Krzysztof Bartosz

Mircea Sofonea

THE ROTHE METHOD FOR VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO CONTACT MECHANICS *

Keywords: variational-hemivariational inequality, Clarke subdifferential, Rothe method, existence and uniqueness, viscoelastic material, frictionless contact, normal compliance, unilateral constraint, weak solution

We consider a new class of first order evolutio nary variational-hemivariational in-equalities for which we prove an existence and uniqueness res ult. The proof is based on a time-discretization method, also known as the Rothe method. It cons ists of considering a discrete version of each inequality in the class, proving its unique solvability, an d recovering the solution of the con-tinuous problem as the time step converges to zero. Then we introduce a quasi-static frictionless problem for Kelvin-Voigt viscoelastic materials in which the con tact is modeled with a nonmonotone normal compliance condition and a unilateral constraint. We prove the variational formulation of the problem cast in the abstract setting of variational-hemivariati onal inequalities, with a convenient choice of spaces and operators. Further, we apply our abstract result in order to prove the unique weak solvability of the problem.

1. Introduction. Variational and hemivariational inequalities play an important role in the study of both the qualitative and numerical analysis of various problems arising in mechanics, physics, and the engineering sciences. Basic references in the field include [START_REF] Baiocchi | Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems[END_REF][START_REF] Brézis | Equations et inéquations non linéaires dans les espaces vectoriels en dualité[END_REF][START_REF] Brézis | Problèmes unilatéraux[END_REF][START_REF] Denkowski | An Introduction to Nonlinear Analysis: Theory[END_REF][START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF][START_REF] Kinderlehrer | An Introduction to Variational Inequalities and Their Applications[END_REF][START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF][START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications,B i r k h äuser[END_REF][START_REF] Panagiotopoulos | Hemivariational Inequalities[END_REF]. Started in the early 1960s, the theory of variational inequalities uses as its main ingredients the arguments of monotonicity and convexity, including properties of the subdifferential of a convex function. In contrast, the theory of hemivariational inequalities is based on properties of the subdifferential in the sense of Clarke, defined for locally Lipschitz functions which may be nonconvex. The numerical analysis of variational inequalities is a broadly developed field, as illustrated in [START_REF] Glowinski | Numerical Methods for Nonlinear Variational Problems[END_REF][START_REF] Glowinski | Numerical Analysis of Variational Inequalities[END_REF][START_REF] Hlaváček | Solution of Variational Inequalities in Mechanics[END_REF] and the references therein. In contrast, there are still very few publications devoted to numerical methods for hemivariational inequalities and, in particular, to evolutionary hemivariational inequalities. The basic reference in the field is the book [START_REF] Haslinger | Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications[END_REF]. The Rothe method represents one of the few discretization methods used in the analysis of variational and hemivariational inequalities. It was considered in [START_REF] Kalita | Convergence of Rothe scheme for hemivariational inequalities of parabolic type[END_REF], for instance, in the study of parabolic hemivariational inequalities. The results obtained there were generalized in [START_REF] Kalita | Semidiscrete Variable Time-Step θ-Scheme for Nonmonotone Evolution Inclusion[END_REF], where a θ-discretization scheme was considered. The Rothe method was also considered in [START_REF] Bartosz | Rothe method for evolution variational-hemivariational inequalities[END_REF], in the study of parabolic variational-hemivariational inequality. References concerning the numerical treatment of hemivariational inequalities arising in contact mechanics include [START_REF] Barboteu | Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact[END_REF][START_REF] Barboteu | An analytical and numerical approach to a bilateral contact problem with nonmonotone friction[END_REF][START_REF] Barboteu | Analysis of a contact problem with normal compliance, finite penetration and nonmonotone slip dependent friction[END_REF].

Variational-hemivariational inequalities represent a special class of inequalities, in which both convex and nonconvex functions are involved. Interest in their study is motivated by various problems in mechanics, as shown in [START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF][START_REF] Panagiotopoulos | Nonconvex problems of semipermeable media and related topics[END_REF]. Recent results in their study have been obtained in [START_REF] Han | A class of variational-hemivariational inequalities with applications to frictional contact problems[END_REF][START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF][START_REF] Sofonea | Numerical analysis of history-dependent variationalhemivariational inequalities with applications to contact problems[END_REF]. The inequalities studied in [START_REF] Han | A class of variational-hemivariational inequalities with applications to frictional contact problems[END_REF] are elliptic. There, an existence and uniqueness result was proved through arguments of surjectivity for pseudomonotone operators and the Banach fixed point. The analysis of a class of history-dependent variational-hemivariational inequalities, including their unique solvability, was provided in [START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF]. There, the proof was based on arguments on pseudomonotone operators, again, combined with a fixed point result for nonlinear operators defined on the space of continuous functions. The analysis of the quasi-variational inequalities introduced in [START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF] was continued in [START_REF] Sofonea | Numerical analysis of history-dependent variationalhemivariational inequalities with applications to contact problems[END_REF], where a continuous dependence result was proved, numerical schemes to solve the inequalities were obtained, and error estimates were derived. The history-dependent variationalhemivariational inequalities considered in [START_REF] Han | A class of variational-hemivariational inequalities with applications to frictional contact problems[END_REF][START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF][START_REF] Sofonea | Numerical analysis of history-dependent variationalhemivariational inequalities with applications to contact problems[END_REF] were formulated in the particular case of Sobolev spaces associated to a bounded domain Ω ⊂ R d and to specific operators like the trace operator, for instance.

The aim of this paper is to study a new class of evolutionary variational-hemivariational inequalities with applications to contact mechanics. With respect to our previous works [START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF][START_REF] Sofonea | Numerical analysis of history-dependent variationalhemivariational inequalities with applications to contact problems[END_REF] this paper has some traits of novelty that we describe in what follows. First, in contrast with the inequalities considered in [START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF][START_REF] Sofonea | Numerical analysis of history-dependent variationalhemivariational inequalities with applications to contact problems[END_REF], the inequalities we study in this current paper are introduced in the context of abstract reflexive Banach spaces. Second, they are evolutionary; i.e., they involve the derivative of the unknown function. In addition, their unique weak solvability is obtained by using arguments different from those in [START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF], based on the Rothe method. And, finally, we apply these results to a new model of contact for Kelvin-Voigt viscoelastic materials.

The rest of the paper is organized as follows. In section 2 we review some preliminary material on nonlinear analysis. In section 3 we introduce the class of evolutionary variational-hemivariational inequalities to be studied, list the assumptions on the data, and state our main existence and uniqueness result, Theorem 3.1. The proof of the theorem is presented in section 5. It is based on the Rothe method that we describe in section 4. Next, in section 6, we introduce a frictionless contact problem in which the material's behavior is modeled with the Kelvin-Voigt viscoelastic constitutive law and the contact conditions are with normal compliance and unilateral constraints. We list the assumptions on the data and derive the weak formulation of the problem, which is in the form of a variational-hemivariational inequality for the displacement field. Finally, we apply our abstract result to prove the unique weak solvability of the viscoelastic contact problem.

Notation and preliminaries.

In this section we briefly present the notation and some preliminary material to be used later in this paper. More details on the material presented below can be found in the books [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF][START_REF] Denkowski | An Introduction to Nonlinear Analysis: Theory[END_REF][START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF][START_REF] Zeidler | Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators[END_REF].

First, we make precise that all linear spaces used in this paper are assumed to be real. Unless it is stated otherwise, in this section we denote by X anormedspacewith the norm • X ,w ed e n o t eb yX * its topological dual, and •, • X * ×X will represent the duality pairing of X and X * .T h e s y m b o l 2 X * is used to represent the set of all subsets of X * . We start with the definition of the subdifferential in the sense of Clarke.

Definition 2.1. Let ϕ : X → R be a locally Lipschitz function. The Clarke generalized directional derivative of ϕ at the point x ∈ X in the direction v ∈ X is defined by

ϕ 0 (x; v) = lim sup y→x,λ↓0 ϕ(y + λv) -ϕ(y) λ .
T h eC l a r k es u b d i ff e r e n t i a lo fϕ at x is a subset of X * given by

∂ Cl ϕ(x)={ ζ ∈ X * : ϕ 0 (x; v) ≥ ζ,v X * ×X for all v ∈ X }.
For a convex function we also recall the definition of its subdifferential in the sense of convex analysis. Definition 2.2. Let Φ:X → R∪{+∞} be a convex functional. Then the convex subdifferential of Φ at x ∈ X is a subset of X * given by

∂ Conv Φ(x)={ ξ ∈ X * :Φ(x + v) -Φ(x) ≥ ξ, v X * ×X for all v ∈ X }.
Now we pass to the definition of pseudomonotonicity, for both single valued and multivalued operators. Definition 2.3. As i n g l ev a l u e do p e r a t o rA :

X → X * is called pseudomonotone if for any sequence {v n } ∞ n=1 ⊂ X, v n → v weakly in X and lim sup n→∞ Av n ,v n -v X * ×X ≤ 0 imply that Av, v -y X * ×X ≤ lim inf n→∞ Av n ,v n -y X * ×X for every y ∈ X.
Definition 2.4. A multivalued operator A : X → 2 X * is called pseudomonotone if the following conditions hold:

(1) A has values which are nonempty, weakly compact, and convex.

(2) A is upper semicontinuous (usc, for short) from every finite dimensional subspace of X into X * endowed with the weak topology.

(3) For any sequence {v n } ∞ n=1 ⊂ X and any v * n ∈ A(v n ), v n → v weakly in X and lim sup n→∞ v * n ,v n -v X * ×X ≤ 0 imply that for any y ∈ X there exists u(y) ∈ A(v) such that u(y),v-y X * ×X ≤ lim inf n→∞ v * n ,v n -y X * ×X .
The next proposition provides the pseudomonotonicity of a multivalued operator corresponding to a superposition of the Clarke subdifferential with a compact operator. For its proof we refer the reader to the proof of Proposition 5.6 in [START_REF] Han | Advances in Variational and Hemivariational Inequalities: Theory, Numerical Analysis and Applications[END_REF].

Proposition 2.5. Let X and U be two reflexive Banach spaces, let ι : X → U be a linear, continuous, and compact operator, and denote by ι * : U * → X * the adjoint operator of ι.L e tJ : U → R be a locally Lipschitz functional, and assume that its Clarke subdifferential satisfies

ξ U * ≤ c (1 + v U ) for all ξ ∈ ∂J(v)
with c>0. Then the multivalued operator M : X → 2 X * defined by

M (v)=ι * ∂J(ιv) for all v ∈ X is pseudomonotone.
Note that, in the statement of Proposition 2.5, U * represents the dual of U and • U and • U * denote the norms on the spaces U and U * , respectively.

We now recall a result providing pseudomonotonicity of the sum of two pseudomonotone operators, which corresponds to Proposition 1.3.68 in [START_REF] Denkowski | An Introduction to Nonlinear Analysis: Theory[END_REF].

Proposition 2.6. Assume that X is a reflexive Banach space and A 1 ,A 2 : X → 2 X * are pseudomonotone operators. Then A 1 + A 2 : X → 2 X * is a pseudomonotone operator.

The next proposition deals with an existence result for an abstract elliptic inclusion and corresponds to Theorem 2.2i n [ 2 4 ] .

Proposition 2.7. Let X be a real reflexive Banach space, let F : D(F ) ⊂ X → 2 X * be a maximal monotone operator, let G : D(G)=X → 2 X * be a multivalued pseudomonotone operator, and let L ∈ X * . Assume that there exist u 0 ∈ X and

R ≥ u 0 X such that D(F ) ∩ B R (0 X ) = ∅ and ξ + η -L, u -u 0 X * ×X > 0 (2.1)
for all u ∈ D(F ) with u X = R and all ξ ∈ F (u), η ∈ G(u). Then there exists at least an element u ∈ D(F ) such that

F (u)+G(u) ∋ L. (2.2)
Note that in the statement of Proposition 2.7 we denote by D(F )a n dD(G) the effective domains of the operators F and G, respectively, 0 X represents the zero element of X,a n dB R (0 X ) represents the ball of radius R and center 0 X .

We now introduce some spaces of vector functions defined on the interval [0,T] where T>0. Let π denote a finite partition of the interval (0,T) by a family of disjoint subintervals σ i =( a i ,b i ) such that [0,T]=∪ n i=1 σi .L e tF denote the family of all such partitions. Then for 1 ≤ q<∞ we define the seminorm of a function x :[0,T] → X by equality x q BV q (0,T ;X) =s u p

π∈F σ i ∈π x(b i ) -x(a i ) q X
and the space BV q (0,T; X)={ x :[0,T] → X; x BV q (0;T ;X) < ∞}.

Assume now that 1 ≤ p ≤∞,1≤ q<∞,a n dX, Z are Banach spaces such that X ⊂ Z with continuous embedding. Denote M p,q (0,T; X, Z)=L p (0,T; X) ∩ BV q (0,T; Z).

Then it is well known that M p,q (0,T; X, Z) is also a Banach space with the norm • L p (0,T ;X) + • BV q (0,T ;Z) . We end this section with the following compactness result obtained in [START_REF] Kalita | Convergence of Rothe scheme for hemivariational inequalities of parabolic type[END_REF].

Proposition 2.8. Let 1 ≤ p, q < ∞.L e tX 1 ⊂ X 2 ⊂ X 3 be real Banach spaces such that X 1 is reflexive, the embedding X 1 ⊂ X 2 is compact, and the embedding X 2 ⊂ X 3 is continuous. Then the embedding M p,q (0,T; X 1 ; X 3 ) ⊂ L p (0,T; X 2 ) is compact.

We shall use Proposition 2.8 in section 6 of this paper.

3. An existence and uniqueness result. In this section, we present the variational-hemivariational inequality in which we are interested, list the assumptions on the data, and state our main existence and uniqueness result, Theorem 3.1.

Let V be a strictly convex, reflexive separable Banach space, and let V * denote its dual. We denote by •, • V * ×V and • V the duality pairing between V and V * and the norm on V , respectively. Let ι : V → U ,w h ereU is a reflexive Banach space, and let •, • U * ×U and • U denote the duality pairing between U and U * and the norm on U , respectively. We also use the notation L(V, V * ) for the space of linear continuous operators from V to V * , and we denote by • L(V,V * ) the norm in space L(V, V * ). Analogously, we introduce the space L(V, U) and the corresponding norm

• L(V,U) .F o rT>0 we use the classical notation for the Lebesgue and Sobolev spaces of function defined on the interval [0,T] with values in a normed space. In addition, we introduce the notation

V = L 2 (0,T; V ), V * = L 2 (0,T; V * ), U = L 2 (0,T; U ), U = L 2 (0,T; U * ), W = { u ∈V |u ′ ∈V * },
where u ′ represents the time derivative of u understood in the sense of distributions. Moreover, we use the symbols •, • V * ×V and •, • U * ×U to denote the duality pairing between V and V * , U and U * , respectively. In addition, C(0; T ; V ) will represent the space of continuous functions defined on [0,T] with values in V , endowed with its usual norm.

Let A, B : V → V * be given operators, let J : U → R,Φ:V → R ∪{+∞} be given functionals, and let f :[ 0 ,T] → V * be a given function. With these data we consider the following problem.

Problem P. Find u ∈W such that u(0) = u 0 and

T 0 Au ′ (t)+Bu(t)+ι * ξ(t) -f (t),v(t) -u(t) V * ×V dt (3.1) + T 0 Φ(v(t)) -Φ(u(t)) dt ≥ 0 for all v ∈V with (3.2) ξ(t) ∈ ∂ Cl J(ιu(t))
for a.e. t ∈ (0,T).

In the study of Problem P we consider the following assumptions on the data: H(A). The operator A : V → V * is linear, bounded, coercive, and symmetric, i.e., the following hold:

(i) A ∈L(V, V * ). (ii) Av, v V * ×V ≥ α v 2 V for all v ∈ V with α>0. (iii) Av, w V * ×V = Aw, v V * ×V for all v, w ∈ V . H(B).
The operator B : V → V * is linear, bounded, and coercive, i.e., the following hold:

(i) B ∈L(V, V * ). (ii) Bv, v V * ×V ≥ β v 2 V for all v ∈ V with β>0. H(J). The functional J : U → R is such that the following hold: (i) J is locally Lipschitz. (ii) ∂ Cl J satisfies the growth condition ξ U * ≤ c (1 + u U ) for all u ∈ U and all ξ ∈ ∂ Cl J(u) with c>0.
(iii) There exists m>0 such that

ξ -η, u -v U * ×U ≥-m u -v 2 U for all u, v ∈ U, ξ ∈ ∂ Cl J(u),η∈ ∂ Cl J(v). H(Φ).
The functional Φ : V → R ∪{+∞} is convex, proper, and lower semicontinuous.

H(ι). The operator ι : V → U is linear, continuous, and compact. Moreover, the associated Nemytskii operator ῑ : M 2,2 (0,T; V, V * ) →U defined by (ῑv)(t)=ι(v(t)) for all t ∈ [0,T] is also compact.

H(0). f ∈ H 1 (0,T; V * ), u 0 ∈ dom(Φ), and the following compatibility condition holds: there exist

ξ 0 ∈ ∂ Cl J(ιu 0 )a n dη 0 ∈ ∂ Conv Φ(u 0 ) such that Bu 0 + ι * ξ 0 + η 0 -f (0) ∈ V.
H(s). Inequality β>mholds, where β and m represent the constants introduced in assumptions H(B)a n dH(J)(iii), respectively.

Note that here and below we use the notation dom(Φ) for the effective domain of the function Φ. Also, we recall that condition H(j)(iii) represents the so-called relaxed monotonicity condition, intensively used in the study of various classes of hemivariational inequalities, as shown in [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF] and the references therein.

The main result of this paper that we state here and prove in section 5 is the following.

Theorem 3.1. Assume that H(A), H(B), H(J), H(Φ), H(ι), H(0),a n dH(s) hold. Then Problem P has a unique solution u ∈ H 1 (0,T; V ).
We end this section with the remark that Problem P is constructed by using the convex function Φ and the nonconvex function J. In addition, it involves the derivative of the solution. For this reason, we refer to Problem P as an evolutionary variational-hemivariational inequality.

4. The Rothe problem. In this section we consider an approximate problem based on time discretization of Problem P. For this discrete problem, also known as the Rothe problem, we prove a result of solvability and we derive error estimates for the solution. To this end, let N ∈ N be fixed and let τ = T N represent the time step. We consider a piecewise constant approximation of f given by

f τ (t)=f k τ := 1 τ kτ (k-1)τ f (s) ds for t ∈ ((k -1)τ, kτ] , for k =1,...,N. (4.1)
Then, using Lemma 3.3 from [START_REF] Carstensen | A theory of discretisation for nonlinear evolution inequalities applied to parabolic Signorini problems[END_REF], we know that

f τ → f strongly in V * as τ → 0. (4.2)
We now formulate the following Rothe problem.

Problem P τ . Find a sequence {u k τ } N k=0 ⊂ V such that u 0 τ = u 0 and 1 τ A(u k τ -u k-1 τ ),v-u k τ V * ×V + Bu k τ ,v-u k τ V * ×V + ξ k τ ,ι(v -u k τ ) U * ×U (4.3) +Φ(v) -Φ(u k τ ) ≥ f k τ ,v-u k τ V * ×V for all v ∈ V with ξ k τ ∈ ∂ Cl J(ιu k τ ) (4.4)
for k =1,...,N.

We have the following existence result.

Lemma 4.1. Assume that H(A),H(B),H(J),H(Φ),a n dH(0) hold. Then there exists τ 1 > 0 such that Problem P τ has at least one solution for all τ ∈ (0,τ 1 ).

Proof. First we observe that Problem P τ can be formulated in the following equivalent way: given

u k-1 τ ∈ V with k =1,...,N -1, find u k τ ∈ V such that 1 τ Au k-1 τ + f k τ ∈ ∂ Conv Φ(u k τ )+ 1 τ Au k τ + Bu k τ + ι * ∂ Cl J(ιu k τ ). (4.5)
In order to establish the solvability of (4.5) we apply Proposition 2.7 with

F (u)= ∂ Conv Φ(u)a n dG(u)= 1 τ Au + Bu + ι * ∂ Cl J(ιu).
To this end we observe that F is a maximal monotone operator, since it represents the subdifferential of the function Φ which is proper, convex, and lower semicontinuous. Moreover, since the operators 1 τ A and B are linear and monotone, they are pseudomonotone. Then, from Propositions 2.5 and 2.6, it follows that G is a pseudomonotone operator.

Let u 0 be the element used as the initial condition in Problem P τ .L e tu, ξ, η be such that u ∈ D(F ), ξ ∈ F (u), η ∈ G(u). We will show that, taking

L = 1 τ Au k-1 τ +f k τ , the inequality (2.1) holds for all u such that u V = R,w h e r eR ≥ u 0 V .W eh a v e η = 1 τ Au + Bu + ι * w with w ∈ ∂ Cl J(ιu). (4.6)
Our aim in what follows is to show that if u V is large enough, then the following inequality holds:

1 τ Au + Bu + ι * w + ξ, u -u 0 V * ×V > 1 τ Au k-1 τ + f k τ ,u-u 0 V * ×V . (4.7)
To this end we note that, using our Definition 2.2 and Lemma 2.5 in [START_REF] Nagase | On an application of Rothe method to nonlinear parabolic variational inequalities[END_REF], there exist

k 1 ,k 2 > 0 such that ξ, u -u 0 V * ×V ≥ Φ(u) -Φ(u 0 ) ≥-k 1 u V -k 2 -Φ(u 0 ) for all u ∈ V. (4.8)
Next, from H(A), H(B), H(J), (4.8), and the inequality ab ≤ εa 2 + 1 4ε b 2 , valid for a, b, ε > 0, we have

1 τ Au + Bu + ι * w + ξ, u -u 0 V * ×V ≥ 1 τ α + β -c ι 2 L(V,U) -3ε u 2 V (4.9) - 1 4ε c 2 ι 2 L(V,U) - 1 4ε 1 τ A L(V,V * ) + B L(V,V * ) + c ι L(V,U) 2 u 0 2 V -c ι L(V,U) u 0 V - k 2 1 4ε -k 2 -Φ(u 0 ).
On the other hand, we have

1 τ Au k-1 τ + f k τ ,u-u 0 V * ×V (4.10) ≤ ε u 2 V + 1 4ε + 1 2 1 τ Au k-1 τ + f k τ 2 V * + 1 2 u 0 2 V .
Let us denote δ(τ

)= 1 τ α + β -c ι 2 L(V,U) and observe that if β ≥ c ι 2 L(V,U) , then δ(τ ) > 0 for all τ>0. Otherwise, if β<c ι 2 L(V,U) ,w eh a v eδ(τ ) > 0 for all τ<τ 1 := α(c ι 2 L(V,U) -β) -1 .
In both cases we see that for ε = 1 8 δ(τ ), the value δ(τ ) -4ε is positive. Thus, there exists R 1 > 0 such that

1 τ α + β -c ι 2 L(V,U) -4ε u 2 V (4.11) > 1 4ε 1 τ A L(V,V * ) + B L(V,V * ) + c ι L(V,U) 2 + 1 2 u 0 2 V + c ι L(V,U) u 0 V + k 2 1 4ε + k 2 +Φ(u 0 )+ 1 4ε c 2 ι 2 L(V,U) + 1 4ε + 1 2 1 τ Au k-1 τ + f k τ 2 V * + 1 2 u 0 2 V for all u ∈ V such that u V ≥ R 1 . From (4.9)-(4.11) we conclude that (4.7) holds for all u ∈ V satisfying u V ≥ R 1 .N e x t ,b yH(0), we have ∂ Conv Φ(u 0 ) = ∅,s o u 0 ∈ D(F ), and therefore D(F ) ∩ B u 0 V (0 X ) = ∅.
We now remark that the constant R =max{ R 1 , u 0 V } satisfies the assumptions of Proposition 2.7. We conclude from here that problem (4.5) has a solution, which completes the proof.

We now derive an appropriate estimate for the solution of the Rothe problem.

u k τ V ≤ c 1 , (4.12) N k=1 u k τ -u k-1 τ 2 V ≤ c 2 , (4.13) max k=1...N u k τ -u k-1 τ τ V ≤ c 3 , (4.14) τ N k=1 u k τ -u k-1 τ τ 2 V ≤ c 4 (4.15)
for all τ ∈ (0,τ 2 ).

Proof. We take v 0 ∈ dom(Φ) as the test function in (4.3) and obtain

1 τ A u k τ -u k-1 τ ,u k τ V * ×V + Bu k τ ,u k τ V * ×V + ξ k τ ,ιu k τ U * ×U +Φ(u k τ ) (4.16) ≤ 1 τ A u k τ -u k-1 τ ,v 0 V * ×V + Bu k τ ,v 0 V * ×V + ξ k τ ,ιv 0 U * ×U +Φ(v 0 )+ f k τ ,u k τ -v 0 V * ×V with ξ k τ ∈ ∂ Cl J(ιu k τ )a n dk =1,...,N.
From H(A)w eo b t a i n

A(u -v),u V * ×V = 1 2 Au, u V * ×V - 1 2 Av, v V * ×V (4.17) + 1 2 A(u -v), (u -v) V * ×V for all u, v ∈ V.
Then, using this identity, assumptions H(B), H(J), and Lemma 2.5 in [START_REF] Nagase | On an application of Rothe method to nonlinear parabolic variational inequalities[END_REF], we obtain

1 τ A u k τ -u k-1 τ ,u k τ V * ×V + Bu k τ ,u k τ V * ×V + ξ k τ ,ιu k τ U * ×U +Φ(u k τ ) (4.18) ≥ 1 2τ Au k τ ,u k τ V * ×V - 1 2τ Au k-1 τ ,u k-1 τ V * ×V + 1 2τ A u k τ -u k-1 τ ,u k τ -u k-1 τ V * ×V + β -c ι 2 L(V,U) -2ε u k τ 2 V - 1 4ε c 2 ι L(V,U) + k 1 -k 2 ,
where k 1 ,k 2 are positive constants which do not depend on τ . Moreover, we have the estimate

1 τ A u k τ -u k-1 τ ,v 0 V * ×V + Bu k τ ,v 0 V * ×V + ξ k τ ,ιv 0 U * ×U +Φ(v 0 ) (4.19) + f k τ ,u k τ -v 0 V * ×V ≤ 1 τ A u k τ -u k-1 τ ,v 0 V * ×V +3ε u k τ 2 V + 1 4ε B 2 L(V,V * ) v 0 2 V + c 2 ι 4 L(V,U) v 0 2 V + 1 4ε + 1 2 f k τ 2 V * +Φ(v 0 )+ 1 2 v 0 2 V .
Next, we combine (4.16)-(4.19) to find that

1 2 Au k τ ,u k τ V * ×V + 1 2 A u k τ -u k-1 τ ,u k τ -u k-1 τ V * ×V + τβ u k τ 2 V (4.20) ≤ A u k τ -u k-1 τ ,v 0 V * ×V + 1 2 Au k-1 τ ,u k-1 τ V * ×V + τ c ι 2 L(V,U) +5ε u k τ 2 V + 1 4ε + 1 2 τ f k τ 2 V * + τ 1 4ε B 2 L(V,V * ) v 0 2 V + c 2 ι 4 L(V,U) v 0 2 V + c 2 ι L(V,U) + k 1 + τ Φ(v 0 )+ 1 2 v 0 2 V + k 2 .
We now introduce the notation

C 1 (ε)= 1 4ε + 1 2 , C 2 (ε)= 1 4ε B 2 L(V,V * ) v 0 2 V + c 2 ι 4 L(V,U) v 0 2 V + c 2 ι L(V,U) + k 1 +Φ(v 0 )+ 1 2 v 0 2 V + k 2 ,
write (4.20) for k =1,...,n ≤ N , add the resulting inequalities, and use assumption

H(A)t oo b t a i n 1 2 α u n τ 2 V + 1 2 α n k=1 u k τ -u k-1 τ 2 V + τ n k=1 β u k τ 2 V ≤ ǫ u n τ 2 V + 1 4ǫ A 2 L(V,V * ) v 0 2 V + 1 2 A L(V,V * ) u 0 τ 2 V + τ n k=1 c ι 2 L(V,U) +5ε u k τ 2 V + C 1 (ε) f τ 2 V * + TC 2 (ε).
This inequality implies that

1 2 α -τc ι 2 L(V,U) -ǫ -5τε u n τ 2 V ≤ τ n-1 k=1 c ι 2 L(V,U) +5ε u k τ 2 V (4.21) + 1 4ǫ A 2 L(V,V * ) v 0 2 V + 1 2 A L(V,V * ) u 0 τ 2 V + C 1 (ε) f τ 2 V * + TC 2 (ε). Let τ 2 = α(2c ι 2 L(V,U) ) -1
, and assume that τ<τ 2 . Then there exist ǫ, ε > 0 such that 1 2 ατc ι 2 L(V,U)ǫ -5τε > 0. On the other hand, (4.2) implies that the sequence {f τ } is bounded in V * as τ → 0. Therefore, we are in a position to apply the discrete Gronwall lemma, i.e., Lemma 7.25 in [START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF]. As a result, from (4.21) we obtain (4.12). Moreover, (4.13) follows from (4.12) and (4.21).

We now proceed with the proof of inequalities (4.14) and (4.15). To this end we take ξ 0 ∈ ∂ Cl J(u 0 ), η 0 ∈ ∂ Conv Φ(u 0 ) and define u -1 τ = u 0 + τ (Bu 0 + ι * ξ 0 + η 0f (0)). Moreover, for k =0,...,N,w ed e n o t eδu

k τ = 1 τ (u k τ -u k-1 τ
). Then it follows that

δu 0 τ = f (0) -Bu 0 -ι * ξ 0 -η 0 . (4.22) Taking v = u k τ in (4.3) we obtain τ Aδu k τ ,δu k τ V * ×V + τ Bu k τ ,δu k τ V * ×V + τ ξ k τ ,ιδu k τ U * ×U +Φ(u k τ ) -Φ(u k-1 τ ) ≤ τ f k τ ,δu k τ V * ×V with ξ k τ ∈ ∂ Cl J(u k τ ). Moreover, replacing k with k -1 in (4.3) and taking v = u k τ yields -τ Aδu k-1 τ ,δu k τ V * ×V -τ Bu k-1 τ ,δu k τ V * ×V -τ ξ k-1 τ ,ιδu k τ U * ×U +Φ(u k-1 τ ) -Φ(u k τ ) ≤ τ f k-1 τ ,δu k τ V * ×V with ξ k-1 τ ∈ ∂ Cl J(u k-1 τ
).

We now add the last two inequalities and use (4.17), H(B), H(J)(iii) to obtain

1 2 Aδu k τ ,δu k τ V * ×V + 1 2 A δu k τ -δu k-1 τ ,δu k τ -δu k-1 τ V * ×V (4.23) + τ (β -m -ε) δu k τ 2 V ≤ 1 2 Aδu k-1 τ ,δu k-1 τ V * ×V + 1 4ε 1 τ f k τ -f k-1 τ 2 V * .
We now write (4.23) for k =1 ,...,n ≤ N ; then we add the resulting inequalities to obtain

1 2 α δu n τ 2 V + n k=1 τ (β -m -ε) δu k τ 2 V ≤ 1 4ε f ′ 2 V * (4.24) +2 A L(V,V * ) f (0) 2 + B 2 L(V,V * ) u 0 2 V + ι 2 L(V,U) ξ 0 2 V + η 0 2 V .
Estimates (4.14) and (4.15) are now a direct consequence of (4. [START_REF] Lea | Range and existence theorem for pseudomonotone perturbations of maximal monotone operators[END_REF] and assumption H(s).

5. Proof of Theorem 3.1. In this section we provide the proof of Theorem 3.1 and, to this end, we use the estimates obtained in section 4 in the study of the Rothe problem. Below we consider various sequences indexed by τ>0 for which we investigate their boundedness and convergence as τ → 0. We start by considering the piecewise linear and piecewise constant interpolant functions u τ :[0,T] → V and ūτ :[0,T] → V , respectively, defined by

ūτ (t)= u k τ ,t∈ ((k -1)τ, kτ] ,k =1,...,N, u 0 τ ,t=0, (5.1)
and

u τ (t)=u k τ + t τ -k (u k τ -u k-1 τ
),t∈ ((k -1)τ, kτ] ,k =1,...,N. (5.2)

Here the sequence {u k τ } N k=0 is a solution of Problem P τ , obtained under the assumptions of Lemma 4.1. In addition, we consider the piecewise constant interpolant ξτ :(0,T] → U * given by ξτ (t)=ξ k τ ,t∈ ((k -1)τ, kτ] ,k =1,...,N, (5.3) where the sequence {ξ k τ } N k=0 satisfies (4.4). Then we note that (4.3) can be written, equivalently, as

Au ′ τ (t)+B ūτ (t) -f τ (t),v(t) -ūτ (t) V * ×V + ξτ (t), ῑ(v(t) -ūτ (t)) U * ×U (5.4) +Φ(v(t)) -Φ(ū τ (t)) ≥ 0 for all v ∈V a.e.

t∈ (0,T).

We now define the Nemytskii operators A, B : V→V * as (Aw)(t)=A(w(t)) and (Bw)(t)=B(w(t)) for all w ∈V and all t ∈ [0,T]. Thus, from (5.4) we get

Au ′ τ + B ūτ -f τ ,v-ūτ V * ×V + ξτ , ῑ(v -ūτ ) U * ×U (5.5) + T 0 Φ(v(t)) -Φ(ū τ (t)) dt ≥ 0 for all v ∈V,
where, recall, ῑ is the Nemytskii operator introduced in assumption H(ι). Let τ 2 be the constant obtained in the proof of Lemma 4.2. We have the following result.

Lemma 5.1. Assume that H(A),H(B),H(J),H(Φ), H(0),a ndH(s) hold. Then, for all τ ∈ (0,τ 2 ), the functions defined by (5.1)- (5.3)

satisfy ūτ L ∞ (0,T ;V ) ≤ d 1 , (5.6) ūτ M 2,2 (0,T ;V,V * ) ≤ d 2 , (5.7) u τ C(0,T ;V ) ≤ d 3 , (5.8) u ′ τ V ≤ d 4 , (5.9)
ξτ U * ≤ d 5 (5.10) with some positive constants d i , 1 ≤ i ≤ 5, not dependent on τ .

Proof. Since the assumptions of Lemma 4.2 are satisfied, the estimates (4.12)-(4.14) hold for all τ ∈ (0,τ 2 ). The estimates (5.6) and (5.8) follow directly from (4.12). Next, we use (5.6) to see that the sequence {ū τ } remains bounded in V. Therefore, in order to establish (5.7), it is enough to estimate ūτ BV 2 (0,T ;V * ) . To this end we consider a division 0 = a 0 <a

1 < ••• <a n = T ,w h e r ea i ∈ ((m i -1)τ, m i τ ]i ss u c h that ūτ (a i )=u m i τ with m 0 =0,m n = N ,a n dm i+1 >m i for i =1,...,N -1. Then ūτ 2 BV 2 (0,T ;V * ) = n i=1 u m i τ -u m i-1 τ 2 V * (5.11) ≤ n i=1 ⎛ ⎝ (m i -m i-1 ) m i k=m i-1 +1 u k τ -u k-1 τ 2 V * ⎞ ⎠ ≤ n i=1 (m i -m i-1 ) ⎛ ⎝ n i=1 m i k=m i-1 +1 u k τ -u k-1 τ 2 V * ⎞ ⎠ = N N k=1 u k τ -u k-1 τ 2 V * = Tτ N k=1 u k τ -u k-1 τ τ 2 V * ≤ CTτ N k=1 u k τ -u k-1 τ τ 2 V .
We now combine (4.15) and (5.11) to see that ūτ 2 BV 2 (0,T ;V * ) is bounded, and we conclude from here that (5.7) holds.

For the proof of (5.9) we observe that

u ′ τ 2 V = τ N k=1 u k τ -u k-1 τ τ 2 V
, and then we use (4.15). Finally, for the proof of (5.10) we note that

ξτ 2 U * = N k=1 τ ξ k τ 2 U * ≤ τ N k=1 c 2 (1+ ιu k τ U ) 2 ≤ 2Tc 2 +2c 2 τ ι 2 L(V,U) N k=1 u k τ 2 V ≤ d 5 .
This completes the proof of the lemma.

We now have all the ingredients to provide the proof of Theorem 3.1.

Proof. First, it follows from Lemma 4.1 that, for τ>0 small enough, there exists a solution of Problem P τ . We consider such a solution, and we use it to construct the functions u τ ,ū τ ,a n d ξτ defined by (5.1)- (5.3). We claim that there exists u ∈V such that, passing to a subsequence again indexed by τ , the following convergences hold: ūτ → u weakly in V, (5.12) ῑū τ → ῑu strongly in U, (5.13) ξτ → ξ weakly in U * , (5.14)

u τ → u weakly in V, (5.15) u ′ τ → u ′ weakly in V. (5.16)
Indeed, it follows from Lemma 4.2, that, for τ>0 small enough, the estimates (5.6)-(5.10) hold. The convergence (5.12) to an element u ∈V follows from the estimate (5.6), the continuous embedding L ∞ (0,T; V ) ⊂V, and the reflexivity of the space V. On the other hand, the estimate (5.7) combined with assumption H(ι)i mplies (5.13). The convergence (5.14) follows from the bound (5.10) and the reflexivity of the space U * . To prove (5.15) we recall that (5.9) implies that the sequence {u τ } is bounded in V, and therefore we can assume that there exists u 1 ∈Vsuch that u τ → u 1 weakly in V as τ → 0. Thus, from (5.12) it follows that ūτu τ → uu 1 weakly in V as τ → 0. We also note that

ūτ -u τ 2 V = N k=1 kτ (k-1)τ (kτ -t) 2 u k τ -u k-1 τ τ 2 V dt = τ 2 3 u ′ τ 2 V ,
and therefore, from the bound (5.9) of {u ′ τ }, we deduce that u = u 1 . We conclude from the above that (5.15) holds. The same bound (5.9) implies the convergence (5.16).

Next, we show that u is a solution of Problem P. To this end we start with passing to the limit in the initial condition. Since the embedding { v ∈V,v ′ ∈V}⊂ C(0,T; V ) is continuous, from (5.15) and (5.16) it follows that u τ → u weakly in C(0,T; V )a sτ → 0. In particular we have u τ (t) → u(t)w e a k l y i n V ′ as τ → 0 for all t ∈ [0,T]. (5.17) Hence, since u τ (0) = u 0 for all τ>0, we get u(0) = u 0 . Now we pass to the limit in (5.5) as τ → 0. For v ∈V we calculate

Au ′ τ ,v-ūτ V * ×V = Au ′ τ ,v V * ×V -Au ′ τ ,u τ V * ×V + Au ′ τ ,u τ -ūτ V * ×V . (5.18)
Since A is linear and continuous, it is also weakly continuous. So from (5.16) we have Au ′ τ →Au τ weakly in V * , i.e., lim

τ →0 Au ′ τ ,v V * ×V = Au ′ ,v V * ×V . (5.19) Moreover, we have lim sup τ →0 -Au ′ τ ,u τ V * ×V (5.20) = lim sup τ →0 1 2 Au τ (0),u τ (0) V * ×V - 1 2 Au τ (T ),u τ (T ) V * ×V = 1 2 Au(0),u(0) V * ×V -lim inf τ →0 1 2 Au τ (T ),u τ (T ) V * ×V .
It is easy to observe that the functional V ∋ v → Av, v V * ×V is continuous and convex and, therefore, that it is weakly lower semicontinuous. Thus, (5.17) yields

Au(T ),u(T ) V * ×V ≤ lim inf τ →0 Au τ (T ),u τ (T ) V * ×V
and, combining this inequality with (5.20), we obtain

lim sup τ →0 -Au ′ τ ,u τ V * ×V (5.21) ≤ 1 2 Au(0),u(0) V * ×V - 1 2 Au(T ),u(T ) V * ×V = -Au ′ ,u V * ×V .
Next, an elementary calculus shows that

Au ′ τ ,u τ -ūτ V * ×V = N k=1 τk τ (k-1) Au ′ τ (t),u τ (t) -ūτ (t) V * ×V dt = N k=1 τk τ (k-1) A 1 τ u k τ -u k-1 τ , t τ -k u k τ -u k-1 τ V * ×V dt = N k=1 1 τ A u k τ -u k-1 τ ,u k τ -u k-1 τ V * ×V τk τ (k-1) t τ -k dt = - N k=1 1 2 A u k τ -u k-1 τ ,u k τ -u k-1 τ V * ×V ≤ 0.
Combining this inequality with (5.18), (5.19), and (5.21) we obtain that lim sup

τ →0 Au ′ τ ,v-ūτ V * ×V ≤ Au ′ ,v-u V * ×V . (5.22) We now estimate lim sup τ →0 B ūτ ,v-ūτ V * ×V ≤ lim sup τ →0 B ūτ ,v V * ×V -lim inf τ →0 B ūτ , ūτ V * ×V . (5.23)
Since the operator B is linear and continuous, it is also weakly continuous. So from (5.16) we have Bu ′ τ →Bu τ weakly in V * , i.e., lim sup

τ →0 Bū τ ,v V * ×V = lim τ →0 B ūτ ,v V * ×V = Bu, v V * ×V . (5.24)
Using now a lower semicontinuity argument, it follows that lim inf

τ →0 B ūτ , ūτ V * ×V ≥ Bu, u V * ×V
and, combining this inequality with (5.23) and (5.24), we obtain that lim sup

τ →0 B ūτ ,v-ūτ V * ×V ≤ Bu, v -u V * ×V . (5.25)
Moreover, from (4.2) and (5.12) we have

(5.26) f τ ,v-ūτ V * ×V → f, v -u V * ×V as τ → 0
and, using (5.13) and (5.14), we see that

(5.27) ξτ , ῑ(v -ūτ ) U * ×U → ξ,ῑ(v -u) U * ×U as τ → 0.
Finally, we define the functional Ψ : V→R ∪{+∞} by

Ψ(v)= T 0 Φ(v(t)) dt for all v ∈V.
We show in what follows that Ψ is lower semicontinuous. Indeed, let u n → u strongly in V. By Lemma 2.5 in [START_REF] Nagase | On an application of Rothe method to nonlinear parabolic variational inequalities[END_REF] there exist

k 1 ,k 2 ∈ R such that for v ∈ V we have Φ(v) ≥ k 1 v V + k 2 , which implies that (5.28) T 0 Φ(u n (t)) dt ≥ k 2 T + k 1 T 0 u n (t) V dt ≥ k 2 T -|k 1 | √ T u n V ≥ k 0 ,
where k 0 is a positive constant which does not depend on n.

Consider now a convergent subsequence of Ψ(u n ), i.e., Ψ(u n ) → M as n →∞ . Then, for a subsequence, we have u n k (t) → u(t) strongly in V for a.e. t ∈ (0,T). From the lower semicontinuity of Φ we have

Φ(u(t)) ≤ lim inf n→∞ Φ(u n k (t)) a.e. t ∈ (0,T).
Inequality (5.28) allows us to use the Fatou lemma. As a result we obtain

T 0 Φ(u(t)) dt ≤ T 0 lim inf n k →∞ Φ(u n k (t)) dt ≤ lim inf n k →∞ T 0 Φ(u n k (t)) dt = M,
which shows that Ψ is a lower semicontinuous function. Moreover, since Φ is convex, then so is Ψ, and, as a consequence, it is weakly sequentially lower semicontinuous. From (5.12) we have Ψ(u) ≤ lim inf τ →0 Ψ(ū τ ), and

(5.29) lim sup τ →0 T 0 (Φ(v(t)) -Φ(ū τ (t)))dt ≤ T 0 (Φ(v(t)) -Φ(u(t)))dt.
We now use (5.22), (5.25)-(5.27), (5.29), and inequality (5.5) to see that (u, ξ) satisfies (3.1). Moreover, since ξτ (t) ∈ ∂ Cl J(ῑū τ (t)) for a.e. t ∈ (0,T), from (5.13), and the convergence theorem of Aubin and Cellina (see [START_REF] Aubin | Set-Valued Analysis,B i r k h äuser[END_REF], for instance) we have η(t) ∈ ∂ Cl J(ῑu(t)) for a.e. t ∈ (0,T).

We deduce from the above that u is a solution of Problem P. The regularity u ∈ H 1 (0,T; V ) follows from (5.15)-(5.16), which concludes the existence part of the theorem.

To prove the uniqueness part we assume in what follows that u 1 and u 2 are two solutions of Problem P, and we let t ∈ [0,T]. We write inequality (3.1)

for u = u 1 with v(s)= u 2 (t)i f s ∈ [0,t], u 1 (t)i f s ∈ [t, T ]
and for u = u 2 with Problem P M . Find a displacement field u :Ω× [0,T] → R d and a stress field σ :Ω× [0,T] → S d such that σ = Cε( u)+Gε (u) in Ω × (0,T), (6.1) Div σ + f 0 = 0 in Ω × (0,T), (6.2) u = 0 on Γ 1 × (0,T), (6.3) σν = f 2 on Γ 2 × (0,T), (6.4)

σ ν = σ 1 ν + σ 2 ν , -σ 1 ν ∈ ∂ Cl j(u ν ), u ν ≤ g, σ 2 ν ≤ 0,σ 2 ν (u ν -g)=0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
on Γ 3 × (0,T), (6.5)

σ τ = 0 on Γ 3 × (0,T), (6.6) u(0) = 0 in Ω. (6.7)
Note that S d represents the space of second order symmetric tensors on R d or, equivalently, the space of symmetric matrices of order d. Recall that the canonical inner products and the corresponding norms on R d and S d are given by

u • v = u i v i , v =(v • v) 1/2 for all u =(u i ), v =(v i ) ∈ R d , σ • τ = σ ij τ ij , τ =(τ • τ ) 1/2 for all σ =(σ ij ), τ =(τ ij ) ∈ S d ,
respectively. Here and below, the indices i, j, k, l run between 1 and d and, unless stated otherwise, the summation convention over repeated indices is used. In (6.1)-(6.7) we use the notation u =(u i ), σ =(σ ij ), and ε(u)=(ε ij (u)) for the displacement vector, the stress tensor, and the linearized strain tensor, respectively, and ν =(ν i ) for the outward unit normal at ∂Ω. Recall that ε ij (u)= 1 2 (u i,j + u j,i ), where an index following a comma indicates a partial derivative with respect to the corresponding component of the spatial variable, denoted by x =(x i ). Moreover, we use the notation v ν and v τ for the normal and tangential components of a vector field v on ∂Ω, i.e., v ν = v•ν and v τ = v-v ν ν. In addition, σ ν and σ τ represent the normal and tangential components of the stress field σ and are defined by σ ν =(σν) • ν and σ τ = σνσ ν ν, respectively.

We now present a short description of the equations and conditions in Problem P M in which, for simplicity, we do not indicate explicitly the dependence of the variables on x. We refer the reader to [START_REF] Eck | Unilateral Contact Problems: Variational Methods and Existence Theorems[END_REF][START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF][START_REF] Sofonea | Mathematical Models in Contact Mechanics[END_REF] for more details on mathematical models in contact mechanics, including additional explanation related to our comments below.

First, (6.1) is the Kelvin-Voigt viscoelastic constitutive law, used in the literature to model the behavior of real bodies like metals, rubbers, and various polymers. Equation (6.2) represents the equation of equilibrium, and we use it since we assume that the process is quasi-static. Next, conditions (6.3) and (6.4) are the displacement and the traction boundary condition, respectively, while (6.6) and (6.7) represent the frictionless condition and the initial condition, respectively.

We turn now to the contact condition (6.5) in which our interest lies and which represents the main trait of novelty of the mechanical model we consider in this paper. This condition is obtained by assuming that the normal stress on the contact surface, denoted by σ ν , has an additive decomposition in two parts, σ 1 ν and σ 2 ν .P a r t σ 1 ν describes the deformability of the obstacle, and therefore it follows a normal compliance condition governed by the subdifferential of a nonconvex potential j.P a r tσ 2 ν describes the rigidity of the obstacle, and therefore it satisfies the Signorini unilateral contact condition. Recall that condition (6.5) models the contact with a foundation which is made by a rigid body covered by a layer made of elastic material, say asperities. It shows that the penetration is restricted, since u ν ≤ g,w h e r eg represents the thickness of the elastic layer. Also, when there is penetration, as far as the normal displacement does not reach the bound g, the contact is described with a nonmonotone normal compliance condition, since in this case -σ ν ∈ ∂j ν (u ν ). Due to the nonmonotonicity of ∂j ν , the condition allows one to describe the hardening or the softening phenomenon of the foundation. Various examples and mechanical interpretation associated with the nonmonotone normal compliance condition can be found in [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF].

In the study of Problem P M we use standard notation for Lebesgue and Sobolev spaces. For all v ∈ H 1 (Ω; R d )w ed e n o t eb yγv the trace of v on ∂Ω and, recall, we use the notation v ν and v τ for its normal and tangential traces. In addition, we introduce spaces V and H defined by

V = { v =(v i ) ∈ H 1 (Ω; R d ) | γv = 0 a.e. on Γ 1 }, H = { τ =(τ ij ) ∈ L 2 (Ω; R d×d ) | τ ij = τ ji , 1 ≤ i, j ≤ d }.
The space H is a real Hilbert space with the canonical inner product given by (σ, τ ) H = Ω σ ij (x) τ ij (x) dx and the associated norm • H .S i n c em(Γ 1 ) > 0, it is well known that V is a real Hilbert space with the inner product

(6.8) (u, v) V =(ε(u), ε(v)) H , u, v ∈ V,
and the associated norm • V . The duality pairing between V and V * is denoted by •, • V * ×V .L e tε ∈ (0, 1 2 )a n dZ = H 1-ε (Ω; R d ). We also define the embedding i

1 : V → Z, the trace operator γ 1 : Z → H 1 2 -ε (Γ 3 ; R d ), and the embedding i 3 : H 1 2 -ε (Γ 3 ; R d ) → L 2 (Γ 3 ; R d ),

and we consider the trace operator

γ : V → L 2 (Γ 3 ; R d ) defined by γ = i 2 • γ 1 • i 1 .
By the Sobolev trace theorem there exists a constant c 0 depending only on the domain Ω, Γ 1 ,a n dΓ 3 such that (6.9)

γv L 2 (Γ 3 ;R d ) ≤ c 0 v V for all v ∈ V.
Let us denote U = L 2 (Γ 3 ) and define the operators ν :

L 2 (Γ 3 ; R d ) → U and ι : V → U by equalities νv = v ν for all v ∈ L 2 (Γ 3 ; R d )a n dι : V → U by ι = ν • γ.
In addition, we consider the spaces V, U,a n dW defined at the beginning of section 2, with the previous notation for V and U . We now consider the following assumptions on the data of Problem P H(j). The normal compliance function j : R → R satisfies for all η ∈ ∂ Cl j(s),s∈ R with d>0;

M . H(C). The viscosity tensor C =(C ijkl ):Ω× S d → S d satisfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (a) C ijkl ∈ L ∞ (Ω), 1 ≤ i, j, k, l ≤ d; (
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (
(c) (η 1 -η 2 )(s 1 -s 2 ) ≥-c 2 |s 1 -s 2 | 2 for all η i ∈ ∂ Cl j(s i ), s i ∈ R,i=1, 2 with c 2 > 0.
H(f ). The densities of forces and traction satisfy

f 0 ∈ H 1 (0,T; L 2 (Ω; R d )), f 2 ∈ H 1 (0,T; L 2 (Γ 2 ; R d )); f 0 (0) ∈ V, f 2 (0) = 0.
In order to derive a variational formulation of Problem P M , we define operators A, B : V → V * by equalities

Au, v V * ×V =(Cε(u), ε(v)) Q for all u, v ∈ V, (6.10) Bu, v V * ×V =(Gε(u), ε(v)) Q for all u, v ∈ V. (6.11)
We also define the functions J : L 2 (Γ 3 ) → R and f :(0,T) → V * by equalities (6.12) Assume now that (u, σ) represent a regular solution of the contact problem (6.1)-(6.7), and let v ∈V. Then, using a standard procedure based on the Green formula and notation (6.10), (6.11) +Φ(v(t)) -Φ(u(t)) ≥ 0a .e.t∈ (0,T).

J(w)= Γ 3 j(w) dΓf o r w ∈ L 2 (Γ 3 ), f (t), v V * ×V = Ω f 0 (t) • v dx + Γ 2 f 2 (t) • v dΓ (6.
We now integrate (6.19) on [0,T] and combine the resulting inequality with inclusion (6.18) and the initial condition (6.7) to obtain the following variational formulation of Problem P M in term of the displacements.

Problem P M V . Find a displacement field u ∈Wsuch that u(0) = 0 and, moreover, The main result of this section states the unique solvability of Problem P M and is formulated as follows.

Theorem 6.1. Assume that H(C), H(G), H(j),a n dH(f ) hold and, moreover, that β>c 2 . (6. [START_REF] Haslinger | Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications[END_REF] Then Problem P V M has a unique solution u ∈ H 1 (0,T; V ).

Proof. We apply Theorem 3.1. To this end we note that assumptions H(C)a n d H(G) imply that the operators A and B defined by (6.10) and (6.11) satisfy the hypotheses H(A)a n dH(B), respectively. Also, assumption H(j)i m p l i e st h a tt h e functional J defined by (6.12) satisfies the assumption H(J). The functional Φ defined by (6.14) satisfies the hypothesis H(Φ). Since u 0 = 0 V ,w eh a v eu 0 ∈ K = dom(Φ). Moreover, it is easy to see that 0 V ∈ ∂ Conv Φ(u 0 ) and, by the basic properties of the Clarke subdifferential, that the set ∂ Cl J(ιu 0 )i sn o n e m p t y . B yH(f ) and (6.13) we obtain f (0) ∈ V ,a n db yH(C) and (6.11) we have Bu 0 = 0 V ∈ V . This implies that condition H(0) is satisfied with some ξ 0 ∈ ∂ Cl J(ιu 0 )a n dη 0 = 0 V . Finally, (6.19) implies that condition H(s) is satisfied, too. Now we will show that the operator ι satisfies H(ι). To this end we consider a bounded sequence {v n }⊂M 2,2 (0,T; V, V * ). We show that for a subsequence, still denoted by {v n },w eh a v e ῑv n → u strongly in U as n →∞ (6.20) with u ∈U. To this end we note that, since the embedding i 1 : V → Z is compact and the embedding Z ⊂ V * is continuous, from Proposition 2.8, it follows that for a subsequence, still denoted by {u n },w eh a v e We combine this inequality with (6.21) to see that (6.20) holds. We conclude from here that condition H(ι) is satisfied. Theorem 6.1 is now a direct consequence of Theorem 3.1.

A couple of functions (u, σ) such that u is a solution of Problem P V M and σ is given by (6.1) is called a weak solution to Problem P M . We conclude that under the assumptions of Theorem 6.1 contact problem P M has a unique weak solution. Moreover, the solution has the regularity u ∈ H 1 (0,T; V )a n dσ ∈ L 2 (0,T; H).

Lemma 4 . 2 .

 42 Assume that H(A),H(B),H(J),H(Φ),H(0),a n dH(s) hold. Let the sequence {u k τ } N k=0 be a solution of Problem P τ , obtained in Lemma 4.1.T h e n there exist τ 2 > 0 and some positive constants c 1 , c 2 , c 3 , c 4 independent on τ such that max k=1,...,N

  b) Cσ • τ = σ •Cτ for all σ, τ ∈ S d a.e. in Ω; (c) Cτ • τ ≥ α|τ | 2 for all τ ∈ S d a.e. in Ω with α>0. H(G). The elasticity tensor G =(G ijkl ):Ω× S d → S d satisfies G ijkl ∈ L ∞ (Ω), 1 ≤ i, j, k, l ≤ d; (b) Gσ • τ = σ •Gτ for all σ, τ ∈ S d a.e. in Ω; (c) Gτ • τ ≥ β|τ | 2 for all τ ∈ S d a.e. in Ω with β>0.

  a) j is locally Lipschitz; (b) ∂ Cl j satisfies the growth condition |η|≤c 1 (1 + |s|)

  [START_REF] Eck | Unilateral Contact Problems: Variational Methods and Existence Theorems[END_REF] for all v ∈ V and all t ∈ [0,T].Finally, let Φ : V → R ∪{+∞} be an indicator function of the setK = {v ∈ V, v ν ≤ g a.e. on Γ 3 }, i.e., Φ(v)= 0i f v ∈ K,+∞ otherwise.(6.14)

T 0 Au 0 Φ

 00 ′ (t)+Bu(t)+ι * ξ(t)f (t), v(t)u(t) V * ×V dt + T (v(t)) -Φ(u(t)) dt ≥ 0 for all v ∈V with ξ(t) ∈ ∂ Cl J(ιu(t))for a.e. t ∈ (0,T)

T 0 i 1 1 2i 1

 0111 u n (t)z(t) 2 Z dt → 0 with n →∞,(6.21) where z ∈ L 2 (0,T; Z). Let u ∈U be given by u =(ν• i 2 • γ 1 )z.W eh a v e ῑv nu 2 U = T 0 ιv n (t)u(t) 2 U dx ≤ ν • i 2 • γ u n (t)z(t) 2 Z dx.

  , (6.13), we deduce thatAu ′ (t)+Bu(t)f (t), v(t)u(t) V * ×V -(t)(v ν (t)u ν (t))dΓa .e.t∈ (0,T).

	(6.15)			Γ 3	σ 1 ν (t)(v ν (t) -u ν (t)) dΓ
	= ν On the other hand, (6.6) and notation (6.14) imply that Γ 3 σ 2
	(6.16)	Γ 3	σ 2 ν (v ν (t) -u ν (t)) dΓ ≥ Φ(u(t)) -Φ(v(t))	a.e.t∈ (0,T).
	Next, we denote
	(6.17)		-σ ν (t)=ξ(t)
	and use again (6.6) and notation (6.12) to deduce that
	(6.18)		ξ(t) ∈ ∂ Cl J(ιu(t))	a.e.t∈ (0,T).

It follows now from (6.15)-(6.17) that

Au ′ (t)+Bu(t)+ι * ξ(t)f (t), v(t)u(t) V * ×V

† Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Computer Science, 30348 Krakow, Poland (krzysztof.jakub.bartosz@gmail.com). This author's research was partially supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant W111/7.PR/2012.‡ Corresponding author. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 66860 Perpignan, France (sofonea@univ-perp.fr).

* This work was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under grant agreement 295118, by the National Science Center of Poland under Maestro Advanced Project DEC-2012/06/A/ ST1/00262, and by the project Polo nium "Mathematical and Numerical Analysis for Contact Problems with Friction" 2014/15 between Jagiellonian University and Universite de Perpignan Via Domitia.

then we add the resulting inequalities. In this way we obtain t 0 A(u ′ 1 (s)u ′ 2 (s)),u 1 (s)u 2 (s) V * ×V ds (5.30)

where ξ i (s) ∈ ∂ Cl J(ιu i (s)) for a.e. s ∈ (0,t), i =1, 2. Using now H(A), H(B),H(J), and (5.30) yields

Recall also that u 1 (0) = u 2 (0) = u 0 , and therefore (5.31) implies that

We now use the Gronwall lemma to conclude that u 1 (t)=u 2 (t) for all t ∈ (0,T), which completes the proof.

A frictionless contact problem.

A large number of quasi-static contact problems with viscoelastic materials lead to evolutionary variational-hemivariational inequalities of the form (3.1), (3.2) in which the unknown is the displacement field. For a variety of such inequalities, Theorem 3.1 can be applied. In this section we illustrate this point on a viscoelastic frictionless contact problem.

The physical setting is the following. A viscoelastic body occupies, in its reference configuration, a regular domain Ω ⊂ R d (d = 2, 3) with boundary ∂Ω. The boundary is partitioned into three disjoint measurable parts Γ 1 ,Γ 2 ,a n dΓ 3 such that the measure of Γ 1 , denoted by m(Γ 1 ), is positive. The body is clamped on Γ 1 ,a n ds o the displacement field vanishes there. Time-dependent surface tractions of density f 2 act on Γ 2 , and time-dependent volume forces of density 0 act on Ω. The body is in frictionless contact on Γ 3 with an obstacle, the so-called foundation. The foundation is made of a perfectly rigid material, covered by a layer of deformable material of thickness g>0. Therefore, we model the contact with a normal compliance condition associated to a unilateral contact condition. The process is quasi-static, and the time interval of interest is [0,T], where T>0. Then the mathematical model of the contact problem (that we state here and explain below in this section) is the following.


Adrien Gougeon

Franc ¸ois Lemercier

Anne Blavette
email: anne.blavette@ens-rennes.fr

Anne-Cécile Orgerie

Modeling the End-to-End Energy Consumption of a Nation-Wide Smart Metering Infrastructure

Several countries have deployed, or have started the deployment of a smart metering infrastructure in order to enable the Smart Grid. This infrastructure aims to provide new services to grid users and grid operators relying on several communication technologies. One of the goals of this infrastructure is to improve energy consumption, for instance by increasing the awareness of the users, or by enforcing energy management policies. Yet, this infrastructure also consumes energy. The objective of this work is to accurately characterize the energy consumption of each part of the smart metering infrastructure, at a nation-wide scale. We also explore several consumption scenarios highlighting the impact of legacy technologies on the energy consumption of the smart metering infrastructure.

I. INTRODUCTION

In order to achieve a climate neutral economy, a largescale roll-out of smart meters is strongly encouraged worldwide and in particular in the European Union [START_REF] Tractebel | Benchmarking smart metering deployment in the EU-28[END_REF]. The objectives of this roll-out are multiple: (1) having a finer temporal granularity of consumption data for the consumers, grid operators and energy providers, in order to implement energy saving schemes, reduce the consumers energy bill, increase the consumption share from renewables and improve network planning, (2) allowing remote automated consumption data collection, (3) enabling remote adjustment of the contract power level, (4) providing two-way data transfer for maintenance and control. The first point underlines one of the main purposes of using a smart metering infrastructure: reducing the grid users energy consumption [START_REF] Mogles | How smart do smart meters need to be?[END_REF].

While smart metering aims at saving energy, it also induces additional energy consumption to power the ICT infrastructure which collects measurements from smart meters and presents them to end-users.

In this context, the purpose of this study is to provide an overview of the whole ICT architecture, model and evaluate the energy consumption of an entire smart metering infrastructure, at a nation-wide scale. This is a first step toward a full analysis of the gains and costs in terms of energy consumption related to smart grid metering. To obtain concrete results, we study the use-case of the deployment in France. Yet, the methodology that we propose in this paper can be applied to other countries. Given the recent deployment of the smart grid infrastructure and the expected lifetime of smart meters (around 30 years), this use case represents a good candidate to realistically assess the impact of smart metering in the next years. Our evaluation takes into account the end-to-end consumption, from the smart meter measuring the consumers individual consumption, to the utility storing and processing the collected data. We also consider the energy cost of the communication networks involved in the system. This paper's contributions can be summarized as follows:

• investigating the end-to-end architecture of advanced metering infrastructures (AMI) for Smart Grids • proposing a complete modeling of the energy consumption of the ICT metering infrastructure of a large-scale electrical grid • applying our model on the French use-case to evaluate the energy consumption of the smart metering infrastructure at a nation-wide scale. The paper is organised as follows. Section II presents the state of the art. Section III provides details about the smart metering infrastructure. Section IV presents the consumption models. Section V deals with the evaluation of our models presenting the parameters we selected, the results obtained and some suggestions to tackle the issues unveiled by our energy model. Finally, Section VI concludes.

II. STATE OF THE ART

Studies on energy-efficiency of smart grid communication technologies mainly tackle optimization problems on routing protocols [START_REF] Ullah | Energy and congestion-aware routing metric for smart grid ami networks in smart city[END_REF], energy harvesting from the smart-meter pointof-view [START_REF] Bera | Energy-efficient smart metering for green smart grid communication[END_REF] or data aggregation to reduce the data volume. The literature is rich on how to optimize the energy efficiency of this ICT system [START_REF] Erol-Kantarci | Energy-Efficient Information and Communication Infrastructures in the Smart Grid: A Survey on Interactions and Open Issues[END_REF], for either actual deployment or envisioned optimized ones [START_REF] Uddin | Energy-aware optimal data aggregation in smart grid wireless communication networks[END_REF], but scarce on assessing its energy consumption. Preisel et al. accurately measured the electricity consumption of various smart meter devices [START_REF] Preisel | Energy Consumption of Smart Meters[END_REF]. These measurements are used by Malmodin et al. to estimate the gains and costs in terms of energy consumption of a large-scale smart metering infrastructure [START_REF] Malmodin | Assessing ICT's enabling effect through case study extrapolation -The example of smart metering[END_REF]. They outline the fact that previous small-case studies tend to be biased towards overly optimistic results, and that: "the impact of the smart metering system itself, while being typically left out in most former studies reviewed, [...] may be significant in low and even medium reduction scenarios". Yet, their study only considers the smart meters themselves and not the overall ICT infrastructure required to collect and process the data produced by the smart meters. Ghasempour et al. present in [START_REF] Ghasempour | Finding the optimal number of aggregators in machine-to-machine advanced metering infrastructure architecture of smart grid based on cost, delay, and energy consumption[END_REF] an energy consumption model of an Advanced Metering Infrastructure (AMI), proposing a new metric that takes into account the product of cost and energy to determine the adequate number of concentrators needed. Despite interesting results verified by simulation on the optimum number of concentrators, this study only consider a 100 km 2 area with a uniform distribution of the meters.

To the best of our knowledge, proposing an end-to-end energy model of a smart metering infrastructure has not been addressed in the literature. In this context, we propose to evaluate the energy consumption of an entire smart metering infrastructure, at a nation-wide scale.

III. SMART METERING ARCHITECTURE

Thirty-five million smart meters are deployed in France [START_REF] Enedis | Programme de R&D et d'Innovation d'Enedis[END_REF], scattered in clusters, each being connected to one of the 770,000 PLC (PowerLine Communication) concentrators [START_REF] Enedis | Programme de R&D et d'Innovation d'Enedis[END_REF]. Concentrators collect metering data, alarms and control data of smart meters once a day. They forward these data to a central utility using cellular communication to reach the closest relay antenna. The communication is then relayed using the WAN to the core Internet. Finally, the servers are connected to the Internet via wired communication as depicted in Figure 1.

A smart meter routine is composed of two tasks: collect local data, and communicate through PLC with its concentrator. Consequently, from an energy consumption perspective, each smart meter is either in idle state, when it does not communicate (either idle or collecting data) or active, when it communicates with its concentrator. The active state duration depends on the PLC technology considered, either G1-PLC [START_REF] Iec | Distribution automation using distribution line carrier systems[END_REF] or G3-PLC [START_REF] Itu | Narrowband orthogonal frequency division multiplexing power line communication transceivers for G3-PLC networks[END_REF], the latter being an evolution of the former and offering better performance with Orthogonal Frequency-Division Multiplexing (OFDM). PLC technologies are also heavily impacted by the quality of the network and by electromagnetic interference, increasing the number of re-transmissions and the overall communication time. Depending on the type of electrical network, some smart meters belong to small clusters with low network variability while the others belong to large clusters that are more prone to collisions. Collected data is forwarded each day, or on demand, by smart meters to the concentrator to which they are associated.

Concentrators are in charge of collecting data from smart meters, and forward aggregated data through the cellular network to the central utility. They must maintain connectivity with their smart meters at all times, either for contacting a specific smart meter on demand, or for being reachable by a smart meter triggering an alarm. As a consequence, concentrators are always active, and this state corresponds to a fixed power consumption. They also present an additional energy consumption, due to the cellular communication network they employ to reach the core network. The cellular technology in use has evolved with time, formerly mainly based on GPRS, while most concentrators use 3G nowadays. Consequently, we consider that a majority of the concentrators has transitioned to 3G while the others still use GPRS.

The concentrators send data over the Internet using cellular networks which rely on Base Transceiver Stations (BTS) for GPRS communication, and on Node B for 3G communication. The energy consumption of cellular network devices is slightly impacted by data traffic [START_REF] Arnold | Power Consumption Modeling of Different Base Station Types in Heterogeneous Cellular Networks[END_REF], in consequence, the total consumption of the cellular network is close to the static consumption of all the devices. Hence, as Guegan et al presented in [START_REF] Guegan | Estimating the End-to-End Energy Consumption of Low-Bandwidth IoT Applications for WiFi Devices[END_REF], we consider that the static consumption of network devices can be imputed to the users of said devices. The share of energy consumed is based on the duration and capacity usage in comparison to average load of the cellular network devices. Finally, as GPRS and 3G technologies paradigms are different, for instance a BTS provides time slots to a GPRS communication, while a Node B provides bandwidth for a 3G communication, we use two different ways to model the energy consumption of each technology.

Once the data transmitted by the concentrators reaches a relay antenna of the cellular network, it goes through the core network, passing through several network devices to reach its destination. Similarly to the cellular network and in accordance with [START_REF] Guegan | Estimating the End-to-End Energy Consumption of Low-Bandwidth IoT Applications for WiFi Devices[END_REF], the share of the energy consumed due to concentrators traffic depends on the relative use of the network devices.

Finally, the transiting data reaches the central utility servers. The data is stored using the utility operator servers that are replicated for security and availability purposes. The overall energy consumption of the servers is considered, including, among others, the energy needed to cool the servers, using the PUE factor (Power Usage Effectiveness) of the data center. The number of servers depends on the user requests on the website, with a high variability of the requests rate, as often, and on the data flows from the concentrators.

The energy consumption models of the overall infrastructure detailed above are presented in the next section.

IV. NETWORK POWER MODELS

In the remainder of the paper, our models consider the energy consumption over a typical day.

A. Smart meters

The smart meters consumption highly depends on the cluster size (small or large) in which they are and on the PLC technology (G1 or G3) they rely on. Thus, four categories can be distinguished for the daily smart meters consumption:

E smallG1 SM = N small SM × R G1 × [P active SM × T small active × F G1 + P idle SM × (T day -T small active ) × F G1 )] (1)
E smallG3 SM = N small SM × (1 -R G1 ) × [P active SM × T small active + P idle SM × (T day -T small active )] (2)
E largeG1 SM = N large SM × R G1 × [P active SM × T large active × F G1 + P idle SM × (T day -T large active ) × F G1 )] (3)
E largeG3 SM = N large SM × (1 -R G1 ) × [P active SM × T small active + P idle SM × (T day -T large active )] (4)
where P active SM and P idle SM are the power consumption in active and idle state, respectively, T small active and T large active are the active time of smart meters in small and large clusters, respectively, N small SM and N large SM are the number of smart meters in small and large clusters, respectively, R G1 is the fraction of G1 smart meters in proportion to all smart meters in the infrastructure, F G1 is a factor increasing the active time due to the low data rate of G1-PLC, T day is the duration of a day.

B. Concentrators

Concentrators' consumption depends on their static consumption and on the additional consumption induced by their communication over GPRS and 3G networks. As such, the concentrators daily energy consumption is divided into three categories: their static consumption, their consumption due to GPRS communication and the one due to 3G communication:

E static CT = P static CT × T day × N CT (5)
E GP RS CT = N CT × R GP RS × P GP RS × DV CT DR GP RS (6)
E 3G CT = N CT × (1 -R GP RS ) × P 3G × DV CT DR 3G (7)
where P static CT is the static power consumption of a concentrator, N CT is the number of concentrators, P GP RS and P 3G are the power consumption delta when using GPRS or 3G communication, respectively, R GP RS is the proportion of concentrators using GPRS, DV CT is the data volume a concentrator sends each day, DR GP RS and DR GP RS are the GPRS and 3G data rate, respectively.

C. Cellular network

Data from concentrators transiting through the cellular network corresponds only to part of the traffic handled by GPRS and 3G network devices. We employ a proportional model to attribute the daily energy consumption induced by concentrators on the cellular infrastructure:

EGP RS = PBT S × N timeslots CT N timeslots BT S × LoadBT S × (DVCT × NCT × RGP RS ) DRGP RS (8) E3G = PNB × DR3G DRNB × LoadNB × (DVCT × NCT × (1 -RGP RS )) DR3G (9)
where P BT S and P N B are the power consumption of a BTS and a Node B, respectively, N timeslots CT is the number of time slots used by a concentrator, N timeslots BT S is the number of time slots on a BTS, B CT is the cellular bandwidth usage of a concentrator, B N B is the maximum bandwidth of a Node B, Load BT S and Load N B are the average load of a BTS and a Node B, respectively.

D. Core network

Similarly to the cellular network, data transiting through the core network relies on specific devices, such as switches and routers. Thus, we evaluate the consumption of the core network that can be imputed to their usage by the smart metering infrastructure using a model from [START_REF] Guegan | Estimating the End-to-End Energy Consumption of Low-Bandwidth IoT Applications for WiFi Devices[END_REF]:

E device static = P device static × B CT B device × Load device × T (10)
E device dynamic = E device byte ×N bBytes+E device pkt ×N bP kts [START_REF] Iec | Distribution automation using distribution line carrier systems[END_REF] where P device static is the power consumption of a core network device, B CT is the bandwidth used by a concentrator, B device is the bandwidth of a core network device, Load device is the average load of a core network device, E device byte and E device pkt are the energy consumed to transfer a byte and a packet, respectively, N bBytes and N bP kts are the number of bytes and packets to transfer, respectively.

E. Servers

Data storage and availability to the users has an energy cost through the usage of servers, whose daily energy consumption represents:

E servers = N servers × N replicas × P U E × P server × T day (12) N servers = max(N writing servers , N requests servers ) (13)
N writing servers = Bandwidth Disk W ritingSpeed (14)
N requests servers =

W eb requests × P eakF actor server requests (15)

where N servers is the number of servers, N replicas is the number of replicas of the servers, P server is the power consumption of a server, N writing servers is the number of servers required to store collected data, N requests servers is the number of servers required to answer the web requests on the website, Bandwidth is the bandwidth at which collected data arrive to the servers, Disk W ritingSpeed is the writing speed of a hard drive disk, W eb requests is the number of web requests per seconds, P eakF actor is a factor to reflect the variability at which web requests arrive, server requests is the number of web requests per second a server can handle. We determine the minimum number of servers needed by considering the number of servers required to respond to the peak of requests from web users and the servers required to store all the data from the meters.

V. EVALUATION

We evaluate the daily consumption of the smart metering infrastructure using the consumption models detailed in the previous section. These models require coherent instantiation of their variables. We first detail the parameter values taken from literature, and then presents the obtained results. These parameters can be modified freely on the web interface we developed for this work 1 . A. Parameters estimation 1) Smart meters: The power consumption of a smart meter depends on its current state. A smart meter idle power is evaluated at around 0.2W [15], [16]. Its active power, when actively communicating through PLC, is highly variable and depends on the impedance of the line to which it is connected. This active power is evaluated around 1W [15], [16]. To assess the consumption of a smart meter, we analyzed the most important electronic components of a French Linky meter, as shown in Figure 2. Concerning the proportion of G1-based meters, this technology was exclusively deployed until 2017 with a total of 7 million smart meters deployed at this moment [START_REF] Enedis | Plus de 7 millions de familles équipées d'un compteur Linky, et alors ?[END_REF]. Afterward and up to the present day, only G3-based meters were deployed. Hence, around 20% of the smart meters still use the G1 PLC technology.

The active time of a smart meter during a typical day can greatly varies from one meter to another. It depends on the data the smart meter has to transmit each day to its concentrator and also on the maintenance operations of the concentrator. This active time is heavily impacted by the PLC technology in use, as G1 data rate is around 14 times lower than G3 [START_REF] Iec | Distribution automation using distribution line carrier systems[END_REF], [START_REF] Itu | Narrowband orthogonal frequency division multiplexing power line communication transceivers for G3-PLC networks[END_REF]. The active time depends also on the number of smart meters in the same cluster. The PLC technology suffers indeed severely from collision when there are multiple communications on the same channel. To determine the ratio of smart meters in large clusters, we use population statistics, assuming that 80% of the French population is located in dense urban areas [START_REF] Costemalle | Toujours plus d'habitants dans les unités urbaines[END_REF], and the concentrators are expected to be spread evenly to ensure a full coverage of the territory, inducing large clusters in urban environment and small clusters in rural environment. Consequently, among the 35 million smart meters deployed [START_REF] Enedis | Programme de R&D et d'Innovation d'Enedis[END_REF], 28 million are in large clusters, and 7 million in small clusters. The active time in small and large clusters is assumed to be in the order of minutes and in the order of hours, respectively.

2) Concentrators: Concentrators must maintain connectivity with their cluster and are consequently always in active 1 Available here: https://smart-grid-network.herokuapp.com/ state. We assume that their static consumption is around 15W. In addition to their static consumption, we also take into account the energy consumed by cellular communication. This consumption depends on the additional power consumption during communication, estimated at around 1.4W for GPRS [START_REF]AirPrime HL6528 Embedded Module[END_REF] and 2.1W for 3G [START_REF] Airprime | HL8528 and HL8529 Embedded Modules[END_REF]. It also depends on the duration of the communication. This duration depends on the data volume to transmit, assumed to be 150kB per concentrator on average, considering that concentrators aggregate on average data from 50 smart meters, each one producing 3kB of data each day. The duration also depends on the upload data rate of the cellular technologies. The GPRS data rate considered is 24kbps [START_REF] Pfitzinger | Network-Wide Measurement of GPRS Bandwidth and Latency[END_REF] with the coding scheme CS-2 and using two time slots, while we consider a data rate of 350kbps for 3G [START_REF] Anwar | Performance Analysis of 3G Communication Network[END_REF]. We expect that the transition from GPRS to 3G followed the transition from G1 to G3, with the production of more modern devices. Thus, 20% of the concentrators are expected to use GPRS while the rest uses 3G.

Finally, there are 770,000 concentrators deployed in the country [START_REF] Enedis | Programme de R&D et d'Innovation d'Enedis[END_REF].

3) Cellular network: A Base Transceiver Station (BTS) site, including an antenna and the necessary pieces of equipment such as batteries and lighting, is assumed to have an average power consumption of 1,430W [START_REF] Arnold | Power Consumption Modeling of Different Base Station Types in Heterogeneous Cellular Networks[END_REF]. We consider 3 transceivers per BTS, with 8 time slots each [START_REF] Jukka | Radio Interface System Planning for GSM/GPRS/UMTS[END_REF]. The average load of a BTS is assumed to be similar to a Node B and equal to approximately 30% [START_REF] Jukka | Radio Interface System Planning for GSM/GPRS/UMTS[END_REF], [START_REF] Chen | Precise Estimation of WCDMA Downlink Pole Capacity in Multipath Propagation Channel by Monte Carlo Method[END_REF]. A Node B has an average power consumption of 1,450W [START_REF] Arnold | Power Consumption Modeling of Different Base Station Types in Heterogeneous Cellular Networks[END_REF] and an available bandwidth of 1,361kbps [START_REF] Chen | Precise Estimation of WCDMA Downlink Pole Capacity in Multipath Propagation Channel by Monte Carlo Method[END_REF]. 4) Core network: According to [START_REF] Guegan | Estimating the End-to-End Energy Consumption of Low-Bandwidth IoT Applications for WiFi Devices[END_REF], a typical data transfer has 9 hops to reach the core network, going through 8 edge switches and 1 core router. Network devices parameters are listed in Table I.

5) Servers:

According to Enedis, the website offering users their detailed consumption has on average 540,000 visits per month, with on average 2.38 webpages viewed per visit. We assume a conservative client-server architecture where the server can handle 200 concurrent requests per second, and the peak factor in web requests is set to 10 [25]. The writing speed of the server's HDD are set to 150MBps. A storage oriented server has an estimated power consumption of 108W [START_REF] Inoue | Power consumption and processing models of servers in computation and storage based applications[END_REF] and the PUE of a small scale cloud is estimated at 1.7 [START_REF] Li | End-to-end energy models for Edge Cloud-based IoT platforms: Application to data stream analysis in IoT[END_REF]. Finally, the number of replicas is set to 3.

B. Results

Figure 3 presents the overall consumption of the smart metering infrastructure using values defined in Table I and Table II. The overall consumption of the smart metering infrastructure is estimated at 405MWh per day. If we assume a static power of 1W per meter for the previous generation of meters, it would represent 840MWh per day. In this case, the smart metering infrastructure would consume less than half of the consumption of the previous generation meters. The concentrators energy consumption represents most of the smart metering infrastructure overall consumption (68%). The additional energy consumption by the concentrators during cellular communication is negligible and accounts for only 0.002% of the concentrators consumption. This is due to their high static power consumption, and to the low amount of data to transfer, inducing a consumption of only 70J per day and per concentrator using GPRS, and 7J per day and per concentrator using 3G.

The smart meters account for 31% of the total consumption, with 96% of their consumption due to large clusters. It means that while 20% of the smart meters are in small clusters, they only account for 4% of the consumption of the smart meters. This result highlights a strong impact of PLC in large clusters representative of a dense urban environment in our model. We also observe the influence of the lower data rate of G1. Large G1 clusters account for 62% of the consumption of the smart meters while large G3 clusters only account for 18%, although there are 4 times more smart meters in large G3 clusters than in large G1 clusters. The energy consumed by the cellular network is low in comparison to the concentrators and the smart meters with, respectively, 0.21% and 0.18% of the total energy consumed due to the 3G and GPRS network.

Finally, the energy consumed by the servers and the core network is negligible, with only 0.003% of the total energy consumed.

Using our energy model, we also explored several other cases, as listed in Table III. The corresponding results are shown in Figure 4.

1) G1-only and G3-only: These cases highlight the interest of newer PLC technologies, improving data rate and drastically reducing the active time of smart meters. We see with the G1-only case that using the G1 technology exclusively would have almost doubled the energy consumed by the infrastructure. On the other hand, upgrading the remaining 20% of G1 smart meters to G3 would reduce the energy consumed by smart meters by 91MWh (72% less energy consumed by the smart meters, 22% of the total).

2) GPRS-only and 3G-only: These cases investigate the interest of switching from the old GPRS technology to 3G. Sticking to GPRS would have increased the energy consumed by 2.7MWh according to our model. On the other hand, upgrading the current 20% concentrators from GPRS to 3G would reduce the energy consumed by 0.7MWh. This represents a reduction of 58% in the energy consumption by the cellular network, but this translates into a reduction of only 0.2% in the overall energy consumption. Hence, although the 3G technology consumes indubitably less energy than GPRS for this given traffic, in comparison to the considerable energy consumed by smart meters and concentrators, the interest of switching from GPRS to 3G is low from an energy consumption point of view, especially when considering the gray energy required to replace the GPRS smart meters. Furthermore, the financial cost may be higher.

3) Energy reduction: Using the end-to-end energy model, one can also explore the possibility of using modified smart meters able to directly communicate through 3G. Such smart meters remove the need for concentrators, as they could directly send their data to the central utility using cellular networks. The increase in power consumption while using the 3G module is the same as with a concentrator, 2.1W. A smart meter now consumes 2.3W while active (previously 1W), but with a datarate of 350kbps. At such a datarate less than 1s is necessary to send a smart meter daily payload. The energy consumed by the cellular network, the core and the servers is the same as the total data volume to transfer remains similar. This case results in a total daily energy consumption of 169MWh, 2.4 times less than the current energy consumption. Yet, the mobile monthly subscription with this option may be prohibitive if fully supported by the grid operator.

Results show the large energy consumption of the concentrators and the smart meters, accounting for respectively 68% and 31% of the energy consumed, making the consumption from the cellular network, the core network and the servers negligible in comparison. They also show that using newer technologies effectively reduces the energy consumed. It is especially visible with the transition from G1 to G3 smart meters, saving 74% of the energy consumed of the smart meters in comparison to the case where the transition did not happen.

C. Discussion

Several benefits are expected from a smart metering infrastructure. From the user point of view, the main benefit consists in the possibility to have energy consumption data at a smaller temporal resolution. This facilitates changes in energy usage behavior. However, the metering granularity and the important delay before the availability of the data online -every day -may degrade this benefit to a nonnegligible extent. From the grid operator point of view, the main benefit is to improve its network planning, control and maintenance, while the energy provider may provide more dynamic energy price schemes to its clients. Nevertheless, the still coarse metering granularity may not be sufficient to get the full benefit of the smart metering infrastructure. The communication technologies are largely limiting the capabilities of the infrastructure. First, PLC is not adapted to transfer data at a high granularity. Secondly, higher granularity would increase the data volume transmitted and the GPRS network could become another bottleneck due to its low data rate.

The smart metering infrastructure has been imagined, developed, and deployed with the idea of a highly communicating smart grid. The role of smart meters, in the long term, would consist not only to collect consumption data but to introduce dynamic, and possibly real-time, energy management mechanisms, either through indirect or direct remote control of smart devices inside each equipped home.

Such features would drastically increase the smart meters active time, and therefore their energy consumption. To explore the energy impact of such new features on the infrastructure, we defined a case (SG-services on Figure 4) where smart meters are active for 30 minutes and 12 hours in small and large clusters respectively. In this case, the energy consumed by smart meters is multiplied by 4, and the total consumption by 1.9, slightly exceeding the G1-only case. Yet, this coarse-grain analysis do not guarantee that such a data traffic, corresponding to about ten data gathering per day, would be feasible in all large clusters because of a high potential of collisions.

Based on the observations presented in this study, some solutions can be considered to reduce the energy consumption of the smart metering infrastructure. The main insight of this work is that concentrators and smart meters account for more than 99% of the infrastructure consumption. Thus, a first option consists in extensively using a low power state, or sleep mode, for concentrators and smart meters. Such sleeping mechanisms have been studied for PLC networks [START_REF] Yoon | Performance analysis of power saving strategies for power line communications[END_REF] and show that the proposed sleeping strategy performance highly depends on the delay and availability of the nodes. In this sense, the current PLC routing policy requires nodes to be available at all times, which may thus need to be adapted. In addition, maintaining the connectivity between concentrators and smart meters, combined with channel congestion, induces an increase in active time, especially in dense urban areas. In that sense, another option consists in shifting from PLC communication to cellular or long range wireless communication in areas with sufficient coverage. This solution reduces the active time of concentrators and smart meters as it is less subject to channel congestion and provides a significant increase in datarate in comparison to PLC communication.

In addition, long range wireless communication would allow to manage more smart meters per concentrators, at least in some areas, or even allow direct connectivity between smart meters and a central utility. Thus, a significant amount of concentrators could be removed from the smart metering infrastructure, inducing a significant reduction in energy consumption. Recent studies have shown the suitability of such long range wireless communication in a smart grid context, as presented in [START_REF] Gallardo | Lora iot-based architecture for advanced metering infrastructure in residential smart grid[END_REF]. However, shifting from one communication technology to another implies a replacement of the current infrastructure, thus inducing gray energy consumption, in addition to installation costs for the grid operator. Further analyses are necessary to evaluate the short and long term benefits of this potential communication technology shift.

VI. CONCLUSION

In this work, we evaluate the end-to-end energy consumption of a smart metering infrastructure at a nation-wide scale. We propose an energy consumption model for each part of the infrastructure. The instantiation of the models parameters is a tough task as data in this domain is scarce. For this reason, we evaluate the daily energy consumption of the infrastructure using the most coherent values found in literature. We also propose an interface that allows the reader to explore our models with its own set of parameters.

For a scenario where data from smart meters are collected once a day, the results show that concentrators and smart meters account for respectively 68% and 31% of the energy consumed, making cellular network, core network and servers' consumption negligible in comparison. Our study also highlight the effectiveness of the newer generation (G3-PLC) in reducing the energy consumption. Indeed, G3-PLC saves 74% of the energy consumed by the G1-PLC smart meters employed in the first phase of smart grid deployment.

These technologies allow the French smart metering program to meet the requirements of an Advanced Metering Infrastructure service, i.e. satisfying 99.99% of the reading rates. However, the granularity offered for measurement purposes, and potentially for control purposes, remains coarse, which may restrain the benefits for energy providers, grid operators and consumers. Moreover, such an improvement in the granularity may be limited by the current infrastructure. These issues may incite smart grid actors to switch to different communication technologies and rethink the whole smart metering infrastructure.

Fig. 1 .

 1 Fig. 1. Communication path from a smart meter cluster to the central utility in the French case.

Fig. 2 .

 2 Fig. 2. Simplified Electronic Architecture of a Linky Smart meter.

Fig. 3 .

 3 Fig. 3. Daily energy consumption of the smart metering infrastructure.

Fig. 4 .

 4 Fig. 4. Exploration of various consumption cases of the smart metering infrastructure for one day.

TABLE II

 II

	MODELS PARAMETERS
	Parameter	Value	Source
	N small SM , N large SM P idle SM , P active SM T active small , T active large	7M, 28M 0.2W, 1W 600s, 3600s	[18] [15], [16] see Section V-A1
	R G1 , R GP RS	0.2, 0.2	[17]
	F G1	14	[11], [12]
	N CT	770,000	[10]
	P static CT	15W	see Section V-A2
	P GP RS	1.4W	[19]
	P 3G	2.1W	[20]
	DV CT	150kB	see Section V-A2
	DR GP RS	24kbps	[21]
	DR 3G	350kbps	[28]
	P BT S	1,430W	[13]
	N timeslots CT N timeslots BT S	2 24	see Section V-A2 [23]
	Load BT S , Load N B	0.3	[24], [23]
	DR N B	1,361kbps	[24]
	P N B	1,450W	[13]
	N replicas	3	see Section V-A5
	P U E	1.7	[27]
	Pserver	108W	[26]
	Disk W ritingSpeed	150MBps	see Section V-A5
	W ebRequests	1.5M/month see Section V-A5
	P eakF actor	10	see Section V-A5
	CP Urequests	200/s	see Section V-A5

ACKNOWLEDGMENT

This project has received financial support from the CNRS Momentum project RI/RE through the MITI interdisciplinary programs


Guillaume Sallé

Pierre-Henri Conze

Nicolas Boussion

Julien Bert

Dimitris Visvikis

Vincent Jaouen

Synthetic tumor insertion using one-shot generative learning for cross-modal image segmentation

Unpaired cross-modal translation with cyclic loss is being increasingly used for a large variety of medical imaging applications such as e.g. segmentation. However, finer-scale details like tumors may be lost during translation, which is a critical limitation in oncological imaging. In this paper, we propose to address the problem of vanishing tumors for cross-modal segmentation. First, we propose a new method to insert realistic tumors in 3-D images using a deep generative model trained on a single 2-D image. Second, we leverage the proposed model using a new unpaired-then-paired two-stage I2I architecture to better penalize the suppression of tumors in cross-modal segmentation.

In our experiments, we validate our model on the ongoing MICCAI crossMoDa tumor segmentation challenge, where we demonstrate superior performance over CycleGAN-based models.

I. INTRODUCTION II. INTRODUCTION

T HE domain shift (DS) refers to differences in data properties during machine learning (ML) model training and deployment [START_REF] De Castro | Causality matters in medical imaging[END_REF]. DS is a major challenge for medical imaging applications as it reduces the generalization power of ML models. For this reason, numerous efforts have recently been devoted to the mitigation of DS in medical imaging through advanced processing techniques, referred to as domain adaptation (DA) methods [START_REF] Hognon | Standardization of Multicentric Image Datasets with Generative Adversarial Networks[END_REF]. A popular application of DA in medical imaging is cross-modal image segmentation, whereby a model is trained on a different domain (i.e. different modality, vendor, etc.) than the one seen during deployment [START_REF] Huo | SynSeg-Net: Synthetic Segmentation Without Target Modality Ground Truth[END_REF].

Deep image-to-image (I2I) translation architectures such as CycleGAN-based models [START_REF] Zhu | Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks[END_REF] allow to learn complex relationships between source and target image distributions in an unpaired setting, i.e. without requiring co-registration between target and source images. Due to this appealing property, unpaired approaches are being increasingly used for DA in medical imaging [START_REF] Huo | SynSeg-Net: Synthetic Segmentation Without Target Modality Ground Truth[END_REF], [START_REF] Hoffman | CyCADA: Cycle-Consistent Adversarial Domain Adaptation[END_REF], [START_REF] Chartsias | Adversarial Image Synthesis for Unpaired Multimodal Cardiac Data[END_REF]. CycleGAN-based models have however raised concerns due to their tendency to learn a global intensity mapping that may not preserve image features at smaller scales [START_REF] Jiang | Tumor-Aware, Adversarial Domain Adaptation from CT to MRI for Lung Cancer Segmentation[END_REF], [START_REF] Cohen | How to Cure Cancer (in images) with Unpaired Image Translation[END_REF]. This drawback is a critically limiting factor in cancer imaging, where a major risk is to hinder the visibility of the pathology (i.e. tumors) in the target domain and thus to reduce the potential interest of these approaches in oncology.

In this paper, we propose to address the problem of vanishing tumors in unpaired image-to-image translation for crossmodal segmentation by combining two contributions. First, we develop a new method to insert plausible tumors in 3D images using a modified single-image SinGAN model [START_REF] Shaham | SinGAN: Learning a Generative Model from a Single Natural Image[END_REF]. The SinGAN model has seldomly been used in medical imaging and to our knowledge never in the context of 3D imaging and fake pathology synthesis. Second, we exploit the proposed model in a new two-stage I2I network with pixel loss to better penalize the suppression of tumors in the context of unpaired cross-modal segmentation.

In our experiments, we validate our model by showing improved performances over CycleGAN-based cross-modal segmentation models. We propose to learn the transfer of the tumor content of I ∈ A to the cycleGAN-generated image J = ϕ 0 AB (I) with realistic harmonization with respect to the style of domain B using a modified SinGAN model [START_REF] Shaham | SinGAN: Learning a Generative Model from a Single Natural Image[END_REF]. SinGAN is a multistage generative approach where patches from N given image scales are used to train a series of generators within a single image. A global image generator is decomposed into N scalespecific generators, each of them responsible for learning the distribution D l at a given scale l in a coarse-to-fine fashion. We harmonize a naively pasted tumor on a given slice by selecting one generator scale n ≤ N and using every generator linked to

(ω) = ϕ AB (I(ω)) = (G N • G N -1 • ... • G n )(I k (ω)).
To this end, the SinGAN original implementation was adapted to accept varying masks and source images during inference with pre-trained generators. We leverage our SinGAN-based fake tumor insertion method in the context of unpaired cross-modal translation, where the target modality B does not possess segmentation labels. The proposed workflow is shown in Fig. 1. We first train a segmentation network S A on modality A with ground truth labels and we seek to obtain the best achievable mapping ϕ BA (B) between B and A to reduce the domain shift at inference on B. To this end, we learn an approximate mapping ϕ 0 AB with CycleGAN (with vanishing tumors). We then enforce tumor presence in B by naively pasting the tumors of A to ϕ 0 AB (A) that we then harmonize using the proposed SinGANbased tumor harmonization model B . Finally, we learn a more efficient tumor-preserving mapping ϕ B'A (B ) using a conditional GAN with pixel-loss [START_REF] Isola | Image-to-image translation with conditional adversarial networks[END_REF] between B and A. The unlabelled images to segment from B are then passed through ϕ B'A (B) before segmentation by the S A network.

IV. EXPERIMENTS AND RESULTS

We focused our experiments on the ongoing crossMoDA cross-modal MR segmentation challenge between labelled contrast T1-weighted imaging and unlabelled high resolution T2weighted images [START_REF]CrossMoDA challenge[END_REF]. A nnU-Net segmentation model [START_REF] Isensee | nnU-Net: a self-configuring method for deep learningbased biomedical image segmentation[END_REF] was used to learn S A . 105 unpaired patients were used for training in each modality and 32 other were used for online validation. We compared our model to CycleGAN-based crossmodal translation followed by segmentation by S A without fake tumor insertion, using Dice similarity (DSC) and average symmetric surface distance (ASSD). Quantitative results using these indices are summarised in Table 1.

Fig. 2d shows an example fake tumor insertion on the crossMoDa dataset. A naively pasted tumor (Fig. 2c) is added instead of the "washed out" tumor obtained from T1-to-T2 CycleGAN transfer (Fig. 2b) and harmonized with the proposed modified SinGAN approach (Fig. 2d). Figure 3 shows example two segmentation results. Table 1 shows quantitative results achieved in the validation set, with a clear impact of the proposed SinGAN-based two-stage approach. In 9/32 patients we achieved a DSC greater than 0.8 compared to only 3 for our baseline comparison. V. CONCLUSION In this paper, we have proposed a new method for realistic fake tumor insertion in 3D medical images using a modified SinGAN generative model requiring a single 2D image at the training stage. The method was validated in the challenging context of cross modality image segmentation without target modality ground truth where we demonstrated superior tumor preservation over cycleGAN-based models. Future experiments will focus on validating the approach in other anatomical locations and using combined PET-MR data. Optimization, limitations and potential failure cases will also be further studied.

Fig. 1 :

 1 Fig. 1: Training workflow for cross domain segmentation

Fig. 2 :

 2 Fig. 2: Fake tumor insertion in T2-weighted MR imaging. (a) initial T1-w image (b) vanishing tumor after CycleGAN translation (c) naively pasted tumor on T2-w (d) proposed harmonized result using our modified SinGAN architecture.

Fig. 3 :

 3 Fig. 3: Test images for vestibular Schwanomma segmentation. (a) example T2 MR image (b,c) results on two subjects. The baseline result is shown in red, ours in green.

Table 1 :

 1 Quantitative scores for the CrossMoDA validation set.

	DSC (25%/50%/75%)	ASSD (25%/50%/75%)

Baseline 0.46 ± 0.32 (0.13, 0.61, 0.70) 8.4 ± 14.5 (4.8, 1.6, 1.0) Ours 0.58 ± 0.31 (0.37, 0.68, 0.81) 7.5 ± 17.9 (2.1, 1.0, 0.6)

The authors are with LaTIM, UMR Inserm 1101, Université de Bretagne Occidentale, IMT Atlantique, Brest, France. J. Bert and N. Boussion are also with CHRU Brest University Hospital, Brest, France.Corresponding address: guillaume.salle@univ-brest.fr