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Titrate:
Unilateral contact by conditions kinematics
Date:
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Key:
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Organization (S): EDF/MTI/MN
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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 operator STAT_NON_LINE. One uses conditions kinematics of
not interpenetration which is dualisées. The formulation used is of main type slave (node-facet 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 constraints within each iteration of the total method of Newton
of operator STAT_NON_LINE.
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1 Introduction
Key word CONTACT of command AFFE_CHAR_MECA makes it possible to define conditions of contact
unilateral which is treated (in command STAT_NON_LINE) in initial or reactualized geometry,
in a nodal formulation or node-facet, adapted better to the incompatible grids and to
slip of surfaces one compared to the other. It will replace the key word in the long term
LIAISON_UNIL_NO which is valid only for compatible grids undergoing of small
slips.
One presents here an algorithm based on the method of the active constraints [bib2]. It is that which is
used by defect and which corresponds to METHODE:“CONTRAINTE” of key word CONTACT. Another
algorithm is available under METHODE:“LAGRANGIEN”. 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 locate the entities between which one writes them (it is what is called pairing).
In Code_Aster, the use of key word CONTACT makes it possible 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 ingredients in an algorithm of processing of the unilateral contact:
· the location: 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 constraints 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
Zones and surfaces of contact
One considers the 3 solids of the figure [Figure 1.2-a], represented in 2D. 3 possible zones were defined
of interpenetration enters the solids: a zone enters the solid A and the solid B, and two zones between
solid B and the solid C. the user, who defines these zones in the command file, supposes here
that apart from these zones, there is no risk of interpenetration, taking into account the loading.
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Solid A
surface 1
zone 3
surface 2
surface 2
zone 1
Solid B
surface 1
Solid C
surface 1
surface 2
zone 2
Appear 1.2-Error! Argument of unknown switch. : Definition of 3 zones of contact
Each zone of contact is defined in operator AFFE_CHAR_MECA by an occurrence of the key word
CONTACT. A zone is composed by definition of two surfaces which one seeks to prevent
interpenetration: first is defined under key word GROUP_MA_1 (or MAILLE_1), the second
under 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 grid 2D, TRIA3, TRIA6,
QUAD4, QUAD8 or QUAD9 for a grid 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 grid.
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 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 (modelled 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 key word GROUP_MA_2 (meshs slaves).
section of the beam can then be taken into account by the use of key word DIST_2 (cf [§3.3]).
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C
N
With
D
N
N
T
With
B
With
C
B
B
B
B
With
With
C
With
B
T
N
N
N
C
D
2D
3D
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 zones of contact, i.e. not having no node in
commun run.
Chapter 2 details the method of pairing for the formulations node-facet and nodal:
the establishment of the potential couples of contact is made zone by zone. 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 constraints, described in
chapter 4: the resolution of the problem obtained is total, i.e. it takes into account them
couples of all the zones simultaneously.
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2
Establishment of the couples of contact
2.1 Principle
Potential surfaces of contact were defined in operator AFFE_CHAR_MECA like
specified in [§1.2]. The effective processing of the contact is done, him, in operator STAT_NON_LINE.
The resolution of a nonlinear problem in operator STAT_NON_LINE is described in detail in
document [R5.03.01]. We recall here briefly the principal phases of them, for a calculation
comprising two steps of time:
1st step 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 step of load and not
during iterations.
One calls “master key of contact” each occurrence of processing of the contact.
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2.1.1 Formulation
node-facet
This formulation, chosen by key word APPARIEMENT: “MAIT_ESCL”, does not grant a role
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 S2 slave
*
*
*
*
surface main S1
Appear 2.1.1-a: Surface Master and surface 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 make it possible to give,
zone by zone, 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
APPARIEMENT: “MAIT_ESCL_SYME” which 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 (APPARIEMENT: “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 grids (nodes “opposite”) which remain 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-facet 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.
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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.
Key word RECHERCHE: “NOEUD_BOUCLE” starts the examination of all the main nodes
belonging to the same zone 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 RECHERCHE: “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 stripes 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: Territoire covered by RECHERCHE: “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 step of time to the other.
One thus can has minimum to use option RECHERCHE: “NOEUD_VOISIN”.
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2.3
Search of the mesh nearest (node-facet 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 A 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.
One seeks the parametric co-ordinate of the point M, projection of the node slave P according to
entering normal NR with mesh AB, defined by:
AM = AB
MP

= NR
(AB, NR) =

0
where (,) indicates the scalar product.
slave
P
P'
NR
NR
Master
Me B M
With
Be reproduced 2.3.1-a: Projection on a segment
(AP, AB)
One a: =
.
(AB, AB)
The point M belongs to mesh AB if 0
[;1]. If > 1 (case of P' projected in Me), one brings back
projection of A by posing = 1; if < 0, one bring back projection out of B by 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:
yB - teststemyà
-

NR =
L
,
2
2
with L
length AB
of: L = (xB - teststemxà) + (yB - teststemyà)
X
.
B -

teststemxà

L

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The value of (PM, NR) is called the play between the node slave and the mesh Master. The 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 vector PM, and
thus play by the standard of PM. However, 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 = (10­15; 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 grid. 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 slave
C
NR
*
With
M
Master
B
Be reproduced 2.3.2-a: Projection on a triangle
One seeks the parametric co-ordinates 1 and 2 of the point M, projection of the node slave P
according to the normal NR entering to triangular mesh ABC (one uses the normal with the mesh, but in
the direction slave towards Master), defined by:
AM = AB
1
+ AC

2
PM

NR = 0
that is to say:

((AP NR), A)
C

1 = -

(AB
)
AC
- AB AC

with NR =
((AP NR), AB)

(entering unit normal).


AB AC
2 =

(AB

)
AC
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If one poses 3 = 1 - 1 - 2, the values of the three parametric co-ordinates 1,2,3 allow
to find if the point M belongs or not to triangle ABC, like illustrates it the figure [Figure 2.3.2-b].
1 <0
3 <0
6
C
1
1 <0
3
0
3 <0
1 >0
2> 0
3> 0
With
1<0
2 <0
5
2 <0
B
2
2 <0
3 <0 4
Appear 2.3.2-b: Zones possible 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 or BC). If it is in sectors 4, 5 or 6, one brings back it on the corresponding point
(not B, A or C). That amounts cancelling the parametric co-ordinates which are negative.
Let us take the example of sector 1 where 1 < 0. One brings back the point M to the point Me, defined by:
AM'
'
= AC

2

AM = AB
1
+ AC
2
(
AM', ME) = 0


AB AC
One finds: '
(
,
)
2 = 2 + 1
.
(
,
AC
)
AC
AB AC
In sector 2, an identical reasoning gives: '
(
,
)
1 = 1 + 2
and AM'
'
= AB
(AB, AB)
1
.
-
'
1 (AB,
)
BC + 1
(-) (
,
AC
)
BC
In sector 3, there are AM'
'
AB (
'
=
2
1
+ 1 -) AC
1
with 1 =
.
(
,
BC
)
BC
The play is calculated like the scalar product between vector PM and the normal NR entering to the mesh
Master.
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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: Découpage 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 MMN 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 grid on the edges of the structures in contact.
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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 key word REAC_GEOM_INTE of the 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 step 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 step of load given. With convergence of this last,
a geometrical reactualization then a new pairing are
carried out. One does not pass to the step of load according to but one begins again
the same step 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
It is a generalization of the preceding case. Within the same step of
(n>2)
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 step 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 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 his 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 Mr. Plaçons us if the mesh Master is a SEG2 for
to simplify the talk. One has then:
U
= (M) U + (M) U
M
With
With
B
B
The relation of nonlinearized penetration consists in saying that relative displacement between 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]).
B
surface main
M
With
NR
P
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.N 0, is P-M- .N + (U
M - U
) .N
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) .N P-M- .N
-
-
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) .N
-
P
M
D,
maybe, by using the relation U
= (M) U + (M) U
M
With
With
B
B:
[U - ((M) U + (M) U)].N -
P
With
With
B
B
D
The extension of the formula for a mesh comprising Master nmaît noted nodes Bj, is immediate:
N

maît


U - (M) U. NR
-
P
B
B
D
J
J

J

=1


If one writes such a relation for all the couples of contact, one obtains the geometrical conditions
of nonpenetration in matric form:
With D
Note:
The effective play in the configuration U + U is d0 - A (U + U), is D - With. The condition
of nonpenetration thus expresses that the effective play remains positive or null in any configuration.
Matrix A, 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]:
D
B
L/2 M1
M
NR
*
2
L/4
C
*
NR
d1
2
D
P
Q
Be reproduced 3.1-b: Ecriture of the matrix of contact A on an example
If one notes for example uB the increment of displacement of the node B according to direction X, and
X
D and D
1
the 2 current plays for the two couples, the two relations of nonpenetration are written
matriciellement:
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U


Px


U
P
y
U
Qx


U
Q
y
NR
NR
0
0
-.
0 NR
5
-.
0 NR
5
-.
0 NR
5
-.
0 NR
5
0
0
U D
X
y
X
y
X
y
B

X
1



0
0
NR
NR
0
0
-.
0 75N
-.
0 75N
-.
0 25N
-.
0 25N
U

D
X
y
X
y
X
y
B

y
2
U
Cx


U
Cy
U
Dx


U


D y
Note:
One considered here only the degrees of freedom of the nodes implied in the contact; in
reality, matrix A is hollower.
3.2
Coefficients of the relation of nonpenetration
The relation of nonpenetration is written:
nmaît



U -
M U
NR D -
()
.
P
B
B
J
J
J

=1

One gives below the values of the functions of form B (M) of the main nodes to the point M
J
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:
(M) = 1 -
With

(M) =
B
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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 this one is curved. Nevertheless, one calculates the values of
functions of form associated to the SEG3 (nodes A and B, node medium C) starting from the co-ordinate
parametric paid to the SEG2:
1


TO (M) = 2 (1 -) (-)

2
1


B (M) = 2 (-)

2

C (M) = 4 (1 -)


3.2.3 Meshs
TRIA3
One explained in [§2.3.2] how one finds the co-ordinates parametric 1 and 2 of
projection M of the node slave in the triangle. The values of the functions of form are in fact those
parametric co-ordinates:

TO (M) = 1 - 1
- 2



B (M) = 1



C (M) = 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
To = 1 - 1 - 2, B = 1, C = 2) to deduce the values from them from the functions of form to the 6
nodes (A, B, C nodes, D, E, F nodes mediums respectively on the sides AB, BC and CA):

WITH (M) = A (2A -)
1

B (M) = B (2B -)

1

C (M) = C
(2C -)
1

D (M) = 4AB

E (M) = 4BC


F (M) = 4 C
With
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3.3
Introduction of a fictitious play
One can want to model the contact between structures whose grid did not take account of
certain characteristics (“hole” or “bump” nonwith a grid (cf [Figure 3.3-a])).
NB: N = ­ NR
N
real structure with a bump
grid
grid
real structure with a hole
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) .N -
P
M
D - (d1 + d2)
where D and D
1
2 are given by the user respectively under key words DIST_1 and DIST_2 for
each zone of contact. These “distances” have a sign: they represent the translation to be applied to
node of the grid in the direction of outgoing normal N to obtain the point of the real structure
(cf [Figure 3.3-b]).
d1 = 0
d1 = 0
N
N
d2 > 0
d2 < 0
Appear 3.3-Error! Argument of unknown switch. : Definition of D and D
1
2
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 and D
1
2 are worth then the half-thickness of the hulls (values
positive) (cf [Figure 3.3-c]).
half-thickness e1
surface average
real edge of the hull
d1 + d2 = e1 + e2 > 0
real edge of the hull
half-thickness e2
surface average
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 grids. If D + D
1
2 < 0, the code will be able to announce contact whereas
visualization will show two interpenetrated grids.
Help memory:
To remember the signs, to think of:
D > 0 or D > 0: “matter addition” compared to the grid,
1
2
D < 0 or D < 0: “ablation” of matter compared to the grid.
1
2
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
+ ATΜ = F
With
D
µ
0

J, (With D)

J µj = 0
The first line expresses the equilibrium equations: vector AT 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 J is nonnull, there is contact and
thus the 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 step of time I:
ui-1
prediction
~u0i
processing of the contact
u0i
update:
u0
U
u0
= - +
I
I 1
I
iteration of Newton number 1
~u1i
processing of the contact
u1i
update:
u1
u0
u1
=
+
I
I
I

iteration of Newton number N
~uni
processing of the contact
linked
update:
one
un-1
one
=
+
I
I
I

When the contact is not treated, the systems to be solved are as follows (with the notations of
[R5.03.01]):
K
BT u~0 Lméca

+ Lther


0
I
I
I

~ =

with the phase of prediction
B
0
0
ud


I
I
K n-1 BT u~n
méca
N
T
N

-
-
-

I
I
L
R
I
(U 1i) B 1i

~ =
with the nth iteration of Newton
B
0
N


I
0
One can write the generic form of the system to be solved when the contact is not treated:
CU = F,
where U gathers the degrees of freedom of displacement U and the associated multipliers of Lagrange
with boundary conditions of Dirichlet (the ~ indicates 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 (D is the initial play, measured on the grid),
0
0
or: With D = d0 - 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
+ AT µ = F
WITH U



D
where A is the complete matrix of the conditions of contact, µ the vector of the multipliers of Lagrange
associated the contact (they must be positive or null), AT 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 d0 - With
D =
=

0
0

4.2
Method of the active constraints
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 constraints 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 ui-1 when one
n-1
draft the contact at the end of the phase of prediction, and ui
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. One seeks the increment U with
to add with U to obtain u0
N
I or ui.
The method of the active constraints is an iterative method uncoupled from the iterations of Newton: with
each iteration of active constraints, the starting solution is noted the U.K., and the increment added by
new iteration is noted K +1. One thus has in theory: The U.K. +1
The U.K.
K +
=
+
1

, and
U = U + U +
+
K
K
1. One starts from U0 = C-1F, which is the increment obtained without treating the contact
(U0 = U0
N
I given by the prediction, or U0 = Ui given by the nth iteration of Newton) and one
carry out the iterations of active constraints until clean convergence of this algorithm.
convergence within the meaning of the active constraints 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 constraints, 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 constraints to satisfy the conditions of contact. Thus, in elasticity, one
will necessarily converge for each step 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] - A
J
K (U + u0)
for all the connections J,
J
by taking account of U0 displacement = C-1F 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 the d0 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 constraints can be decreased by it, as it is the case with the key word
LIAISON_UNIL_NO.
If one notes Ak the matrix of contact reduced to the active connections with the iteration K (one keeps only them
lines corresponding to the active connections), one a:
C

U
K + C k+
1 + AT µ = F
K
WITH U


- + A the U.K.
K +1
K
K
+ A
= D
K
0
or:

k+1
- 1
K
- 1 T
= C F - U

- C Ak µ

,
- 1 T
- 1
-
-

Ak C Ak µ = D0 - AkC F - Ak U
maybe, by taking account of C-1F = U0:

-
- 1 T
-
With C.A. µ = D
0
K
K
- A U


K
K +1

= U0

- The U.K.

-


-
C 1ATk µ

with D = D
, where D = D
0
0 - With ku- is the updated play corresponding to the field of displacement
U.
The first equation gives the values of the multipliers of Lagrange µ associated the relations of
not penetration for the active constraints, and the second equation gives the value of the increment
k+1 of the unknown factors for the kth iteration of the method of the active constraints.
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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 jième component of a vector, i.e. that
corresponding to the connection J). Three situations are possible:
1) relative displacement compensates for the initial play: [A (U
the U.K.
K +
+
+ 1)] = [D
K
0]
J
J
2) relative displacement is lower than the initial play: [A (U
the U.K.
K +
+
+ 1)] < [D
K
0]
J
J
3) relative displacement is higher than the initial play: [A (U
the U.K.
K +
+
+ 1)] > [D
K
0]
J
J
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 of the 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
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:
[
-
K
-
K
d0 - Ak (U + U
)]
D - With U
J
[
K
]

J
J =
[
=
K +1
K +
With
1
K
]
With
J
[K] J
· if
[A K
K +1] is negative, the play for the connection J will increase, and thus the supposed connection
J
not remainder in this state activates when the U.K. +1 is written
The U.K.
K +
=
+
1

,
· if
[A K
K +1] is positive,
J
J should be higher strictly than 1 for a nonactive connection
(situation (b)). One thus examines = Min J on the whole of the connections J declared not
J
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
K +1
K
K +1
J, and U is written
= U +
(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) does not take place to be. In this case, one poses = 1
and one passes directly to the checking (II).
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Checking (II): (is the whole of the active connections too large?)
One places oneself now if 1: the U.K. +1 is taken
The U.K.
K +
=
+
1

.
· if no connection is active, the method converged towards a state without contact,
· if there are presumedly active connections:
-
if all the multipliers of Lagrange µ are positive or null, one also converged
towards a state with effective contact,
-
if there are negative multipliers of Lagrange µ, the 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 step of time
On the theoretical level, the convergence of the method of the active constraints 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 steps of load carried out are too important, of
undesirable phenomena can appear:
· stamp singular contact A C-1AT,
K
K
· oscillation of the method of the active constraints: a node is detected alternatively
“stuck” then “unstuck”.
To mitigate these difficulties, the following strategy was adopted. If:
· the matrix of contact A C-1AT is singular,
K
K
· the iteration count of active constraints is higher than a limit which depends on the number
potential connections
Then one redécoupe the step of time i.e one returns to the preceding step of load and instead of testing
to reach the level of loading following in a step as one has just done it, one does several of them
(For more precise details on this functionality of operator STAT_NON_LINE, to see documentation
[U4.51.03]).
In practice, this functionality is shown very useful for the user.
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5 Precautions
of use
An example of use and the associated consultings are given in [bib6].
The principal 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 command 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 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 command 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 Algorithme”, Documentation
of Référence of Code_Aster n° [R5.03.01].
[2]
G. Dumont, “method of the active constraints applied to the unilateral contact”, Note
intern EDF n° HI-75/93/016
[3]
G. Dumont, “Algorithme of active constraints and unilateral contact without friction”, Revue
European of the finite elements, Vol. 4 n°1/1995, pp. 55-73
[4]
I. Vautier, “Quelques methods to deal with the problems of unilateral contact involved
great slips “, internal Note EDF n° HI-75/97/013
[5]
I. Vautier, “Évaluation of the difficulties of modeling of the unilateral contact for grids
3D quadratic “, Account-returned intern EDF n° MMN/97/023
[6]
I. Vautier, “Exemple of use of the functionalities of contact in great displacements in
Code_Aster “, internal Note EDF n° HI-75/97/034/A
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