Fixed point strategies for elastostatic frictional contact problems Patrick LABORDE1 Yves RENARD2

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Fixed point strategies for elastostatic frictional contact problems. Patrick LABORDE1 , Yves RENARD2 Abstract Several fixed point strategies and Uzawa algorithms (for classical and augmented Lagrangian for- mulations) are presented to solve the unilateral contact problem with Coulomb friction. These meth- ods are analyzed, without introducing any regularization, and a theoretical comparison is performed. Thanks to a formalism coming from convex analysis, some new fixed point strategies are presented and compared to known methods. The analysis is first performed on continuous Tresca problem and then on the finite dimensional Coulomb problem derived from an arbitrary finite element method. Keywords : unilateral contact, Coulomb friction, Tresca problem, Signorini problem, bipotential, fixed point, Uzawa algorithm. Introduction The main goal of this paper is to introduce a formalism to deal with contact and friction of deformable bodies, focusing on fixed point algorithms. We restrict the study to the elastostatic case, the so-called Signorini problem with Coulomb friction (or simply the Coulomb problem) introduced by Duvaut and Lions [12], whose interest is to be very close to the incremental formulation of an evolutionary friction problem. The unilateral contact problem without friction was first considered by Signorini who shown the uniqueness of the solution. Fichera [14] proved an existence result using a quadratic minimization formu- lation. When friction is included, the nature of the problem changes due to the non self-adjoint character of the Coulomb friction condition.

  • coulomb friction

  • fixed point

  • frictional contact

  • let ?

  • ?n ?n

  • tresca problem

  • lagragian formulation

  • elastic forces

  • point algorithm

  • existence result


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Fixedpointstrategiesfor elastostaticfrictionalcontactproblems.
1 2Patrick LABORDE , Yves RENARD
Abstract
SeveralfixedpointstrategiesandUzawa algorithms(forclassical and augmentedLagrangianfor-
mulations) are presented to solve the unilateral contact problem with Coulomb friction. These meth-
ods are analyzed, without introducing any regularization, and a theoretical comparison is performed.
Thanks to a formalism coming from convex analysis, some new fixed point strategies are presented
and compared to known methods. The analysis is first performed on continuous Tresca problem and
thenonthefinitedimensionalCoulombproblemderivedfromanarbitraryfinite elementmethod.
Keywords: unilateralcontact,Coulombfriction,Trescaproblem,Signoriniproblem,bipotential, fixed
point, Uzawaalgorithm.
Introduction
The main goal of this paper is to introduce a formalism to deal with contact and friction of deformable
bodies, focusing on fixed point algorithms. We restrict the study to the elastostatic case, the so-called
Signorini problem with Coulomb friction (or simply the Coulomb problem) introduced by Duvaut and
Lions [12], whose interest is to be very close to the incremental formulation of an evolutionary friction
problem.
The unilateral contact problem without friction was first considered by Signorini who shown the
uniqueness ofthesolution. Fichera[14]proved anexistence resultusing aquadratic minimization formu-
lation. When friction is included, the nature of the problem changes due to the non self-adjoint character
of the Coulomb friction condition. This problem no longer has a potential. Until now, only a partial
uniqueness result has been obtained for the continuous (nonregularized) problem (see [26]). However,
existence result have been established for asufficiently smallfriction coefficient (see [25]for instance).
Weintroduce newfixedpoints formulations thanks toMoreau-Yosida resolvent andregularization us-
ing an approach similar to the proximal point algorithm. Wefirst analyze the self-adjoint Tresca problem
in which the friction threshold is assumed to be known. The properties obtained for the fixed points are
independent of any spatial discretization, which is not the case for the most used algorithms in practice.
As a second step, the analysis is performed on the Coulomb friction problem in finite dimension for an
arbitrary finite element method. The De-Saxc bipotential for friction problem is revisited and adapted to
the continuous framework inorder toobtain new fixedpoint formulations.
Thepaper isoutlined asfollows.
• Section1: thestrongformulation oftheproblem isrecalled andthentheclassical weakformulation
of Duvaut and Lions is presented. The Neumann to Dirichlet operator is introduced in order to
simplify the expression offriction problems.
1MIP,Universite´ PaulSabatier, 118routede Narbonne, 31062 Toulousecedex 4,France, laborde@mip.ups-tlse.fr
2Corresponding author, MIP, INSAT, Complexe scientifique de Rangueil, 31077 Toulouse, France, Yves.Renard@insa-
toulouse.fr
1• Section 2: a classical fixed point method for the continuous Tresca problem is analyzed. This
method is deduced from the Uzawa algorithm on the classical Lagragian formulation. This is a
fixed point on the contact and friction stresses. The convergence properties of this fixed point is
compared to theone obtained by using anaugmented Lagragian formulation.
• Section3: wepresentanadaptationofthisfixedpointmethodforthecontinuousCoulombproblem.
Anequivalence result isproved.
• Section 4: the analysis is done onthe Signorini problem withCoulomb friction infinite dimension,
using an arbitrary finite element method and a particular discretization of contact and friction con-
ditions allowing us to obtain uniform estimates. As in [15], but still for an arbitrary finite element
method, uniqueness is obtained for a sufficiently small friction coefficient and existence for any
friction coefficient.
• Section 5: a convergence analysis of the discretization method introduced in section 4 is done for
the Tresca problem.
• Section6: anewfixedpointoperatoronthecontactboundarydisplacementispresented. Itisproved
that ithas the same contraction property than the classical one.
• Section7: theDeSaxce´’sbipotentialtheoryisusedandajustificationispresentedinthecontinuous
framemork. Twonew fixedpoints operators are derived.
• Section8: finally,theclassicalfixedpointonthefrictionthresholdiscomparedtothepreviousones.
1 TheCoulombproblem
1.1 Strongformulation
.
Γ
D
n
Γ ΓN NΩ
Γ
C
Rigid foundation
.
Figure 1: Elastic bodyΩ infrictional contact.
dLet Ω⊂R (d =2 or 3) be a bounded domain representing the reference configuration of a linearly
elastic body submitted to a Neumann condition on Γ , a Dirichlet condition on Γ . On Γ , a unilateralN D C
contact with static Coulomb friction condition between the body and aflat rigid foundation is prescribed.
2
Theproblem consists in finding the displacement field u(x) satisfying:
−divσ(u)= f, in Ω, (1)
σ(u)=Aε(u), inΩ, (2)
σ(u)n=g, onΓ , (3)
N
u=0, onΓ , (4)D
where Γ , Γ and Γ are nonoverlapping open parts of ∂Ω, the boundary of Ω, σ(u) is the stress tensor,
N D C
ε(u)isthelinearized strain tensor,A istheelastic coefficient tensorwhichsatisfies classical conditions of
symmetry and ellipticity, nis the outward unit normal toΩ on∂Ω, and f,gare the given external loads.
OnΓ , it is usual to decompose the displacement and the stress vector in normal and tangential com-C
ponents:
u =u.n, u =u−u n,
N T N
σ (u)=(σ(u)n).n, σ (u)=σ(u)n−σ (u)n.
N T N
1To give a clear sense to this decomposition, we assume Γ to have the C regularity. Prescribing also
C
that there is no initial gap between the solid and the rigid foundation, the unilateral contact condition is
expressed bythe following complementary condition:
u ≤0, σ (u)≤0, u σ (u)=0. (5)
N N N N
Denoting byF ≥0 thefriction coefficient, the static Coulomb friction condition reads as:
if u =0 then |σ (u)|≤−Fσ (u), (6)
T T N
u
Tif u =0 then σ (u)=Fσ (u) . (7)
T T N |u |
T
1.2 Classicalweakformulation
Letusintroduce the following Hilbert spaces
1 dV ={v∈H (Ω;R ),v=0onΓ },
D
d1/2X ={v :v∈V}⊂H (Γ ;R ),
C|Γ
C
X ={v :v∈V}, X ={v :v∈V},
N TN | T |Γ Γ
C C
′ ′ ′ ′and their topological dual spacesV , X , X and X . It is assumed thatΓ is sufficiently smooth such that
CN T
d−1 d−11/2 1/2 ′ −1/2 ′ −1/2 1/2X ⊂H (Γ ),X ⊂H (Γ ;R ),X ⊂H (Γ )andX ⊂H (Γ ;R ). Classically, H (Γ )
C C C C CN T N T
1 −1/2isthespaceoftherestrictiononΓ oftraceson∂ΩoffunctionsofH (Ω),andH (Γ )isthedualspaceC C
1/2 1/2of H (Γ ) which is the space of the restrictions on Γ of functions of H (∂Ω) vanishing outside Γ .
C C C00
Werefer to[23]and [1]for acomplete discussion on trace operators.
Theset ofadmissible displacements is defined as
K ={v∈V,v ≤0onΓ }. (8)
N C
Thefollowing maps Z
a(u,v) = Aε(u):ε(v)dx,
Ω
3
6Z Z
l(v)= f.vdx+ g.vdΓ,
Ω Γ
N
j(s,v )=−hs,|v |i
T T ′X ,X
N N
represent the virtual work of elastic forces, the external load and the “virtual work” of friction forces
respectively. Weassume standard hypotheses:
a(.,.) bilinear symmetric continuous coercive form onV×V :
2∃α>0,∃ M>0,a(u,u)≥αkuk ,a(u,v)≤Mkuk kvk ∀u,v∈V, (9)
V VV
l(.)linear continuous form onV, (10)
F Lipschitz-continuous nonnegative function onΓ . (11)
C
The latter condition ensures that j(Fλ ,v ) is linear continuous on λ and also convex lower semi-
N T N
′continuous on v when λ is a nonpositive element of X (see for instance [3]). Problem (1) – (7) is
T N N
then formally equivalent tothe following inequality formulation (Duvaut and Lions [12]):

Find u∈K satisfying
(12)

a(u,v−u)+ j(Fσ (u),v )− j(Fσ (u),u )≥l(v−u), ∀ v∈K.
N T N T
Existence results for this problem can be found in Necˇas, Jarucˇek and Haslinger [25] for a two-
dimensional elastic strip, assuming that the coefficient of friction is small enough and using a shifting
technique, previously introduced by Fichera, and later applied to more general domains by Jarucˇek [20]
[21]. Recently,EckandJarucˇek[13]havegivenadifferentproofusingapenalization method. Weempha-
size that most results on existence forfrictional problems involve acondition of smallness for the friction
coefficient (and a compact support onΓ ). As far as weknow, it does not exist a global uniqueness resultC
for the continuous problem. A partial uniqueness result is presented in [26] and some multi-solutions for
alarge friction coefficient are presented by P.Hildin[17,18].
The major difficulty about (12) is due to the coupling between the friction threshold and the contact
pressure σ (u). The consequence is that this problem does not represent a variational inequality, in the
N
sense that there does not exist apotential for the Coulomb friction force.
1.3 NeumanntoDirichletoperator
Now, we introduce the Neumann to Dirichlet operator on Γ which allows to restrict the problem on Γ .
C C
′Letλ=(λ ,λ )∈X then, there exists aunique solution uto
N T

Find u∈V satisfying
(13)

a(u,v)=l(v)+hλ,vi ′ ∀ v∈V,X ,X
under hypotheses (9)and (10) (see [12]). So, itispossible to define theoperator
′E:X −→ X
λ −→u|Γ
C
Thisoperatorisaffineandcontinuous. Moreover,itisinvertibleanditsinverseiscontinuous. Itispossible
−1
toexpressE as follows: forw∈X,letube the solution to the Dirichlet problem

Findu∈V satisfying u =wand |Γ
C (14)

a(u,v)=l(v), ∀v∈V,v =0,|Γ
C
4−1 ′thenE (w)is equal toλ∈X defined by
hλ,vi =a(u,v)−l(v), ∀v∈V.′X ,X
−1 −1Inaweak sense, onehas therelationE (u)=σ(u)nonΓ . Thecontinuity ofEandE isgivenbythe
C
following lemma.
Lemma1 Under hypotheses (9) and (10), thefollowing estimates hold:
2C1 2 1 1 2kE(λ )−E(λ )k ≤ kλ −λ k (15)
X ′Xα
−1 −11 2 2 1 2kE (u )−E (u )k ≤MC ku −u k (16)′ 2 XX
whereC is the continuity constant of the trace operator on Γ , α the coercivity constant of the bilinear1 C
forma(.,.),M isthecontinuityconstantofa(.,.)andC >0isthecontinuity constantofthehomogeneous2
Dirichlet problem corresponding to(14) (i.e. with l(v)≡0).
1 2 ′ 1 2Proof. Letλ andλ be given inX and u , u the corresponding solutions to(13), then
T
1 11 2 2 1 2 1 2 1 2 1 2ku −u k ≤ a(u −u ,u −u )= hλ −λ ,u −u i
′V X ,Xα α
C1 1 2 1 2≤ kλ −λ k ku −u k′ VXα
and consequently
C11 2 1 2ku −u k ≤ kλ −λ k (17)′V Xα
which gives the first estimate using again the continuity of the trace operator onΓ . The second estimate
C
can beperformed asfollows:
−1 −11 2hE (u )−E (u ),wi
′−1 1 −1 2 X ,XkE (u )−E (u )k = sup′X kwkw∈X X
w=0
!
1 2a(u −u ,v)
= sup inf
kwkw∈X X{v∈V:v =w}
w=0 |Γ
C !
kvk1 2 V≤ Mku −u k sup inf
V
w∈X kwk
X{v∈V:v =w}
w=0 |Γ
C
1 2≤ Mγku −u k (18)
V
kvk
Vwhereγ=sup inf . Sinceγ≤C , this gives (16).2
w∈X kwkX{v∈V:v =w}w=0 |Γ
C
5
66662 AclassicalfixedpointmethodfortheTrescaproblem
2.1 TheTrescaproblem
Letusintroduce theso-called Trescaproblem, whichisastaticfriction problem withaprescribed friction
threshold−sdefined onΓ satisfying
C
′s∈X , snonpositive in the weaksense: hs,vi ≥0, ∀v∈K .NN ′X ,X
N N
TheTresca problem can be written as follows:

Find u∈K satisfying
(19)

a(u,v−u)+ j(s,v )− j(s,u )≥l(v−u), ∀v∈K.T T
′Ofcourse, findingasolution totheCoulombfriction problem isfindings∈X andasolution to(19)such
N
that s=Fσ (u). TheTresca problem corresponds toa variational problem. DenotingN
1
J(u)= a(u,u)−l(u)+ j(s,u )+I (u),KT2
where I is the indicator function of K (if u∈ K then I (u) = 0, else I (u) = +∞), Problem (19) isK K K
equivalent to 
 Findu∈V satisfying
(20)
 J(u)= infJ(v).
v∈V
Under classical assumptions (9) (10) (11) the functional J isstrictly convex, coercive and lowersemicon-
tinuous. Thus, J admits aunique minimizer (see [23]for instance) inV.
2.2 ClassicalLagrangianforTrescaproblem
Theset ofadmissible normal stresses onΓ can bedefined asC
′Λ ={f ∈X :hf ,v i ≥0, ∀v ∈K }.
N N N N N NN ′X ,X
N N
∗This is the opposite of K the polar cone to K . Let us also introduce the set of admissible tangentialNN
stresses onΓ :
C
′Λ (s)={f ∈X :−hf ,w i +hs,|w |i ≤0, ∀w ∈X }.
T T T T ′ T ′ TT TX ,X X ,X
T T N N
n−12 2Remark1 Whens∈L (Γ )thens≤0a.e. onΓ andΛ (s)={λ ∈L (Γ ,R ):|λ |≤−sa.e. onΓ }.
C C T T C T C
Using these definitions, it is classical to consider the following Lagrangian for the Tresca Problem
(see [23], [2]for instance)
1
L(u,λ)= a(u,u)−l(u)−hλ,ui −I (λ )−I (λ ).′ ΛX ,X Λ (s) T NNT2
6Thefollowing saddle point problem is then equivalent toProblem (19):

′Findu∈V andλ∈X satisfying
(21)
 L(u,λ)= inf supL(v,µ).
v∈V ′µ∈X
We choose here to express the constraints on L(u,λ) thanks to indicator functions. This Lagrangian
problem corresponds to a dualization of the indicator function in the expression of J(u), in the sense of
Rockafellar [28]. Optimality conditions of Problem (21) are

a(u,v)=l(v)+hλ ,w i +hλ ,w i ∀v∈V, N N T T ′ ′X ,X X ,X
N N T T
(22)u +N (λ )∋0,ΛN NN u +N (λ )∋0,
T Λ (s) T
T
which isclassicaly equivalent to Problem (19).
We will now formulate the classical Uzawa algorithm on Problem (21) (see [24] for instance) in the
continuous framework. It corresponds to a gradient with projection algorithm onλ. In order to define the
′projection step, weintroduce the following duality mapfrom X to X :
N N
′i :X −→ X ,
N N N
λ −→v defined byhλ ,w i =(v ,w ) ∀w ∈X .
N N N N N N X N′ NX ,X N
N N
′and the duality mapfrom X toX :
T T
′i :X −→ X ,
T T T
λ −→v defined byhλ ,w i =(v ,w ) ∀w ∈X ,T T T T T T X T′ TX ,X TT T
where (·,·) and (·,·) are the inner products of X and X respectively. These two duality maps are
X X N TN T
isometries. Forthe sake of convenience i(λ)willstand for the pair (i (λ ),i (λ )).
N N T T
˜ ˜ ˜ ˜Denotingλ =i (λ ),λ =i (λ ),Λ =i (Λ )andΛ (s)=i (Λ (s)),theUzawaalgorithm canbe
N N N T T T N N N T T T
written asfollows:

0 0 0 0 0 •Step0: λ =(λ ,λ )withλ ∈Λ andλ ∈Λ (s) arbitrary chosen.
N TN T N T
n n n n+1