Lori Badea

Marius Cocou
email: cocou@lma.cnrs-mrs.fr

Internal and subspace correction approximations of implicit variational inequalities

Keywords: Evolution inequalities, implicit inequalities, contact problems, existence results, internal approximation, subspace correction methods

The aim of this paper is to study the existence of solutions and some approximations for a class of implicit evolution variational inequalities that represents a generalization of several quasistatic contact problems in elasticity. Using appropriate estimates for the incremental solutions, the existence of a continuous solution and convergence results are proved for some corresponding internal approximation and backward difference scheme. To solve the fully discrete problems, general additive subspace correction algorithms are considered, for which global convergence is proved and some error estimates are established.

Introduction

This paper concerns the mathematical and numerical analysis of a system of evolution variational inequalities that represents a generalization of several quasistatic frictional contact problems in small deformation elasticity.

The results presented here are based on a unified approach, which contains in particular the contact problems studied in [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF], [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF], and can also be applied to various quasistatic problems, including unilateral or bilateral contact with nonlocal friction, normal compliance conditions with friction or more complex 1 interaction laws, as, for example, interface laws coupling unilateral contact, adhesion and nonlocal friction between two elastic bodies [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF].

The uniqueness of the solution does not hold in general, as was shown in [START_REF] Ballard | A counter-example to uniqueness in quasistatic elastic contact problems with small friction[END_REF] for a quasistatic frictional contact problem, even for an arbitrarily small coefficient of friction.

In this paper, we consider a class of implicit evolution variational inequalities involving a nonlinear operator, for which approximation and existence results are proved by using a semi-discrete scheme. The case of a linear operator was investigated in [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF]. Also, for a problem which represents a particular case of the one considered here, some similar results were announced in [START_REF] Badea | Approximation results and subspace correction algorithms for implicit variational inequalities[END_REF].

To solve the general quasi-variational inequalities of the second kind that are obtained by the previous incremental procedure, two additive subspace correction algorithms are introduced, for which global convergence is proved and error estimates are established.

In [START_REF] Badea | One-and two-level Schwarz methods for inequalities of the second kind and their application to frictional contact problems[END_REF] (see also [START_REF] Badea | One-and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind: Part I -general convergence results[END_REF] and [START_REF] Badea | One-and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind: Part II -frictional contact problems[END_REF]), one-and two-level multiplicative methods have been introduced for variational and quasi-variational inequalities of the second kind. Also, their application to contact problems with friction has been analyzed. The additive methods, even if they are a little slower than the multiplicative ones, have the advantage of being totally parallelizable. In the present paper, we use the same techniques as in the multiplicative case, but the convexity of the functionals plays a more important role here. Some details in these proofs are omitted, but they can be found in [START_REF] Badea | One-and two-level additive methods for variational and quasivariational inequalities of the second kind[END_REF], [START_REF] Badea | One-and two-level Schwarz methods for inequalities of the second kind and their application to frictional contact problems[END_REF] or [START_REF] Badea | One-and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind: Part I -general convergence results[END_REF]. In a subsequent paper, we shall use these results to prove the convergence of the one-and two-level additive methods for contact problems with friction.

The paper is organized as follows. In Section 2, the formulation of a general system of evolution inequalities and some examples are presented. Using an implicit time discretization scheme and some estimates, strong convergences are proved and an existence result is established.

In Section 3, convergence results are proved for a method based on a semi-discrete internal approximation.

In Section 4, two subspace correction algorithms, of additive type, are proposed for the solution of the problem discretized in time. These algorithms are introduced in a general framework and in a Hilbert space. We suppose here an assumption on the convex set and the correction subspaces, which will be essential in the proof of the convergence of the algorithms. Mainly, this hypothesis refers to the decomposition of the elements in the convex set, and introduces a constant C 0 which will play an important role in the writing of the convergence rate. Another hypothesis is made on the non differentiable term in the inequality. Under these assumptions, we prove that the two subspace correction algorithms are convergent and give an estimation of the convergence rate. As for the multiplicative algorithms introduced in [START_REF] Badea | One-and two-level Schwarz methods for inequalities of the second kind and their application to frictional contact problems[END_REF], the introduced assumptions are satisfied in the particular case of the frictional contact problems if the coefficient of friction is sufficiently small.

Existence results for a system of evolution inequalities

Let (V, . , . ), (H, (•, •) H ) be two real Hilbert spaces with the associated norms • and • H , respectively. Let K be a closed convex cone contained in V with its vertex at 0 and (K(g)) g∈V be a family of nonempty closed subsets of K satisfying the following conditions: 0 ∈ K(0) and

if g n → g in V, v n ∈ K(g n ) and v n → v in V then v ∈ K(g). (1)
Consider a functional F : V → R differentiable on V , and assume that its derivative F : V → V is strongly monotone and Lipschitz continuous, that is there exist two constants α, β > 0 for which

α v -u 2 ≤ F (v) -F (u), v -u (2)
and

F (v) -F (u) V ≤ β v -u (3)
for all u, v ∈ V . Using the relations

F (v) -F (u) = 1 0 F (u + r(v -u)), v -u dr = F (u), v -u + 1 0 F (u + r(v -u)) -F (u), v -u dr
and ( 2), (3), it is easily seen that for all u, v ∈ V it results

F (u), v -u + α 2 v -u 2 ≤ F (v) -F (u) ≤ F (u), v -u + β 2 v -u 2 . ( 4
)
We remark that since F satisfies (4), it follows that F is strictly convex and sequentially weakly lower semicontinuous on V .

Let γ : V × K → H be an operator such that γ(0, 0) = 0,

if g n → g in V, v n ∈ K and v n v in V then γ(g n , v n ) → γ(g, v) in H (5)
and for all

g i ∈ V , v i ∈ K, i = 1, 2, γ(g 1 , v 1 ) -γ(g 2 , v 2 ) H ≤ k 1 ( g 1 -g 2 + v 1 -v 2 ), (6)
where k 1 is a positive constant. Let j : V ×K ×V → R be a functional satisfying the following conditions:

j(g, v, •) is sequentially weakly continuous on V ∀ g ∈ V, v ∈ K, (7)
j(g, v, •) is sub-additive for all g ∈ V, v ∈ K, that is (8) j(g, v, w 1 + w 2 ) ≤ j(g, v, w 1 ) + j(g, v, w 2 ) ∀ g, w 1,2 ∈ V, v ∈ K, j(g, v, •) is positively homogeneous for all g ∈ V, v ∈ K, (9)
that is j(g, v, θw) = θj(g, v, w) ∀ g, w ∈ V, v ∈ K, θ ≥ 0, j(0, 0, w) = 0 ∀ w ∈ V, (10)
and there exists k 2 > 0 such that

|j(g 1 , v 1 , w 2 ) + j(g 2 , v 2 , w 1 ) -j(g 1 , v 1 , w 1 ) -j(g 2 , v 2 , w 2 )| ≤ k 2 ( g 1 -g 2 + γ(g 1 , v 1 ) -γ(g 2 , v 2 ) H ) w 1 -w 2 ∀ g i , w i ∈ V, v i ∈ K, i = 1, 2. ( 11
)
We assume that k 1 and k 2 satisfy the following property:

k 1 k 2 < α. (12)
For each

g ∈ V , let b(g, •, •) : K(g) × V → R be a functional such that ∀ v ∈ K(g), b(g, v, •) is linear on V, (13)
|b(g, v, w)| ≤ k 3 ( g + v ) w ∀ v ∈ K(g), ∀ w ∈ V, (14)
where

k 3 is a positive constant, if g n → g in V, v n ∈ K(g n ), v n → v in V and w n → w in V, then b(g n , v n , w n ) → b(g, v, w). ( 15
)
From the above properties of F, j and K and by a classical argument, it follows that for all g ∈ V, d ∈ K, w ∈ K(g) the elliptic variational inequality

u ∈ K F (u), v -u + j(g, w, v -d) -j(g, w, u -d) ≥ 0 ∀ v ∈ K
has a unique solution, so that we can define the mapping S g,d : K(g) → K by S g,d (w) = u. We assume that for all g ∈ V, d ∈ K K(g) is stable under S g,d i.e. S g,d (K(g)) ⊂ K(g).

For all g ∈ V, d ∈ K, we consider the following problems:

( P )        u ∈ K(g) F (u), v -u + j(g, u, v -d) -j(g, u, u -d) ≥ b(g, u, v -u) ∀ v ∈ V, b(g, u, z -u) ≥ 0 ∀ z ∈ K, ( Q) u ∈ K(g) F (u), v -u + j(g, u, v -d) -j(g, u, u -d) ≥ 0 ∀ v ∈ K,
and we assume that if u is a solution of ( Q), then u is a solution of ( P ).

Remark 2.1. If u satisfies ( P ), then u obviously satisfies ( Q).

Remark 2.2. i) From ( 11) and ( 6) it follows that

|j(g 1 , v 1 , w 2 ) + j(g 2 , v 2 , w 1 ) -j(g 1 , v 1 , w 1 ) -j(g 2 , v 2 , w 2 )| ≤ ((k 1 + 1)k 2 g 1 -g 2 + k 1 k 2 v 1 -v 2 ) w 1 -w 2 ∀ g i , w i ∈ V, v i ∈ K, i = 1, 2. ( 18
)
ii) Since, by (9) j(•, •, 0) = 0, from [START_REF] Nečas | Mathematical Theory of Elastic and Elasto-Plastic Bodies. An Introduction[END_REF], for w 2 = 0, w 1 = w, it results that

|j(g 2 , v 2 , w) -j(g 1 , v 1 , w)| ≤ k 2 ( g 1 -g 2 + γ(g 1 , v 1 ) -γ(g 2 , v 2 ) H ) w ≤ ((k 1 + 1)k 2 g 1 -g 2 + k 1 k 2 v 1 -v 2 ) w (19)
∀ g i , w ∈ V, v i ∈ K, i = 1, 2.
iii) As j satisfies (8), (9), j(g, v, •) is convex, and from [START_REF] Capatina | Internal approximation of quasi-variational inequalities[END_REF] and [START_REF] Mandel | A multilevel iterative method for symmetric, positive definite linear complementarity problems[END_REF], with

g 2 = g, v 2 = v, g 1 = v 1 = 0, it follows that |j(g, v, w 1 ) -j(g, v, w 2 )| ≤ ((k 1 + 1)k 2 g + k 1 k 2 v ) w 1 -w 2 ∀ g ∈ V, v ∈ K, ∀ w i ∈ V, i = 1, 2.
iv) Using ( 5) and [START_REF] Mandel | Etude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre[END_REF] we have

if g n → g in V, v n ∈ K and v n v in V then j(g n , v n , w) → j(g, v, w) ∀ w ∈ V. (20)
Let f ∈ W 1,2 (0, T ; V ) be given. Using the hypotheses (2), (3), ( 16) and [START_REF] Mandel | A multilevel iterative method for symmetric, positive definite linear complementarity problems[END_REF], it follows that S f (0),0 : K(f (0)) → K(f (0)) is a contraction if the condition [START_REF] Zeidler | Nonlinear Functional Analysis and its Applications[END_REF] holds. Thus, the following implicit elliptic variational inequality has a unique solution u 0 ∈ K(f (0)) (see, e.g. [START_REF] Capatina | Internal approximation of quasi-variational inequalities[END_REF]):

F (u 0 ), w -u 0 + j(f (0), u 0 , w) -j(f (0), u 0 , u 0 ) ≥ 0 ∀ w ∈ K.
We consider the following evolution problem involving an implicit variational inequality.

Problem P: Find u ∈ W 1,2 (0, T ; V ) such that (P )              u(0) = u 0 , u(t) ∈ K(f (t)) ∀ t ∈]0, T [, F (u(t)), v -u(t) + j(f (t), u(t), v) -j(f (t), u(t), u(t)) ≥ b(f (t), u(t), v -u(t)) ∀ v ∈ V a.e. on ]0, T [, b(f (t), u(t), w -u(t)) ≥ 0 ∀ w ∈ K, ∀ t ∈]0, T [.
The quasistatic unilateral contact problems with nonlocal friction in linearized elasticity studied in [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF], [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF] can be considered as particular cases of problem P corresponding to a quadratic functional F .

We shall briefly present two examples involving nonlinear operators F . We consider an elastic body occupying an open, bounded, connected set Ω ⊂ R d , d = 2, 3 with a Lipschitz continuous boundary Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 , where Γ 1 , Γ 2 , Γ 3 are open and disjoint parts of Γ with meas(Γ 1 ) > 0. Assume the small deformation hypothesis and that the inertial effects are negligible.

We denote by u the displacement field, by ε the infinitesimal strain tensor and by σ the stress tensor, with the components u = (u i ), ε = (ε ij ) and σ = (σ ij ), respectively. We use the classical decompositions

u = u N n + u T , u N = u • n, σn = σ N n + σ T , σ N = (σn) • n,
where n is the outward normal unit vector to Γ with the components n = (n i ). Also, the usual summation convention will be used for i, j, k, l = 1, . . . , d.

Assume that in Ω a body force

ϕ 1 ∈ W 1,2 (0, T ; [L 2 (Ω)] d ) is prescribed, on Γ 1 the displacement vector equals zero and on Γ 2 a traction ϕ 2 ∈ W 1,2 (0, T ; [L 2 (Γ 2 )] d ) is applied.
The displacement fields will be assumed to belong to the following space:

V 0 := {v ∈ [H 1 (Ω)] d ; v = 0 a.e. on Γ 1 }.
The body can be in contact with a support on Γ 3 , so that the displacements and the stress vector on this part of the boundary will satisfy some frictional contact conditions. The initial gap between the contact surface and the support is assumed to be zero. Consider an elastic body satisfying the following nonlinear Hencky-Mises constitutive equation (see [START_REF] Nečas | Mathematical Theory of Elastic and Elasto-Plastic Bodies. An Introduction[END_REF], [START_REF] Zeidler | Nonlinear Functional Analysis and its Applications[END_REF]):

σ(u) = ψ(u) = (k - 2 3 μ(γ(u)))(tr ε(u)) I + 2 μ(γ(u)) ε(u),
where k is the constant bulk modulus, μ is a continuously differentiable function in [ 0, +∞[ satisfying

0 < μ0 ≤ μ(r) ≤ 3 2 k, 0 < μ1 ≤ μ(r) + 2 ∂ μ(r) ∂r r ≤ μ2 , ∀ r ≥ 0, γ(u) := γ(u, u), γ(u, v) = - 2 3 ϑ(u) ϑ(v) + 2 ε(u) • ε(v), with ϑ(u) := tr ε(u) = div u ∀ u, v ∈ V 0 .
Assume Signorini contact conditions with a nonlocal friction law on Γ 3 . The classical formulation of the quasistatic contact problem is as follows. Problem P 1 c : Find a displacement field u = u(x, t) which satisfies the initial condition u(0) = u 0 in Ω such that for all t ∈] 0, T [

(P 1 c )                                    div σ(u) = -ϕ 1 in Ω, σ(u) = ψ(u) in Ω, u = 0 on Γ 1 , σn = ϕ 2 on Γ 2 , u N ≤ 0, σ N ≤ 0, u N σ N = 0 on Γ 3 , |σ T | ≤ µ|(Rσ) N | on Γ 3 and |σ T | < µ|(Rσ) N | ⇒ uT = 0, |σ T | = µ|(Rσ) N | ⇒ ∃λ ≥ 0, uT = -λσ T ,
where µ ∈ L ∞ (Γ 3 ) with µ ≥ 0 a.e. on Γ 3 is the coefficient of friction and Rσ is a regularization of the stress tensor that will be presented below.

To obtain a variational formulation for this problem, we use the following notations:

K 0 := {v ∈ V 0 ; v N ≤ 0 a.e. on Γ 3 }, (• , •) := (• , •) [H 1 (Ω)] d , H 1 2 (Γ 3 ) := {w : Γ 3 → R; w ∈ H 1 2 (Γ), w = 0 a.e. on Γ 1 }, ∀ L ∈ V 0 S L := {v ∈ V 0 ; Ω ψ(v) • ε(φ)dx = (L, φ) ∀ φ ∈ V 0 such that φ = 0 a.e. on Γ 3 }. We assume that R : [L 2 sym (Ω)] d 2 → [H 1 (Ω)] d 2 is a regularization operator satisfying R0 = 0 and if v n ∈ K 0 and v n v in V 0 then R ψ(v n ) → R ψ(v) in [H 1 (Ω)] d 2 , R ψ(v 1 ) -R ψ(v 2 ) [H 1 (Ω)] d 2 ≤ k 4 v 1 -v 2 [H 1 (Ω)] d ∀ v i ∈ K 0 , i = 1, 2,
where k 4 is a positive constant. Similar regularizations were described in [START_REF] Kuttler | Dynamic bilateral contact with discontinuous friction coefficient[END_REF], [START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF] for a viscoelastic body. For each L ∈ V 0 and v ∈ S L , we define the stress vector

σ(v)n ∈ ([H 1 2 (Γ 3 )] d ) by ∀ w ∈ [H 1 2 (Γ 3 )] d σ(v)n, w Γ 3 = Ω ψ(v) • ε( w)dx -(L, w),
where • , • Γ 3 denotes the duality pairing on ([H

1 2 (Γ 3 )] d ) × [H 1 2 (Γ 3 )] d
, w ∈ V 0 satisfies w = w a.e. on Γ 3 , and we define the normal component of the stress vector σ N (v) ∈ (H

1 2 (Γ 3 )) by ∀ w ∈ H 1 2 (Γ 3 ) σ N (v), w Γ 3 = Ω ψ(v) • ε( w)dx -(L, w),
where • , • Γ 3 denotes also the duality pairing on (H

1 2 (Γ 3 )) × H 1 2 (Γ 3
), w ∈ V 0 satisfies wT = 0 a.e. on Γ 3 , wN = w a.e. on Γ 3 . It is easy to verify that for all v ∈ S L the definitions of σ(v)n and of σ N (v) do not depend on the choices of w having the above properties, respectively.

Let

J 1 : K 0 × V 0 → R be a functional defined by J 1 (v, w) = Γ 3 µ|(R ψ(v)) N | |w T |ds ∀ v ∈ K 0 , w ∈ V 0 ,
and L ∈ V 0 be given by the relation

(L, v) = (ϕ 1 , v) [L 2 (Ω)] d + (ϕ 2 , v) [L 2 (Γ 2 )] d ∀ v ∈ V 0 . (21)
We have the following variational formulation of (P 1 c ).

Problem P 1 v : Find u ∈ W 1,2 (0, T ; V 0 ) such that (P 1 v )                u(0) = u 0 , u(t) ∈ K 0 ∀ t ∈] 0, T [, Ω ψ(u(t)) • ε(v -u(t))dx + J 1 (u(t), v) -J 1 (u(t), u(t)) ≥ (L(t), v -u(t)) + σ N (u(t)), v N -uN (t) Γ 3 ∀ v ∈ V 0 a.e. on ] 0, T [, σ N (u(t)), z N -u N (t) Γ 3 ≥ 0 ∀ z ∈ K 0 , ∀ t ∈] 0, T [.
We define the functional F : V 0 → R by

F (v) = 1 2 k Ω ϑ 2 (v)dx + 1 2 Ω γ(v) 0 μ(r)dr dx ∀ v ∈ V 0 .
One can verify, see, e.g. [START_REF] Nečas | Mathematical Theory of Elastic and Elasto-Plastic Bodies. An Introduction[END_REF], Ch. 8, that F is differentiable on V 0 and for all

u, v ∈ V 0 (F (u), v) = Ω [(k - 2 3 μ(γ(u))) ϑ(u) ϑ(v) + 2 μ(γ(u)) ε(u) • ε(v)]dx. Taking V = V 0 , K = K 0 , H = L 2 (Γ 3 ), f = L, K(f ) = K 0 ∩ S L , γ(L, v) = |(R ψ(v)) N |, j(L, v, w) = J 1 (v, w) -(L, w), b(L, v, w) = σ N (v), w N Γ 3 ,
it results that (P 1 v ) can be written in the form (P ), where L is defined by [START_REF] Kornhuber | Monotone multigrid methods for elliptic variational inequalities II[END_REF]. It is easily seen that if the coefficient of friction is sufficiently small then the condition ( 12) is satisfied.

Example 2. (Frictional contact with normal compliance)

We consider a linearly elastic body and we denote by E the elasticity tensor, with the Cartesian coordinates E = (a ijkl ) satisfying the usual properties of symmetry and ellipticity.

Let us define a :

V 0 × V 0 → R by a(v, w) = Ω a ijkl ε ij (v)ε kl (w)dx = Ω σ(v) • ε(w)dx ∀ v, w ∈ V 0 .
We suppose that the frictional contact between Γ 3 and the support is described by a normal compliance law, see, e.g. [START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and references therein for more general laws.

The classical formulation of the contact problem is as follows. Problem P 2 c : Find a displacement field u = u(x, t) which satisfies the initial condition u(0) = u 0 in Ω such that for all t ∈] 0, T [

(P 2 c )                                    div σ(u) = -ϕ 1 in Ω, σ(u) = E ε(u) in Ω, u = 0 on Γ 1 , σn = ϕ 2 on Γ 2 , σ N = -C N (u N ) + on Γ 3 with C N > 0, |σ T | ≤ µ|σ N | on Γ 3 and |σ T | < µ|σ N | ⇒ uT = 0, |σ T | = µ|σ N | ⇒ ∃λ ≥ 0, uT = -λσ T . Let J 2 : V 0 × V 0 → R and p N : V 0 × V 0 → R be two functionals defined by J 2 (v, w) = Γ 3 µ C N (v N ) + |w T |ds ∀ v, w ∈ V 0 , p N (v, w) = Γ 3 C N (v N ) + w N ds ∀ v, w ∈ V 0 .
A variational formulation of (P 2 c ) is as follows. Problem P 2 v : Find u ∈ W 1,2 (0, T ; V 0 ) such that u(0) = u 0 and

(P 2 v ) a(u(t), v -u(t)) + p N (u(t), v -u(t)) + J 2 (u(t), v) -J 2 (u(t), u(t)) ≥ (L(t), v -u(t)) ∀ v ∈ V 0 a.e. on ] 0, T [,
where L ∈ V 0 is given by the relation [START_REF] Kornhuber | Monotone multigrid methods for elliptic variational inequalities II[END_REF].

Let us define the functional F : V 0 → R by

F (v) = 1 2 a(v, v) + C N 2 Γ 3 (v N ) 2 + ds ∀ v ∈ V 0 . F is differentiable on V 0 and for all u, v ∈ V 0 (F (u), v) = a(u, v) + p N (u, v). Taking V = K = K(f ) = V 0 , H = L 2 (Γ 3 ), f = L, γ(L, v) = C N (v N ) + , j(L, v, w) = J 2 (v, w) -(L, w), b(L, v, w) ≡ 0,
it results that (P 2 v ) can be written in the form (P ). We remark that C N can be chosen arbitrarily positive and that if the coefficient of friction is sufficiently small then the condition ( 12) is satisfied. Now, we shall prove the existence of a solution to problem P by using an implicit time discretization scheme and its convergence properties.

For ν ∈ N * , we set ∆t := T /ν, t i := i ∆t and K i := K(f (t i )), i = 0, 1, ..., ν. If θ is a continuous function of t ∈ [0, T ] valued in some vector space, we use the notations θ i := θ(t i ) unless θ = u, and if ζ i , ∀ i ∈ {0, 1, ..., ν}, are elements of some vector space, then we set

∂ζ i := ζ i+1 -ζ i ∆t ∀ i ∈ {0, 1, ..., ν -1}.
We denote u 0 := u 0 and we approximate (P ) using the following sequence of incremental problems (P i ν ) i=0,1,...,ν-1 .

Problem P i ν : Find u i+1 ∈ K i+1 such that (P i ν )        F (u i+1 ), v -∂u i + j(f i+1 , u i+1 , v) -j(f i+1 , u i+1 , ∂u i ) ≥ b(f i+1 , u i+1 , v -∂u i ) ∀ v ∈ V, b(f i+1 , u i+1 , w -u i+1 ) ≥ 0 ∀ w ∈ K.
By ( 17) and Remark 2.1 for g = f i+1 , d = u i , it is easily seen that for all i ∈ {0, 1, ..., ν -1} the problem P i ν is equivalent to the following quasivariational inequality.

Problem Q i ν : Find u i+1 ∈ K i+1 such that (Q i ν ) F (u i+1 ), w -u i+1 + j(f i+1 , u i+1 , w -u i ) -j(f i+1 , u i+1 , u i+1 -u i ) ≥ 0 ∀ w ∈ K.
From the hypotheses (2), ( 12), ( 16) and [START_REF] Mandel | A multilevel iterative method for symmetric, positive definite linear complementarity problems[END_REF], it follows that S f i+1 ,u i : K i+1 → K i+1 is a contraction. Therefore (Q i ν ) has a unique solution which is equally the unique solution of (P i ν ), for all i ∈ {0, 1, ..., ν -1}. Remark 2.3. i) Since K is a cone with the vertex at 0, the solutions u of (P ) and u i+1 of

(P i ν ) obviously satisfy b(f (t), u(t), u(t)) = 0 on [0, T ] and b(f i+1 , u i+1 , u i+1 ) = 0 ∀ i ∈ {0, 1, ..., ν -1}.
ii) It follows also that (see, e.g. [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF])

b(f (t), u(t), u(t)) = 0 a.e. on [0, T ].
Let us define the following functions:

                 u ν (0) = ûν (0) = u 0 , f ν (0) = f 0 and ∀ i ∈ {0, 1, ..., ν -1}, ∀ t ∈]t i , t i+1 ], u ν (t) = u i+1 , ûν (t) = u i + (t -t i )∂u i , f ν (t) = f i+1 .
Then for all ν ∈ N * the sequence of inequalities (P i ν ) i=0,1,...,ν-1 is equivalent to the following incremental formulation: for almost every t ∈ [0, T ]

(P ν )            u ν (t) ∈ K(f ν (t)), F (u ν (t)), v - d dt ûν (t) + j(f ν (t), u ν (t), v) -j(f ν (t), u ν (t), d dt ûν (t)) ≥ b(f ν (t), u ν (t), v - d dt ûν (t)) ∀ v ∈ V, b(f ν (t), u ν (t), w -u ν (t)) ≥ 0 ∀ w ∈ K.
Also, the sequence (Q i ν ) i=0,1,...,ν-1 implies the following inequality: for almost every

t ∈ [0, T ] (R ν ) F (u ν (t)), w -u ν (t) + j(f ν (t), u ν (t), w -u ν (t)) ≥ 0 ∀ w ∈ K,
which, by [START_REF] Ballard | A counter-example to uniqueness in quasistatic elastic contact problems with small friction[END_REF], is clearly equivalent to the following inequality: for almost every t ∈ [0, T ]

( Rν ) F (w) -F (u ν (t)) + j(f ν (t), u ν (t), w -u ν (t)) ≥ α 2 w -u ν (t) 2 ∀ w ∈ K.
Using straightforward generalizations of the linear case presented in [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF] one can prove the following two lemmas.

Lemma 2.1. For all ν ∈ N * we have

u ν (t) ≤ M 0 F (0) + M 1 f C([0,T ];V ) ∀ t ∈ [0, T ], (22)
u ν (s) -u ν (t) ≤ M 1 min{t+∆t,T } s ḟ (τ ) dτ ∀ s, t ∈ [0, T ], s < t, u ν (t) -ûν (t) ≤ T ν d dt ûν (t) a.e. t ∈ [0, T ], u ν -ûν L 2 (0,T ;V ) = T ν √ 3 d dt ûν L 2 (0,T ;V ) , (23)
d dt ûν L 2 (0,T ;V ) ≤ M 1 ḟ L 2 (0,T ;V ) ,
where

M 0 = 1 α -k 1 k 2 , M 1 = (k 1 + 1)k 2 α -k 1 k 2 .
Lemma 2.2. There exist a subsequence of (u ν , ûν ) ν , denoted by (u νp , ûνp ) p , and an element u ∈ W 1,2 (0, T ; V ) such that

u νp (t) u(t) in V ∀ t ∈ [0, T ], (24)
ûνp u in W 1,2 (0, T ; V ), d dt ûνp u in L 2 (0, T ; V ). ( 25
)
Also, we have

lim inf p→∞ T 0 j(f νp (t), u νp (t), d dt ûνp (t)) dt ≥ T 0 j(f (t), u(t), u(t)) dt. ( 26
)
Note that since F is nonlinear, it is not possible to use a simple extension of the linear case presented in [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF] in order to pass to the limit for the terms involving F or F . Now, we prove the following strong convergence and existence result.

Theorem 2.1. Under the assumptions (1)-( 3), ( 5)-( 17) every convergent subsequence of (u ν , ûν ) ν , still denoted by (u ν , ûν ) ν , and its limit u ∈ W 1,2 (0, T ; V ), given by lemma 2.2, satisfy the following properties:

u ν (t) → u(t) in V ∀ t ∈ [0, T ], ( 27
) ûν → u in L 2 (0, T ; V ), ( 28
)
and u is a solution of problem P.

Proof. Let (u ν ) ν be the subsequence given by lemma 2.2 and u its weak limit. Taking w = u(t) in ( Rν ), we obtain for all t ∈ [0, T ]

F (u(t)) -F (u ν (t)) + j(f ν (t), u ν (t), u(t) -u ν (t)) ≥ α 2 u(t) -u ν (t) 2 .
Since F is sequentially weakly lower semicontinuous, by using the previous relation, ( 19), ( 24), ( 5) and ( 7), we have that for all t ∈ [0, T ] lim sup

ν→∞ α 2 u(t) -u ν (t) 2 ≤ F (u(t)) + lim sup ν→∞ (-F (u ν (t))) + lim ν→∞ (j(f ν (t), u ν (t), u(t) -u ν (t)) -j(f (t), u(t), u(t) -u ν (t)) + lim ν→∞ j(f (t), u(t), u(t) -u ν (t)) ≤ F (u(t)) -lim inf ν→∞ F (u ν (t)) + lim ν→∞ k 2 ( f ν (t) -f (t) + γ(f ν (t), u ν (t)) -γ(f (t), u(t)) H ) u(t) -u ν (t) + lim ν→∞ j(f (t), u(t), u(t) -u ν (t)) ≤ 0,
which proves (27). By applying Lebesgue's dominated convergence theorem, it follows from ( 27) and ( 22) that

u ν → u in L 2 (0, T ; V ), (29)
and using [START_REF] Badea | Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities[END_REF] we obtain (28). It remains to prove that the limit u is a solution of problem P. First, by ( 1) we obtain that u(t) ∈ K(f (t)) ∀ t ∈]0, T [ and by ( 15) one can pass to the limit in the second inequality of (P ν ) for all t ∈]0, T [ so that the second inequality of (P ) is satisfied. Second, integrating both sides in the first inequality of (P ν ) over [0, T ] and passing to the limit, by the relations (27), (29), ( 25), (3), ( 20), ( 26) and Remark 2.3 i), we have for all v ∈ L 2 (0, T ; V )

T 0 F (u(t)), v(t) -u(t) dt + T 0 j(f (t), u(t), v(t))dt - T 0 j(f (t), u(t), u(t))dt ≥ T 0 b(f (t), u(t), v(t))dt. ( 30
)
By Lebesgue's theorem and Remark 2.3 ii), it follows that u is a solution of the first inequality of problem P.

Internal approximation and convergence results

In this section we shall consider a semi-discrete approximation of (P ), which extends the classical internal approximations as presented in, e.g. [START_REF] Glowinski | Analyse numérique des inéquations variationnelles[END_REF], [START_REF] Glowinski | Numerical Methods for Nonlinear Variational Problems[END_REF].

We shall prove a convergence result for a method based on an internal approximation and a backward difference scheme that represents a generalization of the case studied in [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF], where F is linear and there is no internal approximation for the functional j, which is a limitation on the numerical realization of the discrete problem. For a positive parameter h converging to 0, let (V h ) h be an internal approximation of V , that is a family of finite-dimensional subspaces of V which satisfies:

there exists U ⊂ V such that U = V and ∀ v ∈ U, ∃ v h ∈ V h for each h, such that v h → v in V. (31)
Let (K h ) h be a family of closed convex cones with their vertices at 0 such that K h ⊂ V h for all h and (K h ) h is an internal approximation of K, i.e.

if v h ∈ K h for all h and v h v then v ∈ K, (32)
∀ v ∈ K, ∃ v h ∈ K h for each h, such that v h → v in V. (33)
Let (K h (g)) g∈V be a family of nonempty closed subsets of K h such that 0 ∈ K h (0) for all h, satisfying the following conditions:

if g n → g in V, v hn ∈ K h (g n ) and v hn → v h in V h then v h ∈ K h (g), ( 34
) if v h ∈ K h (g) for all h and v h → v then v ∈ K(g) ∀ g ∈ V. (35)
We assume that there exists an operator γ h : V × K h → H such that γ h (0, 0) = 0 and for all

g i ∈ V , v hi ∈ K h , i = 1, 2, γ h (g 1 , v h1 ) -γ h (g 2 , v h2 ) H ≤ k 1 ( g 1 -g 2 + v h1 -v h2 ). ( 36
) Let j h : V × K h × V h → R be a functional satisfying the following conditions for all g ∈ V : if v h ∈ K h for all h, v h v in V and w h w in V then lim h→0 j h (g, v h , w h ) = j(g, v, w), (37)
for all h and

v h ∈ K h j h (g, v h , •) is sub-additive, (38)
for all h and v h ∈ K h j h (g, v h , •) is positively homogeneous,

j h (0, 0, w h ) = 0 ∀ w h ∈ V h , (39)
and

if v h (t) ∈ K h for all h and t ∈ [0, T ], v h v in W 1,2 (0, T ; V ) then lim inf h→0 T 0 j h (g(t), v h (t), vh (t)) dt ≥ T 0 j(g(t), v(t), v(t)) dt (40)
for all g ∈ C([0, T ]; V ),

|j h (g 1 , v h1 , w h2 ) + j h (g 2 , v h2 , w h1 ) -j h (g 1 , v h1 , w h1 ) -j h (g 2 , v h2 , w h2 )| ≤ k 2 ( g 1 -g 2 + γ h (g 1 , v h1 ) -γ h (g 2 , v h2 ) H ) w h1 -w h2 (41)
∀ g i ∈ V, v hi ∈ K h , w hi ∈ V h , i = 1, 2.
From the properties of F, j h and K h , it follows that for all g ∈ V, d h ∈ K h , w h ∈ K h (g), the elliptic variational inequality:

u h ∈ K h F (u h ), v h -u h + j h (g, w h , v h -d h ) -j h (g, w h , u h -d h ) ≥ 0 ∀ v h ∈ K h
has a unique solution. Hence we can define a mapping S h g,d h :

K h (g) → K h by S h g,d h (w h ) = u h . We suppose that for all g ∈ V, d h ∈ K h S h g,d h (K h (g)) ⊂ K h (g). ( 42
)
For all g ∈ V, d h ∈ K h , we consider the following problems:

( Ph )

       u h ∈ K h (g), F (u h ), v h -u h + j h (g, u h , v h -d h ) -j h (g, u h , u h -d h ) ≥ b(g, u h , v h -u h ) ∀ v h ∈ V h , b(g, u h , z h -u h ) ≥ 0 ∀ z h ∈ K h ,

and

( Qh )

u h ∈ K h (g), F (u h ), v h -u h + j h (g, u h , v h -d h ) -j h (g, u h , u h -d h ) ≥ 0 ∀ v h ∈ K h .

Assume that

if u h is a solution of ( Qh ), then u h is a solution of ( Ph ). (43)

Remark 3.1. If u h satisfies ( Ph ), then u h obviously satisfies ( Qh ).

Remark 3.2. Using similar arguments as in Remark 2.2, one can easily prove the following properties:

|j h (g 1 , v h1 , w h2 ) + j h (g 2 , v h2 , w h1 ) -j h (g 1 , v h1 , w h1 ) -j h (g 2 , v h2 , w h2 )| ≤ ((k 1 + 1)k 2 g 1 -g 2 + k 1 k 2 v h1 -v h2 ) w h1 -w h2 ∀ g i , w hi ∈ V h , v hi ∈ K h , i = 1, 2, |j h (g 2 , v h2 , w h ) -j h (g 1 , v h1 , w h )| ≤ ((k 1 + 1)k 2 g 1 -g 2 + k 1 k 2 v h1 -v h2 ) w h ∀ g i ∈ V, w h ∈ V h , v hi ∈ K h , i = 1, 2, |j h (g, v h , w h1 ) -j h (g, v h , w h2 )| ≤ ((k 1 + 1)k 2 g + k 1 k 2 v h ) w h1 -w h2 ∀ g ∈ V, v h ∈ K h , ∀ w hi ∈ V h , i = 1, 2, if g n → g in V, v hn ∈ K h and v hn v h in V h then j h (g n , v hn , w h ) → j h (g, v h , w h ) ∀ w h ∈ V h .
We introduce the following semi-discrete approximation of problem P.

Problem P h : Find u h ∈ W 1,2 (0, T ; V h ) such that (P h )              u h (0) = u 0h , u h (t) ∈ K h (f (t)) ∀ t ∈]0, T [, F (u h (t)), v h -uh (t) + j h (f (t), u h (t), v h ) -j h (f (t), u h (t), uh (t)) ≥ b(f (t), u h (t), v h -uh (t)) ∀ v h ∈ V h a.e. on ]0, T [, b(f (t), u h (t), z h -u h (t)) ≥ 0 ∀ z h ∈ K h , ∀ t ∈]0, T [, where u 0h ∈ K h (f (0)) is the unique solution of the variational inequality F (u 0h ), w h -u 0h + j h (f (0), u 0h , w h ) -j h (f (0), u 0h , u 0h ) ≥ 0 ∀ w h ∈ K h .
By (4), it follows that (P h ) implies the following relation:

( Rh )      F (w h ) -F (u h (t)) + j h (f (t), u h (t), w h -u h (t)) ≥ α 2 w h -u h (t) 2 ∀ w h ∈ K h a.e. on ]0, T [.
The full discretization of (P h ) is obtained by using an implicit scheme as for (P ). For u 0 h := u 0h and i ∈ {0, 1, ..., ν -1}, we define u i+1 h as the solution of the following problem.

Problem P i hν : Find u i+1 h ∈ K i+1 h such that (P i hν )        F (u i+1 h ), v h -∂u i h + j h (f i+1 , u i+1 h , v h ) -j h (f i+1 , u i+1 h , ∂u i h ) ≥ b(f i+1 , u i+1 h , v h -∂u i h ) ∀ v h ∈ V h , b(f i+1 , u i+1 h , z h -u i+1 h ) ≥ 0 ∀ z h ∈ K h ,
where

K i+1 h := K h (f i+1
). As in Section 2, it follows that for all i ∈ {0, 1, ..., ν -1} the problem (P i hν ) is equivalent to the following variational inequality:

find u i+1 h ∈ K i+1 h such that (Q i hν ) F (u i+1 h ), w h -u i+1 h + j h (f i+1 , u i+1 h , w h -u i h ) -j h (f i+1 , u i+1 h , u i+1 h -u i h ) ≥ 0 ∀ w h ∈ K h .
From ( 2), ( 36), ( 41), ( 12) and ( 42) it results that the mapping S h f i+1 ,u i h :

K i+1 h → K i+1 h
is a contraction, so that (Q i hν ) has a unique solution which is also the unique solution of (P i hν ), for all i ∈ {0, 1, ..., ν -1}. If we define the functions

           u hν (0) = ûhν (0) = u 0h and ∀ i ∈ {0, 1, ..., ν -1}, ∀ t ∈]t i , t i+1 ], u hν (t) = u i+1 h , ûhν (t) = u i h + (t -t i )∂u i h ,
then for all ν ∈ N * the sequence of inequalities (P i hν ) i=0,1,...,ν-1 is equivalent to the following incremental formulation: for almost every t ∈ [0, T ]

(P hν )            u hν (t) ∈ K h (f ν (t)), F (u hν (t)), v h - d dt ûhν (t) + j h (f ν (t), u hν (t), v h ) -j h (f ν (t), u hν (t), d dt ûhν (t)) ≥ b(f ν (t), u hν (t), v h - d dt ûhν (t)) ∀ v h ∈ V h , b(f ν (t), u hν (t), w h -u hν (t)) ≥ 0 ∀ w h ∈ K h .
We have the following analogue to theorem 2.1 in the finite-dimensional case.

which implies (47).

As one can easily prove that

u 0h → u 0 in V, from (48)
it follows that u(0) = u 0 . For each t ∈ [0, T ], by (33) one can find (û h ) h such that ûh ∈ K h and ûh → u(t) in V . Taking w h = ûh in ( Rh ), we have for all h

F (û h ) -F (u h (t)) + j h (f (t), u h (t), ûh -u h (t)) ≥ α 2 ûh -u h (t) 2 .
Since F is strongly continuous and sequentially weakly lower semicontinuous, by using the previous relation, ( 48) and (37), we obtain lim sup

h→0 α 2 ûh -u h (t) 2 ≤ lim h→0 F (û h ) -lim inf h→0 F (u h (t)) + lim h→0 j h (f (t), u h (t), ûh -u h (t)) ≤ F (u(t)) -F (u(t)) + j(f (t), u(t), 0) = 0,
which proves (45). From ( 44) and (45), by applying Lebesgue's dominated convergence theorem, we obtain (46). Now, we shall prove that the limit u is a solution of problem P. Using (33), ( 45), (35) and passing to the limit in the second inequality of (P h ) for all t ∈]0, T [, we obtain that u(t) ∈ K(f (t)) ∀ t ∈ [0, T ] and the second inequality of (P ).

Let

π h v be the projection of v ∈ V on V h defined by π h v, w h = v, w h ∀ w h ∈ V h . Thus if v ∈ L 2 (0, T ; V ) then π h v ∈ L 2 (
0, T ; V h ) and using (33) we have π h v(t) → v(t) in V a.e. on [0, T ]. For all v ∈ L 2 (0, T ; V ), integrating both sides in the first inequality of (P h ) over [0, T ] with v h = π h v and passing to the limit, by the relations (3), (45), ( 46), (49), (37) and (40) it follows that (30) is satisfied, which, by Lebesgue's theorem, implies that u is a solution to the first inequality of problem P.

Using theorems 3.1 and 3.2, we obtain the following main approximation result.

Theorem 3.3. Under the assumptions of theorem 3.2, there exists a subsequence of (u hν ) hν such that

u hν (t) → u(t) in V ∀ t ∈ [0, T ], (50)
uhν u in L 2 (0, T ; V ), (51)
where u ∈ W 1,2 (0, T ; V ) is a solution of (P ). Furthermore every cluster point of (u hν ) hν is a solution of (P ).

Subspace correction approximation

The aim of this section is to give, for a fixed time step i, two additive subspace correction algorithms for problem Q i ν , prove their global convergence and estimate the convergence rate.

The methods we deal here generalize the projected multilevel relaxation method suggested in [START_REF] Mandel | A multilevel iterative method for symmetric, positive definite linear complementarity problems[END_REF] and [START_REF] Mandel | Etude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre[END_REF] for complementarity problems. This method was later developed in [START_REF] Kornhuber | Monotone multigrid methods for elliptic variational inequalities I[END_REF]- [START_REF] Kornhuber | Adaptive monotone multigrid methods for nonlinear variational problems[END_REF] and named as monotone multigrid method. On the other hand, the application of this method to other types of convex sets in general abstract spaces and monotone minimizing functionals have been investigated in [START_REF] Badea | Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities[END_REF], [START_REF] Badea | Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals[END_REF] and [START_REF] Badea | Additive Schwarz method for the constrained minimization of functionals in reflexive Banach spaces[END_REF], for instance. Also, the case where the inequality contains extra terms which do not stem from the minimization of a functional has been investigated in [START_REF] Badea | Schwarz methods for inequalities with contraction operators[END_REF]. Additional non-linear terms have also to be considered in the case of quasi-variational, or implicit, inequalities.

As in the previous section, we consider a Hilbert space V and let V 1 , . . . , V m , m ≥ 2, be some closed subspaces. We also consider a closed convex subset K ⊂ V and assume that the following assumption is satisfied. Assumption 4.1. There exists a constant C 0 > 0 such that for each w, v ∈ K there exist v ι ∈ V ι , ι = 1, . . . , m, which satisfy

v ι ∈ K -w for ι = 1, . . . , m, (52)
v -w = m ι=1 v ι , (53)
and

m ι=1 v ι ≤ C 0 v -w . ( 54
)
This assumption is satisfied for various convex sets in Sobolev spaces and will be used in the proofs, where v is the exact solution and w is the current approximation. Now, we consider a functional ϕ : V × V → R and we assume that it is convex and lower semicontinuous with respect to the second variable, and

|ϕ(v 1 , w 2 ) + ϕ(v 2 , w 1 ) -ϕ(v 1 , w 1 ) -ϕ(v 2 , w 2 )| ≤ k 1 k 2 v 1 -v 2 w 1 -w 2 ∀ v 1 , v 2 , w 1 , w 2 ∈ V. (55)
Also, we suppose Assumption 4.2.

m ι=1 ϕ(u, w + v ι ) ≤ (m -1)ϕ(u, w) + ϕ(u, v) (56)
for each u ∈ K and for v, w ∈ K and v ι ∈ V ι , ι = 1, . . . , m, as in Assumption 4.1.

This assumption has been introduced for technical proof reasons. As in the previous section, let F : V → R be a differentiable functional satisfying (2), ( 3) and ( 12), and we consider the problem of finding û ∈ K, the solution of the following quasi-variational inequality

F (û), v -û + ϕ(û, v) -ϕ(û, û) ≥ 0 ∀ v ∈ K. ( 57
)
Since ϕ satisfies (55), with similar arguments as for problem Q i ν , we can prove that problem (57) admits a unique solution.

Evidently, since ϕ is convex with respect to the second variable, F is differentiable and satisfies (4), problem (57) is equivalent to the problem

û ∈ K F (û) + ϕ(û, û) ≤ F (v) + ϕ(û, v) ∀ v ∈ K.
Also, in view of (4), we see that the solution û of (57) satisfies

α 2 v -û 2 ≤ F (v) -F (û) + ϕ(û, v) -ϕ(û, û) ∀ v ∈ K. (58)
A first algorithm corresponding to the subspaces V 1 , . . . , V m and the convex set K is the following. Algorithm 4.1. We start the algorithm with an arbitrary u 0 ∈ K. At iteration n+1, having u n ∈ K, n ≥ 0, we simultaneously solve the inequalities

w n+1 ι ∈ V ι ∩ (K -u n ) F (u n + w n+1 ι ), v ι -w n+1 ι + ϕ(u n + w n+1 ι , u n + v ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι ) ≥ 0 ∀ v ι ∈ V ι ∩ (K -u n ), (59)
for ι = 1, . . . , m, and then we update

u n+1 = u n + r m m ι=1 w n+1 ι
, where r is a fixed constant such that 0 < r ≤ 1.

A simplified variant of Algorithm 4.1 can be written as Algorithm 4.2. We start the algorithm with an arbitrary u 0 ∈ K. At iteration n + 1, having u n ∈ K, n ≥ 0, we solve the inequalities

w n+1 ι ∈ V ι ∩ (K -u n ) F (u n + w n+1 ι ), v ι -w n+1 ι + ϕ(u n , u n + v ι ) -ϕ(u n , u n + w n+1 ι ) ≥ 0 ∀ v ι ∈ V ι ∩ (K -u n ), (60)
for ι = 1, . . . , m, and then we update

u n+1 = u n + r m m ι=1 w n+1 ι
, where r is a fixed constant such that 0 < r ≤ 1.

The following theorem proves that if k 1 k 2 is small enough in comparison with α, then Algorithms 4.1 and 4.2 are convergent.

> χ 0 > 0, such that if k 1 k 2 α ≤ χ 0 (61)
then the two algorithms are globally convergent and we have the following error estimations

F (u n ) + ϕ(û, u n ) -F (û) -ϕ(û, û) ≤ C 1 C 1 + 1 n F (u 0 ) + ϕ(û, u 0 ) -F (û) -ϕ(û, û) , (62)
u n -û 2 ≤ 2 α C 1 C 1 + 1 n F (u 0 ) + ϕ(û, u 0 ) -F (û) -ϕ(û, û) , (63)
where the constant C 1 > 0 depends on α, β, k 1 , k 2 , the number of subspaces m and on the constant C 0 introduced in Assumption 4.1.

Remark 4.1. Condition ( 12) is an existence and uniqueness condition for the solution of problems P i ν and Q i ν in Section 2. Theorem 4.1 proves that the algorithms introduced in this section for problem Q i ν are convergent if condition (61) is satisfied. We see that the two conditions are similar, but the convergence condition is more restrictive than the existence and uniqueness one. This fact seems to be natural, and, roughly speaking, these conditions hold for frictional contact problems if the coefficient of friction is sufficiently small.

Proof. Since the proof of the theorem is almost the same for both algorithms, we prove the theorem only for Algorithm 4.1. Also, in view of (58), we notice that (63) can be obtained from (62). Now, we prove (62). Let us denote

ũn+1 = u n + m ι=1 w n+1 ι . (64)
Applying Assumption 4.1 for w = u n and v = û, we get a decomposition u n 1 , . . . , u n m of û -u n . From (52), we can replace v ι by u n ι in (59), and in view of (4) and the convexity of F , we obtain

F (u n+1 ) -F (û) + ϕ(û, u n+1 ) -ϕ(û, û) + r m α 2 û -ũn+1 2 ≤ (1 - r m ) [F (u n ) -F (û)] + r m F (ũ n+1 ) -F (û) + α 2 û -ũn+1 2 +ϕ(û, u n+1 ) -ϕ(û, û) ≤ (1 - r m )[F (u n ) -F (û)] + r m F (ũ n+1 ), ũn+1 -û + ϕ(û, u n+1 ) -ϕ(û, û) ≤ (1 - r m )[F (u n ) -F (û)] + r m m ι=1 F (u n + w n+1 ι ) -F (ũ n+1 ), u n ι -w n+1 ι + r m m ι=1 [ϕ(u n + w n+1 ι , u n + u n ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] +ϕ(û, u n+1 ) -ϕ(û, û).
Consequently, we have

F (u n+1 ) -F (û) + ϕ(û, u n+1 ) -ϕ(û, û) + r m α 2 û -ũn+1 2 ≤ (1 - r m )[F (u n ) -F (û) + ϕ(û, u n ) -ϕ(û, û)] + r m m ι=1 F (u n + w n+1 ι ) -F (ũ n+1 ), u n ι -w n+1 ι + r m m ι=1 [ϕ(u n + w n+1 ι , u n + u n ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] + r m [ϕ(û, u n ) -ϕ(û, û)] + ϕ(û, u n+1 ) -ϕ(û, u n ). (65)
Using ( 3), (54) and the Hölder inequality, we get (see (3.21) in [START_REF] Badea | One-and two-level additive methods for variational and quasivariational inequalities of the second kind[END_REF], for p = q = 2),

m ι=1 F (u n + w n+1 ι ) -F (ũ n+1 ), u n ι -w n+1 ι ≤ βm 1 + C 0 (1 + 1 2ε 1 ) m ι=1 w n+1 ι 2 + βC 0 ε 1 2 û -ũn+1 2 (66)
for each ε 1 > 0. In view of the convexity of ϕ in the second variable, we have

ϕ(û, u n+1 ) ≤ (1 -r)ϕ(û, u n ) + r m m ι=1 ϕ(û, u n + w n+1 ι ).
From this equation, by (56), ( 55) and (54), we obtain

r m m ι=1 [ϕ(u n + w n+1 ι , u n + u n ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] + r m [ϕ(û, u n ) -ϕ(û, û)] + ϕ(û, u n+1 ) -ϕ(û, u n ) ≤ r m m ι=1 [ϕ(u n + w n+1 ι , u n + u n ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] + r m m ι=1 ϕ(û, u n + w n+1 ι ) - r m [(m -1)ϕ(û, u n ) + ϕ(û, û)] ≤ r m m ι=1 [ϕ(u n + w n+1 ι , u n + u n ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] + r m m ι=1 [ϕ(û, u n + w n+1 ι ) -ϕ(û, u n + u n ι )] ≤ r m k 1 k 2 m ι=1 u n + w n+1 ι -û w n+1 ι -u n ι ≤ r m k 1 k 2 ũn+1 -û + m ι=1 w n+1 ι m ι=1 w n+1 ι + u n ι ≤ r m k 1 k 2 ũn+1 -û + m ι=1 w n+1 ι • C 0 ũn+1 -û + (1 + C 0 ) m ι=1 w n+1 ι . Therefore, r m m ι=1 [ϕ(u n + w n+1 ι , u n + u n ι ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] + r m [ϕ(û, u n ) -ϕ(û, û)] + ϕ(û, u n+1 ) -ϕ(û, u n ) ≤ r m k 1 k 2 [C 0 + (1 + 2C 0 ) ε 2 2 ] ũn+1 -û 2
C 1ε = m -r r + C 4ε m r , C 2ε = α 2 -k 1 k 2 (1 + 1 2ε 3 )m, C 3ε = α 2 -βC 0 ε 1 2 -k 1 k 2 C 0 + 1 + 2C 0 2 ε 2 -k 1 k 2 ε 3 2 C 4ε , C 4ε = m C 2ε β 1 + C 0 (1 + 1 2ε 1 ) + k 1 k 2 1 + C 0 + 1 + 2C 0 2ε 2 , (72)
then, from (68) and (71), on the condition C 2ε > 0, we get F (u n+1 ) -F (û) + ϕ(û, u n+1 ) -ϕ(û, û) + C 3ε û -ũn+1 2 ≤ C 1ε F (u n ) -F (u n+1 ) + ϕ(û, u n ) -ϕ(û, u n+1 ) .

We can easily see that C 3ε , as a function of ε 1 , ε 2 , and ε 3 , reaches its maximum value for

ε 1 = ε 2 = ε 3 = k 1 k 2 m α 2 -k 1 k 2 m , ( 74
)
and this is

C 3max = α 2 -k 1 k 2 C 0 -[βC 0 + k 1 k 2 (1 + 2C 0 )] k 1 k 2 m α 2 -k 1 k 2 m -(1 + C 0 )(β + k 1 k 2 ) k 1 k 2 2 m 2 ( α 2 -k 1 k 2 m) 2 . Condition C 3max ≥ 0 is satisfied if ( 1 2 -C 0 k 1 k 2 α ) α β ≥ (1 + 3C 0 ) k 1 k 2 α m 1 2 - k 1 k 2 α m + 2(1 + C 0 ) ( k 1 k 2 α ) 2 m 2 ( 1 2 - k 1 k 2 α m) 2
.

Writing χ = k 1 k 2 α , we see that equation

( 1 2 -C 0 χ) α β = (1 + 3C 0 ) χm 1 2 -χm + 2(1 + C 0 ) χ 2 m 2 ( 1 2 -χm) 2 , ( 75
)
which corresponds to the above inequality, has at least solution in (0, 1 2C 0 ), and if χ 0 is the smallest one, then by taking k 1 k 2 α ≤ χ 0 , we have C 3max ≥ 0.

The value of C 2ε for ε 3 in (74) is

C 2 max = 1 2 α 2 -k 1 k 2 m .
Since we can always take C 0 ≥ m and the above solution χ 0 of equation ( 75)

satisfies χ 0 < 1 2m , then C 2 max > 0 if k 1 k 2 α ≤ χ 0 .

Example 1 .

 1 (Unilateral contact with nonlocal friction for a nonlinear Hencky material)

Theorem 4 . 1 .

 41 Let us assume that Assumptions 4.1 and 4.2 are satisfied. If û is the solution of problem (57), u n , n ≥ 0, are its approximations obtained from one of Algorithms 4.1 or 4.2, there exists a constant χ 0 satisfying 1

+rk 1 k 2 [ 1 +each ε 2 > 2 -βC 0 ε 1 2 -k 1 k 2 [ 2 ] û -ũn+1 2 ≤ 1 ) + k 1 k 2 m 1 +, u n + w n+1 ι ) ≥ α 2 w n+1 ι 2 .F 2 ≤each ε 3 > 2 -k 1 k 2

 212222221122322 0. Consequently, from (65)-(67), we haveF (u n+1 ) -F (û) + ϕ(û, u n+1 ) -ϕ(û, û) + α C 0 + (1 + 2C 0 ) ε 2 m -r r [F (u n ) -F (u n+1 ) + ϕ(û, u n ) -ϕ(û, u n+1 )] + βm 1 + C 0 (1 + 1 2εfor every ε 1 , ε 2 > 0. From (59) and (4), we get that, for each n ≥ 0 and ι = 1, . . . , m,F (u n ) -F (u n + w n+1 ι ) + ϕ(u n + w n+1 ι , u n ) -ϕ(u n + w n+1 ιUsing the convexity of F and the above relation, we haveF (u n+1 ) ≤ (1 -r)F (u n ) + r m m ι=1 (u n + w n+1 ι ) ≤ F (u n ) F (u n ) -F (u n+1 ) +ϕ(û, u n ) -ϕ(û, u n+1 ) + r m m ι=1 [ϕ(u n + w n+1 ι , u n ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )] -ϕ(û, u n ) + ϕ(û, u n+1 ). u n + w n+1 ι , u n ) -ϕ(u n + w n+1 ι , u n + w n+1 ι )]-ϕ(û, u n ) + ϕ(û, u n+1 ) 0. In view of (69) and (70), we getα n ) -F (u n+1 ) + ϕ(û, u n ) -ϕ(û, u n+1 ) +k 1 k 2 ε 3 2 ũn+1 -û 2(71)for each ε 3 > 0. If we write

Acknowledgments

The authors acknowledge the support of the "Laboratoire Européen Associé CNRS Franco-Roumain de Mathématiques et Modélisation"LEA Math-Mode, for this work.

Theorem 3.1. Assume that (2), (3), ( 12)-( 15), (34), (36), (38), (39), (41)-(43) hold. Then there exists a subsequence of (u hν , ûhν ) ν , still denoted by (u hν , ûhν ) ν , such that

where u h is a solution of (P h ).

By similar arguments as in lemma 2.1 and passing to the limit by using the previous theorem, we find the following a priori estimates for the solutions of (P h ) which are limits of subsequences of (u hν ) ν .

where

We have the following convergence and existence result.

Theorem 3.2. Under the assumptions (2), ( 3), ( 12)-( 15), (31)-(43) there exists a subsequence of (u h ) h such that

where u is a solution of (P ).

Proof. From lemma 3.1 it follows that there exists a subsequence of (u h ) h and an element u ∈ W 1,2 (0, T ; V ) such that

Now, (62) is obtained from (73), where

is the value of C 1ε for ε 1 , ε 2 and ε 3 given in (74).

Constant C 1 has a similar expression in the case of Algorithm 4.2.

The above general result will be used in a subsequent paper to prove the convergence of the one-and two-level methods for contact problems with nonlocal friction. In this case, the constant C 0 will be explicitly written in terms of the mesh parameters.


Marius Cocou

A class of implicit evolution inequalities and applications to dynamic contact problems

Keywords: Implicit inequalities, dynamic problems, variational solutions, adhesion, friction, viscoelasticity Mathematics Subject Classification (2010) : 35K85, 35R35, 49J40, 74A55, 74D05, 74H20

This paper deals with the analysis of a class of implicit variational inequalities which generalizes some dynamic contact problems coupling adhesion and friction between two viscoelastic bodies of Kelvin-Voigt type. Existence and uniqueness results are proved for a general system of evolution inequalities that constitutes a unified approach to study some complex dynamic surface interactions, including rebonding, debonding and friction conditions. The proofs are based on incremental formulations, several estimates, compactness arguments and a fixed point technique. These results are applied to dynamic frictional contact conditions with reversible adhesion and the coefficient of friction depending on the slip velocity.

Introduction

The aim of this paper is to study a system of evolution inequalities that generalizes some interaction laws including dynamic contact, recoverable adhesion and friction between two viscoelastic bodies, when the coefficient of friction depends on the slip velocity, which represents a more realistic model than the one described in [START_REF] Cocou | A dynamic unilateral contact problem with adhesion and friction in viscoelasticity[END_REF].

An interface model coupling unilateral contact, irreversible adhesion and local friction for elastic bodies was considered in the quasistatic case in [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF] and its mathematical analysis has been provided in [START_REF] Cocou | Existence results for unilateral quasistatic contact problems with friction and adhesion[END_REF].

Nonlocal friction laws, given by appropriate regularizations of the normal component of the stress vector which occurs in the Coulomb friction conditions, and normal compliance models have been considered by several authors in the (quasi)static case, see, e.g. [START_REF] Kikuchi | Contact Problems in Elasticity : A Study of Variational Inequalities and Finite Element Methods[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and references therein.

Dynamic frictional contact problems with normal compliance laws for a viscoelastic body have been studied in [START_REF] Martins | Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws[END_REF][START_REF] Kikuchi | Contact Problems in Elasticity : A Study of Variational Inequalities and Finite Element Methods[END_REF][START_REF] Kuttler | Dynamic friction contact problems for general normal and friction laws[END_REF][START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF] and dynamic unilateral or bilateral contact problems with friction for viscoelastic bodies have been considered in [START_REF] Jarušek | Dynamic contact problems with given friction for viscoelastic bodies[END_REF][START_REF] Eck | Unilateral Contact Problems -Variational Methods and Existence Theorems[END_REF][START_REF] Kuttler | Dynamic bilateral contact with discontinuous friction coefficient[END_REF][START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF][START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF].

1 More recently, dynamic frictionless problems with adhesion were studied in [START_REF] Chau | Dynamic frictionless contact with adhesion[END_REF][START_REF] Kuttler | Existence and regularity for dynamic viscoelastic adhesive contact with damage[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF] and a dynamic viscoelastic problem coupling unilateral contact, recoverable adhesion and nonlocal, mathematically consistent, friction was analyzed in [START_REF] Cocou | A dynamic unilateral contact problem with adhesion and friction in viscoelasticity[END_REF].

Some interesting relaxed unilateral contact condition with pointwise friction have been proposed in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] in the static case and extended in [START_REF] Cocou | Sur la modélisation mathématique de conditions unilatèrales en mécanique du contact[END_REF] to the quasistatic case.

This paper is organized as follows. First, existence and uniqueness results are proved for a general system of evolution inequalities that constitutes a unified approach to study some complex dynamic surface interactions, including rebonding, debonding and friction conditions.

Second, the classical formulation of a dynamic contact problem with adhesion and friction is presented and a corresponding variational formulation is given as a system which contains an implicit evolution variational inequality coupled with a parabolic variational inequality.

Finally, based on the previous abstract results, the existence and uniqueness of variational solutions are analyzed.

Analysis of a system of evolution inequalities

Let (H 0 , |.|, (. , .)), (V 0 , . , . , . ), (U 0 , . U 0 ) and (Π 0 , |.| Π 0 , (. , .) Π 0 ) be four Hilbert spaces such that V 0 ⊂ U 0 ⊆ H 0 , the imbedding from V 0 into U 0 is compact and V 0 dense in H 0 .

Let Λ 0 be a closed convex set in Π 0 such that 0 ∈ Λ 0 . Suppose also that Λ 0 is bounded, to simplify the estimates.

Define two bilinear and symmetric forms, a 0 , b 0 : V 0 × V 0 → R and the mapping γ 0 :

V 0 × Π 0 × Π 0 → R such that ∃ m a , m b > 0 a 0 (u, v) ≤ m a u v , b 0 (u, v) ≤ m b u v , (2.1)
∃ A, B > 0 a 0 (v, v) ≥ A v 2 , b 0 (v, v) ≥ B v 2 ∀ u, v ∈ V 0 , (2.2) ∀ u ∈ V 0 , γ 0 (u, •, •) is a bilinear and symmetric form, (2.3)
∃ m γ > 0 such that ∀ u 1,2 ∈ V 0 , ∀ δ 1,2 ∈ Λ 0 , ∀ η ∈ Π 0 , |γ 0 (u 1 , δ 1 , η) -γ 0 (u 2 , δ 2 , η)| ≤ m γ ( u 1 -u 2 + |δ 1 -δ 2 | Π 0 ) |η| Π 0 , (2.4) γ 0 (u, η, η) ≥ 0 ∀ u ∈ V 0 , ∀ η ∈ Π 0 . (2.5) Let φ 0 : [0, T ] × Λ 0 × V 3 0 → R
φ 0 (t, η, u, v, θw) = θ φ 0 (t, η, u, v, w), (2.8)
φ 0 (0, 0, 0, 0, w) = 0, (2.9)

∀ t ∈ [0, T ], ∀ η ∈ Λ 0 , ∀ u, v, w, w 1,2 ∈ V 0 , ∀ θ ≥ 0, ∃ m φ > 0 such that ∀ t 1,2 ∈ [0, T ], ∀ η 1,2 ∈ Λ 0 , ∀ u 1,2 , v 1,2 , w 1,2 ∈ V 0 , |φ 0 (t 1 , η 1 , u 1 , v 1 , w 1 ) -φ 0 (t 1 , η 1 , u 1 , v 1 , w 2 ) + φ 0 (t 2 , η 2 , u 2 , v 2 , w 2 ) -φ 0 (t 2 , η 2 , u 2 , v 2 , w 1 )| ≤ m φ (|t 1 -t 2 | + |η 1 -η 2 | Π 0 + u 1 -u 2 U 0 + v 1 -v 2 U 0 ) w 1 -w 2 .
(2.10)

Assume that L 0 ∈ W 1,∞ (0, T ; V 0 ), u 0 , u 1 ∈ V 0 , β 0 ∈ Λ 0 and that the following compatibility condition holds:

∃ l 0 ∈ H 0 such that ∀ w ∈ V 0 (l 0 , w) + a 0 (u 0 , w) + b 0 (u 1 , w) + φ 0 (0, β 0 , u 0 , u 1 , w) = L 0 (0), w . (2.11)
Consider the following problem.

Problem Q : Find u ∈ W 2,2 (0, T ; H 0 )∩W 1,2 (0, T ; V 0 ), β ∈ W 1,∞ (0, T ; Π 0 ) such that u(0) = u 0 , u(0) = u 1 , β(0) = β 0 , β(τ ) ∈ Λ 0 for all τ ∈ ]0, T [, and a.e. t ∈ ]0, T [ (ü, v -u) + a 0 (u, v -u) + b 0 ( u, v -u) (2.12) +φ 0 (t, β, u, u, v) -φ 0 (t, β, u, u, u) ≥ L 0 , v -u ∀ v ∈ V 0 , ( β, η -β) Π 0 + γ 0 (u, β, η -β) ≥ 0 ∀ η ∈ Λ 0 . (2.13)
Define the set

X 0 = {η ∈ C 0 ([0, T ]; Π 0 ) ; η(0) = β 0 , η(t) ∈ Λ 0 ∀ t ∈ ]0, T ]},
where the Banach space C 0 ([0, T ]; Π 0 ) is endowed with the norm

η k = max t∈[0,T ] [exp(-kt) |η(t)| Π 0 ] for all η ∈ C 0 ([0, T ]; Π 0 ), k ≥ 0.
The existence and uniqueness of the solution of the problem Q will be proved by using the following lemmas and a fixed point argument, see also [START_REF] Cocou | A dynamic unilateral contact problem with adhesion and friction in viscoelasticity[END_REF] for the particular case when the coefficient of friction is slip rate independent.

Lemma 2.1 For each β ∈ X 0 there exists a unique

u β ∈ W 2,2 (0, T ; H 0 ) ∩ W 1,2 (0, T ; V 0 ), solution of the inequality (2.12) such that u β (0) = u 0 , uβ (0) = u 1 .
The proof is based on some incremental formulations, see, e.g. [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF], and on a useful estimate, see [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF] or [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF], which, when applied to the spaces V 0 ⊂ U 0 ⊆ H 0 , implies the following result: for every > 0 there exists C > 0 such that

u U 0 ≤ u + C |u| ∀ u ∈ V 0 . (2.14)
The full proof will be presented in a forthcoming paper.

Lemma 2.2 Let β 1 , β 2 ∈ X 0 and let u β 1 , u β 2 be the corresponding solutions of (2.12) with the same initial conditions u 0 , u 1 , respectively. Then there exists a constant

C 1 > 0, independent of β 1 , β 2 , u β 1 , u β 2 , such that for all t ∈ [0, T ] | uβ 1 (t) -uβ 2 (t)| 2 + u β 1 (t) -u β 2 (t) 2 ≤ C 1 t 0 |β 1 (s) -β 2 (s)| 2 Π 0 ds. (2.15)
Proof. Let u β 1 , u β 2 be the solutions of (2.12) corresponding to

β 1 , β 2 ∈ X 0 . Taking in each inequality v = uβ 2 and v = uβ 1 , respectively, for a.e. s ∈ ]0, T [ it follows that (ü β 1 -üβ 2 , uβ 1 -uβ 2 ) + a 0 (u β 1 -u β 2 , uβ 1 -uβ 2 ) + b 0 ( uβ 1 -uβ 2 , uβ 1 -uβ 2 ) ≤ φ 0 (s, β 1 , u β 1 , uβ 1 , uβ 2 ) -φ 0 (s, β 1 , u β 1 , uβ 1 , uβ 1 ) +φ 0 (s, β 2 , u β 2 , uβ 2 , uβ 1 ) -φ 0 (s, β 2 , u β 2 , uβ 2 , uβ 2 ) ≤ m φ ( |β 1 -β 2 | Π 0 + u β 1 -u β 2 U 0 + uβ 1 -uβ 2 U 0 ) uβ 1 -uβ 2 ,
where the second inequality follows by (2.10). For all t ∈ [0, T ], as the solutions u β 1 , u β 2 verify the same initial conditions, integrating between 0 and t yields

1 2 | uβ 1 (t) -uβ 2 (t)| 2 + 1 2 a 0 (u β 1 (t) -u β 2 (t), u β 1 (t) -u β 2 (t)) + t 0 b 0 ( uβ 1 -uβ 2 , uβ 1 -uβ 2 ) ds ≤ m φ t 0 |β 1 -β 2 | Π 0 uβ 1 -uβ 2 ds +m φ t 0 ( u β 1 -u β 2 U 0 uβ 1 -uβ 2 + uβ 1 -uβ 2 U 0 uβ 1 -uβ 2 ) ds.
Using (2.14), Young's inequality for the last three terms with appropriate constants, V 0 -ellipticity of a 0 , b 0 and Gronwall's inequality, the estimate (2.15) follows.

Now, for every element u ∈ W 1,2 (0, T ; V 0 ), consider the inequality (2.13) with the initial condition β 0 , the solution of which is denoted by β u . The existence and uniqueness results for this parabolic inequality follow by classical references, see, e.g. [START_REF] Brézis | Problèmes unilatéraux[END_REF], [START_REF] Barbu | Semigroups and Differential Equations in Banach Spaces[END_REF], [START_REF] Barbu | Nonlinear Differential Equations of Monotone Types in Banach Spaces[END_REF], and a direct proof is presented in [START_REF] Cocou | A dynamic unilateral contact problem with adhesion and friction in viscoelasticity[END_REF].

Lemma 2.3 For each u ∈ W 1,2 (0, T ; V 0 ) there exists a unique solution β u ∈ X 0 ∩ W 1,∞ (0, T ; Π 0 ) of the inequality (2.13).

Lemma 2.4 Let u 1 , u 2 ∈ W 1,2 (0, T ; V 0 ) and let β u 1 , β u 2 ∈ X 0 be the corresponding solutions of (2.13) with the same initial condition β 0 , respectively. Then there exists a constant

C 2 > 0, independent of u 1 , u 2 , β u 1 , β u 2 , such that for all t ∈ [0, T ] |β u 1 (t) -β u 2 (t)| 2 Π 0 ≤ C 2 t 0 u 1 (s) -u 2 (s) 2 ds. (2.16)
Proof. Let β u 1 , β u 2 be the solutions of (2.13) corresponding to u 1 , u 2 .

Taking in each inequality η = β u 2 , η = β u 1 , respectively, for all t ∈ ]0, T [, integrating over [0, t], using (2.4) and some elementary inequality yield

1 2 |β u 1 (t) -β u 2 (t)| 2 Π 0 ≤ t 0 [γ 0 (u 2 , β u 2 , β u 1 -β u 2 ) -γ 0 (u 1 , β u 1 , β u 1 -β u 2 )] ds = t 0 [γ 0 (u 2 , β u 2 , β u 1 -β u 2 ) -γ 0 (u 2 , β u 1 , β u 1 -β u 2 )] ds + t 0 [γ 0 (u 2 , β u 1 , β u 1 -β u 2 ) -γ 0 (u 1 , β u 1 , β u 1 -β u 2 )] ds ≤ m γ t 0 |β u 1 -β u 2 | 2 Π 0 ds + m γ t 0 u 1 -u 2 |β u 1 -β u 2 | Π 0 ds ≤ m γ 2 t 0 u 1 (s) -u 2 (s) 2 ds + 3 m γ 2 t 0 |β u 1 (s) -β u 2 (s)| 2 Π 0 ds.
By Gronwall's inequality the estimate (2.16) is established. Now we can prove the following existence and uniqueness result for the abstract problem Q. Proof. For every β ∈ X 0 let u β ∈ W 2,2 (0, T ; H 0 ) ∩ W 1,2 (0, T ; V 0 ) be the solution of the inequality (2.12) corresponding to β such that u β (0) = u 0 , uβ (0) = u 1 and let β u β ∈ X 0 ∩ W 1,∞ (0, T ; Π 0 ) be the solution of the inequality (2.13) corresponding to u β . Define the mapping T : X 0 → X 0 as ∀ β ∈ X 0 T β = β u β . We shall prove that T : X 0 → X 0 has a unique fixed point, which is equally the solution of the problem Q.

For all β 1 , β 2 ∈ X 0 , for all t ∈ [0, T ], (2.16) and (2.15) imply that

|T β 1 (t) -T β 2 (t)| 2 Π 0 ≤ C 2 t 0 u β 1 (s) -u β 2 (s) 2 ds ≤ C 1 C 2 t 0 s 0 exp(-2kr) • exp(2kr) |β 1 (r) -β 2 (r)| 2 Π 0 dr ds ≤ C 1 C 2 β 1 -β 2 2 k t 0 exp(2ks) 2k ds ≤ C 1 C 2 4k 2 • exp(2kt) β 1 -β 2 2 k .
Then

T β 1 -T β 2 k = max t∈[0,T ] [exp(-kt) |T β 1 (t) -T β 2 (t)| Π 0 ] ≤ √ C 1 C 2 2k β 1 -β 2 k .
Hence, for all

β 1 , β 2 ∈ X 0 T β 1 -T β 2 k ≤ √ C 1 C 2 2k β 1 -β 2 k ,
so that if k is sufficiently large it follows that T is a contraction and its fixed point is the solution of the problem Q.

A dynamic contact problem with adhesion and friction

Consider two viscoelastic bodies, characterized by a Kelvin-Voigt constitutive law, which occupy the reference domains Ω α of R d , d = 2 or 3, with Lipschitz continuous boundaries Γ α = ∂Ω α , α = 1, 2. Assume the small deformation hypothesis. Let Γ α 1 , Γ α 2 and Γ α 3 be three open disjoint sufficiently smooth parts of Γ α such that Γ

α = Γ α 1 ∪ Γ α 2 ∪ Γ α 3
and, to simplify the estimates, meas(Γ α 1 ) > 0, α = 1, 2. Let y α (x α , t) be the position at time t ∈ [0, T ], where T > 0, of the material point represented by x α in the reference configuration and u α (x α , t) := y α (x α , t)x α be the displacement vector of x α at time t, with the Cartesian coordinates

u α = (u α 1 , ..., u α d ) = (u α , u α d ).
Let ε α , with the Cartesian coordinates (ε ij (u α )), and σ α , with the Cartesian coordinates (σ α ij ), be the infinitesimal strain tensor and the stress tensor, respectively, corresponding to Ω α , α = 1, 2. The usual summation convention will be used for i, j, k, l = 1, . . . , d.

Assume that the displacement U α = 0 is prescribed on Γ α 1 × ]0, T [ , α = 1, 2, and, to simplify, that the densities of both bodies are equal to 1. Let f = (f 1 , f 2 ) and F = (F 1 , F 2 ) denote the given body forces in Ω 1 ∪ Ω 2 and tractions on Γ 1 2 ∪ Γ 2 2 , respectively. The initial displacements and velocities of the bodies are denoted by u 0 = (u 1 0 , u 2 0 ), u 1 = (u 1 1 , u 2 1 ). Suppose that the solids can be in contact between the potential contact surfaces Γ 1 C and Γ 2 C that can be parametrized by two

C 1 functions, ϕ 1 , ϕ 2 , defined on an open subset Ξ of R d-1 such that ϕ 1 (ξ) -ϕ 2 (ξ) ≥ 0 ∀ ξ ∈ Ξ and each Γ α C is the graph of ϕ α on Ξ that is Γ α C = { (ξ, ϕ α (ξ)) ∈ R d ; ξ ∈ Ξ}, α = 1, 2. Let m α : Ξ → R d , with m 1 (ξ) := (∇ϕ 1 (ξ), -1), m 2 (ξ) := (-∇ϕ 2 (ξ), 1), ∀ ξ ∈ Ξ, be the outward normal to Γ α C , α = 1, 2.
Since the displacements, their derivatives and the gap are assumed small, by using a method similar to the one presented in [START_REF] Boieri | Existence, uniqueness, and regularity results for the two-body contact problem[END_REF] (see also [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF]) the following unilateral contact condition at time t on the set Ξ is obtained:

0 ≤ ϕ 1 (ξ) -ϕ 2 (ξ) + u 1 d (ξ, ϕ α (ξ), t) -u 2 d (ξ, ϕ α (ξ), t) -∇ϕ 1 (ξ) • u 1 (ξ, ϕ 1 (ξ), t) + ∇ϕ 2 (ξ) • u 2 (ξ, ϕ 2 (ξ), t) ∀ ξ ∈ Ξ.
Using the definition of m 1 , m 2 , this relation can be written under the following form: for all ξ ∈ Ξ

m 1 (ξ) • u 1 (ξ, ϕ 1 (ξ), t) + m 2 (ξ) • u 2 (ξ, ϕ 2 (ξ), t) ≤ ϕ 1 (ξ) -ϕ 2 (ξ). (3.1)
Let n α := m α /|m α | denote the unit outward normal vector to Γ α C , α = 1, 2, and define the initial normalized gap between the two contact surfaces by

g 0 (ξ) := ϕ 1 (ξ) -ϕ 2 (ξ) 1 + |∇ϕ 1 (ξ)| 2 ∀ ξ ∈ Ξ.
Let the normal and tangential components of a displacement field v α , α = 1, 2, of the relative displacement corresponding to v := (v 1 , v 2 ), including the initial gap g 0 in the normal direction, and of the stress vector σ α n α on Γ α C be given by

v α := v α (ξ, t) = v α (ξ, ϕ α (ξ), t), v α N := v α N (ξ, t) = v α (ξ, ϕ α (ξ), t) • n α (ξ), v α T := v α T (ξ, t) = v α -v α N n α , [v N ] := [v N ](ξ, t) = v 1 N + v 2 N -g 0 , [v T ] := [v T ](ξ, t) = v 1 T -v 2 T , σ α N := σ α N (ξ, t) = (σ α n α ) • n α , σ α T := σ α T (ξ, t) = σ α n α -σ α N n α , (3.2)
for all ξ ∈ Ξ and for all t

∈ [0, T ]. Let g := -[u N ] = g 0 -u 1 N -u 2 N
be the gap corresponding to the solution u := (u 1 , u 2 ). Assuming that ∇ϕ 1 (ξ) ∇ϕ 2 (ξ), it follows that the unilateral contact condition (3.1) at time t can be written as

[u N ] (ξ, t) = -g(ξ, t) ≤ 0 ∀ ξ ∈ Ξ. (3.3)
Let β denote an internal state variable, see, e.g. [START_REF] Frémond | Adhérence des solides[END_REF]- [START_REF] Frémond | Non-Smooth Thermodynamics[END_REF], representing the intensity of adhesion or the adhesion field (β = 1 means that the adhesion is total, β = 0 means that there is no adhesion and 0 < β < 1 is the case of partial adhesion).

Classical formulation

Let A α , B α denote two fourth-order tensors, the elasticity tensor and the viscosity tensor corresponding to Ω α , with the components (A α ijkl ) and (B α ijkl ), respectively. Assume that these components satisfy the following classical symmetry and ellipticity conditions:

C ijkl = C jikl = C klij ∈ L ∞ (Ω α ), ∀ i, j, k, l = 1, . . . , d, ∃ α C > 0 such that C ijkl τ ij τ kl ≥ α C τ ij τ ij , ∀ τ = (τ ij ) verifying τ ij = τ ji , ∀ i, j = 1, . . . , d, where C ijkl = A α ijkl , C = A α or C ijkl = B α ijkl , C = B α ∀ i, j, k, l = 1, . . . , d, α = 1, 2.
We choose the following state variables: the infinitesimal strain tensor (ε

1 , ε 2 ) = (ε(u 1 ), ε(u 2 )) in Ω 1 ∪ Ω 2 , the relative normal displacement [u N ] = u 1 N + u 2 N -g 0 , the relative tangential displacement [u T ] = u 1 T -u 2
T , and the intensity of adhesion β in Ξ.

Let µ = µ(ξ, [ uT ]) ≥ 0 be the slip rate dependent coefficient of friction and assume that µ : Ξ × R d → R + is a bounded function such that for a.e. ξ ∈ Ξ, µ(ξ, •) is Lipschitz continuous with the Lipschitz constant independent of ξ and for any v ∈ R d , µ(•, v) is measurable.

Define a truncation operator ϑ = ϑ l 0 by ϑ : R → R, ϑ(s) = -l 0 if s ≤ -l 0 , ϑ(s) = s if |s| < l 0 and ϑ(s) = l 0 if s ≥ l 0 , where l 0 > 0 is a given characteristic length (see, e.g. [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF]).

Let κ : R × [0, 1] → R be a bounded Lipschitz continuous function such that κ(0, 0) = 0. Note that various normal compliance conditions, friction and adhesion with damage laws can be obtained by choosing particular functions as κ, see [START_REF] Kikuchi | Contact Problems in Elasticity : A Study of Variational Inequalities and Finite Element Methods[END_REF][START_REF] Kuttler | Dynamic friction contact problems for general normal and friction laws[END_REF][START_REF] Kuttler | Existence and regularity for dynamic viscoelastic adhesive contact with damage[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF].

We consider the following classical formulation of the dynamic contact problem coupling adhesion and friction.

Problem P c : Find (u 1 , u 2 ), β such that u(0) = u 0 = (u 1 0 , u 2 0 ), u(0) = u 1 = (u 1 1 , u 2 1 ) in Ω 1 × Ω 2 , β(0) = β 0 in Ξ and, for all t ∈]0, T [, üα -div σ α (u α , uα ) = f α in Ω α , (3.4)
σ α (u α , uα ) = A α ε(u α ) + B α ε( uα ) in Ω α , (3.5
)

u α = 0 on Γ α 1 , σ α n α = F α on Γ α 2 , α = 1, 2, (3.6
)

σ 1 n 1 + σ 2 n 2 = 0 in Ξ, (3.7
)

σ N = κ([u N ], β) in Ξ, (3.8)
| σ T | ≤ µ([ uT ]) | σ N | in Ξ and (3.9) | σ T | < µ([ uT ]) | σ N | ⇒ [ uT ] = 0, | σ T | = µ([ uT ]) | σ N | ⇒ ∃ θ ≥ 0, [ uT ] = -θσ T , β ∈ [0, 1] in Ξ and (3.10) b β ≥ w if β = 0, b β = w -C N ϑ([u N ] 2 ) β if β ∈ ]0, 1[, b β ≤ w -C N ϑ([u N ] 2 ) if β = 1,
where

β 0 ∈ [0, 1] in Ξ, C N > 0, b > 0, w > 0, σ α = σ α (u α , uα ), α = 1, 2, σ N := σ 1 N , σ T := σ 1 T , σ := σ 1 .

Variational formulation

We adopt the following notations:

H s := [H s (Ω 1 )] d × [H s (Ω 2 )] d ∀ s ∈ R, v, w -s,s = v 1 , w 1 H -s (Ω 1 ),H s (Ω 1 ) + v 2 , w 2
H -s (Ω 2 ),H s (Ω 2 ) ∀ v = (v 1 , v 2 ) ∈ H -s , ∀ w = (w 1 , w 2 ) ∈ H s .
Define the Hilbert spaces (H, |.|) with the associated inner product denoted by (. , .), (V , . ) with the associated inner product (of H 1 ) denoted by . , . and the set Λ as follows:

H := H 0 = L 2 (Ω 1 ) d × L 2 (Ω 2 ) d , V = V 1 × V 2
, where V α = {v α ∈ H 1 (Ω α ) d ; v α = 0 a.e. on Γ α 1 }, α = 1, 2, Λ = {η ∈ L 2 (Ξ) ; η ∈ [0, 1] a.e. in Ξ}.

Assume that F α ∈ W 1,∞ (0, T ; [L 2 (Γ α

2 )] d ), f α ∈ W 1,∞ (0, T ; [L 2 (Ω α )] d ), α = 1, 2, u 0 , u 1 ∈ V and β 0 ∈ Λ.

Let a, b be two bilinear, continuous and symmetric mappings defined on H 1 × H 1 → R as a(v, w) = a 1 (v 1 , w 1 ) + a 2 (v 2 , w 2 ), b(v, w) = b 1 (v 1 , w 1 ) + b 2 (v 2 , w 2 ) ∀ v = (v 1 , v 2 ), w = (w 1 , w 2 ) ∈ H 1 , where, for α = 1, 2,

a α (v α , w α ) = Ω α A α ε(v α )•ε(w α ) dx, b α (v α , w α ) = Ω α B α ε(v α )•ε(w α ) dx.
Consider L as an element of W 1,∞ (0, T ; H 1 ) such that ∀ t ∈ [0, T ]

L, v = α=1,2 Ω α f α • v α dx + α=1,2 Γ α 2 F α • v α ds ∀ v = (v 1 , v 2 ) ∈ H 1 .
One can easily verify the properties (2.1)- (2.11). Thus, by Theorem 2.1 there exists a unique solution of the problem P v .

Note that the same method can be used to study the dynamic contact problem with irreversible adhesion (see, e.g. [START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF]), for which the evolution of the intensity of adhesion is governed by a differential equation.

Theorem 2 . 1

 21 Assume that conditions (2.1)-(2.11) hold. Then there exists a unique solution of the problem Q.

We define the following mappings:

Assume the following compatibility condition:

A variational formulation of the problem P c is the following.

The formal equivalence between the variational system (3.12),(3.13) and the classical problem (3.4)-(3.10) can be easily proved by using Green's formula.

Existence and uniqueness of variational solutions

The following existence and uniqueness result holds.

Theorem 3.1 Under the above assumptions, there exists a unique solution of the Problem P v .

Proof. We apply Theorem 2.1 to


Franziska Schmidt

Xiao Yi Zhou

François Toutlemonde

A peaks-over-threshold analysis of extreme traffic load effects on bridges

For purposes of design of new structures or safety assessment of existing structures, extreme effects need to be assessed, for example through calculation of characteristic values of high quantile.

Indeed, these values make it possible to determinate stresses for extreme events (extreme weather impacts like temperatures or rain, but also extreme traffic events). This paper focuses on modeling and estimating tail parameters of bridge traffic load effects by generalized Pareto distribution (GPD) to obtain these characteristic values. Its main advantage is that it makes use of all relevant data on the high load effect induced by the passage of traffic, not just the daily maxima used for generalized extreme value (GEV) distribution. However, two critical problems of parameter estimation and threshold selection complicate the use of GPD in practice. A Kolmogorov-Smirnov (KS) statistic based threshold selection method is used to choose the optimal threshold, and the maximum likelihood estimator is adopted to estimate the parameters in this study. Results show that the GPD method is a theoretically well supported technique for fitting a parametric distribution to the tail of an unknown underlying distribution.

INTRODUCTION

A bridge structure should have the capacity to withstand the load expected with a given probability over the design life: it is stated in Eurocode 1 [START_REF] Afnor | NF EN 1991-1-2, Eurocode 1 : Actions sur les structures[END_REF] that the structures should be able to resist to the characteristic values for 1000-year return period, which means events with a probability of exceedance of 5% in 50 years.

But it is impossible to collect enough data to determine the characteristic value of load effect expected over lifetime of loading. Even getting sufficient data for the short return period would require several samples of short period data in order to assess the corresponding probability, which is not currently available. Therefore, some form of statistical projection should be performed.

In development of live load model of AASHTO, Nowak [START_REF] Nowak | Bridge live-load models[END_REF] proposed a method that uses normal probability paper. This method had and still has great influence on other American researchers, for example [START_REF] Kozikowski | WIM based live load model for bridge reliability[END_REF], [START_REF] Sivakumar | Protocols for collecting and using traffic data in bridge design[END_REF]. Straight lines are fitted to the tails of the load effect distributions plotted on normal probability paper. The extreme value of load effect can be easily determined by the inversed normal distribution, and the cumulative distribution of maximum load effect is calculated by raising the fitted distribution to a certain power according to the extreme value theory. Although the method is extensively used, the choice of the tail fraction is empirical and subjective: Nowak [START_REF] Nowak | Live load model for highway bridges[END_REF] does not state how he selects the beginning of the tail and Sivakumar uses the upper 5%, see [START_REF] Sivakumar | Protocols for collecting and using traffic data in bridge design[END_REF].

In the background studies for developing the traffic load model of Eurocode 1, five methods have been proposed to predict extreme traffic load or traffic load effect. These are: Among these methods, Rice's formula is different from the others as it uses the full history of the load effect. It has been proved that if the process is Gaussian and stationary, the up-crossing intensity can be expressed by Rice's formula [START_REF] Oswald | Mathematical analysis of random noise[END_REF]. In practice, the load effect is a mixed Gaussian process, therefore the level crossing rate has a Gaussian tail. Cremona [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF] proposed an automatic method to select the optimal threshold from which the tail can be expressed by Rice formula. But the mathematical assumption of Gaussian stationary process is a strong requirement, as the required length of the influence line of interest is large compared to the length occupied by the single vehicles [START_REF] Ditlevsen | Traffic loads on large bridges modeled as white-noise fields[END_REF]. The approach may be not suitable to common load effect of short-to medium-span bridge as it has sharply varying nonzero influence area.

In recent years, the extreme value theory has been used for modeling traffic load effect. Many authors approach the problem by identifying the maxima load effect recorded during a loading event or in a reference period such as a day or a week, and then fit these maxima to an extreme value distribution. Bailey and Bez [START_REF] Bailey | Site specific probability distribution of extreme traffic action effects[END_REF] determined that the Weibull distribution is the most appropriate to model the extreme traffic load effect. Caprani et al. [START_REF] Caprani | Characteristic traffic load effects from a mixture of loading events on short to medium span bridges[END_REF] classified the loading events by the number of trucks involved in the maximal load effect, and then the maxima of each loading event are modeled by GEV distribution. This block maxima approach is wasteful of data as only one data point in each block is taken. The second highest value in one block may be larger than the highest of another block and this is generally not accounted for.

A peaks-over-threshold (POT) approach exists that avoids this drawback, by using the Generalized Pareto distribution. Indeed, due to the advances in extreme value theory, the GPD emerged as a natural family for modeling exceedances over a high threshold. The POT approach has shown its importance and success in a number of statistical analysis problems relating to finance, insurance, hydrology, geographical phenomena, and other domains [START_REF] Holmes | Application of the generalized Pareto distribution to extreme value analysis in wind engineering[END_REF]. However, despite the sound theoretical basis and wide applicability, fitting of this distribution in practice is not a trivial exercise.

Two main factors, the choice of threshold and the choice of the parameter estimator, affect the accuracy of the estimations of the return values. An extensive discussion of the various parameter estimation methods has been given by Zea Bermudez and Kotz [START_REF] De | Parameter estimation of the generalized Pareto distribution -part i[END_REF], [START_REF] De | Parameter estimation of the generalized Pareto distribution -part ii[END_REF]. Scarrott and MacDonald [START_REF] Scarrott | A review of extreme value threshold estimation and uncertainty quantification[END_REF] reviewed the threshold estimation methods. While each methodology has its advantages and disadvantages, traditional parameter estimator such as maximum likelihood is undefined in some regions of the parameter space. Due to these difficulties, the POT approach has not gotten much attention for modeling extreme bridge load effects [START_REF] James | Analysis of Traffic Load Effects on Railway Bridges[END_REF].

The POT approach makes full use of all the data present in the tail as Nowak's approach, but it approximates the tail through extreme value distribution. Therefore, the POT approach is a natural family for modeling exceedances over a high threshold. The purpose of this paper is to introduce an efficient procedure for using POT approach to determine the extreme bridge traffic load effect. The rest of the article is organized as follows: In Section 2, we provide key distributional properties of the GPD. In Section 3, we fit the GPD to the bridge traffic load effects and summarize the results.

Discussions are carried out and conclusions are drawn in Section 4.

THE PEAKS-OVER-THRESHOLD METHOD AND GENERALIZED PARETO DISTRIBUTION

Let be a sequence of independent and identically distributed random variables with common distribution function F , and let's n M denote its maximum:

In theory the distribution of can be derived exactly for all values of n :

But in most cases, the distribution function F is unknown. The value of n F can be approximated by extreme value distribution if the maximum, , belongs to a maximum domain of attraction. In this case:

where , and are shape , scale and location parameters, respectively. The events that exceed some high threshold can be regarded as extreme events and the behavior of these extreme events is given by the conditional probability:

For large enough u, the distribution of , conditional on , is approximated with the extreme value theory by:

In extreme value theory, the previous distribution belongs to the generalized Pareto family. The cumulative distribution function (CDF) of the GPD with shape, scale and location parameters , and , respectively, is defined as:

The probability density function of the GPD is given by:

APPLICATION OF THE GPD TO THE ANALYSIS OF TRAFFIC LOAD EFFECTS

Bridge Traffic Load Effect

Traffic load effects can be measured directly on a bridge, or can be calculated and simulated through a convolution of traffic data with the influence lines of the effect of interest. This last possibility is what has been done here.

Traffic Data

Traffic data, taken from a piezo-ceramic weigh-in-motion system on the A9 motorway near Saint Jean de Vedas, in the South-East of France, is used to validate the POT method on the estimation of characteristic bridge traffic load effects. Weight and dimensional data were collected for trucks travelling in the slow and fast lanes in one direction of the 6-lane motorway from January 2010 to May 2010. The reliable loading data are useful for estimating extreme traffic load effect on bridge.

The unreliable data (trucks with single axle, "0" axle weight and etc) can be eliminated from records with some chosen criteria [START_REF] Sivakumar | Protocols for collecting and using traffic data in bridge design[END_REF]. For operational reasons, on some days the WIM system may not be active for some hours, it is important to exclude these days to ensure having continuous traffic.

581011 trucks for 86 days were drawn from the original data excluding errors, weekends and system inactive days. The histogram of gross vehicle weight of the remaining trucks is presented in Figure 1: the quality of the records is good, as it can be seen at the high percentage of trucks with GVW around the weight limit. The traffic composition plot shows that 5-axle trucks are the most frequently trucks used in France, and the characteristic of this truck affects the load effect on bridges.

Bridges supporting bi-directional four lanes are common in France. An artificial synthesis approach is used to create a sample with four lanes traffic from the records measured on these two lanes.

Influence lines

Previous studies have demonstrated that the critical influence lines for developing load models the bending moment at mid-span of a simply supported bridge, the bending moment at first mid-span of two-span continuous bridge and the hogging moment at middle support of the former. In this study, these three types of load effect are studied. In addition to these, three load effects for the three-span continuous bridge are studied as it is an usual type of bridge in France. A complete list of the influence lines are indicated in Table 1.

Thus three types of bridge structure are investigated: simply supported, two-span continuous with equal span length, and three-span continuous with span configuration of 0.75 L + L + 0.75 L. In the calculations four span lengths were considered: 20, 30, 40, and 50 m.

Preliminary study

In this section, we fit the GPD model to the bridge traffic load effects resulting from combination of traffic data and influence lines, which has been extensively studied in the literature by using other extrapolation methods (see, e.g. [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF]). In the first stage, we detail the application of GPD to the moment induced by the four lanes of traffic at mid-span of a simply supported bridge with span of 50 m. The standard exponential Quantile-Quantile (Q-Q) plot in FIGURE 2(a) gives evidence of lighter than exponential (LTE) distribution nature of the hourly maximum traffic load effects as the curve has a convex shape. This becomes even more apparent in the Mean Excess (ME) plot given in FIGURE 2(b) with a negative slope ratio. This asymptotic linear tail on the ME plot indicates a Pareto type distribution can reasonably fit to the tail from the turn point.

Estimation of The GPD Parameters

The last decade has seen development of a number of approaches for parameter estimation in Gross vehicle weight (tons)

Probability density

Weight limit GPD applications. The maximum likelihood (ML), the probability weighted moments (PWM) and the method of moments (MM) are often used. Previous studies have demonstrated that the maxima of traffic load effect converge to Gumbel or Weibull distribution [START_REF] Bailey | Site specific traffic load models[END_REF]. Therefore, the three extensively used estimators are available for traffic load effects. An extensive discussion of the various methods has been presented by de Zea Bermudez [START_REF] De | Parameter estimation of the generalized Pareto distribution -part i[END_REF]. The main advantages and disadvantages of each one of these methods mentioned are thus known. Moharram et al [START_REF] Moharram | A comparative study for the estimators of the generalized Pareto distribution[END_REF], Singh and Ahmad [START_REF] Singh | A comparative evaluation of the estimators of the three-parameter generalized Pareto distribution[END_REF], and Mackay et al [START_REF] Edward | A comparison of estimators for the generalised Pareto distribution[END_REF] utilized Monte Carlo simulations to evaluate the performance of the estimators for GPD. All of the authors concluded that no method is preferable to another one for all simulated numerical samples. The ML estimator is asymptotically efficient, but for small samples the ML estimators do not always exist and when it does exist, it has a larger bias and variance than other estimators. The PWM and MM estimators are straightforward to compute, but the existence of PWM estimator is restricted to shape parameter less than 0.5 for the PWM estimator and 1.0 for the MM estimator.

Choice of The Optimal Threshold

The complex shape of the ME plot in FIGURE 2 indicates a variation of the properties for excesses over a high threshold. We need a refined method for choosing an optimal threshold. The determination of such a threshold still remains a delicate point of the whole set-up of the GPD-fitting to distribution tails. The recommendation to use the Mean Excess function for this purpose is not suitable here. The automatic threshold selection method by KS test proposed by Cremona [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF] has shown its success in the optimal extrapolation for the level crossing method. It may be a good solution Turn points for the choice of optimal threshold for the use of the POT approach. The Kolmogorov Distance (KD) between the tail histogram and the fitted GPD is thus used as a value to choose the optimal threshold. This statistical test compares two probability functions corresponding respectively to the reference and the tested distribution. If and are the probability functions, the KS-test studies the statistics of the variation D :

For a value d, the probability is approximated by the Kolmogorov function :

Where is the total number of observations blocks. This is the discrete Kolmogorov-Smirnov test.

Figure 3 shows the estimates of the shape parameter and the value of P using the ML, MM and PWM estimators with threshold value. It is clear that the PWM estimator is very sensitive to the threshold value as shown in Figure 3, as the value of the shape parameter shows great changes. The ML estimator is slightly less sensitive that the PWM and MM estimators, displaying a slightly lower variation of shape parameter with threshold value. At a threshold of =11000 u the shape parameter appears to reach a plateau. The corresponding value of P with a KS test keept at a high level over 0.8 means the GPD model fits the excesses well. The fitting is quite sensitive to the endpoints, as the threshold closing to the tail, the shape parameter and value of P fluctuate display some irregular fluctuations. Therefore, the optimal threshold should be selected from the plateau around =11000 u .

Table 2 summarizes the results of the application. The results include the optimal thresholds selected for each parameter estimator, corresponding estimates of return level for 1000 years return period.

Results

The availability of GPD has been illustrated in the previous preliminary study, and then we extend its application to other types of load effect and span lengths. The load effects are determined by combining a four lane free flowing traffic and influence lines for span length from 20 to 50 m.

Characteristic values of the load effects for 1000-year return period are extrapolated using the GPD.

The results are compared with those found by the conventional GEV method, and also compared with the design values calculated from LM 1 of Eurocode 1. The GEV is fitted to daily maxima to avoid correlation, while the GPD is applied to hourly maxima ensuring more data can be involved. In order to reduce possible difference arising from parameter estimator, parameter estimator of ML is used to GPD and GEV.

A summary of the results is presented in Table 3. So one first comment is that the predictions by using GEV and POT methods are less than the design effects in all cases. Interestingly, the differences compared to design values for various type of load effect are around 40% (columns 6 and 7 of Table 3). The predictions obtained by fitting the daily maxima to the GEV, are slightly less than those obtained by the GPD approach. Comparisons of the POT results with the 1000-year GEV results are given in Figure 4.

DISCUSSION AND CONCLUSIONS

In this article, we have introduced the Peaks-over-threshold method for statistical analysis of traffic load effects on bridges. It is based on the excesses over a high threshold which can be modeled by the generalized Pareto distribution. Its main advantage is that it uses all relevant data of the high tail of the effects induced by the passage of traffic, and not just daily or annual maxima. The two difficulties of threshold selection and parameter estimation for using this method have been discussed in the case of bridge traffic load effects. Some remarks can be made:  The automatic approach based on KS test can efficiently identify the optimal threshold for the use of POT method.  The GPD model has shown its success on predicting characteristic bridge traffic load effects as it captures the tail behavior very well.  The almost same differences indicate the Load model 1 (LM1) of Eurocode 1 has a consistent safety margin for various types of load effect.  Based upon the results it is reasonable to conclude that the LM1 of Eurocode is sufficient for modern free flowing traffic.

  i. a half-normal curve fitted to the end of the histogram, ii. a Gumbel distribution fitted to the tail of the histogram, iii. the asymptotic extreme distribution obtained with extreme value theory, iv. Rice's formula for stationary Gaussian processes, v. Monte-Carlo simulation of artificial traffic and Gumbel extrapolation.

FIGURE 1

 1 FIGURE 1 Traffic composition and histogram of GVW TABLE 1 Influence lines used in calculation of load effectItemDescription RepresentationI1Bending moment at mid-span of a simply supported bridge

FIGURE 2 (

 2 FIGURE 2 (a) Standard Q-Q plot and (b) Mean Excess plot for hourly maximum bending moment

FIGURE 3

 3 FIGURE 3 (a) Threshold vs. P-value and (b) threshold vs. shape parameter

FIGURE 4

 4 FIGURE 4 Difference

TABLE 2 ML, MM and PWM estimates

 2

	Method	Threshold	No. excesses	P-value	Shape	Return level
	ML	11012.07	117	0.99046	-0.00087	22575.21
	MM	11012.07	117	0.994234	0.003436	23004.72
	PWM	11012.07	117	0.993113	0.012504	23960.33

TABLE 3 Comparison of POT results

 3

	Load effect	Span length (m)	Characteristic load effect (kN.m) LM1 GEV POT	Percentage difference (%) GEV a POT a POT b
		20	8115	4527	4481	-44.21	-44.79	-1.03
	I1	30 40	14209 21540	7801 12267	8254 11915	-45.10 -43.05	-41.91 -44.68	5.82 -2.86
		50	30109	17386	22575	-42.26	-25.02	29.84
		20	-4775	-2779	-5081	-41.80	6.40	82.83
	I2	30 40	-9027 -14514	-4689 -6960	-6432 -7589	-48.06 -52.05	-28.75 -47.71	37.17 9.05
		50	-21239	-9355	-9671	-55.95	-54.46	3.38
		20	-1769	-1101	-1249	-37.74	-29.40	13.41
	I3	30 40	-3121 -4782	-1874 -2853	-1875 -3361	-39.97 -40.35	-39.94 -29.72	0.05 17.81
		50	-6752	-3888	-3806	-42.42	-43.63	-2.10
		20	6402	3847	3771	-39.91	-41.09	-1.95
	I4	30 40	11157 16841	6140 8735	6810 9211	-44.96 -48.14	-38.96 -45.31	10.91 5.45
		50	23455	12477	13431	-46.80	-42.74	7.64
		20	6452	5226	3804	-19.00	-41.04	-27.21
	I5	30 40	11325 17188	4457 6815	4051 7480	-60.64 -60.35	-64.23 -56.48	-9.12 9.77
		50	24042	7386	7607	-69.28	-68.36	3.00
		20	-4763	-1951	-2098	-59.04	-55.95	7.54
	I6	30 40	-8886 -14162	-3376 -5446	-5904 -5738	-62.01 -61.54	-33.56 -59.48	74.87 5.36
		50	-20592	-7122	-8877	-65.41	-56.89	24.65
		20	5338	3482	3433	-34.76	-35.70	-1.43
	I7	30 40	9290 13986	4667 7287	4762 7641	-49.77 -47.90	-48.74 -45.37	2.04 4.87
		50	19426	9980	10333	-48.62	-46.81	3.54
	a Related to load effects from LM1 of EC1.
	b Related to GEV predictions.