Oana Iosifescu
email: iosifescu@math.univ-montp2.fr

Pongpol Juntharee
email: juntharee@lmgc.univ-montp2.fr

Christian Licht
email: licht@lmgc.univ-montp2.fr

Gérard Michaille

A mathematical model for a pseudo-plastic welding joint

An elementary situation in welding involves the perfect assembly of two adherents and a strong adhesive occupying a thin layer. The bulk energy density of the hyperelastic adherents grows superlinearly while the one of the pseudo-plastic adhesive grows linearly with a stiffness of the order of its thickness ε. We propose a simplified but accurate model by studying the asymptotic behavior, when ε goes to zero, through variational convergence methods: at the limit, the intermediate layer is replaced by a pseudoplastic interface which allows cracks to appear.

Introduction

Motivated by the mathematical modeling of a problem of welding, we revisit previous studies ( [START_REF] Acerbi | Thin inclusions in linear elasticity: a variational approach[END_REF], [START_REF] Bessoud | Multi-material with strong interface: Variational modelings[END_REF]) devoted to the asymptotic behavior of a structure made of two adherents connected by a thin and strong adhesive layer. In [START_REF] Bessoud | Multi-material with strong interface: Variational modelings[END_REF] the adherents and the adhesive were modeled as hyperelastic by bulk energy densities with the same growth exponent p laying in (1, +∞), the stiffness of the adhesive being of the order of the inverse of its thickness. Here, our first attempt to account for some fracture phenomena in soldered joint is to model the adhesive as pseudo-plastic, that is to say its behavior is described by a bulk energy density with linear growth. Hence, from the mathematical point of view, two difficulties appear: the growths of the the bulk energy in the adhesives and the adherent are different and the linear growth in the adherent will imply to work in spaces of displacement fields with free discontinuities

The paper is organized as follows. In Section 2, we describe a model problem with a simplified geometry directly connected to the study [START_REF] Bessoud | Multi-material with strong interface: Variational modelings[END_REF] where we assume that the bulk energy density of the adherents is quasiconvex and the one of the adhesive is convex. In Section 3, a variational convergence result, when the thickness of the adhesive layer goes to zero, justifies our proposal of simplified but accurate enough model. The adhesive layer is replaced by a material pseudoplastic surface. The case when f is not quasiconvex and g is not convex is considered in Section 4. In Section 5, we use the previous results to model a more realistic situation of welding. Eventually, in the spirit of [START_REF] Bessoud | Variational convergence of energy functionals for elastic materials with ε-thin strong inclusions growing as p(ε), p(ε) → 1 +[END_REF], [START_REF] Temam | Problèmes Mathématiques en Plasticité[END_REF], we consider a variational regularization of the limit functional involved by our model in Section 6.

Description of the model

We make no difference between R 3 and the three dimensional euclidean physical space whose orthogonal basis is denoted by (e 1 , e 2 , e 3 ), Greek coordinate indexes will run in {1, 2} and Latin ones in {1, 2, 3}. For all ζ = (ζ 1 , ζ 2 , ζ 3 ) of R 3 , ζ stands for (ζ 1 , ζ 2 ). Let S be a domain of R 2 with a Lipschitz-continuous boundary ∂S and r a positive number. The cylindrical domain Ω := S×(-r, r) is the reference configuration of a structure made of two adherents and an adhesive which respectively occupies Ω ± ε := {x ∈ Ω : ±x 3 > ε/2} and B ε := {x ∈ Ω :

|x 3 | < ε/2}. We set Ω ε := Ω + ε ∪ Ω - ε .
The structure is clamped on a part Γ 0 of the boundary Γ of Ω with a positive H 2 -measure and we assume that there exists ε 0 > 0 such that dist( Γ0 , Bε0 ) > 0. The structure is subjected to body forces of density Φ and to surface forces of density ϕ on the complementary part Γ ϕ of Γ 0 . We assume that the supports of ϕ and Φ lay outside of Bε0 . Obviously, one can consider other type of boundary conditions (e.g. a combination of some components of the stress vector and of the displacement). At last, adhesive and adherents are assumed to be perfectly stuck together. In section 5 we will consider a more realistic structure (see figure 1 for the two geometrical strucures).

The stiffness of the material occupying the thin layer B ε is assumed to be of order 1/ε so that the strain in B ε is expected to be small and we will use the framework of small perturbations to model the behavior of the adhesive. To account for possible fracture phenomena inside B ε , we consider the adhesive as pseudo-plastic. Hence the behavior of the adhesive is described by a bulk energy density like 1/ε g(e(u)) where g is a convex function with linear growth of the linearized strain e(u) i.e., the symmetric part (∇u) s of the gradient displacement ∇u. By contrast, the deformations in the adherents may be large and they are modeled as hyperelastic with a continuous quasiconvex bulk energy density f , function of the gradient displacement. More precisely, we assume that there exists p > 1, and two positive constants α, β such that f, g : M 3×3 → R;

α|ξ| p ≤ f (ξ) ≤ β(1 + |ξ| p ) for all ξ ∈ M 3×3 ; (1) α|ξ| ≤ g(ξ) ≤ β(1 + |ξ|) for all ξ ∈ M 3×3 s .
(

) 2
Here and in the sequel M n×n and M n×n s stand for the set of n × n matrices and n × n symmetric matrices with real entries respectively. It is well known that f and g satisfy the following locally Lipschitz conditions: there exists a positive constant L such that

|f (ξ) -f (ξ ′ )| ≤ L|ξ -ξ ′ | 1 + |ξ| p-1 + |ξ ′ | p-1 for all ξ, ξ ′ ∈ M 3×3 ; (3) |g(ξ) -g(ξ ′ )| ≤ L|ξ -ξ ′ | for all ξ, ξ ′ ∈ M 3×3 s . (4)
Thus, if F ε (u) := Ωε f (∇u) dx + 1 ε Bε g(e(u) dx and L(u) := Ω Φ.u dx + Γϕ ϕ.u dH 2 respectively denote the total stored energy and the work of the external loading, determining the equilibrium configurations leads to the problem

inf Ωε f (∇u) dx + 1 ε Bε g(e(u)) dx -L(u) : u ∈ A ε with A ε := u ∈ LD(Ω, R 3 ) : u ⌊Ωε ∈ W 1,p Γ0 (Ω ε , R 3 ) ; W 1,p Γ0 (Ω ε , R 3 ) := u ∈ W 1,p (Ω ε , R 3 ) : u = 0 on Γ 0 ; LD(Ω, R 3 ) := u ∈ L 1 (Ω, R 3 ) : e(u) ∈ L 1 (Ω, M 3×3 s .
We aim to propose a simplified but accurate model where qualitative and quantitative analysis are able to be done in an easier way than with the starting problem. For this, we will consider ε as a parameter going to zero and determine the asymptotic behavior of (approximate) solutions of the previous minimization problem by identifying the Γ-limit of F ε extended into the fixed space L 1 (Ω, R 3 ). More precisely, we still denote by F ε its extension outside A ε given by:

F ε : L 1 (Ω, R 3 ) → R ∪ {+∞}, F ε (u) :=    Ωε f (∇u) dx + 1 ε Bε g(e(u)) dx if u ∈ A ε +∞ otherwise.
Clearly the previous minimization problem is equivalent to the following

(P ε ) inf F ε (u) -L(u) : u ∈ L 1 (Ω, R 3 )}.
We will use the classical spaces

W 1,p Γ0 (Ω, R 3 ) := u ∈ W 1,p (Ω, R 3 ) : u = 0 on Γ 0 ; BD(Ω, R 3 ) := v ∈ L 1 (Ω, R 3 ) : e(v) ∈ M(Ω, M 3×3 s ; BD(S, R 2 ) := v ∈ L 1 (S, R 2 ) : e(v) ∈ M(S, M 2×2
s and the set of "horizontal rigid motions" on S, i.e.

R H := v ∈ BD(S, R 2 ) : e αβ (v) = 0 = v : v(x) = (a 1 -bx 2 , a 2 + bx 1 ), (a 1 , a 2 ) ∈ R 2 , b ∈ R .
For reasons clarified in Lemma 2 below, we define the limit admissible set A 0 by

A 0 := u ∈ W 1,p Γ0 (Ω, R 3 ) : γ S (û) ∈ BD(S, R 2
) and its subspace A 1 0 made of smooth elements

A 1 0 := u ∈ A 0 : γ S (u) ∈ C 1 ( S, R 3 ) .
For simplicity of notation γ S will denote indifferently the trace operator from

W 1,p (Ω, R i ) into W 1-1 p ,p (S, R i ) for i ∈ {1, 2, 3}. For all ξ ∈ M 3×3 s , define ξ ∈ M 2×2
s by ( ξ) αβ = ξ αβ and consider the function

g 0 : M 2×2 s → R defined by g 0 (ζ) := min g(ξ) : ξ ∈ M 3×3 s , ξ = ζ .
It is easily seen that g 0 is a convex function on M 2×2 s and it will be sometimes convenient to express g 0 as stated in the next lemma:

Lemma 1. For all 3 × 2-matrix ξ, inf λ∈R 3 g((ξ|λ) s ) = g 0 ((ξ αβ ) s ).
Proof. The conclusion is a straightforward consequence of the calculation

min λ∈R 3 g((ξ|λ) s ) = min λ∈R 3 g   ξ 11 ξ12+ξ21 2 λ1+ξ31 2 ξ12+ξ21 2 ξ 22 λ2+ξ32 2 λ1+ξ31 2 λ2+ξ32 2 λ 3   = min λ∈R 3 g   ξ 11 ξ12+ξ21 2 λ 1 ξ12+ξ21 2 ξ 22 λ 2 λ 1 λ 2 λ 3  
and the definition of g 0 .

In Section 3, we establish the Γ-convergence of the sequence (F ε ) ε>0 to the functional defined by:

F 0 : L 1 (Ω, R 3 ) → R ∪ {+∞} F 0 (u) :=    Ω f (∇u) dx + S g 0 (e(γ s (û)) if u ∈ A 0 +∞ otherwise when L 1 (Ω, R 3
) is equipped with its strong topology. For the defintion of the scalar measure g 0 (e(v)), v ∈ BD(S, R 2 ), we refer the reader to [START_REF] Demengel | Convex functions of measures and applications[END_REF], [START_REF] Temam | Problèmes Mathématiques en Plasticité[END_REF], [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]. We recall that the integral over S of the measure g 0 (e(v)) is given by

S g 0 (e(v)) = S g 0 (e a (v)) dx + S g ∞ 0 e s (v) |e s (v)| |e s (v)|
where e(v) = e a (v) dx + e s (v) is the Lebesgue decomposition of e(v), |e s (v)| denotes the total variation of the singular measure e s (v), es(v) |es(v)| its Radon-Nikodym derivative, and ξ → g ∞ 0 (ξ) := lim t→+∞ g 0 (tξ)/t is the recession function of g 0 . In [START_REF] Ambrosio | Fine Properties of Functions in BD[END_REF], Ambrosio, Coscia and Dal Maso proved that the singular measure e s (v) has the following structure: there exists a rectifiable set S v ⊂ S with normal ν v and traces v ± on both sides of S v such that

e s (v) = 1 2 (v + -v -) ⊗ ν v + ν v ⊗ (v + -v -) H 1 ⌊S v + Cv,
with Cv singular with respect to the Lebesgue measure and vanishing on Borel sets of σ-finite H 1 -measure. We will also denote by [v] ⊗ s ν v the symetrical tensor product (v

+ -v -) ⊗ ν v + ν v ⊗ (v + -v -).
The set S v will represent the macroscopic cracks whereas the support of C v deals with the diffuse defects or fractal cracks in S towards the layer shrinks.

We start by establishing a compactness result which justifies the introduction of the limit set A 0 of admissible functions. As usual the arrows → and ⇀ will denote strong and weak convergences respectively. Lemma 2 (Compactness lemma). Let (u ε ) ε>0 be a sequence in L 1 (Ω, R 3 ) such that sup ε>0 F ε (u ε ) < +∞. Then, there exists u ∈ L 1 (Ω, R 3 ) and a subsequence not relabelled such that i)

u ε ⇀ u in BD(Ω, R 3 ) and u ∈ W 1,p Γ0 (Ω, R 3 ); ii) u ε ⇀ u in W 1,p Γ0 (Ω η , R 3 ) for every η > 0; iii) γ S (û) ∈ BD(S, R 2 ); iv) ∃r ε ∈ R H such that 1 ε ε 2 -ε 2 ûε dx 3 + r ε ⇀ γ S (û) in BD(S, R 2 ).
Proof. From now on, we do not relabel the various subsequences obtained in the proof, and C will denote a positive constant which may vary from line to line. We divide the proof into two steps.

Step 1. We establish (i) and (ii). According to the coerciveness condition [START_REF] Acerbi | Thin inclusions in linear elasticity: a variational approach[END_REF] and because u ε = 0 on Γ 0 , (u ε ) ε>0 is clearly bounded in LD(Ω, R 3 ) so that there exist u ∈ BD(Ω, R 3 ) and a subsequence satisfying u ε ⇀ u in BD(Ω, R 3 ) and u ε → u in L 1 (Ω, R 3 ). The weak convergence of u ε to u in W 1,p Γ0 (Ω η , R 3 ) for every η > 0 is obvious. We are going to prove that u ∈ W 1,p Γ0 (Ω, R 3 ). We extend every function w ∈ W 1,p Γ0 (Ω ε , R 3 ) by reflexion on S ×(±r, ±2r∓ ε 2 ) so that the extended function, denoted by w, belongs to W 1,p (S × (± ε 2 , ±2r ∓ ε 2 ), R 3 ) and satisfies

S×(± ε 2 ,±2r∓ ε 2 )
|∇ w| p dx ≤ 2

Ω ± ε |∇w| p dx.
Set Ω ± := Ω ∩ [±x 3 > 0] and, for every function

w ∈ W 1,p Γ0 (Ω ε , R 3 ), define its ε-translate T ε w in W 1,p (Ω \ S, R 3 ) by T ε w(x, x 3 ) = w(x, x 3 + ε/2), if x ∈ Ω + w(x, x 3 -ε/2), if x ∈ Ω -. Because sup ε>0 Ω\S |∇T ε u ε | p dx ≤ sup ε>0 S×( ε 2 ,2r-ε 2 )∪S×(-2r+ ε 2 ,-ε 2 ) |∇ũ ε | p dx ≤ 2 sup ε>0 Ωε |∇u ε | p dx < +∞,
Poincaré's inequality implies that there exist z ∈ W 1,p (Ω \ S, R 3 ) and a subsequence of (u ε ) ε>0 such that

T ε u ε ⇀ z in W 1,p (Ω \ S, R 3 ) and u ε → z in L p (Ω, R 3 ). Actually u = z (so that u ∈ W 1,p (Ω \ S, R 3 )) since for all ψ ∈ D(Ω \ S, R m ), Ω\S u.ψ dx = lim ε→0 Ω\S u ε .ψ(x, x 3 - ε 2 ) dx = lim ε→0 Ω\S T ε u ε .ψ dx = Ω\S z.ψ dx.
For all w ∈ W 1,p (Ω \ S, R 3 ), we will denote the traces on S of w considered as a function of W 1,p (Ω ± , R 3 ) by w ± and its jump accross S by [w] := w + -w -. Take θ ∈ C ∞ c (S)), Green's formula yields

2 Bε e α3 (u ε ) θ dx = S θ[T ε u ε α ] dx + I ε (5)
where

I ε := - Bε ∂u ε 3 ∂x α θ dx.
The left hand side term of ( 5) tends to 0 by coercivity condition (2), and from

u ε 3 (x, ±|x 3 |) = u ε 3 (x, ±ε/2) + ±|x3| ±ε/2 ∂u ε 3 ∂x 3 (x, t) dt = (T ε u ε 3 ) ± (x) + ±|x3| ±ε/2 ∂u ε 3 ∂x 3 (x, t) dt,
we deduce

Bε |u ε 3 (x, x 3 )| dx ≤ ε S |(T ε u ε 3 ) + (x)| + |(T ε u ε 3 ) -(x)| dx + ε Bε | ∂u ε 3 ∂x 3 | dx ≤ Cε(1 + ε) (6)
so that

|I ε | = Bε u ε 3 ∂θ ∂x α dx ≤ Cε(1 + ε),
and the right hand side term of ( 5) tends to S θ[u α ] dx. Going to the limit on ε in ( 5) yields [u α ] = 0 a.e. on S. Similarly, by letting ε → 0 in

Bε ∂u ε 3 ∂x 3 θ dx = S θ[T ε u ε 3 ] dx,
and by using coercivity condition (2), we obtain [u 3 ] = 0 a.e. on S.

Step 2. We establish (iii) and (iv). Coercivity condition (2) implies

sup ε>0 1 ε Bε | e(u ε )dx| < +∞ so that there exists a subsequence of v ε := 1 ε ε 2 -ε 2 ûε dx 3 , r ε ∈ R H and v ∈ BD(S, R 2 ) such that v ε + r ε ⇀ v in BD(S, R 2 ). It remains to prove that v = γ S (û). For arbitrary θ ∈ C ∞ c (S) and x 3 ∈ (-ε 2 , ε 2 ), one has u ε α (x, ±|x 3 |)θ(x) = (T ε u ε α ) ± (x))θ(x) + ±|x3| ±ε/2 ∂u ε α ∂x 3 (x, t)θ(x) dt so that 1 ε ε 2 -ε 2 S u ε α (x, x 3 )θ(x) dx = 1 2 S (T ε u ε α ) + (x) + (T ε u ε α ) + (x))θ(x) dx + J ε (7)
where

J ε := 1 ε 0 -ε 2 S -|x3| -ε/2 ∂u ε α ∂x 3 (x, t)θ(x) dt dx+ 1 ε ε 2 0 S |x3| ε/2 ∂u ε α ∂x 3 (x, t)θ(x) dt dx.
We claim that lim ε→0 J ε = 0. Indeed

S x3 ε/2 ∂u ε α ∂x 3 θ(x) dtdx = S x3 ε/2 2e 3α (u ε ) - ∂u ε 3 ∂x α θ dtdx = S x3 ε/2 2e 3α (u ε )θ + u ε 3 ∂θ ∂x α dtdx, thus |J ε | ≤ C Bε |e 3α (u ε )| + |u ε 3 )| dx,
and the claim follows from ( 6) and coercivity condition [START_REF] Hafsa | Interchange of infimum and integral[END_REF]. Letting ε → 0 in [START_REF] Bessoud | Variational convergence of energy functionals for elastic materials with ε-thin strong inclusions growing as p(ε), p(ε) → 1 +[END_REF], we obtain

S v α θ dx = S γ S (u α )θ dx thus v α = γ S (u α ) a.e.
in S since θ is arbitrary.

The main convergence result

The main result of this section is the following theorem.

Theorem 1. The sequence (F ε ) ε>0 Γ-converges to the functional F 0 .

The proof consists in establishing Proposition 1 and Proposition 2 below, corresponding to the lower bound and the upper bound in the definition of the Γ τs -convergence.

Proposition 1 (Lower bound). For all u and all sequence (u ε ) ε>0 in L 1 (Ω, R 3 ) such that u ε τs → u, the following inequality holds:

F 0 (u) ≤ lim inf ε→0 F ε (u ε ). ( 8
)
Proof. Clearly one may assume that lim inf ε→0 F ε (u ε ) < +∞ so that from Lemma 2, u belongs to A 0 , and

e 1 ε ε 2 -ε 2 ûε dx 3 = 1 ε ε 2 -ε 2 e(u ε ) dx 3 ⇀ e(γ S (û)) in M(S, M 2×2 s ). Thus from Jensen's inequality lim inf ε→0 F ε (u ε ) = lim inf ε→0 Ωε f (∇u ε )dx + 1 ε Bε g(e(u ε ))dx ≥ lim inf ε→0 Ωε f (∇u ε )dx + lim inf ε→0 1 ε Bε g(e(u ε ))dx ≥ lim inf ε→0 Ωε f (∇u ε )dx + lim inf ε→0 1 ε Bε g 0 ( e(u ε ))dx ≥ Ω f (∇u)dx + S g 0 (e(γ S (û)).
In the last inequality, we used the weak lower semicontinuity of the second integral functional with respect to the weak convergence of measures. For the convergence of the first integral, we proceeded as follows: take η > ε, write

Ωε f (∇u ε )dx ≥ Ωη f (∇u ε )dx, and apply the lower semicontinuity of the integral functional u → Ωη f (∇u)dx for the weak convergence in W 1,p (Ω η , R 3 ). Then let η → 0.

For proving the upper bound, we need to establish the following relaxation result.

Lemma 3 (Relaxation). The functional F 0 is the l.s.c. regularization for the strong topology of the functional defined on L 1 (Ω, R 3 ) by

F0 (u) :=    Ω f (∇u) dx + S g 0 (e(γ S (û)) dx if u ∈ A 1 0 , +∞ otherwise. Proof.
Step 1. By using standard lower semicontinuous arguments, it is easily seen that for every sequence (u n ) n∈N strongly converging to u in L 1 (Ω, R 3 ), one has

F 0 (u) ≤ lim inf n→+∞ F0 (u n ). Step 2. We assume u ∈ A 0 and we construct a sequence (u δ ) δ>0 in A 1 0 such that              u δ → u strongly in W 1,p (Ω, R 3 ); γ S (û δ ) = v δ → v := γ S (û) strongly in L 1 (S, R 2 ); lim δ→0 Ω f (∇u δ ) dx = Ω f (∇u) dx; lim δ→0 S g 0 (e(v δ )) dx = S g 0 (e(v)).
Consider the open cylinder Ω := S × (-r, r) containing Ω, where S is an open set of R 2 strictly containing S and extend u into a function ũ in W 1,p ( Ω, R 3 ). We also extend v into a function ṽ in BV( S, R 3 ). More precisely, let P 3 denote the extension operators

P 3 : W 1,p (Ω, R 3 ) → W 1,p ( Ω, R 3 ),
and γ S the trace operator associated with the Sobolev space W 1,p ( Ω, R 3 ). We define ṽ in BV( S, R 3 ) by ṽ := γ S • P 3 (û), i.e., ṽ = γ S ( û).

For all δ > 0, consider the open ball B η(δ) (0) of R 2 centered at 0 with a radius η(δ) > 0 small enough so that S + B η(δ (0) ⊂ S. Let φ in C ∞ c ( S) with φ = 1 on a neighborhood S +B η(δ (0) included in S and ρ η(δ) a standard mollifier with support Bη(δ) (0). Set v δ := ρ η(δ) * (φṽ), u δ := ρ η(δ) * (φũ) and choose η(δ) small enough so that:

v δ -v L 1 (S,R 2 ) < δ; (9) R 2 g 0 (ρ η(δ) * (φe(ṽ))) - R 2 g 0 (φe(ṽ)) < δ; (10)
u δ -u W 1,p (Ω,R 3 ) < δ. (11)
Estimates ( 9) and ( 11) are standard (note that the mollification of ũ takes place only on the x argument). Estimate ( 10) is a straightforward consequence of the narrow convergence of the measure g 0 (ρ η(δ) * (φe(ṽ))) to the measure g 0 (φe(ṽ)) in M + (R 2 ) (see for instance Lemma 5.2 and Remark 5.1 in [START_REF] Temam | Problèmes Mathématiques en Plasticité[END_REF], or, if g 0 is positively homogeneous of degree 1, use Reshetnyak's continuity theorem, Theorem 2.39 in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]).

Clearly

v δ ∈ C ∞ ( S, R 2 ) and, from (9), v δ → v in L 1 (S, R 2 )
. Moreover, from [START_REF] Chipot | Equilibrium configurations of crystals[END_REF], and noticing that φ = 1 on S + B η(δ) (0),

S g 0 (e(v δ )) - S g 0 (e(v)) ≤ R 2 g 0 (ρ η(δ) * (φe(ṽ)) - R 2 g 0 (φe(ṽ)) < δ, so that lim δ→0 S g 0 (e(v δ )) = S g 0 (e(v)).
From [START_REF] Maso | An introduction to Γ-convergence[END_REF],

u δ → u in W 1,p (Ω, R 3 ) and, from (3), lim δ→0 Ω f (∇u δ ) dx = Ω f (∇u) dx.
According to the definition of v δ , u δ , and from the fact that ṽ = γ S ( û), one has

γ S (û δ ) = v δ .
The sequence (u δ ) δ>0 fulfills all the conditions of the set A 1 0 except the boundary condition. By using De Giorgi's slicing method in a neighborhood of Γ 0 (see for instance Theorem 11.2.1 in [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]), one can modify u δ into a new function ũδ ∈ A 1 0 which has the same trace as its weak limit u on ∂Ω, and satisfies lim sup

δ→0 Ω f (∇ũ δ ) dx ≤ Ω f (∇u δ ) dx, thus, finally lim δ→0 Ω f (∇ũ δ ) dx = Ω f (∇u) dx.
Note that u δ is not affected on a neighborhood of S by this modification because dist(Γ 0 , ∂B ε ∩ Γ) > 0. Thus ũδ that we denote now by u δ is the expected sequence.

Proposition 2 (Upper bound). The following inequality holds in

L 1 (Ω, R 3 ) (Γ -lim sup F ε ) ≤ F 0 . ( 12
)
Proof.

Step 1. We establish Γ -lim sup F ε ≤ F0 . Take u ∈ A 1 0 . In what follows we set v := γ S (û). For every fixed ξ in D(S, R 3 ), consider the function

v ε in W 1,p (B, R 3 ), B = S × (-1 2 , 1 2 ) defined by v α ε (x, x 3 ) := v α (x) + εx 3 ξ α (x) v 3 ε (x, x 3 ) := εu 3 (x, 0) + ε 2 x 3 ξ 3 (x)
and set

u ε (x, x 3 ) :=          u(x, x 3 - ε 2 ) + ε 2 ξ(x) in Ω + ε v ε (x, x 3 ε ) in B ε u(x, x 3 + ε 2 ) - ε 2 ξ(x) in Ω - ε .
Clearly u ε ∈ A ε except the boundary condition and

u ε → u in L 1 (Ω, R 3
). An easy calculation and the local Lipschitz conditions (3), ( 4), yield

lim ε→0 Ωε f (∇u ε ) dx = Ω f (∇u) dx and lim ε→0 1 ε Bε g(e(u ε )) dx = lim ε→0 B g ( ∇v ε | 1 ε ∂v ε ∂x 3 ) s dx = S g ( ∇v|ξ) s dx, so that lim ε→0 F ε (u ε ) = Ω f (∇u) dx + S g ( ∇v|ξ s ) dx.
By using again De Giorgi's slicing method in a neighborhood of Γ 0 , one can modify u ε into a function still denoted by u ε , satisfying the boundary condition on Γ 0 and

lim ε→0 F ε (u ε ) = Ω f (∇u) dx + S g ( ∇v|ξ s ) dx. (13)
According to a well known interchange result between infimum and integral (see [START_REF] Hafsa | Interchange of infimum and integral[END_REF]) we have

S g 0 (e(v)) dx = inf ξ∈D(S,R 3 ) S g ( ∇v|ξ) s dx. (14)
By taking the infimum over all ξ ∈ D(S, R 3 ) in [START_REF] Kohn | The relaxation of a double-well energy[END_REF] and by using Lemma 1 we deduce inf lim sup

ε→0 F ε (u ε ) : u ε → u in L 1 (Ω, R 3 ) ≤ F0 (u), i.e. Γ -lim sup F ε ≤ F0 .
Step 2. Taking the lower semicontinuous envelope of each two functionals for the strong topology of L 1 (Ω, R 3 ), the conclusion then follows from the lower semicontinuity of Γ -lim sup F ε and from Lemma 3.

According to Lemma 2 and to variational properties of the Γ-convergence, we obtain : Corollary 1. Let ūε be a solution of (P ε ). Then there exist a subsequence of (ū ε ) ε>0 and ū in W 1,p Γ0 (Ω, R 3 ) such that

ūε ⇀ ū in BD(Ω, R 3 ); ūε ⇀ ū in W 1,p Γ0 (Ω η , R 3 ) for every η > 0; γ S ( û) ∈ BD(S, R 2 ).
Moreover ū is solution of the minimization problem

(P) min F 0 (u) -L(u) : u ∈ L 1 (Ω, R 3 )
and

min F ε (u) -L(u) : u ∈ L 1 (Ω, R 3 ) → min F 0 (u) -L(u) : u ∈ L 1 (Ω, R 3 ) .
Thus, in this simplified case (see Section 5 for a realistic geometry), our proposal of model is given by the limit problem (P) which describes the equilibrium of a structure made of two adherents perfectly stuck to a material surface. The reference configuration of the adherents are Ω ± := Ω ∩ [±x 3 > 0] while the one of the material surface is S. The adherents are hyperelastic with bulk energy density f and the material surface is pseudo-plastic with surface density g 0 . Due to the linear growth of g 0 , the displacement field solution of (P) may present discontinuities in S which may be interpreted in terms of cracks. It is worthwhile to note that this situation with strong adhesive layer is completely different from the one considered in [START_REF] Licht | A modelling of elastic adhesive bonding joints[END_REF] with a soft adhesive: in the asymptotic model, the soft adhesive layer is replaced by a mechanical constraint between the adherents, whereas the strong adhesive layer is replaced by a material surface perfectly stuck to adherents. Another strategy proposed in [START_REF] Bessoud | Variational convergence of energy functionals for elastic materials with ε-thin strong inclusions growing as p(ε), p(ε) → 1 +[END_REF] leads to a similar model. [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] The case when f is not quasiconvex and g is not convex

In this section, we drop the quasiconvex and convex assumptions on the density functions f and g respectively. This is the case when the materials undergo reversible solid/solid phase transformations, for which the density functions present a multi-well structure (for f in the large deformation setting see [START_REF] Chipot | Equilibrium configurations of crystals[END_REF], for g in the setting of small perturbations see [START_REF] Kohn | The relaxation of a double-well energy[END_REF]). However we assume that f and g satisfy the locally Lipschitz conditions (3), ( 4) and that g is positively 1-homogeneous. In this more general situation, we would like to show that the limit energy functional is given by

F 0 : L 1 (Ω, R 3 ) → R ∪ {+∞} F 0 (u) :=    Ω Qf (∇u) dx + S SQg 0 (e(γ s (û)) if u ∈ A 0 ,

+∞ otherwise

where Qf is the quasiconvex envelope of f and SQg 0 : M 2×2 s → R is the symmetric quasiconvex envelope of g 0 defined by

SQg 0 (ζ) := inf 1 | D| D g 0 (ζ + e(ϕ)) dx : ϕ ∈ C ∞ 0 ( D, R 2 ) . Let denote the operator ζ → ζ s from M 2×2 into M 2×2 s by S 2 . Thus, for every ζ ∈ M 2×2 s , SQg 0 (ζ) = Q(g 0 •S 2 )(ζ) where Q(g 0 •S 2 )
is the quasiconvex envelope of g 0 • S 2 . Note that the right hand side term does not depend on the choice of the cube D of R 2 and that SQg 0 is 1-homogeneous. With the notation of Section 2, the integral over S of the measure SQg 0 (e(v)) is given by

S SQg 0 (e(v)) = S SQg 0 (e a (v)) dx + S SQg 0 (e s (v) |e s (v)| |e s (v)|.
Like in Section 2, we prove Proposition 3 and Proposition 5 below, corresponding to the lower and the upper bound in the definition of the Γ-convergence. Unfortunately, we establish the lower bound when u belongs to the subset Ã0 of A 0 defined by

Ã0 := u ∈ W 1,p Γ0 (Ω, R 3 ) : γ S (û) ∈ SBD(S, R 2 )
where SBD(S, R 2 ) denotes the set of the elements u of BD(S, R 2 ) whose the Cantor part of the strain tensor e(u) is zero. Then, the main result of this section is Theorem 2. The restriction of the Γ-limit of F ε to the set Ã0 is given by

F 0 (u) = Ω Qf (∇u) dx+ S SQg 0 (e a (γ S (û))) dx+ S SQg 0 ([γ s (û)]⊗ s ν γ S (û) )dH 1 .
Remark 1. If we assume that an approximate minimizer of (P ε ) strongly converges to some ū in L 1 (Ω, R 3 ) whose distributional gradient has no Cantor part, according to the variational nature of the Γ-convergence, we deduce that ū is a solution of the limit problem

(P) min F 0 (u) -L(u) : u ∈ L 1 (Ω, R 3 ) and inf F ε (u) -L(u) : u ∈ L 1 (Ω, R 3 ) → min F 0 (u) -L(u) : u ∈ Ã0 .
Therefore, under the asumption that some approximate minimizer is regular in the sense above, problem (P) is a good model in the sense of Section 3, where the density functions are now Qf and SQg 0 .

Remark 2. In the case when the deformations in the adhesive may be large, they are modeled as hyperelastic together with the deformations in the adherents.

In this particular case, we obtain a complete description of the Γ-limit F 0 in the set

Ã′ 0 := u ∈ W 1,p Γ0 (Ω, R 3 ) : γ S (u) ∈ BV(S, R 3 ) .
More precisely F 0 is defined in Ã′ 0 by Proposition 3 (Lower bound). For all u in Ã0 and all sequence (u ε ) ε>0 in L 1 (Ω, R 3 ) such that u ε → u, the following inequality holds:

F 0 (u) = Ω Qf (∇u) dx + S Qg 0 (∇(γ S (u))) dx + S (Qg 0 ) ∞ D s γ S (u) |D s γ S (u)| |D s γ S (u)
F 0 (u) ≤ lim inf ε→0 F ε (u ε ). ( 15
)
Proof. One has

lim inf ε→0 F ε (u ε ) = lim inf ε→0 Ωε f (∇u ε )dx + 1 ε Bε g(e(u ε ))dx ≥ lim inf ε→0 Ωε f (∇u ε )dx + lim inf ε→0 1 ε Bε g(e(u ε ))dx ≥ lim inf ε→0 Ωε f (∇u ε )dx + lim inf ε→0 1 ε Bε g 0 ( e(u ε ))dx. ( 16
)
For a.e. x in B, set vε (x) := vε (x, εx 3 ) and

v 3 ε (x) = εv 3 ε (x, x 3 ). Set B := S × (-1 2 , 1 2 ). We have lim inf ε→0 1 ε Bε g 0 ( e(u ε ))dx ≥ lim inf ε→0 B SQg 0 (( e(v ε ))dx. ( 17
)
Consider the function h : M 3×3 s → R defined for every ξ ∈ M 3×3 s by h(ξ) := SQg 0 ( ξ). It is easily seen that h is symmetric quasiconvex, i.e. satisfies the inequality:

h(ξ) ≤ 1 |D| D h(ξ + e(ϕ)) dx.
for every ϕ ∈ C ∞ c (D, R 3 ) where D is any cube of R 3 (see [START_REF] Barroso | A relaxation Theorem in the space of Functions of Bounded Deformation[END_REF] for the definition). Moreover (17) can be written

lim inf ε→0 1 ε Bε g 0 ( e(u ε ))dx ≥ lim inf ε→0 B h(e(v ε ))dx. ( 18
)
Let denote by BD(B, R 3 ) the space of bounded deformation on B and by R H the set of rigid motions on B. According to coercivity condition (2) we have

sup ε>0 B e αβ (v ε ) 1 ε e α3 (v ε ) 1 ε e 3α (v ε ) 1 ε 2 ∂v ε 3 ∂v3 dx ≤ 1 α sup ε>0 1 ε Bε g(e(u ε ))dx < +∞. ( 19
)
Thus, by using the arguments of the proof of Lemma 2, one can easily establish the existence of v ∈ BD(B, R 3 ) and

r ε ∈ R H such that v ε + r ε ⇀ v in BD(B, R 3 ), v = γ S (û)
, v 3 = 0 and ∂v ∂x3 = 0. Combining ( 16), ( 18), a classical results in relaxation theory (Theorem 11.2.1 in [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]) and a relaxation result in BD(S, R 2 ) (see [START_REF] Barroso | A relaxation Theorem in the space of Functions of Bounded Deformation[END_REF]), we infer

lim inf ε→0 F ε (u ε ) ≥ Ω Qf (∇u) dx + S SQg 0 (e a (γ S (û)) dx + S SQg 0 ([γ s (û)] ⊗ s ν γ S (u) (x))dH 1 (x)
which ends the proof.

For proving the upper bound, we need to establish the following relaxation result.

Proposition 4 (Relaxation). The functional F 0 is the l.s.c. regularization for strong topology of the functional defined on L 1 (Ω, R 3 ) by

F0 (u) :=    Ω f (∇u) dx + S g 0 (e(γ S (û)) dx if u ∈ A 1 0 , +∞ otherwise.
In the proof of Proposition 4 we will use the following lemma.

Proof. From the integral representations

SQ(η| . | p + g 0 )(ζ) = inf φ∈C ∞ c (D,R 2 ) 1 |D| D (η| . | p + g 0 )(ζ + e ( φ))dx, and
SQg 0 (ζ) = inf φ∈C ∞ c (D,R 2 ) 1 |D| D g 0 (ζ + e(φ))dx,
where D is an arbitrary cube of R 2 , we can write

lim η→0 inf φ∈C ∞ c (D,R 2 ) 1 |D| D (η| . | p + g 0 )(ζ + e(φ))dx = inf η>0 inf φ∈C ∞ c (D,R 2 ) 1 |D| D (η| . | p + g 0 )(ζ + e(φ))dx = inf φ∈C ∞ c (D,R 2 ) inf η>0 1 |D| D (η| . | p + g 0 )(ζ + e(φ))dx = inf φ∈C ∞ c (D,R 2 ) lim η→0 1 |D| D (η| . | p + g 0 )(ζ + e(φ))dx .
We complete the proof by using the dominated convergence theorem.

Proof of Proposition 4. Denote by F0 the l.s.c. regularization of F0 . By using standard l.s.c. results on integral functionals defined on Sobolev or BV -spaces, one can easily prove F 0 ≤ F0 (see for instance [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF], chapter 10). We only establish the converse inequality F0 ≤ F 0 . Its proof is not easy because of the condition u ∈ A 0 and the fact that f and g do not fulfill the same growth conditions. Let u ∈ L 1 (Ω, R 3 ) such that F 0 (u) < +∞. Then u ∈ A 0 . We have to exhibit u n in A 1 0 strongly converging to u in L 1 (Ω, R 3 ) such that lim n→+∞ F0 (u n ) = F 0 (u). We proceed into two steps.

Step 1. We prove the thesis when u ∈ A 1 0 . For shorten notation we write v := γ S (û).

Let η > 0 intended to go to 0 and denote the constant involved in Korn's inequality by K:

S |∇w| p dx ≤ K S |e(w)| p + |w| p dx
for all function w in W 1,p (S, R 2 ). From relaxation theory in Sobolev spaces, there exists a sequence of smooth functions (u n , v n ) n∈N in W 1,p Γ0 (Ω, R 3 )×W 1,p (S, R 2 ) such that (see for instance [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF], Theorem 11.2.1 and Theorem 11.4.2)

             u n ⇀ u in W 1,p Γ0 (Ω, R 3 ), (|∇u n | p ) n∈N uniformly integrable; v n ⇀ v in W 1,p (S, R 3 ); lim n→+∞ Ω f (∇u n )dx = Ω Qf (∇u)dx; lim n→+∞ S (3Kβη| . | p + g 0 )(e(v n ))dx = S SQ(3Kβη| . | p + g 0 )(e(v))dx.
The additional condition that (|∇u n | p ) n∈N may be assumed to be uniformly integrable comes from the following consideration. Consider the sequence (ũ n ) n∈N whose gradients generate the same Young measure µ and such that (|∇ũ n | p ) n∈N is unifomly integrable (Lemma 11.4.1 in [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]). By using lower semicontinuity and continuity properties of Young measures (Proposition 4.3.3 and Theorem 4.3.3 in [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]), and standard lower semicontinuity results in Sobolev spaces, we have

Ω Qf (∇u) dx = lim n→+∞ Ω f (∇u n ) dx ≥ Ω×M 3×3 f (λ) dµ = lim n→+∞ Ω f (∇ũ n ) dx ≥ Ω Qf (∇u) dx, so that lim n→+∞ Ω f (∇ũ n ) dx = lim n→+∞ Ω f (∇u n ) dx = Ω Qf (∇u) dx
which proves the thesis. In what follows, we still denote by (u n ) n∈N the sequence (ũ n ) n∈N . We start by modifying the function u n near S so that γ S (u

n α ) = v n α . Set Σ η := S × (-η, η), Σ 2η := S × (-2η, 2η), consider a cut-off function ϕ η in C 1 (R) satisfying ϕ η = 1 on Ω \ Σ 2η , ϕ η = 0 on Σ η , 0 ≤ ϕ η ≤ 1, | dϕ η dx 3 | ≤ 1 η ,
and define the function u n,η by

u n,η α := ϕ η (u n α -v n α ) + v n α ; u n,η 3 := u n 3 .
Clearly u n,η ∈ W 1,p Γ0 (Ω, R 3 ) and γ S (u n,i α ) = v n α . From the growth condition satisfied by f , we have

Ω f (∇u n,η ) dx = Ση f (∇u n,η ) dx + Σ2η\Ση f (∇u n,η ) dx + Ω\Σ2η f (∇u n ) dx ≤ β Ση (1 + |∇v n | p ) dx + β Σ2η\Ση (1 + |∇v n | p ) dx + C Σ2η |∇u n | p dx + 1 η p Σ2η |û n -v n | p dx + Ω f (∇u n )dx = 3βη S |∇v n | p dx + C η + Σ2η |∇u n | p dx + 1 η p Σ2η |û n -v n | p dx + Ω f (∇u n )dx.
Thus, according to Korn's inequality (note that sup n∈N S |v n | p dx < +∞),

Ω f (∇u n,η ) dx ≤ 3βKη S |(e(v n )| p dx + C η + Σ2η |∇u n | p dx + 1 η p Σ2η |û n -v n | p dx + Ω f (∇u n )dx. which yields Ω f (∇u n,η) )dx + S g 0 (e(v n ))dx ≤ Ω f (∇u n )dx + S (3βKη| | p + g 0 )(e(v n ))dx +C η + Σ2η |∇u n | p dx + 1 η p Σ2η |û n -v n | p dx and, from (4), lim sup n→+∞ F0 (u n,η ) ≤ Ω Qf (∇u)dx + S SQ(3βKη| | p + g 0 )(e(v n ))dx +C η + sup n∈N |∇u n | p dx + 1 η p Σ2η |û -v| p dx But since γ S (û) = v, clearly one has Σ2η |û -v| p dx ≤ η p Σ2η | ∂u ∂x 3 | p dx so that lim sup n→+∞ F0 (u n,η ) ≤ Ω Qf (∇u)dx + S SQ(3βKη| | p + g 0 )e(v)) dx +C η + sup n∈N Σ2η |∇u n | p dx + Σ2η | ∂u ∂x 3 | p dx
By letting η → 0, from the uniform integrability of (|∇u n | p ) n∈N , Lemma 4 and Lebesgue's dominated convergence theorem, we obtain lim sup

η→0 lim sup n→+∞ F0 (u n,η ) ≤ Ω Qf (∇u)dx + S SQg 0 (e(v)) dx.
By using a standard diagonalization argument, there exists a map n → η(n) such that, setting ũn :

= u n,η(n) , lim sup n→+∞ F0 (ũ n ) ≤ F 0 (u).
It is easily seen that ũn → u in L 1 (Ω, R 3 ). Since classically lim inf n→+∞ F0 (ũ n ) ≥ F 0 (u), the proof of step 1 is complete.

Step 2. We end the proof as the step 2 of the proof of Lemma 3. We only have to substitute

R 2 SQg 0 (ρ η(δ) * (φe(ṽ)) dx - R 2 SQg 0 (φe(v) < δ (20)
for [START_REF] Chipot | Equilibrium configurations of crystals[END_REF]. Estimate (20) is a straightforward consequence of the weak convergence of the measure

ρ η(δ) * (φe(v))) to the measure φe(v)) in M(R 2 ) together with lim n→+∞ R 2 |ρ η(δ) * (φe(v)))| = R 2 |φe(v))|
, and Reshetnyak's continuity theorem (see Theorem 2.39 in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]).

Then proceeding exactly like in the proof of Proposition 2 and using Lemma 4, one has Proposition 5 (Upper bound). The following inequality holds in

L 1 (Ω, R 3 ) (Γ -lim sup F ε ) ≤ F 0 . (21)

A modeling of a welding assembly

An elementary situation in welding can be described as follows. Let Σ + , Σ - and S three domains of R 2 with Lipschitz-continuous boundaries such that S = Σ + ∩ Σ -. Let r and ε two positive numbers such that ε << r and

Ω ± ε := Σ ± × (±ε/2, ±r), Ω ε = Ω + ε ∪ Ω - ε , B ε = S × (-ε/2, ε/2), and
S ± ε = S ± ± ε/2 e 3 . Then O ε := Ω ε ∪ S + ε ∪ S - ε ∪ B ε
is the reference configuration of a structure made of two adherents and an adhesive (the soldered joint) which respectively occupies Ω ± ε and B ε (see Figure 1). The structure is clamped on a part Γ 0 of the boundary Γ of Ω with a positive H 2 -measure and we assume that there exists ε 0 > 0 such that dist( Γ0 , Bε0 ) > 0. The structure is subjected to body forces of density Φ and to surface forces of density ϕ on the complementary part Γ ϕ of Γ 0 . We assume that the supports of ϕ and Φ lay outside of Bε0 . Obviously one can consider other type of boundary conditions (e.g. a combination of some components of the stress vector and of the displacement). At last, adhesive and adherents are assumed to be perfectly stuck together along S ± .

The adherents and the adhesive are modeled as in section 2 so that determining the equilibrium configuration leads to the problem

(P ε ) inf F ε (u) -L(u) : u ∈ A ε
where F ε and L have the same expression as in Section 2 but with the new definitions of Ω ε and B ε , whereas A ε now reads as:

A ε := u ∈ LD(O ε , R 3 ) : u ⌊Ωε ∈ W 1,p Γ0 (Ω ε , R 3 ) .
Again, to propose a simplified but accurate model we consider ε as a parameter and study the asymptotic behavior, when ε goes to zero, of (approximate) solution of (P ε ). The essential difference from the model problem of Section 2 is that here the structure occupies a domain O ε which varies with ε, which from

!"#$ !"%$ &' ( ( !"#$ !"%$ ) $*+ &' ( ($ !,% !,# ) -$*+ !,%$ !,#$ Figure 1: the reference configuration O ε . The reference configuration involved in Section 2 is Ω + 2ε ∪ Ω - 2ε ∪ B ε ∪ S + ε ∪ S - ε .
the mathematical point of view is only of technical nature by simply modifying the kinds of convergences. That is why we have preferred to consider the model problem in whole details and to confine to state the sole results about this realistic problem of welding.

Let Ω := Σ + × (0, r) ∪ S ∪ Σ -× (-r, 0) the set to which O ε "converges". Let

I ε := F ε -L and I 0 := F 0 -L with F 0 : L 1 (Ω, R 3 ) → R ∪ {+∞}, F 0 (u) :=    Ω f (∇u) dx + S g 0 (e(γ s (û)) if u ∈ A 0 +∞ otherwise,
where we keep the same definition for γ S and A 0 but with the new definition of Ω. Doing the same for the definition of the operator T ε , our asymptotic model is supplied by: Theorem 3. Let ūε be a solution of (P ε ). Then there exist a subsequence of (ū ε ) ε>0 and ū in W 1,p Γ0 (Ω, R 3 ) such that

T ε ūε → ū weakly in W 1,p (Ω \ S, R 3 ); ūε⌊S×(-r,r) → ū weakly in BD(S × (-r, r), R 3 ); γ S ( û) ∈ BD(S, R 2 ).
Moreover ū is solution of the minimization problem

(P) min F 0 (u) -L(u) : u ∈ L 1 (Ω, R 3 ) and min F ε (u) -L(u) : u ∈ L 1 (Ω, R 3 ) → min F 0 (u) -L(u) : u ∈ L 1 (Ω, R 3 ) .
Sketch of the proof. The proof follows the line of the proof of Section 2 by considering simultaneously T ε ūε and its restriction to the fixed domain S × (-r, r).

For the mechanical interpretation see the end of Section 3.

A variational regularization of the limit functional

For the numerical solving of the optimization problem inf F 0 (u) -L(u)

obtained in Section 3, we approximate, in a variational way, the functional of measure u → S g 0 (e(γ S (û)) by a suitable functional defined in the Sobolev space W 1,q (S, R 2 ), where q is close to 1 in the spirit of Norton-Hoff regularisation (cf [START_REF] Temam | Problèmes Mathématiques en Plasticité[END_REF]). The mathematical technics used here is an adaptation from that of [START_REF] Bessoud | Variational convergence of energy functionals for elastic materials with ε-thin strong inclusions growing as p(ε), p(ε) → 1 +[END_REF]. In order to simplify the proofs, we assume that S is a finite union of cubes in R 2 . We denote the limit density g 0 by h that we assume to be positively 1homogeneous and fulfilling the growth conditions α|ξ| ≤ h(ξ) ≤ β|ξ|. We consider a sequence (h q ) q∈(1,p) satisfying the following three conditions: i q ) h q : M 2×2 s → R + is convex and positively homogeneous of degree q; ii q ) h q → h pointwise in M 2×2 s ; iii q ) there exists a > 0 such that for all q > 1 close enough to 1,

h q (ξ) ≥ h(ξ) for all ξ ∈ M 2×2 s , |ξ| ≥ a. (22)
For instance, when h = |.|, h q = |.| q satisfies these conditions with a = 1.

Another natural example consists in choosing h q := h q . Condition iii q ) is then satisfyed by taking a = 1 α . In these two examples, h q satisfies uniform growth conditions with respect to q (for the second example, α 2 |ξ| q ≤ h q (ξ) ≤ 2β|ξ| q for q closed to 1). Note that, according to iii) q , h q fulfills the equi-coerciveness condition:

h q (ξ) ≥ α|ξ| ∀ξ, |ξ| ≥ a. (23)
In what follows, the function h q is not assumed to satisfy a uniform upper growth condition.

We consider the functional F q : W 1,p Γ0 (Ω, R 3 ) → R + ∪ {+∞} defined by:

F q (u) =    Ω f (∇u) dx + S h q (e(γ S (û)))dx if u ∈ B q ,
+∞ otherwise where

B q := u ∈ W 1,p Γ0 (Ω, R 3 ) : h q • (e(γ S (û))) ∈ L 1 (Ω)
. We are going to establish the Γ-convergence of F q when q → 1, when the space W 1,p Γ0 (Ω, R 3 ) is equipped with its weak topology. The expected limit is the functional F 0 : W 1,p Γ0 (Ω, R 3 ) → R + ∪ {+∞} defined in the previous section, more precisely its restriction to W 1,p Γ0 (Ω, R 3 ) defined by

F 0 (u) =    Ω f (∇u) dx + S h(e(γ S (û))) if u ∈ B +∞ otherwise, where B := u ∈ W 1,p Γ0 (Ω, R 3 ) : γ S (û) ∈ BD(S, R 2 ) .
Lemma 5 (Compactness lemma). Consider a sequence (u q ) q∈]1,p] in W 1,p Γ0 (Ω, R 3 ) such that sup q∈(1,p) F q (u q ) < +∞. Then there exist a subsequence of (u q ) q∈]1,p] and u ∈ B such that (u q , γ S (û q )) ⇀ (u, γ S (û)) in W 1,p Γ0 (Ω, R 3 ) × BD(S, R 2 ) when q goes to 1.

Proof. Since sup q∈(1,p) F q (u q ) < +∞, one has u q ∈ B q and

F q (u q ) = Ω f (∇u q )) dx + S h q (e(v q )dx, v q := γ S (û q ). Thus, from (23) sup q∈(1,p) S |e(v q )| dx ≤ sup q∈(1,p) a|S| + [|e(vq)|≥a] |e(v q )| dx ≤ a|S| + 1 α sup q∈(1,p) S h q (e(v q )) dx < +∞. ( 24
)
On the other hand from the coercivity condition fulfilled by f , there exists a subsequence and u ∈ W 1,p Γ0 (Ω, R 3 ) such that u q ⇀ u in W 1,p Γ0 (Ω, R 3 ). According to the continuity of the trace operator γ S , we deduce that v q → γ S (û) in L p (S, R 3 ), thus strongly in L 1 (S, R 2 ) which, combined with (24), yields v q ⇀ v = γ S (û) in BD(S, R 2 ) and u ∈ B.

Proposition 6 (lower bound). For every sequence ((u

q ) q∈]1,p] converging to u in W 1,p Γ0 (Ω, R 3 ), we have F 0 (u) ≤ lim inf q→1 F q (u q ). Lemma 6. Let V q ∈ L q (S, M 2×2 s
) and assume that the measure µ q = V q dx weakly converges to µ in M(S, M 2×2 s ) and that |µ q | = |V q | dx weakly converges to ν in M + (S). Then for all open subset ω of S such that ν(∂ω) = 0, one has lim inf

q→1 ω h q (V q ) dx ≥ h ω dµ .
Proof. From Jensen's inequality and since h q is positively q-homogeneous,

ω h q (V q ) dx ≥ |ω|h q 1 |ω| ω V q dx = |ω| 1-q h q ω V q dx .
It remains to establish

lim inf q→1 h q ω V q dx ≥ h ω dµ . (25)
One may assume ω dµ > 0, otherwise ω dµ = 0 and (25) is trivially satisfied.

Since ν(∂ω) = 0, one has lim q→1 ω V q dx = ω dµ (cf Corollary 4.2.1 in [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]).

Then for 0 < σ < ω dµ , there exists q 0 , 1 < q 0 ≤ p such that, for all q, 1 < q ≤ q 0 , | ω V q dx| ≥ σ. One has

h q ω V q dx = σ a q h q ( a σ ω V q dx
with a σ ω V q dx ≥ a, so that from assumption (22),

h q ω V q dx ≥ σ a q h( a σ ω V q dx = σ a q-1 h( ω V q dx .
Letting q → 1, (25) follows from the lower semicontinuity of h and the fact that lim q→1 ω V q dx = ω dµ.

Proof of Proposition 6. One may assume F q (u q ) < +∞ so that, from Lemma 5, one has u ∈ B. We write v q for γ S (û q ) and v for γ S (û). According to a standard lower semicontinuity result in Sobolev spaces we have

lim inf q→1 F q (u q ) ≥ Ω f (∇u)dx + lim inf q→1 S
h q (e(v q )) dx and it remains to establish lim inf q→1 S h q (e(v q )) dx ≥ S h(e(v)).

For δ > 0 intended to go to 0, consider a standard mollifier ρ δ and θ δ in C ∞ c (S) satisfying 0 ≤ θ δ ≤ 1, θ δ → 1 a.e. in S. For a subsequence not relabelled on q, clearly ρ δ * θ δ |e(v q )| dx weakly converges to some measure ν δ in M + (S). Moreover ρ δ * θ δ e(v q ) dx weakly converges to the measure ρ δ * θ δ e(v) in M(S, M 2×2 Such a family exists (use Lemma 4.2.1 of [START_REF] Attouch | Variational analysis in Sobolev and BV space: application to PDEs and Optimization[END_REF]). Note that this family depends on δ. By using the q-homogeneity of h q , Jensen's inequality and standard convex duality principle, we have S h q (e(v q )) dx ≥ S θ q δ h q (e(v q )) dx = S h q (θ δ e(v q )) dx ≥ S h q (ρ δ * (θ δ e(v q )) dx ≥ i∈Iη ωi h q (ρ δ * (θ δ e(v q ))) dx.

Thus, from Lemma 6, 1-homogeneity and Lipschitz property of h, lim inf q→1 S h q (e(v q )) dx ≥ lim inf q→1 i∈Iη ωi h q (ρ δ * (θ δ e(v q ))) dx The first term of the second member is a Riemann sum. Since moreover ρ δ * θ δ e(v) is a smooth function, by letting η → 0, we obtain lim inf q→1 S h q (e(v q )) dx ≥ S h(ρ δ * θ δ e(v)).

Noticing that ρ δ * θ δ e(v) ⇀ e(v) in M(S, M 2×2 s ), estimate (26) is obtained by letting δ → 0.

Proposition 7 (Upper bound). For every u ∈ W 1,p Γ0 (Ω, R 3 ) there exists u q weakly converging to u in W 1,p Γ0 (Ω, R 3 ) such that lim sup q→1 F q (u q ) ≤ F 0 (u), or, equivalently, Γ -lim sup F q ≤ F 0 .

Proof. Consider the following subset B 1 of B: Such a sequence exists from step 1 of the proof of Proposition 4 in previous section and u n belongs to B q . Writting e(v n ) = i∈In a i,n 1 Si,n where (S i,n ) i∈In is a finite partition of S, and a i,n ∈ M 2×2 s , the following estimate holds: Then, by using a standard diagonalization argument, there exists a map q → n(q) such that lim q→1 F q (u n(q) ) = F0 (u); u n(q) ⇀ u, which implies inf lim sup p→1 F q (u q ) : u q ⇀ u in W 1,p Γ0 (Ω, R 3 ) ≤ F0 (u) for all u ∈ W 1,p Γ0 (Ω, R 3 ). Thus Γ -lim sup F q ≤ F0 . The conclusion of Proposition 7 follows by taking the lower semi-continuous envelop of each two functionals for the weak topology of W 1,p Γ0 (Ω, R 3 ) and by using Lemma 3.

lim q→1 Ω f (∇u n ) dx + S h q (e(v n )) dx = Ω f (∇u n ) dx + lim
Corollary 2. Assume that h q satisfies the additional coerciveness condition: there exists α q > 0 such that α q |ξ| q ≤ h q (ξ) for all ξ ∈ M 2×2 s . Then

i) The problem min F q (u) -L(u) possesses at least a solution ūq ;

|dH 1 ,

 1 where Du = ∇u dx + D s u is the Lebesgue decomposition of the distributional derivative Du, g 0 (ζ) = min{g(ξ) : ξ ∈ M 3×3 , ξ = ζ} for every ζ ∈ R 3 , and ζ → (Qg 0 ) ∞ (ζ) := lim t→+∞ Qg 0 (tζ)/t is the recession function of Qg 0 . The proof uses the relaxation theorem, Theorem 11.3.1 in [6], instead of Proposition 4 below and follows point by point the claims of Propositions 3, 5 below.

Lemma 4 .

 4 Let η > 0, then for every ζ ∈ M 2×2 s , lim η→0 SQ(η| . | p + g 0 )(ζ) = SQg 0 (ζ).

  η > 0, consider a finite family (ω i ) i∈Iη of pairwise disjoint open subsets of S, |ω i | < η, such that |S \ ∪ i∈Iη ω i | = 0 and a family (ω η ) i∈Iη satisfying |S \ i∈Iη ωi | ≤ η; ωi ⊂ ω i ; ν δ (∂ ωi ) = 0.

i |h 1

 1 |ω i | ωi ρ δ * (θ δ e(v)) -L|ρ δ * (θ δ e(v))|(S \ i∈Iη ωi ).

B 1 :F0f

 1 = u ∈ W 1,p Γ0 (Ω, R 3 ) : γ S (u) ∈ C 1 ( S, R 3 )and the functional F0 :W 1,p Γ0 (Ω, R 3 ) → R + {+∞} defined by (∇u) dx + S h(e(γ S (u))) dx if u ∈ B 1 +∞ otherwise.Take u ∈ B 1 and set v := γ s (u). Let (v n ) n∈N * be a sequence of continuous piecewise affine functions satisfying v n -v W 1,1 (S,R 2 ) ≤ 1/n, and consider a sequence (u n ) n∈N weakly converging tou in W 1,p Γ0 (Ω, R 3 ) satisfying lim n→+∞ Ω f (∇u n ) dx = Ω f (∇u) dx; γ S (û n ) = v n .

  q→1 i∈In h q (a i,n )|S i,n | = Ω f (∇u n ) dx + i∈In h(a i,n )|S i,n | = Ω f (∇u n ) dx + S h(e(v n )) dx. (27)Letting n → +∞, (27) yieldslim n→+∞ lim q→1 Ω f (∇u n ) dx + S h q (e(v n )) dx) = Ω f (∇u) dx + Sh(e(v)) dx.


M Marciante

C Champenois

J Pedregosa-Gutierrez

A Calisti

M Knoop

Parallel ion strings in linear multipole traps

Additional radio-frequency (rf) potentials applied to linear multipole traps create extra field nodes in the radial plane which allow to confine single ions, or strings of ions, in totally rf field-free regions. The number of nodes depends on the order of the applied multipole potentials and their relative distance can be easily tuned by the amplitude variation of the applied voltages. Simulations using molecular dynamics show that strings of ions can be laser cooled down to the Doppler limit in all directions of space. Once cooled, organized systems can be moved with very limited heating, even if the cooling process is turned off.

Radio-frequency (rf) traps are very useful tools for a wide range of research interests such as high resolution spectroscopy [1], frequency standards [2], quantum information processing [3] and quantum simulations [4]. The ability to trap and cool atomic or molecular species in a well defined manner allows to control and study the quantum behaviour of these systems. Most experiments require a totally perturbation-free dynamics of the confined ions, and the micro-motion due to the rf trapping field can sometimes be a disturbing factor. In these cases, cold ions are confined in a string configuration along the longitudinal axis of a linear quadrupole trap, where the rf field vanishes. For a number of applications, in particular in quantum logic and quantum simulations [3,4], it could be desirable to create more sites inside a trap where the ions do not undergo the rf parametric excitation. Linear traps of higher order, here called multipole traps, offer larger regions of low rf electric fields [START_REF] Gerlich | Inhomogeneous rf fields: a versatile tool for the study of processes with slow ions[END_REF] which allow to trap large samples with a reduced driven micromotion, as compared to the same sample trapped in a linear quadrupole trap [START_REF] Prestage | Proceedings of the 1999 Joint EFTF-IFCS[END_REF]. Local minima can be induced in the trapping potential by adding static voltages to these multipole rf voltages [START_REF] Wester | [END_REF]8] but the ions then undergo a rf driven motion which is detrimental when very low temperatures need to be reached. The superimposition of a lower order rf field onto the main trapping field generates minima to the trapping potential where the rf field is nulled. In such a configuration and depending on their number, the ions can settle in each minimum as individual ions or as parallel strings of ions expanding in the axial direction. Due to the absence of a local rf electric field, laser cooling can reduce the temperature of the ions as low as for a chain of ions in a quadrupole trap.

In this letter we first give an example of principle of the proposed method by combining a quadrupole potential with the potential created by a linear octupole trap. To demonstrate how to reach the Doppler limit by Doppler laser cooling we use molecular dynamics (m.d.) simulations of a 10-ion system, in a double line configuration. In a second part, we discuss the general case of two com- * mathieu.marciante@etu.univ-provence.fr bined rf potentials of different order. In these two parts, we assume an ideal electric field for each superimposed rf field. In the third part of this letter, the effect of a nonideal geometry to generate the lowest order rf field [9] is taken into account by combining m.d. simulations with an analytic fit of the actual electric field. This analysis shows that, providing an extra voltage which compensates for the imperfections of the electric fields, the same properties are observed and that our proposal still holds.

First, we describe the principle of the creation of extra rf field-free minima within the linear octupole trap geometry. In order to create additional nodes in the rf electric field, a rf quadrupole potential, with identical phase and frequency as the octupole potential, is superimposed onto the former. By the superposition theorem, the resulting rf electric potential Φ rf in the radial plane is given by

Φ rf = Φ 8rf + Φ 4rf , (1)
where the expression of the 2k-pole electric potentials Φ (2k)rf using the polar coordinates (see Fig. 1) can be approximated by [START_REF] Gerlich | Inhomogeneous rf fields: a versatile tool for the study of processes with slow ions[END_REF]:

Φ (2k)rf (R, φ, t) = -V 2k R k cos (kφ) cos (Ωt) (2)
with Ω and V 2k respectively the frequency and the amplitude of the applied rf potentials, and R = r/r 0 the relative distance to the trap center, scaled to the inner radius of the trap r 0 . The components of the electric field E = -∇Φ rf are:

E r = + 2 r 0 2V 8 R 3 cos (4φ) + V 4 R cos (2φ) cos (Ωt) , (3)
E φ = - 2 r 0 2V 8 R 3 sin (4φ) + V 4 R sin (2φ) cos (Ωt) .( 4
)
With our choice of initial phase and frame reference (sketched on Fig. 1), it is straightforward from Eq. 4 that E φ can be cancelled for any direction defined by φ n = nπ/2, with n ∈ Z. The relative variation of the cosines in Eq. 3 makes then possible a total rf field cancellation at the relative radial positions R 0f for φ ±1 = ±π/2 (the y-axis in our case, see Fig. 1). This relative radial position depends on the ratio of the two rf voltages as :

R 2 0f = V 4 /2V 8 . (5)
x ŷ rφ In the adiabatic approximation, where the rf period is much smaller than the time scale of the motion induced by the spatial variation of the electric field amplitude (the so-called macro-motion) [10], the dynamics of an ion of charge q e and mass m inside the trap can be understood using the pseudopotential (psp) Ψ, which results from the time-average of the rf electric field Ψ = q 2 e E 2 /2mΩ 2 , where the overline holds for the average over one rf period. The psp does not obey the superposition theorem and in our case of superimposed rf fields, the resulting psp writes as :

Ψ (R, φ) = Ψ 8 (R) + Ψ 4 (R) + 2 Ψ 8 (R)Ψ 4 (R) cos (2φ) (6)
where the Ψ 2k (R) are the usual 2k-polar psp, 2k being the number of electrodes [START_REF] Gerlich | Inhomogeneous rf fields: a versatile tool for the study of processes with slow ions[END_REF]:

Ψ 2k (R) = k 2 4 q 2 e V 2 2k m r 2 0 Ω 2 R 2k-2 = ψ 2k R 2k-2 . ( 7
)
By definition of the psp, the pseudo-potential wells exactly match the nodes of the rf electric field. Eq. 6 shows that two extra psp minima, compared to the simple Ψ 8 case, are expected for R = 0 if cos(2φ) = -1.

However, the total effective trapping potential U trp responsible for the ion confinement results from the psp itself and the static potential required to trap along the symmetry axis Oz and which can be approximated by a harmonic confinement mω 2 z z 2 /2. To obey the Laplace equation, this trapping potential has a de-confining counterpart -mω 2 z r 2 /4 which must be taken into account [8,11]. In the case of an extra static octupole potential V st applied to the electrodes, the effective trapping potential now reads

U trp = Ψ+q e V st R 4 cos (4φ)+ 1 2 m ω 2 z z 2 - R 2 2 r 2 0 ( 8
)
To make sure that the total potential minima still match the rf field-free positions, it is mandatory to compensate the radial de-confining force by an appropriate choice of the static potential V st . The matching of these two forces at the field free positions (R 0f , φ ±1 ) is obtained under the condition:

q e V st = m ω 2 z r 2 0 / 8R 2 0f (9)
To remain as general as possible, we have considered the extra octupole static voltage to compensate for the deconfining effect of the axial confinement. The use of a quadrupole voltage leads to a similar condition which theoretically does not depend on R 0f (cf. discussion of the realization below). We would like to point out that the rf field cancellation condition Eq. 5 as well as the matching condition Eq. 9 do not depend on the mass of the trapped particles and the process described here can apply to a multi-species system (because ω 2 z ∝ 1/m). The lowest order expansion of the effective trapping potential around its minima (R 0f , φ ±1 ) allows the definition of characteristic frequencies for the motion in the radial and angular directions, if Eq. 9 is satisfied :

ω 2 r = ω 2 z 8ψ 4 m r 2 0 ω 2 z + 1 = 4ω 2 u + ω 2 z , (10)
ω 2 φ = ω 2 z 8ψ 4 m r 2 0 ω 2 z -2 = 4ω 2 u -2ω 2 z , (11)
where ω u is the frequency of motion in the radial plane, when only the rf quadrupole potential is applied. The local steepness of the potential, given in Eq. 10 and 11, is controlled by the quadrupole potential V 4 and the axial confinement ω z , thus ruling the morphology of a cold sample in the local 3D anisotropic harmonic potential (ω r , ω φ , ω z ) [12][13][14]. A close look at Eq. 11 shows that it takes q e V 4 > mr 2 0 ω z Ω/2 to reach this local 3D harmonic confinement. The local potential being independent of the octupole potential V 8 , this last parameter can be tuned to independently control the distance between the two minima.

In a second step, we show that ions can effectively be cooled to the Doppler limit and localized in the two extra minima in a controlled manner. For this purpose we carry out m.d. simulations of a set of 10 calcium ions, Doppler laser-cooled along the three directions of space in an octupole trap. The m.d procedure for laser cooling takes into account the absorption and emission processes of photons on the involved atomic transitions. Details of its modeling as well as the definition of temperature used are explained in [14]. The rf electric fields obey Eq. 3 and 4 and at t = 0, the rf quadrupole potential V 4 and the static octupole potential V st are off. The amplitude of the rf octupole potential V 8 is such that cold ions form a ∼ 20 µm radius ring in the z = 0 plane for ω z = 2π × 1 MHz, shown on figure 2a. The stability of such a structure has been studied elsewhere [8,15], where it is shown that the Doppler limit can be reached for the motion along the longitudinal direction and that the cooling of the motion in the radial plane is limited by rf heating [8]. At t = 2 ms, the rf amplitude V 4 is switched on, breaking the cylindrical symmetry of the trapping potential and producing the separation of the ring in two sets of five ions, as shown on Fig. 2b. From t = 2 ms to t = 4.4 ms, the amplitude V 4 is linearly increased, increasing the distance between the two sets of ions and the steepness of the local potential to reach a separation of 0.22r 0 . For t = 4.4 ms, the ions form two strings along the z-axis (Fig. 2c and Fig. 2d). At this time, V 4 is kept unchanged and V st is switched on from 0 to the value fulfilling Eq. 9 (1.7 V) in 1 ms, matching the equilibrium positions of the strings of ions and the field-free positions and reaching characteristic frequencies ω r = 2π × 4.2 MHz and ω φ = 2π × 3.8 MHz. The signature of this position matching is the abrupt drop of T y , the temperature of the motion along Oy, from 0.1 K to the Doppler limit (see Fig. 2e). Another signature of this position matching and of the interest of this new ion organization is the lack of rf-heating observed in the simulations when the cooling lasers are switched off, once the parallel lines are formed. Furthermore, tuning the amplitude V 8 and the static potential V st accordingly, the ion strings can be moved along the Oy axis. This relative position tuning can be performed with very limited heating as long as the Coulomb repulsion between the two chains is negligible. When it is not the case, this repulsion has to be incorporated in the force compensation condition and a larger static voltage than the one given in Eq. 9 is required.

It is possible to generalize the concept introduced previously. We now assume a 2k-pole linear trap on which an rf potential V 2k is applied. An additional V 2p rf potential with identical frequency and phase is superimposed on 2p electrodes to generate a rf contribution with a p-order of rotational symmetry. Each rf contribution builds its own psp Ψ 2k and Ψ 2p (see Eq. 7) and the resulting psp can be written as :

Ψ = Ψ 2k + Ψ 2p + 2 Ψ 2k Ψ 2p cos ((k -p) φ) . (12)
The extra psp minima created by cancellation of the rf electric fields are located where

φ n = (2n + 1)π/(k -p) (for cos((k -p)φ) = -1) and R = R 0f such that R k-p 0f = pV 2p /kV 2k . (13)
To compensate for the deconfining contribution of the axial confinement at the psp minima, an additional static potential with a 2π/(k -p) rotational symmetry is required to produce the same confining contribution along all the φ n = (2n + 1)π/(k -p) lines. Depending on the choice of k and p, this extra static potential may have a different order of symmetry but a cancellation condition equivalent to Eq. 9 can always be found. As an example, four parallel strings can be formed by the superimposition of an rf quadrupole potential to an rf dodecapole potential, their location match the rf-field nodes providing a static voltage with an octupole symmetry is applied (see figure 3 for different configurations). The order of symmetry s of the static potential V st must be a multiple of (k -p). For the case introduced in the first part of this letter, we used s = 2(k -p) = 4. This choice has an influence on the characteristic frequencies of the motion of the ions in the extra minima. Indeed, a lowest order expansion of the total trapping potential around these positions results in frequency expressions which are generalizations of Eq. 10 and 11 :

+ + + + + + + + + (a) (b) (c)
ω 2 r = 2 (k -p) 2 ψ 2p m r 2 0 R 2p-4 0f + (s -2) ω 2 z 2 , (14)
ω 2 φ = 2 (k -p) 2 ψ 2p m r 2 0 R 2p-4 0f -s ω 2 z 2 . ( 15
)
The choice of an order s higher than strictly required by the rotational symmetry may then be justified by the need for high frequencies ω r compared to ω φ , which can be used to control the morphology of the subsystems (string, zig-zag...) , at the expense of the static voltage itself, which scales as 1/(sR s-2 0f ). Until now, we have assumed equations for the rf electric field and its psp which correspond to the ideal case of perfectly machined electrodes. Even if the trap design is as ideal as possible, it can not be ideal for both the highest and the lowest order potential at the same time. To illustrate such an effect, let us go back to our first example of an extra quadrupole potential applied to the octupole trap electrodes and let's assume the linear octupole trap has round electrodes designed to generate an rf electric field as close as possible as the ideal equation [16]. On the quadrupole point of view, these electrodes are too small to produce an ideal quadrupole field inside the trap [9] and most importantly, the four non-connected electrodes are equivalent to grounded rods which induce a large deviation from the ideal rf field. Using SIMION software [17], we have calculated the potential created by a quadrupole-like voltage connection on the octupole trap used for the previous m.d. simulations. Its relevant characteristics are an inner radius of r 0 = 0.2 mm and an electrode radius of r 0 /3, which corresponds to the optimisation of the octupole potential [16].

The calculation shows that the actual quadrupole potential deviates from the ideal one but keeps the quadrupole symmetry. In figure 4 is drawn the normalized difference of the actual quadrupole field component E r with respect to the ideal one, E id , along the radial direction under the angle φ ±1 . This value is constant in the central part of the trap (r < r 0 /5) where, providing an E r amplitude 20% higher, the fields can be considered identical. As an example, the ion ring in Figure 2 fills a tenth of the actual trap dimensions. Figure 4 also shows that Eq. 9 becomes position-dependent for an extra quadrupole static voltage, and thus justifies the use of the optimized static octupole voltage if more distant positions were required.

The m.d. simulations using the calculated potential, fitted by a polynomial equation [18] show that the same procedure to create two strings, to bring them to the rf nodes and to laser cool them to the Doppler limit can be performed.

We have shown that a superimposed lower-order rf potential onto the main trapping one adds supplementary nodes in the rf field. An ensemble of ions can be separated in several sub-sets in a controlled manner and lasercooled to the Doppler limit. Once cooled, they can be considered as in a 3D anisotropic harmonic potential and controlled like in a linear quadrupole trap. This opens the way to create parallel ion strings with a large choice of geometries. The distance between the potential minima can be continuously tuned to reach conditions where the Coulomb repulsion between the subsets implies a strong enough correlation to influence their relative equilibrium positions. The different patterns formed by neighboring ions seem specifically interesting for quantum simulations [19,20], but may certainly find interesting applications in other domains.

FIG. 1 .

 1 FIG.1. Connection of a quadrupole rf electric potential onto the electrodes of an octupolar trap. Black and hatched rods show the different polarities of the applied -V4 and +V4 potentials respectively, at time t = 0 s. The broken line rods are only connected to the octupolar potential .

FIG. 2 .

 2 FIG. 2. (Color online) (a)-(d) Pictures of the ion structure at 3 different times (roughly corresponding to the temperature graph below). (e) Temperatures of the different directions of motion for a simulated dynamics detailed in the text. The horizontal dashed line shows the Doppler limit temperature for calcium ions and the vertical dashed lines materialize first, when the quadrupole potential is switched on and second, when the static octupole potential is switched on.

FIG. 3 .

 3 FIG. 3. Diagram of the rf cylindrical rods of different trapswhere the filled and hatched rods are, respectively, the additional -V2p and +V2p at t = 0. Inner crosses stand for the extra field-free positions of (a) octupole trap with additional quadrupole potential, (b) dodecapole trap with hexapole potential, and (c) dodecapole trap with quadrupole potential.

FIG. 4 .

 4 FIG.4. Normalized difference of the quadrupole electric field, Er, and the ideal one, E id , along the radial direction under the angle φ±1.


R Gribonval
email: remi.gribonval@inria.fr

SPARSE DECOMPOSITION OF STEREO SIGNALS WITH MATCHING PURSUIT AND APPLICATION TO BLIND SEPARATION OF MORE THAN TWO SOURCES FROM A STEREO MIXTURE

We develop a method of sparse decomposition of stereo audio signals, and test its application to blind separation of more than two sources from only two linear mixtures. The decomposition is done in a stereo dictionary which we can define based on any standard time-frequency or time-scale dictionary, such as the multiscale Gabor dictionary. A decomposition of a stereo mixture in the dictionary is computed with a Matching Pursuit type algorithm called Stereo Matching Pursuit. We experiment an application to blind source separation with three (mono) sources mixed on two channels. We cluster the parameters of the stereo atoms of the decomposition to estimate the mixing parameters, and recover estimates of the sources by a partial reconstruction using only the appropriate atoms of the decomposition. The method outperforms the best achievable linear demixing by dB to more than dB on our preliminary experiments, and its performance should increase as we let the number of iterations of the pursuit increase. Sample sound files can be found here :

INTRODUCTION

Stereo audio signals can be modeled as a pair of (noisy) mixtures [START_REF] Berger | A method of denoising and reconstructing audio signals[END_REF] of (mono) sources , with an additive (stereo) Gaussian noise and a pair of linear filters. In the natural acoustic mixing that occurs during the simultaneous recording of several sources with a pair of microphones, each pair of filters depends on the spatial location of the source relatively to the sensors. In the anechoic case, they can be modeled as gain-delay filters, and the stereo mixture can be expressed as [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] where is a panpot parameter : corresponds to mixing entirely on the left channel, mixes it to the right channel.

Mono audio sources , which can be considered as vectors in the Hilbert space of finite energy signals, have been shown to have sparse decompositions in a variety of time-frequency dictionaries (e.g. local trigonometric bases [START_REF] Berger | A method of denoising and reconstructing audio signals[END_REF], wavelet or wavepacket bases [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] or the union of them [START_REF] Daudet | Représentations structurelles de signaux audiophoniques : m éthodes hybrides pour des applications à la compression[END_REF], or the Gabor multiscale dictionary [START_REF] Torrésani | Wavelets associated with representations of the affine Weyl-Heisenberg group[END_REF][START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF][START_REF] Qian | Signal representation using adaptive normalized Gaussian functions[END_REF]). By sparse representation we mean that with (we denote the dictionary by ) and the sequence has a fast decay when tends to infinity. It follows that the stereo mixture , which lies in the Hilbert space stereo of finite energy stereo signals, has a representation as [START_REF] Daudet | Représentations structurelles de signaux audiophoniques : m éthodes hybrides pour des applications à la compression[END_REF] in the stereo dictionary stereo of stereo atoms [START_REF] Torrésani | Wavelets associated with representations of the affine Weyl-Heisenberg group[END_REF] where is a (mono) atom, is a panpot parameter, and a delay parameter which can be restricted to where is the maximum delay between channels. Given the stereo mixture , we propose to decompose it on a stereo dictionary using a Matching Pursuit type algorithm [START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF]. After iterations, is decomposed as [START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF] c 2002 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

where is a residual. We define the Stereo Matching Pursuit algorithm in Section 2.

In Section 3 we experiment a simple blind source separation algorithm based on Stereo Matching Pursuit decomposition. The idea, which was exploited in [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF] using the complex spectrogram as a sparse representation of each channel, is that each stereo atom in the decomposition (5) corresponds to an atom in the representation (3) for some , hence every pair from ( 5) yields an estimate of the panpot and relative delay . By clustering one can estimate the number of sources and partition the indexes into classes . As mentioned in [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF], in the case of sources, it is possible to estimate a demixing matrix from the clusters, and the sources can be estimated linearly by applying the demixing matrix. We are more interested in the case of sources : then, no linear demixing can in general completely separate the sources. However by partitioning the decomposition (5), one can obtain a nonlinear estimate of the sources (up to gain and delay) [START_REF] Qian | Signal representation using adaptive normalized Gaussian functions[END_REF] In a way, this extends the complex spectrogram-based blind source separation technique proposed in [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF] by providing a method for adaptively choosing the size of the window.

STEREO MATCHING PURSUIT

We recall the definition of the Matching Pursuit algorithm [START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF] and specialize it to the setting of stereo signals. Given a complete dictionary , i.e. a family of unit vectors in a Hilbert space that spans a dense subspace of (note : it can easily be checked that if is complete in then stereo is complete in stereo ), and an arbitrary number , the Matching Pursuit decomposes any signal into a linear combination of atoms chosen among and a residual term as in [START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF]. The strong convergence of the algorithm was proved by Jones [START_REF] Jones | On a conjecture of Huber concerning the convergence of PP-regression[END_REF] and shows that one can get as good an approximation to as wanted by taking big enough.

Standard algorithm

Standard Matching Pursuit goes as follows. From a decomposition of into atoms, one gets an -atom decomposition in the following way :

1. Compute for all .

2. Select the best atom of the dictionary 3. Compute the new residual [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF] with .

Stereo Matching Pursuit

For any mono atom and delay parameter , the pair of stereo vectors is an orthonormal basis of its linear span in stereo , which we will call a stereo subspace. The orthonormal projection of onto is given by [START_REF] Jones | On a conjecture of Huber concerning the convergence of PP-regression[END_REF] and one can easily check that Assume that for any and , : it follows that Matching Pursuit with stereo goes as :

1. Compute for and all .

2. Compute for all and .

3. Select the best stereo subspace Hence, no exhaustive search over the panpot parameter is needed for the optimization of a stereo atom. The complexity of iterations of Stereo Matching Pursuit for a signal of samples is essentially twice that of standard Matching Pursuit, i.e. [START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF][START_REF]Fast matching pursuit with a multiscale dictionary of Gaussian chirps[END_REF] with the usual discretization of the Gabor dictionary [START_REF] Torrésani | Wavelets associated with representations of the affine Weyl-Heisenberg group[END_REF]. c 2002 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

EXPERIMENTS

Using the Matching Pursuit Package of the LastWave program [START_REF] Bacry | LastWave software[END_REF] we have implemented Stereo Matching Pursuit using the stereo dictionary of real Gabor atoms [START_REF] Torrésani | Wavelets associated with representations of the affine Weyl-Heisenberg group[END_REF][START_REF] Mallat | Matching pursuit with timefrequency dictionaries[END_REF] (10) where is a given window of unit energy, e.g. the Gaussian window, is a scale parameter, a time parameter, a frequency parameter and a normalizing constant. Each real Gabor atom is in the linear span of two conjugated complex Gabor atoms [START_REF] Bergeaud | Repr ésentations adaptatives d'images numériques[END_REF] therefore, similarly to what was done above with the panpot parameter , no exhaustive search over the phase parameter is needed [START_REF] Bergeaud | Repr ésentations adaptatives d'images numériques[END_REF][START_REF] Goodwin | Matching pursuit with damped sinusoids[END_REF][START_REF]Fast matching pursuit with a multiscale dictionary of Gaussian chirps[END_REF].

Stereo Matching Pursuit of real audio signals

We performed experiments on a mixture of three sources : is a recording of cello; is a recording of drums; is a recording of piano. Each source is sampled at kHz and we use seconds of each signal, i.e. samples. Stereo Matching Pursuit was performed using iterations. The computation time on a Pentium III 750 MHz laptop was about minutes. We display on Figure 1 [START_REF]Fast matching pursuit with a multiscale dictionary of Gaussian chirps[END_REF]. We expect the computation time to be divided by about 25, thus enabling more iterations and better approximations.

Source separation of panpot mixture

We tested the source separation capabilities of our decomposition method in the case where the sources are mixed using pure panpot [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF], i.e. (see Equation ( 2)) , , . , . , and . For each value of the difference in intensity between channels is at most decibels, hence each of the three sources is perceived strictly between the two ears in binaural hearing [START_REF] Hugonnet | Th éorie et pratique de la prise de son stéréophonique[END_REF].

Stereo Matching Pursuit was performed with . Figure 2 Table 1. Signal to noise ratio between the original and the estimated sources with the Stereo Matching Pursuit based separation (SNR ) and the best linear demixing (SNR ), and their correlation with the contribution of the source to the stereo mixture and the number of atoms used in the estimation (see Equation ( 6)).

we summarized in Table 1, the norm of its contribution to the mixture, the number of atoms used in its c 2002 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

estimation, the absolute signal to noise ratio (SNR) in decibels SNR and, as a reference, the value SNR of the best SNR attainable by linear demixing. One can notice that the SNR logically increases as the contribution of the source to the stereo mixture increases. Moreover, Stereo Matching Pursuit separation outperforms the best linear demixing by 3 dB (for the piano) to more than dB (for the drums). It seems that the larger the number of atoms used in the estimation of a source, the larger the improvement over linear demixing. Hence, we expect the SNR to improve if we make more iterations of the Stereo Matching Pursuit.

CONCLUSION AND ONGOING WORK

In this paper we presented a new method of decomposition of stereophonic audio signals, using the notion of stereo time-frequency dictionary and a Matching Pursuit approach. With the stereo Gabor dictionary, we performed blind source separation by clustering the decomposition coefficients and partial reconstruction. Among other potential applications of the decomposition method, let us point out the modification of the stereo image (i.e. remixing at the user end) by changing the and parameters before reconstruction, as well as techniques of compression of audio signals where, depending on the available bitrate, we may choose to respect more or less the stereo image by using more or less bits to code and .

At the time of writing this paper, we were about to test an implementation of a fast version of the Stereo Matching Pursuit decomposition, which we expect to multiply the computation speed by 25. Using the fast algorithm, we will make experiments with more iterations of the pursuit, leading to smaller energy of the residue . Therefore, we expect to see an improvement in the SNR of the blind separation application.

Because acoustic stereo recordings generally involve a combination of phase and intensity stereophony, we plan to turn to source separation with nonzero delays . Hence, we will have to deal with the fact that the relative delay is more reliably estimated from short atoms of the decomposition (i.e. with a small scale ), while the phase difference is more reliable for longer atoms. Moreover, we will likely need to cluster simultaneously the panpot, delay and phase difference parameters of the decomposition atoms. We may eventually have to investigate modified decomposition algorithms where some automatic clustering is done adaptively at each iteration of the pursuit, driving the selection of the next atom.

1 Fig. 1 .

 11 Fig. 1. Decay (in decibels) of the relative approximation error as a function of the number of stereo atoms.

Fig. 2 .

 2 Fig. 2. Histogram of from a Stereo Matching Pursuit decomposition of a panpot mixture of three audio signals. One can observe three peaks, centered around the values and (see text).

c 2002 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

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