Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
1/30

Organization (S): EDF-R & D/AMA
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.51

Contact ­ discrete friction in 2D and 3D

Summary:

Discrete modelings of the contact with friction in 2D and 3D, zones of slip being
respectively 1D and 2D, are proposed starting from a mixed variational formulation constraints
displacements. The conditions of contact and friction are treated with the nodes of surfaces of contact of
solids implied with taking into account of great displacements. The law of friction of Coulomb is treated
by operators STAT_NON_LINE and DYNA_NON_LINE after definition of the conditions of contact and of
friction under key word CONTACT of AFFE_CHAR_MECA.

In 2D as in 3D, various methods are usable for the modeling of the problem:

·
taking into account of the contact and friction using multipliers of Lagrange,
·
taking into account of the contact using multipliers of Lagrange and friction using
penalization,
·
taking into account of the contact and friction using penalization.

The subjacent algorithms are inspired by the active constraints [bib2] [bib9] and by the predictor-correctors
usually used in plasticity.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
2/30

Count

matters

1 Introduction ............................................................................................................................................ 3
2 Formulation of the model .......................................................................................................................... 4
2.1 The criterion of Coulomb ..................................................................................................................... 4
2.2 General formulation for various criteria .................................................................................. 6
2.3 Formulation by differential inclusions ........................................................................................ 7
2.4 Resolution of the problem of balance. ................................................................................................ 8
2.5 Variational formulation ............................................................................................................... 9
3 Contact rubbing 2D and 3D in Code_Aster .................................................................................... 11
3.1 Formulation of the problem ............................................................................................................... 11
3.2 Dualisation of the conditions of contact and friction ................................................................. 12
3.3 Dualisation of the conditions of contact, regularization of the conditions of friction ..................... 13
3.4 Regularization of the conditions of contact and friction ........................................................... 14
4 algorithmic Resolution ..................................................................................................................... 15
4.1 Linearization of the various terms ................................................................................................ 15
4.1.1 Forces intern ..................................................................................................................... 15
4.1.2 Forces of contact ................................................................................................................. 15
4.1.3 Forces of slip ........................................................................................................... 15
4.1.4 Forces of adherence .............................................................................................................. 16
4.1.5 Notice ............................................................................................................................. 16
5 Resolution ............................................................................................................................................ 17
5.1.1 Dualisation of the conditions of contact and friction in 2D ............................................. 17
5.1.2 Dualisation of the conditions of contact and regularization of friction in 2D and 3D ............ 18
5.1.3 Regularization of the conditions of contact and friction in 2D and 3D .............................. 20
5.1.4 Dualisation of the conditions of contact and friction in 3D ............................................. 21
5.1.5 Convergence ........................................................................................................................ 23
5.1.6 Notice ............................................................................................................................. 23
6 Compatibility with the boundary conditions of Dirichlet ................................................................... 24
6.1 Writing of the boundary conditions ................................................................................................ 24
6.2 Return to the problem of contact ..................................................................................................... 24
6.3 Illustration on a simple example ................................................................................................. 25
6.4 Notice ...................................................................................................................................... 25
7 Implementation in Code_Aster .................................................................................................... 26
7.1 Algorithms .................................................................................................................................... 26
7.2 Geometrical reactualization ......................................................................................................... 27
7.3 Post processing ............................................................................................................................... 27
7.4 Precautions of use ................................................................................................................. 28
8 Conclusion ........................................................................................................................................... 29
9 Bibliography ........................................................................................................................................ 30
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
3/30

1 Introduction

Discrete modelings of the contact with friction in 2D and 3D, zones of slip being
respectively 1D and 2D, are proposed starting from a mixed variational formulation constraints
displacements. The conditions of contact and friction are treated with the nodes of surfaces of
contact of the solids implied with taking into account of great displacements. The law of friction of
Coulomb is treated by operators STAT_NON_LINE and DYNA_NON_LINE after definition of
conditions of contact and friction under key word CONTACT of AFFE_CHAR_MECA.

In 2D as in 3D, various methods are usable for the modeling of the problem:

·
taking into account of the contact and friction using multipliers of Lagrange,
·
taking into account of the contact using multipliers of Lagrange and friction using
penalization,
·
taking into account of the contact and friction using penalization.

The subjacent algorithms couple the method of the active constraints [bib2] to determine them
areas of contact and an algorithm of resolution of Newton inspired of the methods of the type
predictor-corrector, usually used in plasticity, for friction, in order to determine them
zones of slip [bib1] [bib7] [bib10].

In 2D as in 3D, the user has a whole of methods panachant dualisation and
regularization by penalization:

·
dualisation of the conditions of contact and friction,
·
dualisation of the conditions of contact and regularization of the conditions of friction,
·
regularization of the conditions of contact and friction.

The compared interests of these methods are well-known: the dualisation introduces news
unknown factors but it provides an exact solution; the regularization provides only one approximation of
solution dependant on a parameter chosen by the user but it does not introduce a news
unknown factors.

The document begin with a general presentation from the laws from friction. One presents then
discretization of these laws like their linearization for their integration within the method of
Newton. One details the algorithms then allowing to solve these problems. End of the document
draft of the use practices these methods within Code_Aster and of their post front processing
to approach the conclusions.
Handbook of Référence
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HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
4/30

2
Formulation of the model

2.1
The criterion of Coulomb

Are 2 solids being able to come into rubbing contact:


T
N


The outgoing normal on the surface of contact, U are N = U N
. displacement following this normal,
N
G existing initial play between 2 solids, = N
. N
. the normal force exerted by one of surfaces
N
on the other and = N
. - N shearing.
T
N


Solid
Soli 2
of
N 1 T T
u2
U
T
T
1
2
Solid
Soli 1
of
u1
U


More precisely, for two solids (1) and (2) in contact: the zone of contact is either specific, or
linear is surface. The force of shearing then has as a direction in the zone of contact one
vector T located in the tangent plan (T, T) indicated on the figure above. One defines:
1
2

R =
(N
. ) -
((N
.
N
). ) N = rt, R =
1
1
1
1
1
1
T

the force of shearing exerted by the solid (2) on the solid (1) per unit of area of contact.

Let us write the system of equations and inequations having to be checked by these sizes:

U .n + U .n = U U N
G
1
1
2
2
(-
1
2).


1
0
N
(U .n +u .n - G) = 0
N
1
1
2
2


- µ 0

T
N
U & = U & U & T R
T
(-

2
1)

. =
(- µ) = 0

T
N

0
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
5/30

where N is the outgoing normal with the solid (1) and N the outgoing normal with the solid (2) opposed to N.
1
2
1
first set of equations and inequations corresponds to the management of the contact; it will not be detailed and
we return to [bib9]. The second corresponds to the description of friction obeying the criterion of
Coulomb. It utilizes several fields and binds between them: normal pressure, shearing and
tangent displacement. It can be included/understood as follows:

·
If < µ
,
T
N
-
= 0 and U & = 0
T
·
If = µ
,
T
N
-
> 0 and U & = R

T

One can give following graphic interpretations:


adhéren
adhere T


N
N
slipped
glis NT
its
T
T
2
1
T1

t2


In the space of the constraints, the effort of rubbing contact can be only inside the cone of
Coulomb: if it is strictly inside, the contact is adherent; if it is on the surface of the cone, it
contact is slipping. One can thus give another representation of this criterion for a situation of
contact known:


R
µ N
slipping
member
u&t
slipping
- µ N


Friction induces the concept of threshold; we will see now how to formulate in manner
general of other laws of friction by using this concept.

Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
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:
R5.03.51-C Page:
6/30

2.2
General formulation for various criteria

The selected criteria of friction are form:

G (R) 0

where G (R) is a convex function. The field of nonslip is defined by the interior of the convex one.

Two criteria of friction of the form G (R) 0 are particularly used:

·
the criterion of Tresca where:

G (R) = R - K 0 and K =constante

One notes C the convex disc C of radius K centered in the beginning defined by:

C = {R
R K}.

The condition of nonslip is then defined by the membership of R inside C.
In the event of slip, for R located on the border of C, the direction of slip T of U &
is given by the normal to the criterion in R, as indicated below:


t2
T
R
C
T1
- K
+k
T
+k


·
the criterion of Coulomb where:

G (;
R µ,) = R - K (µ,) 0 and K = µ

N
N
N

The value of K depends on
= (.
N) .n the 0 normal component of the force exerted by
N
one of surfaces on the other and µ, the coefficient of friction of Coulomb. In the event of
slip, for R located on the border of G which is a cone, direction of slip T of
U & is not given by the normal to the criterion in R, but by the normal with the convex disc
C of radius K = µ.
N
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
7/30


N
T
R = µ N
t2
T
T1
T

Determination of the direction of slip for
(T, T
1
2)
the criterion of Coulomb in the reference mark of the vectors


·
the criterion of Mohr-Coulomb where:

G (R, µ, c) = R - K (µ, c) 0 and K = C + µ

N
N
N

It is particularly used to characterize the behavior of interfaces the géomatériaux one (clays
in particular). C is the cohesion of material and µ the coefficient of friction (µ = tan, where is the angle
of friction). Again
= (.
N) .n < 0 so that the contact remains maintained.
N

2.3
Formulation by differential inclusions

One notes V the whole of displacements kinematically acceptable of the problem. The relation enters
the speed of relative slip U & and the shear stress R translates the two possible states of
system: not slip or relative slip following the normal direction to the convex disc C.
For the three criteria presented, the function u& (R) and its reciprocal R (u&) belongs both
with the under-differentials of two combined pseudopotentials, so that one can write:

U & (R) and
*
R (U.
C
&)
C

The appearance of included differential comes from the not-differentiable character of the laws from
contact-friction. Indeed, the indicating function of the convex disc C of radius K indicates,
C
centered at the origin, previously definite. It is such as:

0 if R C
(R) =

C

+ if not
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
8/30

(R) of as a R. It is then the under-differential merges with the normal external with C as a R.
C
C
* U
() =
, where K is the threshold of friction resistance, is combined of Fenchel of
C
&
K u&
indicating function.
*
(U is positively homogeneous degree 1. This function is interpreted
C
&)
C
like the density of power dissipated in the slip. Using the concepts of under-differential,
one can establish the following relations for U & and R associated:

(R) = 0
U & (R)
C

;
C
&.(
U R - R),
0 R
C
U
& V
*
R



C (U
&)
R
(.v & - u&)
*

-

C (v
&)
*
C (U
&)
;
, v & V
*

+
=
=
C (U
&)
C (R)
&.
U R K U &.

Note:

1) The two combined pseudopotentials presented are nondifferentiable.
2) Once known the normal reaction for the criterion of Coulomb, one is reduced locally to
a criterion of friction of Tresca whose threshold is worth K = µ
.
N
3) The adopted local criteria having a circular form one deduces from it that U & (R) implies
C
that there is real positive such as U & = R
.
4) The formulation of the problem of speed suggests an incremental numerical resolution of
problem of friction. The resolution of the problem of balance will thus be presented under
incremental form.

2.4
Resolution of the problem of balance.

One considers two solids of total volume whose surface of contact is. To simplify, one
C
will suppose the existence of a differentiable deformation energy to characterize the answer of
two solids separated with external stresses. In fact, one can show that the results given
hereafter are independent of this assumption. One notes V the whole of the fields of displacement
kinematically acceptable, constrained by the respect of the conditions of contact and friction on
the interface.
The balance of the two solids in the absence of friction is written:

To find U field of displacement kinematically acceptable such as:

U =
min
arg
[((v))- W (v)] {(U) - W (U) (v) - W (v), vV}.

v V



In elasticity, (v) = ((v D
))

is the deformation energy. The function W (v) represents it

work of the external forces. A condition necessary and which becomes sufficient if is strictly
convex so that this balance is checked is that:


D (U) - DW (U) =
D (U) - L
= 0
ext.

where D is the operator derived Gâteaux and L is the linear form associated the external forces.
ext.
Handbook of Référence
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Code_Aster ®
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Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
9/30

With the introduction of friction, the problem must be tackled in incremental form. One is
conduit [bib3] [bib4] with the problem of minimization following on the unit V of the fields
kinematically acceptable constrained by the respect of the conditions of contact and friction on
the interface:

U known, to find U V such as:
U + U
=
min
arg
[((U+ v
))
*
+ (v
) - W (U + v
).
C
T
]
vV

U + U
is thus solution of:



min ((U + v))
D + K v
D
- W (U + v).
T
C


v V




C



where v
is the tangential component of the increment of relative displacement of solid 2 compared to
T
solid 1 along the surface of contact, with the conventions adopted with [§2.1].

Using the relations * (v
) = K v
and * (v
) R. v
if R C one deduces from it that U + U
is
C
T
T
C
T
T
solution of the problem of following MinMax, on space V of the fields kinematically acceptable:

Min Max J (U + v
, R)
v
V
R


where:

J (U + v
, R) =
(
(U + v
))
D + (.
R v
- (R))
D - W (U + v
)



T
C
C

C

The presence of the indicating function in this expression indicates that shearing R on
surface contact belongs to the convex disc of friction C.
C

2.5 Formulation
variational

If is convex, the problem of MinMax to be solved puts in an equivalent way in the form:

To find U V and R C, together of independent variables such as:
J (U + U, R) 0

This amounts solving the system of equations to following balance:

((U+U))
D + R v
D - L v =,
0



T
C
ext.
C


R U
D - (R),
0

T
C
C
C
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Code_Aster ®
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Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
10/30

or in an equivalent way:

((U+U)) d+

(L - L v =
frot
ext.)
,
0

U (R),

T
C
R = (.n T

1
). on .c

As in the preceding section, L is the linear form associated the external forces. The form
ext.
linear L
is associated the forces of shearing exerted by solid 2 on the surface of
frot
contact of solid 1. It will be also noted that the variational formulation makes it possible to find not
only equilibrium equations of the system but also membership of U
with under
T
differential of.
C
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
11/30

3
Contact rubbing 2D and 3D in Code_Aster

We saw the formulation previously continues problem of rubbing contact. We go
now to examine how it is expressed in discrete form.

3.1
Formulation of the problem

With each step of time N, one seeks to check the total balance of the structure:

L (U) = L - L
- L
int
N
ext.
cont
frot
where:
·
L is the operator of calculation of the internal forces
int
·
L is the vector of the external forces
ext.
·
L
is the vector of the forces of contact
cont
·
L
is the vector of the forces of friction
frot

Moreover, the field of displacement U is subjected to a whole of conditions equality and inequality which
N
include/understand connection by connection:

·
With (U) D
nc
N
nc
·
With (U) = D
C
N
C
·
TO (U - U) = 0
sg
N
n-1
·
WITH (U - U) = L
where 0
G
N
n-1
frot
where
·
L: together possible connections of contact (active and nonactive)
·
NC: together nodes of potential surfaces of contact which are not in contact
(nonactive connections)
·
C: together nodes indeed in contact (active connections)
·
SG: together adherent nodes of contact
·
G: together slipping nodes of contact
·
C = SG G, C NC =
I
, L = C NC
·
A is the matrix of the nodes in contact
C
·
A is the matrix of the nodes in adherent contact
sg
·
A is the matrix of the nodes in slipping contact
G

Because of incremental nature of the resolution of balance, one can rewrite these equations and
inequations in the form:

i-1
I
I
I
L (U
+ U

+ U
) = L - L
- L

int
n-1
N
ext.
cont
frot

subjected to:
·
i-1
I
WITH (U
+ U

+U) = D is
I
1
-
WITH U = I
D with
C
n-1
N
C
C
C
I 1
-
I 1
-
I 1
-
i-2
I 1
D
= D - With (U + U)
-
= D - WITH U = D - A U
C
C
C
N 1
-
N
C
C
N
C
C
N
·
WITH (U
i-1 +Ui) = 0 is
I
1
-
WITH U = I
D
sg
N
sg
sg
·
I 1
I
I
WITH (U
- +U) = L

where 0
G
N
frot

We have the discretized formulation of a problem of rubbing contact. We will see them
various manners of taking into account the whole of conditions (or constraints) equality and inequality
who relate to the field of displacements: dualisation or regularization.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
12/30

3.2
Dualisation of the conditions of contact and friction

To take into account the constraints relating to the field of displacements, it is possible of
dualiser i.e to utilize them in balance through multipliers of Lagrange (like
that is done for the boundary conditions kinematics in the code). 3 sets are introduced of
multipliers of Lagrange:

·
µ bearing on the conditions of contact
C
·
µ bearing on the conditions of adherence
sg
·
µ bearing on the conditions of slip
G

Balance is written then in the following form:

i-1
I
T
I
T
I
T
I
L (U
+ U

+ U) + A µ + A µ + A µ = L
int
n-1
N
C
C
sg
sg
G
G
ext.

subjected to:

·
I
1
-
WITH U = I
D
C
C
·
I
1
-
WITH U = I
D
sg
sg
·
I 1
I
I
WITH (U
- +U) = µ where 0 and I
I
I
µ
= K = µ µ
G
N
G
G
G
C

This system allows the following interpretation of the multipliers of Lagrange:

T
·
I
With µ is the whole of the nodal forces of contact
C
C
T
·
I
With µ is the whole of the nodal forces of adherence
sg
sg
T
·
I
With µ is the whole of the nodal forces of slip
G
G

Note:

1) In the expression of balance, the condition of contact became an equality. Indeed,
this equation is written for the nodes really in contact (for the active connections).
It is a logic which takes as a starting point the the method of the active constraints established in
code for the processing of the unilateral contact [bib9]. It is nevertheless imperative to check
condition:
µi > 0
C
2) Indeed: For a connection operator A associates the fields displacements U and
C
1
U the sum relative displacements U N
. + U N
. compared to the normals with
2
1
1
2
2
surface contact is the scalar (U - U .n. The operator T
A associates the scalar µ
1
2) 1
C
C
nodal forces µ N and µ N applying to the solids (1) and (2) respectively. These
C
1
C
2
nodal forces are equivalent to external forces - µ N applying to the solids
C
T
(1) and (2) respectively, which amounts transferring the term
I
With µ of the equation
C
C
of balance from left to right. In the surface scalar term close - µ N is equivalent to
C
N what implies the positivity of µ in the event of contact.
N
C

3) It is also checked that for the nonactive connections one has well:
I
1
-
WITH U I
D.
nc
nc

4) Matrices A, A, A, A vary during reiterated. We will clarify these
nc
C
sg
G
variations more in detail with [§4].
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
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Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
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Author (S):
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:
R5.03.51-C Page:
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3.3
Dualisation of the conditions of contact, regularization of the conditions
of friction

It is possible of dualiser the conditions of contact and to regularize the conditions of friction. One
understands by there the fact of facilitating the processing of friction by removing of its graph the infinite slope
in 0 i.e.:


R
µ N
AND
u&t
- µ N


This graph calls for several observations:

·
the concept of adherence with properly spoken disappeared, all the nodes slip. It is defined
nevertheless by:
node I is “adherent” if, being given R = E U, R < µ

T & T
N
· more the slope T
E is strong, plus the regularized graph approaches the graph not regularized
· in fact of regularization of the conditions of friction, it is rather about regularization of
conditions of adherence

Taking into account the preceding remarks, one rewrites balance in the form:

i-1
I
T
I
T
i-1
I
T
I
L (U
+ U

+U) + A µ + E A A (U

+ U) + A µ = L
int
n-1
N
C
C
T
sg
sg
N
G
G
ext.

subjected to:

·
I
1
-
WITH U = I
D
C
C
·
I 1
I
I
E WITH (U
- +U) < µ µ
T
sg
N
C
·
I 1
I
I
WITH (U
- +U) = µ where 0 and I
I
I
µ
= K = µ µ
G
N
G
G
G
C
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
14/30

3.4
Regularization of the conditions of contact and friction

With same logic that in the preceding paragraph, one defines the graph of the law of contact
regularized:

N
DNN
IN


And one deduces the form from it from balance:

i-1
I
T
I
I
T
- 1
i-1
I
T
I
L (U
+ U

+U) + E A (A U - D) + E A A (U

+ U) + A µ = L
int
n-1
N
NR
C
C
C
T
sg
sg
N
G
G
ext.

subjected to:

·
E WITH (
I 1
-
U +
I
U) < µ E (
I
I 1
-
WITH U - D)
T
sg
N
NR
C
C
·
I 1
I
I
WITH (U
- +U) = µ where 0 and I
µ = I
K = µ E (
I
I 1
-
WITH U - D)
G
N
G
G
G
NR
C
C

Note:

T
The term E A (
I
1
-
WITH U - I
D) is calculated only for the active connections. The use of
NR
C
C
C
X if X 0
+


the operator left positive [X] =
a compact writing of the law of contact allows
0 if not
T
regularized for all the possible connections in the form E A A U
D
. It would be
NR
C [
I
I
-
1
C
C] +
possible to use it to write the law of regularized adherence. We preferred to preserve
partition between the various states of the connections because this presentation is closer to
numerical integration in the code.

Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
15/30

4 Resolution
algorithmic

4.1
Linearization of the various terms

It is the method of Newton which is used in Code_Aster for the resolution of the problems not
linear [bib8]. It is in fact necessary to linearize the various terms appearing in
the expression of balance.

4.1.1 Forces
interns

The linearization of the operator of forces intern compared to
I
U
conduit with:

- 1
- 1
L

I
I
I
int
I
L (U
+ U

+ U
) L (U
+ U

) +
. U

int
n-1
N
int
n-1
N
I
U



I
U
+ - 1
N
U
- 1
N
i-1
I
I
L (U
+ U

) + K. U

int
n-1
N
N

I
K is the coherent tangent matrix which includes non-linearities behavioral and geometrical.
N

4.1.2 Forces of contact

In the absence of regularization of the forces of contact, one does not carry out any linearization. In
contrary case, one with the linearization:

T
I
I 1
- T
I
I 1
- T
I 1
-
I
I 1
- T
I 1
-
With µ A
µ E A
WITH U - E WITH
D
C
C
C
C
NR
C
C
NR
C
C
Note:

·
During the linearization of T I
With µ the index i-1 in the matrix of contact I 1 appeared
-
A. In
C
C
C
effect, at the time of the determination of
I
U

, only the state of contact rubbing with the iteration i-1 is known.
T
·
The term
I 1
-
I 1
-
E WITH
A contributes a new share to the tangent matrix of the problem,
NR
C
C
T
while the term
I 1
-
I 1
-
E WITH
D contributes a new share to the second member.
NR
C
C

4.1.3 Forces of slip

Taking into account the definition of the forces of slip, they can be expressed:

With (
I 1
-
I
U

+ U)
I
I
G
N
µ
= µ µ
G
C
With (
I 1
-
I
U

+ U)
G
N

With (
I 1
-
I
U

+ U)
I
G
N
= kg A (I 1
I
U

+ U)
G
N

They check the conditions indeed:

I 1
I
I
WITH (U
- +U) = µ where 0 and I
I
I
µ
= K = µ µ
G
N
G
G
G
C
and reveal two unknown factors I
µ and
I
U.
C
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
16/30

In practice, however, it is considered that the knowledge of the threshold of slip is acquired with
the preceding iteration, which amounts being brought back to a criterion of Tresca for each iteration. With
convergence the threshold is obviously fixed: there are not thus more differences between the thresholds with the course
iterations. Another approach, used in [R5.03.52], amounts solving, for each state of
contact, the solution with friction and thus to consider that the threshold of slip is a fixed point. Our
approach takes as a starting point this type of method while being less constraining and thus less expensive in
time CPU but can be less robust.
I
µ is thus approximated by:
G
I 1
With - (
I 1
-
I
U

+U)
I
I 1
-
G
N
µ
µ µ
G
C
I 1
With - (
I 1
-
I
U

+U)
G
N

I 1
With - (
I 1
-
I
U

+U)
I 1
-
G
N
kg
I 1
With - (
I 1
-
I
U

+U)
G
N
Linearization of I
µ compared to
I
U
in the expression given previously led to
G
following formulation:

I 1
-
I 1
-
I 1
-
I
I 1
-
I 1
-
I 1
-
I 1
-
I 1
WITH U
WITH U
WITH U
WITH U. -
Ad interim
U
I
I 1
-
G
N
I 1
-
G
I 1
-
µ K
+ K
-
G
N
G
N
G
K

G
G
I 1
-
1
-
G
I
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
WITH U
G
N
G
N
G
N
WITH U
G
N

who is still written:

I 1
-
I 1
-
I 1
-
I
I 1
-
I 1
-
I 1
- T
I 1
- T
I 1
-
WITH U
WITH U
WITH U
U
With
Ad interim
U
I
I 1
-
G
N
I 1
-
G
I 1
-
µ
K
+ K
-
G
N
N
G
G
K
G
G
I 1
-
1
-
G
I
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
WITH U
G
N
G
N
G
N
WITH U
G
N

I 1
-
I
I 1
-
I 1
-
I 1
- T
I 1
- T
I 1
-
WITH U
WITH U
U
With
Ad interim
U
I 1
-
I 1
-
G
I 1
-
µ + K
-
G
N
N
G
G
K
G
G
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
G
N
G
N
WITH U
G
N
Note:

I 1
-
I 1
-
I 1
-
I 1
- T
I 1
- T
I 1
-
With
WITH U
U
With
With
·
The terms I 1
-
G
I 1
-
K
-
G
N
N
G
G
K
news brings
G
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
G
N
G
N
WITH U
G
N
contributions to the tangent matrix of the problem.
I 1
-
I 1
-
WITH U
·
In 2D, I
µ is linearized out of I 1
-
G
N
I 1
-
K
= µ which is independent of
I
U

, there is not thus
G
G
I 1
-
1
-
With
G
I
U
G
N
not new contributions to the tangent matrix.

4.1.4 Forces
of adherence

In the absence of regularization of the forces of adherence, one does not carry out any linearization. In
contrary case, one with the linearization:

T
I
I T
1
-
I
I T
1
-
I 1
-
I 1
-
I T
1
-
I
I
With µ A
µ E A
With
U

+ E A A
1 U
sg
G
sg
G
T
sg
sg
N
T
sg
sg

4.1.5 Notice

The matrices of contact A, slip A and nonslip A are brought to be
C
G
sg
modified during iterations of Newton if contacts change statute or if one
geometrical reactualization takes place: they are thus subscripted
I
With, I
With and I
A. Dans the contrary case
C
G
sg
I
With, I
With and I
A do not vary during not reiterated.
C
G
sg
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
17/30

5 Resolution

The resolution strictly speaking is approached in this part. Will be presented the total system
solved in the framework of the method of Newton and the algorithm of processing of the rubbing contact
(under-iteration). The initial state with the iteration i=1 for all these methods corresponds to the resolution of one
problem without contact nor friction. If there is indeed no detected contact, this initial state
corresponds to the solution of the problem in the linear elastic case for example, if not it is modified
as of this iteration in order to take into account the connections which would violate the conditions of contact
unilateral.

5.1.1 Dualisation of the conditions of contact and friction in 2D
Method of Newton:

·
With iteration I, at the total level, resolution of the system (one voluntarily does not make
to appear the conditions of Dirichlet):

I
I
I 1
-
1
- T
I
I
1
- T
I
I
1
- T
I
I 1
K U = L - L (U)
-
- A
µ - A
µ - A
µ
N
ext.
int
N
C
C
sg
sg
G
G

Algorithm of rubbing contact:

·
Determination of the connections in contact

- Initial State: NC0 = L,
0
C =,
0
SG =,
0
G = 0/. All points of surface
N
N
N
N
potential of contact are lack of contact.
- If i=1 first elastic design without taking into account of the contact. That is to say then
1
C =
. If 1
C = the solution without contact is
N
{connections t.q. 1
0
1
D = D - A U <
nc
nc
nc
} 0
N
valid. If 1
C then
1
1
SG = C, 1
G = 0/and resolution of the system of equations Ci
N
N
N
N
above with the new conditions for iteration 1.
- If geometrical reactualization I
C =
D
,
I
i-1
I
G = G C and
N
{connections t.q I <
nc
} 0 N N
N
I
I
I
SG = C - G.
N
N
N
- If not I
1
-
C = I
C,
I
1
-
SG =
I
SG, I
1
-
G = I
G
N
N
N
N
N
N

·
Resolution of the system

I
I
1
-
T
I
I
I 1
-
1
- T
I
I 1
K U + A
µ
= L - L (U)
-
- A
µ
N
c+sg
c+sg
ext.
int
N
G
G
I 1
-
I
I 1
-
With
U = D
c+sg
c+sg

One carries out for that a resolution per blocks:

1
I 1
I -
-
1 T
I
I
With
K
With
µ
= L - L (I 1
U)
1T
I
I 1
-
- A
µ
c+sg
N
c+sg
c+sg
ext.
int
N
G
G

1
I
I -
U = K (L - L (I 1
U)
1T
I
I 1
-
1 T
I
I
- A
µ - A
µ
)
N
ext.
int
N
G
G
c+sg
c+sg
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
18/30

·
Checking of the state of the sliding joints

T
- If for a connection J I 1
-
µ
. I 1
-
With (
I 1
-
U +
I
U) < 0, Gi = Gi - {}
J and SGi = SGi + {}
J
G
G
N
N
N
N
N
I
With (
I 1
-
I
U

+U)
- If not I
I
G
N
µ = µ µ

G
C
I
With (
I 1
-
I
U

+U)
G
N
- So at least a connection changed state, return to the resolution of the system of equations
for same iteration I, but with the connections not slipping located above.

·
Checking of the state of the adherent connections

I
µ
- If for a connection J
I
I
µ
µ µ,
I
I
sg
µ = µ µ
, SG I = SG I - {}
J and
sg
C
G
C
I
µ
N
N
sg
Gi = Gi + {}
J
N
N

·
Checking of the state of the connections of contact

- If the nonactive connection J supposed is active, the most violated, i.e that whose play is it
more negative, is added with the whole of the active connections, C I = C I + {}
J and
N
N
Gi = Gi + {}
J, return to the resolution of the system of equations for same iteration I,
N
N
but with the connections not slipping located above.
- If for a connection J I
µ < 0, C I = C I - {}
J SGi = SGi - {}
J Gi = Gi - {}
J (in
C
N
N
N
N
N
N
function of the type of the connection)

·
Update
I
U
, I
µ, I
µ, I
µ, I
With
and
I
A.
C
G
sg
c+sg
G

5.1.2 Dualisation of the conditions of contact and regularization of friction in 2D and
3D

For this modeling, the algorithm is the same one for the 2D and the 3D.

Method of Newton:

·
With iteration I, the total level, resolution of the system:

I
I
F
I
I 1
-
1
- T
I
I
1
- T
I
I 1
-
I
F
I 1
(K + K N) U = L
- L (U)
-
- A
µ - A
µ
- K naked
N
ext.
int
N
C
C
G
G
N
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
19/30

Note:

I
·
The matrix
F
K N contains the contributions of terms of slip and of adherence is:
I 1
-
I 1
-
I 1
-
I 1
- T
I 1
- T
I 1
-
I
With
With
U
U
With
With
T


F
I 1
-
I 1
-
I 1
-
G
I 1
-
K N = E A
With
+ K
-
G
N
N
G
G
K

T
sg
sg
G
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
G
N
G
N
WITH U
G
N
·
In the preceding expression, a contribution is preceded by the sign -. The command of the terms
of this contribution is the same one as that of the other terms. The effect of this contribution
is particularly destabilizing for the total behavior of the tangent matrix with
system, more particularly when one is far from balance and thus with the beginning from
resolution with each new step of time. One thus decides to only take it into account
partially by affecting it of a coefficient [
]
1
,
0. One advises to use an initial value
from 0.5 for this coefficient and to decrease it if convergence is not obtained. In the case
where = 0 convergence always seems to be obtained but is particularly slow.
When one is close to the solution, it is on the other hand very useful to have a value of it
coefficient equalizes to 1 in order to accelerate convergence. That is done automatically
in the code when residue RESI_GLOB_RELA is lower than
3
10 -.
I
·
The matrix
F
K N contains the contributions of the new second members of adherence,
that is to say:
I
F
I 1
- T
1
-
K N =
I
E WITH
With
T
sg
sg

Algorithm of rubbing contact:

·
Determination of the connections in contact

- Initial State: NC0 = L,
0
C =,
0
SG =,
0
G = 0/. All points of surface
N
N
N
N
potential of contact are lack of contact.
- If i=1 first elastic design without taking into account of the contact. That is to say then
1
C =
,
1
1
G = C,
1
SG = 0/. If 1
C =
N
{connections t.q. 1
0
1
D = D - A U <
nc
nc
nc
} 0 N N
N
N
solution without contact is valid. If 1
C then
1
1
SG = C, 1
G = 0/and resolution of
N
N
N
N
system with the new conditions for iteration 1.
- If geometrical reactualization I
C =
D
,
I
i-1
I
G = G C and
N
{connections t.q I <
N
} 0
N
N
N
I
I
I
SG = C - G.
N
N
N
- If not I
1
-
C = I
C,
I
1
-
SG =
I
SG, I
1
-
G = I
G
N
N
N
N
N
N

·
Resolution of the system

I
I
F
I
1
- T
I
I
I 1
-
1
- T
I
I 1
-
I
F
I 1
(K + K N) U + A
µ = L
- L (U)
-
- A
µ
- K naked
N
C
C
ext.
int
N
G
G
N
I 1
-
I
I 1
-
WITH U = D
C
C

One carries out like previously a resolution per blocks.

I 1
With - (
I 1
-
I
U

+U)
·
For all the connections, I
I
G
N
µ = µ µ

G
C
I 1
With - (
I 1
-
I
U

+U)
G
N
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A

Code_Aster ®
Version 7.1
Titrate:
Contact ­ discrete friction in 2D and 3D


Date:
08/10/03
Author (S):
NR. TARDIEU, Key P. MASSIN
:
R5.03.51-C Page:
20/30

·
Checking of the state of the connections of contact

- If the nonactive connection J supposed is active, the most violated, i.e that whose play is it
more negative, with the whole of the active connections C I = C is added I + {}
J and
N
N
Gi = Gi + {}
J, return to the resolution of the system of equations for same iteration I,
N
N
but with the connections not slipping located above.
- If for a connection J I
µ < 0, C I = C I - {}
J Gi = Gi - {}
J
C
N
N
N
N

·
Checking of the state of the sliding joints and “adherent”

- If for a connection J
i-1
I 1
I
I
I
E WITH (U
- + U) < µ µ = µ, then
T
G
N
C
G
I
I
µ = E A (
I 1
-
I
U

+ U
), SGi = SGi + {}
J and Gi = Gi - {}
J.
sg
T
sg
N
N
N
N
N

I
I
·
Calculation of the tangent matrices
F
K
F
N and K N (if RESI_GLOB_RELA<1.E-3, = 1. )

·
Update
I
U
, I
µ, I
µ, I
µ, I
With, I
With and I
A.
C
G
sg
C
sg
G

5.1.3 Regularization of the conditions of contact and friction in 2D and 3D

For this modeling, the algorithm is the same one for the 2D and the 3D.

Method of Newton:

·
With iteration I, the total level, resolution of the system:

I
I
F
I
I 1
-
1
- T
I
I 1
-
I
F
I 1
-
1
- T
I
I 1
(K + K N) U = L - L (U)
-
- A
µ - K naked + E A
D
N
ext.
int
N
G
G
N
NR
C
C

Note:

I
·
The matrix
F
K N contains the contributions of terms of contact, slip and
of adherence is:
I 1
-
I 1
-
I 1
-
I 1
- T
I 1
- T
I 1
-
I
With
With
U
U
With
With
T
T


F
I 1
-
I 1
-
I 1
-
I 1
-
I 1
-
G
I 1
-
K N = E A
With
+ E A
With
+ K
-
G
N
N
G
G
K

NR
C
C
T
sg
sg
G
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
G
N
G
N
WITH U
G
N
I
·
The matrix
F
K N contains the contributions of the new second members of adherence,
that is to say:
I
F
I 1
- T
1
-
K N =
I
E WITH
With
T
sg
sg
T
·
The term
I 1
-
1
-
-
I
E WITH
D contains the contributions of the new second members of
NR
C
C
contact.


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Algorithm of rubbing contact:

·
Determination of the connections in contact

- Initial State: NC0 = L,
0
C =,
0
SG =,
0
G = 0/. All points of surface
N
N
N
N
potential of contact are lack of contact.
- If i=1 first elastic design without taking into account of the contact. That is to say then
1
C =
,
1
1
G = C,
1
SG = 0/and
N
{connections t.q. 1
0
1
D = D - A U <
nc
nc
nc
} 0
N
N
N
1
µ = E (1
1
0
WITH U - D).
C
NR
C
C
- If geometrical reactualization I
C =
D
,
I
I
G = C,
I
SG = 0/
N
{connections t.q I <
nc
} 0 N N
N
- If not is I
C =
D
, I
µ = E (I
I
I 1
-
WITH U - D), I
I
G = C,
I
SG = 0/
N
{connections t.q I <
nc
} 0 C NR C
C
N
N
N
I 1
With - (
I 1
-
I
U

+U)
·
For all the connections, I
µ = µ E (I
I
I 1
WITH U - D -)
G
N

G
NR
C
C
I 1
With - (
I 1
-
I
U

+U)
G
N

·
Checking of the state of the sliding joints and “adherent”

- If for a connection J
I 1
-
E WITH (
I 1
-
U +
I
U) < µ E (I
I
I 1
-
WITH U - D), then
T
G
N
NR
C
C
I
I
µ = E A (
I 1
-
I
U

+ U
), SGi = SGi + {}
J and Gi = Gi - {}
J
sg
T
sg
N
N
N
N
N

I
I
·
Calculation of the tangent matrices
F
K
F
N and K N (if RESI_GLOB_RELA<1.E-3, = 1)

·
Update
I
U
, I
µ, I
µ, I
µ, I
With, I
With and I
A.
C
G
sg
C
sg
G

5.1.4 Dualisation of the conditions of contact and friction in 3D

For the resolution of this problem, one defines the statute of the connections using regularization: initially
all the adherent connections are treated by regularization with a term of penalization E
T
determined by the code. One then repeatedly increases the value of E per E = 10th and one
T
T
T
remove the connections not checking the condition
i-1
I 1
I
I
E WITH (U
- +U) < µ µ. When the process
T
G
N
C
is stabilized, the adherent connections and the sliding joints are treated by multipliers of
Lagrange and the penalization does not appear.

Method of Newton:

·
With iteration I, at the total level, resolution of the system (one voluntarily does not make
to appear the conditions of Dirichlet):

I
I
F
I
I 1
-
1
- T
I
I
1
- T
I
I
1
- T
I
I 1
(K + K N) U = L - L (U)
-
- A
µ - A
µ - A
µ
N
ext.
int
N
C
C
sg
sg
G
G

Note:

I
·
The matrix
F
K N contains the contributions of terms of slip is:
I 1
-
I 1
-
I 1
-
I 1
- T
I 1
- T
I 1
-
I
With
WITH U
U
With
With
F
I 1
-
G
I 1
-
K N = K
-
G
N
N
G
G
K

G
I 1
-
1
-
G
I
I 1
-
I 1
-
2
I 1
-
I 1
-
WITH U
WITH U
G
N
G
N
WITH U
G
N
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Algorithm of rubbing contact:

·
Determination of the connections in contact

- Initial State: NC0 = L,
0
C =,
0
SG =,
0
G = 0/. All points of surface
N
N
N
N
potential of contact are lack of contact.
- If
I =1,
1
C =
,
1
1
SG = C,
1
G = 0/,
N
{connections t.q. 1
0
1
D = D - A U <
nc
nc
nc
} 0
N
N
N
E = (Max {diagonal terms of 1
K}) 25
.
0

T
N
- If geometrical reactualization, I
C =
D
,
I
i-1
I
G = G C and
N
{connections t.q I <
nc
} 0 N N
N
I
I
I
SG = C - G.
N
N
N
- If not I
1
-
C = I
C,
I
1
-
SG =
I
SG, I
1
-
G = I
G
N
N
N
N
N
N

·
Resolution of the system

I
I
1
-
T
I
I
I 1
-
1
- T
I
I 1
K U + A
µ
= L - L (U)
-
- A
µ
N
c+sg
c+sg
ext.
int
N
G
G
I 1
-
I
I 1
-
With
U = D
c+sg
c+sg

One carries out for that a resolution per blocks like previously.

·
Checking of the state of the connections of contact

- If the nonactive connection J supposed is active, the most violated, i.e that whose play is it
more negative, is added with the whole of the active connections, C I = C I + {}
J and
N
N
Gi = Gi + {}
J, return to the resolution of the system of equations for same iteration I,
N
N
but with the connections not slipping located above.
- If for a connection J I
µ < 0, C I = C I - {}
J SGi = SGi - {}
J Gi = Gi - {}
J (in
C
N
N
N
N
N
N
function of the type of the connection)

·
Checking of the state of the adherent connections

I
µ
- If for a connection J
I
I
µ
µ µ,
I
I
sg
µ = µ µ
, SG I = SGi - {}
J and
sg
C
G
C
I
µ
N
N
sg
Gi = Gi + {}
J
N
N

·
Checking of the state of the sliding joints

- If for a connection J
i-1
I 1
I
I
E WITH (U
- +U) < µ µ, Gi = Gi - {}
J and
T
G
N
C
N
N
SGi = SGi + {}
J
N
N
I 1
With - (
I 1
-
I
U

+U)
- If not I
I
G
N
µ = µ µ

G
C
I 1
With - (
I 1
-
I
U

+U)
G
N
- So at least a connection changed state, i.e one detected an adherent connection among
slipping supposed connections, then E = 10th and return to the resolution of the system
T
T
equations for same iteration I, but with the connections not slipping located
above.
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I
·
Calculation of the tangent matrices
F
K N (if RESI_GLOB_RELA<1.E-3, = 1).

·
Update
I
U
, I
µ, I
µ, I
µ, I
With
and
I
A.
C
G
sg
c+sg
G

5.1.5 Convergence

The convergence of the algorithm with multipliers of Lagrange for the contact without friction in
a finished number of iterations was shown in [bib2]. For the problems with friction, of
results of convergence with unicity of the solution to the discretized problem are established in
[bib6] for low values of the coefficient of friction of Coulomb. The results are established in
using an algorithm of point fixes associated with a method of multipliers of Lagrange. For
each problem of solved contact, one studies the problem of associated friction. Once this one
solved, one solves a new problem of contact and so on. These methods are however
different from those presented here and one cannot thus have results of convergence
theoretical for these last.

The condition of fastening of the points which come in contact is particularly important for
to ensure the convergence of the method with multipliers of Lagrange. Indeed when a point
returns to the contact during reiterated its tangential displacement remains free. A condition of not
slip would be far too constraining. The algorithm would oscillate then between two states with or without
contact in examples of the type of that presented in [V6.04.105]. The point which is attached is
thus regarded as free from the point of view of the slip. One can then calculate the normal reaction
as well as the tangential reaction by using the assumption of slip and one estimated of the increment
of displacement of initial slip.

The use of the penalization alone makes it possible to avoid these oscillations while making it possible to slacken them
constraints on the preceding system. Coupled with a method of multipliers of Lagrange, one
find pathology announced above.

5.1.6 Notice

The method of usual search linear RECH_LINEAIRE of STAT_NON_LINE is not usable
in this case. Indeed, a correction takes place already on the field of displacement, which is
supposed to be optimal for the realization of the conditions of contact. One thus risks if it is corrected
field during iterations of Newton not to have compatibility between displacements more and
reactions, which makes the method particularly unstable and involves an absence of convergence.

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6
Compatibility with the boundary conditions of Dirichlet

In the case of the methods with multipliers of Lagrange, one can observe incompatibilities
with the fact of imposing boundary conditions of the Dirichlet type. Indeed, it is necessary that physically it
problem has a direction. One cannot deal with problem of contact in the direction of axis Z if all
the points have a null displacement according to Z. As we will see it, to deal with such a led problem
T
with a singularity of the matrices of type A K 1
- A with the processing of the boundary conditions of
C
C
Dirichlet by double lagrange of Code_Aster.

6.1
Writing of the boundary conditions

While taking as a starting point the the reference material [R5.03.01] of STAT_NON_LINE, the dualisation of
boundary conditions of Dirichlet
D
DRUNK = U (T) led to the system of equations following to solve:

I
T
I 1

C U + B = L
- L (-
U)
ext.
int
N

I
D
I 1
-
DRUNK = U - BUn

One notes K then the matrix of rigidity of the system such as:

C BT
K =


B
0

This matrix has a reverse of the form:

E
F
- 1


K =

T


F
G
such as: EBT = 0.
One checks thus that for each boundary condition I one with property EB T = 0.
I

6.2
Return to the problem of contact

T
T
Matrix A K 1
- A can be also written A EA since the vectors of connection A do not make
C
C
C
C
C
to intervene that degrees of freedom of displacement.

·
It from of results whereas if a vector of connection J of matrix A is a linear combination
C
boundary conditions of the Dirichlet type it checks the following property: EA T = 0.
C J
T
stamp A EA is then singular because it has a column of zeros. In
C
C
practical, without particular processing, one finishes in the code on a message of stop of the type
STOP ON MATRIX OF CONTACT-FROTTEMENT SINGULIERE. The detection of these
singular columns was implemented in the code in order to eliminate from the relations from
contact-friction this type of relations and to avoid the stop previously described.
·
It from of results whereas if a vector of connection J of matrix A contains a combination
C
linear of the boundary conditions of the Dirichlet type and A is written
= B + A, it checks
C J
I
I
C J
T
following property:
T
T
EA
= EA
. One can then have a singular matrix A EA
C J
C J
C
C
because it has two identical lines. This detection is not for the moment not available
in the code and one finishes in the code on a message of stop of the stop type on matrix of
contact-friction singular.
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6.3
Illustration on a simple example

The two preceding situations can meet for the same example of study. That is to say one
surface being able to slip into the xOy plan. It is supposed blocked in direction X. If direction of
blocking corresponds to the one of the principal directions of slip which the user can give in
the command file one is found in case 1 of [§7.2]. If the direction of blocking is tilted
compared to the principal directions of slip then one finds oneself in case 2 of [§7.2]. One
of the two directions of slip is of too much to characterize the physical system.

6.4 Notice

This problem of compatibility between contact-friction and the boundary conditions does not appear
with the regularized methods insofar as one adds rigidity with total rigidity and that
one does not make elimination as in the calculation of the lagranges.
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7
Implementation in Code_Aster

The call to the routines for the contact with friction in Code_Aster takes place at the same place in
STAT_NON_LINE that for the unilateral contact without friction. The phase of prediction to the step of
time N is based on the increment of load between the steps of time n-1 and N.

7.1 Algorithms

These developments are accessible under command STAT_NON_LINE. They are activated by
key word CONTACT of the command AFFE_CHAR_MECA with which one defines the zones of contact
possible. The whole of the parameters of the model is provided in CONTACT under the key word
FROTTEMENT:

FROTTEMENT= “WITHOUT”
“COULOMB”

The various algorithms are chosen by key word METHODE according to logic:

“LAGRANGIAN” METHODE=
Dualisation contact friction 2D and 3D
“PENALIZATION”
if indicated E_T, dualisation contact, regularization friction
if indicated E_T and E_N, regularization contact and friction

One introduces also the COULOMB key words for the value of the coefficient of friction of Coulomb, and
COEF_MATR_FROT the coefficient of taking into account of the negative component of the tangent matrix
of friction ranging between 0 and 1. The user can thus define the loading in the following way
:

CHA =AFFE_CHAR_MECA (MODELE= MO,
CONTACT= _F (GROUP_MA_1 = ISOL1, GROUP_MA_2 = ISOL2

If the method used is “LAGRANGIEN”, no other indication is necessary in 2D and it is necessary
to provide COEF_MATR_FROT in 3D. In the case of the penalized methods it is necessary to give the value of
coefficient of penalization E_T, E_N and the COEF_MATR_FROT in all the cases.

The loading thus defined is then used in STAT_NON_LINE:

RESU = STAT_NON_LINE (MODEL = MO, CHAM_MATER = CHMAT,
EXCIT = _F (LOAD = CHA),
NEWTON=_F (REAC_ITER=1),
SOLVEUR = (METHODE = “LDLT”) etc…);

It is noticed that one recomputes the tangent matrix with all the iterations of Newton
(NEWTON=_F (REAC_ITER=1)).

Note:

With GCPC like method of resolution, one can make only contact without friction
for the moment. One does not recommend however to do it because the performances in
term of time calculation are not good with this method. One recommends the employment of
MULT_FRONT with a renumerotation of the type “METIS”.
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7.2
Geometrical reactualization

The geometrical evolution is one of non-linearities constitutive of the rubbing contact. Within
Code_Aster, its taking into account is done by a method of point fixes (for more precise details,
to defer to [bib9]). One points out the use of the key word:

REAC_GEOM= “SANS” not of reactualization
“AUTOMATIQUE” reactualization managed by the code
“Contrôle” N reactualizations at the beginning of step of time, then afterwards
each convergence until the n-1čme. N is defined by:
NB_REAC_GEOM=n

7.3 Post
processing

Different post processing is possible. For example, the calculation of the efforts of contact can be
carried out in command POST_RELEVE_T by calculating the resultant of the nodal forces on
group meshs representing one of surfaces of contact.
One draws the attention to the structure of data VALE_CONT which is produced for each calculation
implying rubbing contact. It is printed as follows in the form of table:

MATABLE=POST_RELEVE_T (ACTION=_F (INTITULE=' INFOS FROTTMNT',
GROUP_NO=' ESCLAVE',
RESULTAT=U,
INST=10.,
TOUT_CMP=' OUI',
NOM_CHAM=' VALE_CONT',
OPERATION=' EXTRACTION',),);

IMPR_TABLE (TABLE=MATABLE);

The information printed in each node slave is as follows:

·
CONT: indicator of rubbing contact
- 0: no the contact
- 1: slipping contact
- 2: adherent contact
·
JEU: value of the play
·
RN: normal reaction of contact normalizes
·
RNX: component according to DX of the normal reaction of contact
·
RNY: component according to DY of the normal reaction of contact
·
RNZ: component according to DZ of the normal reaction of contact
·
GLIX: component according to T1 of the tangential slip (local reference mark)
·
GLIY: component according to t2 of the tangential slip (local reference mark)
·
GLI: normalizes tangential slip
·
RTAX: component according to DX of the tangential force of adherence
·
RTAY: component according to DY of the tangential force of adherence
·
RTAZ: component according to DZ of the tangential force of adherence
·
RTGX: component according to DX of the tangential force of slip
·
RTGY: component according to DY of the tangential force of slip
·
RTGZ: component according to DZ of the tangential force of slip
·
X-ray: component according to DX of the force of rubbing contact (RNX+RTAX+RTGX)
·
RY: component according to DY of the force of rubbing contact (RNY+RTAY+RTGY)
·
RZ: component according to DZ of the force of rubbing contact (RNZ+RTAZ+RTGZ)
·
R: force of rubbing contact normalizes
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Moreover, it is possible to trace this information for a post graphic processing:

IMPR_RESU (MODELE=MO,
RESU=_F (FORMAT=' CASTEM' or “GMSH” or “IDEAS”,
RESULTAT=U,
NOM_CHAM=' VALE_CONT',
NOM_CMP= (“CONT”, “RNX”, “RNY”, “RNZ”, etc.),);

One presents below a cylinder in a boring with interaction of rubbing contact. One traced
normal forces of contact, tangential forces of adherence and the indicator of contact.





7.4 Precautions
of use

These precautions of use are about the same ones as those stated in [R5.03.50]. One them
recall here:

·
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 with the contact friction in quadratic 3D if the meshs of edge are QUAD8
(to avoid using HEXA20 to net volume): to use HEXA27 preferably, or
many PENTA15 whose TRIA6 sides are the meshs of 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 or friction. In others
terms, that wants to say that a calculation made in elasticity with command STAT_NON_LINE without
to treat the contact must pass,
·
in the event of structure “held” only by the contact, one can add a spring of weak
rigidity to maintain it,
·
not to use method of search linear, a STAT_NON_LINE incompatible with
processing of the conditions of contact-friction

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8 Conclusion

Discrete modelings of contact-friction with slip surfaces 1D and 2D were
established in Code_Aster. These modelings usable with STAT_NON_LINE and
DYNA_NON_LINE are accessible under AFFE_CHAR_MECA by the operator from contact with friction
in great displacements CONTACT.

Contrary to [bib5] modelings suggested are not pressed on dedicated finite elements.
They are based on the grids of surfaces coming in contact and make it possible to retranscribe node with
node conditions of contact friction between surfaces after discretization of the formulation
variational corresponding. The method extends then without difficulty of small displacements to the case
great displacements. Indeed, the absence of use of finite elements, between surfaces being able
to come in contact, avoids the great distortion of the latter, in the case of great displacements. One
can then use is conditions of direct connections nodes to nodes for grids
initially compatible, that is to say conditions of connections nodes to nodes balanced according to one
approach by projection of the master-slave type for incompatible grids. The different ones
conditions of connection are developed in documentation [R5.03.50] available on the processing
contact without friction in great displacements.

In the case of slip surfaces 1D one could only develop a using algorithm
multipliers of Lagrange. The finished convergence of this type of algorithm is proven for the contact
unilateral without friction [bib2] and in the case with friction for low values of the coefficient of
friction of Coulomb [bib6]. In the case of slip surfaces 2D, the rubbing contact is treated
either by dualisation or by regularization with various mixings.

One always advises the use of the dualisation on the contact and friction for the 2D:
method does not utilize of new tangent matrices and it does not present difficulties
major in term of use except compatibility with the conditions of Dirichlet, cf [§6]. In 3D,
one always advises to use the dualisation on the contact and friction; nevertheless it can afterwards
blow being interesting to test the method with dualisation of the contact and penalization of friction:
the problems of compatibility with the conditions of Dirichlet are then less and times of
calculation can be reduced. One insists on the other hand on the very strong dependence of the result with
value of the terms of penalization. For systematic studies, one can nevertheless test
validity of the solution regularized compared to the solution with dualisation, used like reference
on a standard study.
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9 Bibliography

[1]
BEN DHIA H., MASSIN P., TARDIEU NR., ZARROUG Mr.: Various algorithms for
problems of contact friction 2D and 3D in great displacements, GIENS 2001.
[2]
DUMONT G.: Algorithm of active constraints and unilateral contact without friction, Revue
European of Eléments Finis, Vol 4, n°1, 1995
[3]
DUVAUT G., LIONS J.L. : Inequations in mechanics and physics, Dunod, Paris, 1972.
[4]
EKELAND I., TEMAM R.: Convex analysis and variational problems, Bordas, 1974.
[5]
G. JACQUART: A finite element of contact with friction, Documentation de Référence of
Code_Aster [R5.03.41].
[6]
LICHT C., PRATT E., RAOUS Mr.: Remarks one has numerical method for unilateral contact
including friction, International Series off Numerical Mathematics, Vol. 101, 1991, pp. 129-144.
[7]
MASSIN P., BEN DHIA H.: 2D and 3D algorithms for frictional problems with small
displacements, Proceedings off the European Congress one Computational Methods in Applied
Sciences and Engineering, ECCOMAS 2000, Barcelona, September 11-14, 2000.
[8]
TARDIEU NR.: Algorithm Non Linéaire Quasi Statique, Documentation de Référence of
Code_Aster [R5.03.01].
[9]
TARDIEU NR.: Unilateral contact by conditions kinematics, Documentation of
Reference of Code_Aster [R5.03.50].
[10]
ZHONG Z.: Finite Element Procedures for Problems Contact-Impact. Oxford University
Near, p.146-148, 1993.

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