase of prediction, and of
U
I
N
-
1
when one treats the contact at the end of
the iteration of Newton number N. Increments of displacements (obtained without taking into account it
contact) calculated before are thus not taken into account in
U
-
. The increment is sought
U
with
to add with
U
-
to obtain
U
I
0
or
U
I
N
.
The method of the active stresses is an iterative method uncoupled from the iterations of Newton: with
each iteration of active stresses, the starting solution is noted
U
K
, and the increment added by
new iteration is noted
K
+
1
. One thus has in theory:
U
U
K
K
K
+
+
=
+
1
1
, and
U
U
U
=
+
+
-
+
K
K
1
. One leaves
U
0
=
C
-
1
F
, which is the increment obtained without treating the contact
(
U
0
=
U
I
0
given by the prediction, or
U
0
=
U
in
given by N
ičme
iteration of Newton) and one
carry out the iterations of active stresses until clean convergence of this algorithm.
convergence within the meaning of the active stresses is obtained when no connection violates the condition
kinematics 2 of [§3.4] and when the associated multipliers of Lagrange are all positive.
In elasticity, at the end of the iterations of active stresses, there is a result converged within the meaning of
Newton. In plasticity or if the geometry is reactualized, it is not the case because several iterations
of Newton are necessary to obtain balance. After each iteration of Newton, one launches
the algorithm of active stresses to satisfy the conditions of contact. Thus, in elasticity, one
will necessarily converge for each pitch in an iteration if
REAC_GEOM_INTE
= 0 or
REAC_GEOM_INTE
= 1 in iterations if
REAC_GEOM_INTE
= N, N > 1.
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4.2.1 Resolution of the system reduced to the active connections
At the beginning of the algorithm, one evaluates the current play
D
0
[]
J
-
With
K
(U
-
+
U
0
)
[
]
J
for all the connections J,
by taking account of displacement
U
0
=
C
-
1
F
estimated without treating the contact. One calls activates one
connection for which this play running is negative, which indicates an interpenetration. One postulates that for
the active connections, the effective play will be null, and that thus the inequality
With
D
0
becomes an equality for
active connections.
Note:
One could leave the old active connection set obtained to convergence of the master key
the preceding one, but if the couples of contact were reactualized, numbers of connections
correspond more inevitably. However, whenever this is licit, the iteration count of
the method of the active stresses can be decreased by it, as it is the case with the key word
LIAISON_UNIL_NO
.
If one notes
With
K
the matrix of contact reduced to the active connections with the iteration
K
(only they are kept
lines corresponding to the active connections), one a:
C U
C
With
F
WITH U
WITH U
With
D
K
K
kT
K
K
K
K K
+
+
=
+
+
=
+
-
+
µ
µ
µ
µ
1
1
0
or:
µ
µ
µ
µ
µ
µ
µ
µ ====
K
K
K
T
K
K
T
K
K
+
-
-
-
-
-
=
-
-
-
-
-
1
1
1
1
0
1
C F
U
A.C.
WITH A.C.
D
WITH C F A U
,
maybe, by taking account of
C
-
1
F
=
U
0
:
-
-
=
-
-
-
-
+
-
WITH A.C.
D
WITH U
U
U
A.C.
K
K
T
K
K
K
kT
1
0
1
0
1
µ
µ
µ
µ ====
µ
µ
µ
µ
with
D
-
=
D
-
0
, where
D
-
=
D
0
-
With
K
U
-
is the updated play corresponding to the field of displacement
U
-
.
The first equation gives the values of the multipliers of Lagrange
µ
µ
µ
µ
associated the relations of
not penetration for the active stresses, and the second equation gives the value of the increment
K
+
1
unknown factors for the kth iteration of the method of the active stresses.
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4.2.2 Validity of the whole of active connections selected
That is to say the connection number
J
(one indicates by
[]
J
the jičme component of a vector, i.e. that
corresponding to the connection
J
). Three situations are possible:
1) relative displacement compensates for the initial play:
[
]
[]
With U
U
D
K
K
K
J
J
(
)
-
+
+
+
=
1
0
2) relative displacement is lower than the initial play:
[
]
[]
With U
U
D
K
K
K
J
J
(
)
-
+
+
+
<
1
0
3) relative displacement is higher than the initial play:
[
]
[]
With U
U
D
K
K
K
J
J
(
)
-
+
+
+
>
1
0
The situation (3) is prohibited: it corresponds to a violation of the condition of noninterpenetration.
situation (1) corresponds to a connection known as active, the situation (2) with a nonactive connection.
At the beginning of the kth iteration of the algorithm, one had postulated a whole of active connections. One has
found an increment possible
K
+
1
unknown factors under these assumptions: one now will check that
this increment is compatible with the assumptions. In practice, that consists in checking:
(I)
that the nonactive supposed connections do not violate the condition of noninterpenetration
(if not one activates one of them);
(II) that the presumedly active connections are associated multipliers of contact
µ
µ
µ
µ
positive or null (if not one decontaminates one of them)
Checking (
I
): (is the whole of the active connections too small?)
One will calculate for all the nonactive supposed connections the quantity:
[
]
[
]
[
]
[
]
J
K
K
J
K K
J
K
K
J
K K
J
=
-
+
=
-
-
+
-
+
D
With U
U
With
D
With U
With
0
1
1
(
)
·
if
[
]
With
K K
J
+
1
is negative, the play for the connection J will increase, and thus the supposed connection
not remainder in this state activates when one writes
U
U
K
K
K
+
+
=
+
1
1
,
·
if
[
]
With
K K
J
+
1
is positive,
J
should be higher strictly than 1 for a nonactive connection
(situation (b)). One thus examines
=
Min
J
J
on the whole of the connections
J
declared not
active. If
<
1
, that indicates that a connection at least is violated (situation (3)) : one adds
then with the list of the active connections the number of the most violated connection, i.e. that which
carry out the minimum of
J
, and one writes
U
U
K
K
K
+
+
=
+
1
1
(that corresponds to a null play
for the added connection). In this case one shunts the checking (
II
).
Note:
If all the connections are active, the checking (
I
) place does not have to be. In this case, one poses
=
1
and one passes directly to the checking (
II
).
Code_Aster
®
Version
5.0
Titrate:
Unilateral contact by conditions kinematics
Date:
17/04/01
Author (S):
NR. TARDIEU, I. VAUTIER
Key:
R5.03.50-B
Page:
23/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Checking (
II
): (is the whole of the active connections too large?)
One places oneself now if
1
: one takes
U
U
K
K
K
+
+
=
+
1
1
.
·
if no connection is active, the method converged towards a state without contact,
·
if there are presumedly active connections:
-
if all multipliers of Lagrange
µ
µ
µ
µ
are positive or null, one also converged
towards a state with effective contact,
-
if there are multipliers of Lagrange
µ
µ
µ
µ
negative, corresponding connections
should not be active: one withdraws from the whole of the active connections the connection of which it
negative multiplier is largest in absolute value.
Note:
One removes and one adds the active connections one by one (and not all those which contradict
assumptions) in order to ensure the convergence of the algorithm in a finished number of iterations,
as shown in [bib2] and [bib3]. However, one could take the risk to add all
the connections which seem active of a blow, or to decontaminate all the connections with multiplier
negative of a blow, in order to accelerate the convergence of the method (it is what is made for
processing of friction, cf [R5.03.51]). Even if convergence is not theoretically ensured,
such an alternative seems to go in practice.
4.3
Recutting of the pitch of time
On the theoretical level, the convergence of the method of the active stresses is ensured in a number
finished iterations. In practice, certain numerical artefacts can return this convergence
delicate. Also a strategy it was developed to ensure the robustness of the algorithm.
During calculations of contact, in particular if the pitches of load carried out are too important, of
undesirable phenomena can appear:
·
stamp contact
WITH A.C.
K
K
T
-
1
singular,
·
oscillation of the method of the active stresses: a node is detected alternatively
“stuck” then “taken off”.
To mitigate these difficulties, the following strategy was adopted. If:
·
the matrix of contact
WITH A.C.
K
K
T
-
1
is singular,
·
the iteration count of active stresses is higher than a limit which depends on the number
potential connections
Then one redécoupe the pitch of time i.e one returns to the preceding pitch of load and instead of testing
to reach the level of loading following in a pitch as one has just done it, one does several of them
(For more precise details on this functionality of the operator
STAT_NON_LINE
, to see documentation
[U4.51.03]).
In practice, this functionality is shown very useful for the user.
Code_Aster
®
Version
5.0
Titrate:
Unilateral contact by conditions kinematics
Date:
17/04/01
Author (S):
NR. TARDIEU, I. VAUTIER
Key:
R5.03.50-B
Page:
24/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
5 Precautions
of use
An example of use and the associated consultings are given in [bib6].
The main consultings or warnings are as follows:
·
to check that the normals on the surfaces of contact are outgoing (to be wary in particular if one has
used operators of symmetrization in the maillor gibi),
·
attention in contact with quadratic 3D if the meshs of edge are QUAD8 (to avoid using
HEXA20 to net volume or to refine “sufficiently”): to use preferably
HEXA27, or many PENTA15 whose TRIA6 sides are the meshs in contact,
·
to remove, by boundary conditions of Dirichlet adapted, the movements of body
rigid: it is not necessary that the structure “holds” only by the contact. In other words, that
wants to say that a calculation made in elasticity with the control
MECA_STATIQUE
(without treating it
contact thus) must pass,
·
in the event of structure “held” only by the contact, one can add a spring of weak
rigidity to maintain it. This rigidity will not have to disturb the field of deformations of
structure supposed no one (since there is rigid movement of body), but to prevent one
displacement ad infinitum. In practice, its choice proves to be delicate and requires a retiming
precondition,
·
to use the key word
SANS_NOEUD
or
SANS_GROUP_NO
to exclude from the list of the future nodes
slaves those which are subjected in addition to boundary conditions of Dirichlet
(
DDL_IMPO
,
FACE_IMPO
,
LIAISON_…
) in the awaited direction of the contact,
·
the calculation of the efforts of contact can be carried out in the control
POST_RELEVE_T
in
calculating the resultant of the nodal forces on the group of meshs representing one of
surfaces of contact,
·
the contact and the linear search for
STAT_NON_LINE
do not do good housework together
when one converges in addition to one iteration. Roughly speaking, that wants to say that one cannot
to use linear search except for elastic designs without reactualization
geometrical, which is rather restricted.
6 Bibliography
[1]
NR. Tardieu, I. Vautier, E. Lorentz, “quasi-static nonlinear Algorithm”, Documentation
of Reference of Code_Aster n° [R5.03.01].
[2]
G. Dumont, “the method of the active stresses applied to the unilateral contact”, Notes
intern EDF n° HI-75/93/016
[3]
G. Dumont, “Algorithm of active stresses and unilateral contact without friction”, Review
European of the finite elements, vol. 4 n°1/1995, pp. 55-73
[4]
I. Vautier, “Some methods to deal with the problems of unilateral contact involved
great slips “, Notes intern EDF n° HI-75/97/013
[5]
I. Vautier, “Evaluation of the difficulties of modeling of the unilateral contact for mesh
3D quadratic “, internal Report EDF n° MN/97/023
[6]
I. Vautier, “Example of use of the functionalities of contact in great displacements in
Code_Aster “, Notes intern EDF n° HI-75/97/034/A