Marco Barchiesi

Antoine Gloria

New counterexamples to the cell formula in nonconvex homogenization

Keywords: homogenization, quasiconvexity, cell formula 2000 Mathematics Subject Classification: 35B27, 49J45, 73B27, 74E30, 74Q05

In this article we show that for the homogenization of multiple integrals, the quasiconvexification of the cell formula is different from the asymptotic formula in general. To this aim, we construct three examples in three different settings: the homogenization of a discrete model, the homogenization of a composite material and the homogenization of a homogeneous material on a perforated domain.

I ε (u) := Ω∩εP W x ε , ∇u(x) dx,
with suitable assumptions (recalled in Section 2.1), the Γ-limit writes

I hom (u) := Ω W hom ∇u(x) dx,
where W hom is obtained by an asymptotic formula on the number of periodic cells considered. If the integrand W (x, •) happens to be convex almost everywhere, then the asymptotic formula reduces to a minimization problem on the unitary cell with periodic boundary conditions, that we denote by W cell . A counterexample due to Stefan Müller in [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF] shows that in general, for quasiconvex nonconvex energy densities, the inequality W cell ≥ W hom can be strict. More recently, Jean-François Babadjian and the first author gave another such example in [START_REF] Babadjian | A variational approach to the local character of G-closure: the convex case[END_REF].

As will be made precise in Section 2 for both examples, the energy density W cell is not rank-one convex. In addition, in both cases, considering the quasiconvex envelop QW cell of W cell surprisingly removes the contradiction which allows to conclude that W cell > W hom . Hence, none of the known examples shows that the inequality QW cell ≥ W hom can be strict, although this is to be expected.

The aim of this paper is twofold: to show that the known counterexamples to the cell formula are not rigid enough to prove that QW cell > W hom , and then to provide some new examples for which the latter strict inequality can be shown. The article is organized as follows. In Section 2, we recall standard results on homogenization as well as the two counterexamples to the cell formula mentioned above. We then show for each example that the methods used by their respective authors to prove the disagreement of W cell with W hom fail to prove the disagreement of QW cell with W hom . The rest of the paper is then dedicated to the construction of three different examples for which there exists a deformation gradient Λ such that QW cell (Λ) > W hom (Λ). The examples are built in dimension two and they are based on the fact that replacing (0, 1) 2 -periodicity by (0, 2) 2 -periodicity is enough to relax significantly the energy to obtain the desired strict inequality. The first example is a discrete example where the keyrole is played by the very strong rigidity of discrete gradients. The second example is based on the same geometry but is written in a continuous setting and exploits the rigidity of the incompatible twowell problem together with an interplay between the geometry, the periodicity and the zero levelset of the energy densities. These examples are presented in Section 3. The last example is the object of Section 4. It relies on the homogenization of a homogeneous material on a perforated domain, for which we prove that the zero levelset of W cell is contained in a quasiconvex set which is strictly contained in the zero levelset of W hom . This is in particular the first example which shows the disagreement of W cell and W hom (as well as QW cell and W hom ) for the homogenization of a homogeneous material on a perforated domain.

Although the main result of this article is technical, we believe the examples are of independent interest. We therefore provide the non-specialist reader with the required background on convexity properties in Section 2.2.

Throughout the paper, we employ the following notation:

• Ω is a bounded open subset of R d ;

• Q = (0, 1) d denotes the unit cell;

• Q n = (0, n) d for all n ∈ N; • Q m = m + Q for all m ∈ Z d ; • χ U is the characteristic function of a subset U of R d ; • M d is the set of d × d real matrices;
• M d sym is the set of d × d symmetric real matrices; • SO d is the family of the elements Λ of M d such that det Λ = 1 and Λ T Λ = I, where I ∈ M d is the identity matrix;

• |Λ| := trace(Λ T Λ) is the Frobenius norm of a matrix Λ ∈ M d ; • L n denotes the n-dimensional Lebesgue measure; • W 1,p per (Q n , R d ) is the space of W 1,p loc (R d , R d ) functions which are Q n -periodic;
• As a general rule, c denotes a constant which may vary from line to line but which is independent of the variables left.

2. Homogenization of multiple integrals and the cell formula 2.1. Continuous and discrete homogenization of nonconvex functionals. In this section, we recall classical results of periodic homogenization of multiple integrals, as well as (less) classical results of periodic homogenization of discrete systems. We refer the reader to the monograph [START_REF] Braides | Homogenization of Multiple Integrals[END_REF] for continuous homogenization and to the article [START_REF] Alicandro | A general integral representation result for the continuum limits of discrete energies with superlinear growth[END_REF] for discrete homogenization.

Definition 1. Let U be a normed space. We say that

I : U → [-∞, +∞] is the Γ-limit of a sequence I h : U → [-∞, +∞],
or that I h Γ-converges to I, if for every u ∈ U the following conditions are satisfied: i) Liminf inequality: for every sequence u h in U such that u h → u,

I(u) ≤ lim inf h→+∞ I h (u h );
ii) Recovery sequence: there exists a sequence u h in U such that u h → u and

I(u) = lim h→+∞ I h (u h ).
Let d ∈ N. We focus on Γ-convergence of integral functionals on the normed space L p (Ω, R d ), p ∈ (1, +∞), in the context of periodic homogenization.

Let a > 0. We denote by W(a, p) the set of all continuous functions W : M d → [0, +∞) satisfying the following coerciveness and growth conditions of order p:

1 a |Λ| p -a ≤ W(Λ) ≤ a(1 + |Λ| p ) for all Λ ∈ M d .
(2.1)

Hypothesis 1. W : R d × M d → [0, +∞) is a Carathéodory function Q-periodic in the first variable such that W (x, •) ∈ W(a, p) for a.e x ∈ Q. Hypothesis 2. P is a Q-periodic and open subset of R d with Lipschitz boundary such that Q \ P ⊂⊂ Q. Note that in particular P is connected.
Under Hypotheses 1 and 2, we consider for any ε > 0 the functional

I ε : L p (Ω, R d ) → [0, +∞] defined by I ε (u) :=    Ω∩εP W x ε , ∇u(x) dx if u| Ω∩εP ∈ W 1,p (Ω ∩ εP, R d ), +∞ otherwise.
Definition 2. We call cell integrand related to (W, P ) the function

W cell : M d → [0, +∞) defined by W cell (Λ) := inf Q∩P W x, Λ + ∇φ(x) dx : φ ∈ W 1,p per (Q, R d ) . (2.2)
If P = R d we simply say that W cell is the cell integrand related to W . We call homogenized integrand related to (W, P ) the function

W hom : M d → [0, +∞) defined by W hom (Λ) := lim n→∞ 1 n d inf Qn∩P W x, Λ + ∇φ(x) dx : φ ∈ W 1,p per (Q n , R d ) .
If P = R d we simply say that W hom is the homogenized integrand related to W .

The following theorem is a standard result (See [

I hom (u) :=    Ω W hom ∇u(x) dx if u ∈ W 1,p (Ω, R d ), +∞ otherwise.
In addition, if W (x, •) is convex for a.e. x ∈ Q, then W hom is also convex and coincides with the cell integrand W cell related to (W, P ).

A result similar to Theorem 1 holds in a discrete setting, as shown in [START_REF] Alicandro | A general integral representation result for the continuum limits of discrete energies with superlinear growth[END_REF]. We give here a simpler version, for which we only consider nearest-neighbors interactions. We also need to slightly extend the result in [START_REF] Alicandro | A general integral representation result for the continuum limits of discrete energies with superlinear growth[END_REF] to take into account volumetric effects, which we will need in Section 3. Yet, the result remains essentially the same as announced in [START_REF] Alicandro | Mathematical derivation of a rubber-like stored energy functional[END_REF], and further details will be given in [START_REF] Alicandro | Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity[END_REF]. Definition 3. Let T be a Q-periodic triangulation of R d and P be the set of vertices of T . We define the couples of nearest neighbors by

N N := (x, y) ∈ (P ∩ Q) 2 : ∃T ∈ T having [x, y] as an edge .
For all ε > 0 and for all bounded open subset U of R d , we define

S ε (U, R d ) := u ∈ C 0 (U, R d ) : u is affine on each element T ∈ εT ∩ U . For ε = 1, we simply write S(U, R d ) = S ε (U, R d ). Moreover, we write S per (Q n , R d ) := u ∈ S(Q n , R d ) : u is Q-periodic .
We are now in position to define energy functionals on discrete systems.

Definition 4. Let f 1 : Q × Q × R d → [0, +∞) and f 2 : Q × R → [0, +∞) be continuous functions and let ε > 0. For any bounded open subset U of R d , we define the energy of u ∈ S ε (U, R d ) as F ε (u, U ) := m∈Z d : εQ m ⊆U F m ε (u),
(with the convention Ø = 0) where, for any m ∈ Z d such that εQ m ⊆ U ,

F m ε (u) := ε d (x,y)∈N N f 1 x, y, u(εm + εx) -u(εm + εy) ε|x -y| +ε d Q f 2 x, det ∇u(εm+εx) dx.
If ε = 1, we simply write F (u, U ) = F ε (u, U ). Finally, we define the functional I ε :

L p (Ω, R d ) → [0, +∞] by I ε (u) := F ε (u, Ω) if u ∈ S ε (Ω, R d ), +∞ otherwise.
As for the continuous setting, we may define a cell integrand and a homogenized integrand as follows.

Definition 5. For all Λ ∈ M d , let ϕ Λ : R d → R d be given by ϕ Λ (x) := Λ • x. We call cell integrand related to (T , F ) the function W cell : M d → [0, +∞) defined by

W cell (Λ) := inf F (ϕ Λ + φ, Q) : φ ∈ S per (Q, R d ) .
We call homogenized integrand related to (T , F ) the function W hom : M d → [0, +∞) defined by

W hom (Λ) := lim n→∞ 1 n d inf F (ϕ Λ + φ, Q n ) : φ ∈ S per (Q n , R d ) .
We have the following result (see [START_REF] Alicandro | A general integral representation result for the continuum limits of discrete energies with superlinear growth[END_REF] and [START_REF] Alicandro | Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity[END_REF]).

Theorem 2. Let T , f 1 , f 2 , I ε and W hom be as in Definitions 3,[START_REF] Babadjian | A variational approach to the local character of G-closure: the convex case[END_REF][START_REF] Ball | Fine mixtures as minimizers of energy[END_REF] Let us further assume that there exist a > 0 and p ∈ (1, ∞) such that

0 ≤ f 2 (x, z) ≤ a(1 + |z| p/d ) for all (x, z) ∈ Q × R, 1 a |w| p -a ≤ f 1 (x, y, w) ≤ a(1 + |w| p ) for all (x, y, w) ∈ Q × Q × R d ,
Then the homogenized integrand W hom associated to (T , F ) is a quasiconvex function satisfying a growth condition (2.1), and for any ε h ց 0 + the sequence I ε h Γ-converges to the functional I hom : L p (Ω, R d ) → [0, +∞] defined by

I hom (u) :=    Ω W hom ∇u(x) dx if u ∈ W 1,p (Ω, R d ), +∞ otherwise.
In addition, if f 2 ≡ 0 and if f 1 (x, y, •) is a convex function for all x, y ∈ Q, then W hom is also convex and coincides with the cell integrand W cell related to (T , F ).

2.2. Short summary of convexity properties. In this section, we recall the notions of polyconvexity, quasiconvexity and rank-one convexity of functions and sets. We refer the reader to [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF], [START_REF] Fonseca | Modern methods in the Calculus of Variations: Sobolev spaces[END_REF], [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF] for details. We also state and prove some elementary lemmas that will be used in the analysis of the counterexamples. Definition 6. (quasiconvex function) A locally bounded and Borel measurable function

W : M d → R is said to be quasiconvex if W (Λ) := inf Q W Λ + ∇φ(x) dx : φ ∈ W 1,∞ 0 (Q, R d ) .
Given W : M d → R we define the quasiconvex envelope of W as

QW (Λ) := sup V (Λ) such that V is quasiconvex and V ≤ W
with the convention sup Ø = -∞. If QW = -∞, then it is quasiconvex function. Moreover, if W is also locally bounded and Borel measurable, then

QW (Λ) = inf - U W Λ + ∇φ(x) dx : φ ∈ W 1,∞ 0 (U, R d ) , (2.3)
where U is any bounded open subset of R d . In particular, the infimum in the formula is independent of the choice of

U . If U = Q, then W 1,∞ 0 (Q, R d ) can be replaced by W 1,∞ per (Q, R d ).
Lemma

ϕ ∈ W 1,p (Q ∩ P, R d ) can be extended to a function ϕ ∈ W 1,p (Q, R d ). As a consequence, {φ |Q∩P : φ ∈ W 1,p per (Q, R d )} is a weakly closed subset of W 1,p (Q ∩ P, R d ).
Definition 7. (polyconvex function) For any matrix Λ ∈ M d , let denote by M(Λ) the vector that consists of all minors of Λ, and denote by δ(d) its length. We can identify M(Λ) with a point of R δ (d) . We say that a function W : M d → R is polyconvex if there exists a convex function g : R δ(d) → R such that for all Λ ∈ M d , W (Λ) = g(M(Λ)).

Definition 8. (rank-one convex function) We say that W :

M d → R is rank-one convex if W (tA + (1 -t)B) ≤ tW (A) + (1 -t)W (B)
for all t ∈ [0, 1] and for all A, B ∈ M d rank-one connected, i.e., such that rank(B -A) = 1.

Given W : M d → R we define the rank-one convex envelope of W as

RW (Λ) := sup V (Λ) such that V is rank-one convex and V ≤ W with the convention sup Ø = -∞. If RW = -∞, then it is rank-one convex function. Lemma 2. Let W : M d → R, then there holds W is convex =⇒ W is polyconvex =⇒ W is quasiconvex =⇒ W is rank-one convex.
One can extend the notions of convexity, polyconvexity, quasiconvexity and rank-one convexity to sets. Definition 9. (polyconvex, quasiconvex and rank-one convex sets) Let K be a compact subset of M d . We define the polyconvex hull K pc , quasiconvex hull K qc and rank-one convex hull K rc of K by

K pc := Λ ∈ M d : f (Λ) = 0 ∀ f : M d → [0, +∞) polyconvex such that f | K ≡ 0 , K qc := Λ ∈ M d : f (Λ) = 0 ∀ f : M d → [0, +∞) quasiconvex such that f | K ≡ 0 , K rc := Λ ∈ M d : f (Λ) = 0 ∀ f : M d → [0, +∞) rank-one convex such that f | K ≡ 0 . The set K is said to be polyconvex if K = K pc , quasiconvex if K = K qc and rank- one convex if K = K rc . We have the inclusions K rc ⊆ K qc ⊆ K pc ⊆ K co ,
where the superscript co denotes the classical convex hull.

We have the following useful characterizations of K qc and K pc .

Lemma 3. ([14, Theorem 4.10]). Given a compact set K ⊆ M d , a matrix A ∈ M d belongs to K qc if and only if there exists a sequence ψ h bounded in W 1,∞ (Q, R d ) such that dist(∇ψ h , K) → 0 in measure; ψ h (x) = A • x for x ∈ ∂Q. Lemma 4. ([12, Lemma 1]). Given a compact set K ⊆ M d , a matrix A ∈ M d belongs to K pc if and only if M(A) lies in {M(Λ) : Λ ∈ K} co .
The sets we will be interested in are the zero-levelsets of energy densities defined as follows.

Definition 10. Let W : M d → [0, +∞) be a continuous function, we define its zero levelset as

W -1 (0) = Λ ∈ M d : W (Λ) = 0 . In particular, if W is a quasiconvex function, then W -1 (0) is a quasiconvex set. Lemma 5. If W ∈ W(a, p), then QW -1 (0) = (W -1 (0)) qc .
Proof. Let K := W -1 (0). The inclusion K qc ⊆ QW -1 (0) is trivial and we only need to prove the opposite one. This proof makes use of Young measures, for which we refer the reader to [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF] for a comprehensive treatment.

Let A ∈ QW -1 (0). By formula (2.3), there exists

φ h ∈ W 1,∞ 0 (Q, R d ) such that 0 = QW (A) = lim h→+∞ Q W A + ∇φ h (x) dx.
As a consequence of the p-coercivity of W and Poincaré's inequality, the sequence

ψ h (x) := A • x + φ h (x) is bounded in W 1,p (Q, R d ). Thus, up to extraction, ∇ψ h generates a Young measure µ : Q ∋ x → µ x ∈ P(M d )
, where P(M d ) denotes the family of probability measures on M d . By the fundamental theorem on Young measures (see [14, Theorem 3.1]), we get

0 = lim h→+∞ Q W A + ∇φ h (x) dx ≥ Q M d W (Λ)dµ x (Λ) dx
and therefore by [6, Lemma 3.3] suppµ x ⊆ K for L d a.e. x ∈ Q. Again by the fundamental theorem, this implies that dist(∇ψ h , K) → 0 in measure.

(2.4)

By using Zhang's lemma (see [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF]Lemma 4.21]), ψ h can be modified on small sets so that its gradient be bounded in L ∞ (Q, M d ), while keeping conditions (2.4) and ψ h (x) = A • x for x ∈ ∂Q. The thesis follows now by Lemma 3.

We will also make use of the following results about the cell integrand. Lemma 6. Assume that W satisfies Hypothesis 1 and that P satisfies Hypothesis 2. Then the cell integrand W cell related to (W, P ) is a continuous function.

Proof. This property is a direct consequence of the following inequality:

W cell (Λ 1 ) ≤ W cell (Λ 2 ) + c(Λ 1 , Λ 2 )|Λ 1 -Λ 2 | for all Λ 1 , Λ 2 ∈ M d , (2.5)
where c(Λ 1 , Λ 2 ) is locally uniformly bounded.

Let us prove inequality (2.5). By Lemma 1 and Remark 1, we have

W cell (Λ) = min Q∩P QW x, Λ + ∇φ(x) dx : φ ∈ W 1,p per (Q, R d ) . (2.6)
Due to the growth condition from above satisfied by QW (x, •), there exists c > 0 such that for every Λ 1 , Λ 2 ∈ M d , there holds

|QW (x, Λ 1 ) -QW (x, Λ 2 )| ≤ c|Λ 1 -Λ 2 |(1 + |Λ 1 | p-1 + |Λ 2 | p-1 )
This property, which is classical for convex functions, holds for rank-one convex functions (see [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Lemma 5.2]). Let now Λ 1 , Λ 2 ∈ M d , and let φ 1 , φ 2 ∈ W 1,p per (Q, R d ) be minimizers associated with Λ 1 and Λ 2 through (2.6). We then have

W cell (Λ 1 ) -W cell (Λ 2 ) = Q∩P QW x, Λ 1 + ∇φ 1 (x) -QW x, Λ 2 + ∇φ 2 (x) dx ≤ Q∩P QW x, Λ 1 + ∇φ 2 (x) -QW x, Λ 2 + ∇φ 2 (x) dx ≤ Q∩P c|Λ 1 -Λ 2 |(1 + |Λ 1 | p-1 + |Λ 2 | p-1 + |Λ 2 + ∇φ 2 (x)| p-1 )dx.
Using the coercivity of QW (lower bound in (2.1)), we may bound Λ 2 + ∇φ 2 p L p from above by the energy, which is less than c(1 + |Λ 2 | p ) using the test function φ ≡ 0 and the upper bound of (2.1). Hence, there exists a constant c > 0 such that the inequality

W cell (Λ 1 ) -W cell (Λ 2 ) ≤ c|Λ 1 -Λ 2 |(1 + |Λ 1 | p-1 + |Λ 2 | p-1 ),
holds for any Λ 1 , Λ 2 ∈ M d , which proves the claim. Lemma 7. Let W ∈ W(a, p) and let P satisfy Hypothesis 2. Assume in addition that W is quasiconvex, and that W -1 (0) is not empty. Then A ∈ M d belongs to the zero levelset of the cell integrand W cell associated to (W, P ) if and only if there exists a Q-periodic

Lipschitz function φ : R d → R d satisfying A + ∇φ(x) ∈ W -1 (0) for a.e. x ∈ Q ∩ P.
Proof. The condition is obviously sufficient. By Lemma 1 and Remark 1, we have

W cell (Λ) = min Q∩P W Λ + ∇φ(x) dx : φ ∈ W 1,p per (Q, R d ) . ( 2

.7)

Let A ∈ W -1 cell (0) and let φ ∈ W 1,p per (Q, R d ) be a minimizer associated with A through (2.7). Then W A + ∇φ(x) = 0 for a.e. x ∈ Q ∩ P. A similar characterization of the levelset of the cell integrand holds in the case of mixtures.

Lemma 8. ([4, Lemma 4.4]). Let W 1 , W 2 ∈ W(a, p) be two quasiconvex functions such that W -1 1 (0) and W -1 2 (0) are not empty. Given a measurable subset U of Q, let set W : R d × M d → [0, +∞) as W (x, Λ) := χ(x)W 1 (Λ) + (1 -χ(x))W 2 (Λ),
where χ is defined by χ := χ U in Q and extended by periodicity to the whole R d . Then A ∈ M d belongs to the zero levelset of the cell integrand W cell associated to W if and only if there exists a Q-periodic Lipschitz function φ :

R d → R d satisfying A + ∇φ(x) ∈ W -1 1 (0) for a.e. x ∈ U W -1 2 (0) for a.e. x ∈ Q \ U . Remark 2.
The previous lemma shows that in the case of a mixture of the type W = χW 1 + (1χ)W 2 , the zero levelset of W cell depends only on the zero levelsets of W 1 , W 2 and not on their global shapes or growths. The same property can be proved for the zero levelset of W hom (see [START_REF] Barchiesi | Loss of polyconvexity by homogenization: a new example[END_REF]Theorem 1.3]). This fact is one of the keys of our counterexamples: we have to introduce suitable zero levelsets first, and only afterwards construct suitable functions.

2.3. Stefan Müller's counterexample. The energy under consideration W η : R 2 × M 2 → [0, +∞), (x, Λ) → χ η (x)W 0 (Λ) models a two-dimensional laminate composite, made of a strong material and a soft material. The coefficient

χ η is the Q-periodic extension on R 2 of χ η (x) := 1 if x 1 ∈ (0, 1/2) η if x 1 ∈ [1/2, 1)
,

where Q ∋ x = (x 1 , x 2 ) and η > 0. The energy density W 0 : M 2 → [0, +∞) is given by W 0 (Λ) = |Λ| 4 + f (det Λ) where f (z) :=        8(1 + a) 2 z + a -8(1 + a) -4 if z > 0 8(1 + a) 2 a -8(1 + a) -4 - 8(1 + a) 2 a 2 z if z ≤ 0
for some a ∈ (0, 1/2).

In particular, W η (x, •) is a nonnegative polyconvex function satisfying a standard growth condition (2.1) of order p = 4. Its zero levelset is SO 2 for all x ∈ Q.

We respectively denote by W η cell and W η hom the cell integrand and the homogenized integrand associated with W η .

Using the one-well rigidity (Liouville theorem) on the unitary cell and using 'bucklinglike' test-functions on several periodic cells (see Figure 2.3), Stefan Müller obtained the following result. However, in view of the following proposition, Theorem 3 does not allow to conclude whether the inequality QW η cell (Λ) ≥ W η hom (Λ) may be strict or not. Note, in particular, that W η cell is not even a rank-one convex (and a fortiori not quasiconvex). Proposition 1. For all λ ∈ (0, 1), there exists c > 0 independent of η such that

RW η cell (Λ) ≤ η c, (2.8)
where Λ := diag(1, λ).

Proof. Let {e 1 , e 2 } be the canonical basis in R 2 . Since, by Jensen's inequality,

RW η cell (Λ) ≤ inf 1 0 W η cell Λ + φ ′ (t)e 1 ⊗ e 2 dt : φ ∈ W 1,∞ per (0, 1)
it is enough to exhibit a test function φ ∈ W 1,∞ per (0, 1) such that the majoration in (2.8) holds. Let φ ∈ W 1,∞ per (0, 1) be such that

φ ′ (t) = χ(t) 1 -λ 2 ,
where χ is the (0, 1)-periodic extension on R of

χ(t) := 1 if t ∈ (0, 1/2) -1 if t ∈ [1/2, 1)
.

We also choose ϕ ∈ L ∞ (0, 1), W 1,∞ per (Q, R 2 ) such that ∇ y ϕ(t, y) = χ(y 1 ) λ -1 0 -χ(t) √ 1 -λ 2 0 ,
Then, in the strong phase ( χ(y 1 ) = 1), the test function Λ + φ ′ (t)e 1 ⊗ e 2 + ∇ y ϕ(t, y) is the rotation

λ ± √ 1 -λ 2 ∓ √ 1 -λ 2 λ ,
and, in the soft phase ( χ(y 1 ) = -1), the deformation gradient is of the form

A := 2 -λ ± √ 1 -λ 2 ± √ 1 -λ 2 λ .
(2.9)

Hence,

RW η cell (Λ) ≤ 1 0 Q W η y, Λ + φ ′ (t)e 1 ⊗ e 2 + ∇ y ϕ(t, y) dy dt ≤ 1 2 ηW 0 (A),
for some A of the form (2.9).

W -1 1 (0) = {O, A} and W -1 2 (0) = {O, I}. We define W : R 2 × M 2 → [0, ∞) by W (x, Λ) := χ(x)W 1 (Λ) + (1 -χ(x))W 2 (Λ),
where χ is given by χ := χ (0,1/2)×(0,1) in Q and extended by periodicity to the whole R 2 .

Then, Jean-François Babadjian and the first author proved in [START_REF] Babadjian | A variational approach to the local character of G-closure: the convex case[END_REF]

, Example 6.1] that W cell (I) = W cell (B) = W hom (I) = W hom (B) = 0 and W cell (C) > 0. Since C ∈ {O, B} rc
and W hom is rank-one convex, this implies that W hom (C) = 0 < W cell (C). However, one also has RW cell (C) = 0.

Counterexamples from composite materials

In this section, we propose two examples for which QW cell (Λ) > W hom (Λ) for some Λ ∈ M 2 . The first one relies on the rigidity of periodic discrete gradients, whereas the second example uses the rigidity of the incompatible two-well problem together with the periodicity constraint. Both examples are based on the same geometry (see Figures 2 and4). Geometry. The triangulation T of R 2 is the Q-periodic replication of the triangulation {T 1 , . . . , T 8 } of Q (see the numerotation of Figure 2) defined as follows.

T 1 := {x ∈ Q : x 2 ≥ x 1 + 1/2}, T 5 := T 4 -(1/2, -1/2), T 3 := {x ∈ Q : x 2 ≤ -x 1 + 1/2}, T 6 := T 3 + (1/2, 1/2), T 2 := {x ∈ Q : x 2 ≥ -x 1 + 3/2}, T 7 := T 2 -(1/2, 1/2), T 4 := {x ∈ Q : x 2 ≤ x 1 -1/2}, T 8 := T 1 + (1/2, -1/2). (3.1)
We will make use of the following notation: for all n ∈ N, T n denotes the restriction of T to Q n , while for all m ∈ Z 2 and τ ∈ {1, . . . , 8}, T m τ = T τ + m. Moreover, for i ∈ {1, 2, 3}, we denote by x m τ,i the i th vertex of the triangle T m τ and we set N m τ := {x m τ,1 , x m τ,2 , x m τ,3 }. For the vertices in ∂Q we use a simpler notation: y 1 := (0, 1/2), y 2 := (1/2, 0), y 3 := (1, 1/2), y 4 := (1/2, 1), y 5 := (0, 1), y 6 := (0, 0), y 7 := (1, 0), and y 8 := (1, 1).

0 0 1 1 0 0 1 1 00 00 11 11 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1
Energy. Let U be a bounded open subset of R 2 . Given u ∈ S(U, R 2 ), m ∈ Z 2 , τ ∈ {1, . . . , 8}, and i ∈ {1, 2, 3}, if U ∩ T m τ = Ø and x m τ,i ∈ U , we set ∇u m τ := ∇u |T m τ (which is constant on T m τ ) u m τ,i := u(x m τ,i ).
Let f 1 , f 2 : R 2 → [0, +∞) be defined by

f 1 (z) :=(z 2 -1) 2 f 2 (z) :=(z -1) 2 .
As in Definition 4, for all η > 0 we consider the energy

F η (u, U ) := m∈Z 2 : Q m ⊆U F η,m (u),
where, for any m ∈ Z 2 such that Q m ⊆ U ,

F η,m (u) := 4 τ =1   1 8 f 2 (det ∇u m τ ) + i,j∈N m τ ,i<j η 2 f 1 |u m τ,i -u m τ,j | |x m τ,i -x m τ,j |   + 8 τ =5   η 8 f 2 (det ∇u m τ ) + i,j∈N m τ ,i<j η 2 f 1 |u m τ,i -u m τ,j | |x m τ,i -x m τ,j |   .
The model satisfies the assumptions of Theorem 2. We respectively denote by W η cell and W η hom the cell integrand and the homogenized integrand associated with (T , F η ).

Theorem 4. For all λ > 1 and η > 0 the following properties hold 1) QW η cell (λI) is bounded from below by (λ 2 -1) 2 /2; 2) there exists c > 0 independent of η such that W η hom (λI) ≤ η c. Therefore the strict inequality QW η cell (λI) > W η hom (λI) holds provided η is small enough.

Proof. Property 1). For all η > 0, let consider the energy

F η (u, U ) := m∈Z 2 : Q m ⊆U F η,m (u),
where, for any m ∈ Z 2 such that Q m ⊆ U ,

F η,m (u) := 4 τ =1 1 8 f 2 (det ∇u τ ) + 8 τ =5 η 8 f 2 (det ∇u τ ).
Let W η cell be the cell integrand associated with (T , F η ). Since we have neglected the contributions of the terms involving f 1 , we have W η cell ≥ W η cell . For all η > 0 and Λ ∈ M 2 invertible, we claim that To prove this assertion, up to multiplying ∇ψ by Λ -1 , it is enough to consider Λ = I. Up to a translation, we can write

W η cell (Λ) = F η (ϕ Λ , Q). Let ψ be an admissible deformation of the form ψ = ϕ Λ + φ, φ ∈ S per (Q, R 2 ).
ψ(y 1 ) = (α 1 , 1/2 + β 1 ) ψ(y 5 ) = (0, 1) ψ(y 3 ) = (1 + α 1 , 1/2 + β 1 ) ψ(y 6 ) = (0, 0) ψ(y 2 ) = (1/2 + α 2 , β 2 ) ψ(y 7 ) = (1, 0) ψ(y 4 ) = (1/2 + α 2 , 1 + β 2 )
ψ(y 8 ) = (1, 1) and the second equation of (3.2) follows now from the fact that Q det(Λ + ∇φ) = det Λ because ± det is quasiconvex.

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 00000000 11111111 2λ λ -z λ Figure 3. Deformation of Q 2 by the Q 2 -periodic competitor ψ. for suitable α 1 , α 2 , β 1 , β 2 ∈ R. A straightforward calculation then shows that det ∇ψ |T 1 = 4 α 1 β 2 -(β 1 -1/2)(α 2 + 1/2) det ∇ψ |T 2 = 4 -α 1 β 2 + (β 1 -1/2)(α 2 -1/2) det ∇ψ |T 3 = 4 -α 1 β 2 + (β 1 + 1/2)(α 2 + 1/2) det ∇ψ |T 4 = 4 α 1 β 2 -(β 1 + 1/2)(α 2 -1/2
Hence, by Jensen's inequality (f 2 is a convex function),

W η cell (Λ) = 1 2 (1 + η)f 2 (det Λ).
Since W η cell is a polyconvex function (hence quasiconvex) not greater than W η cell on M 2 , for all λ ≥ 1 there holds

QW η cell (λI) ≥ W η cell (λI) ≥ 1 2 f 2 (det λI) = 1 2 (λ 2 -1) 2 .
Property 2). Let z λ be a solution of z λ (λ-z λ ) = 1/8. We define a Q 2 -periodic competitor ψ as on Figure 3. The deformation is the symmetrization in Q 2 of the following deformation of the unit cube Q:

ψ(y 1 ) = (0, z λ ) ψ(y 5 ) = (0, λ) ψ(y 3 ) = (λ -z λ , λ) ψ(y 6 ) = (0, 0) ψ(y 2 ) = (λ, λ -z λ ) ψ(y 7 ) = (λ, 0) ψ(y 4 ) = (z λ , λ) ψ(y 8 ) = (λ, λ). T 1 T 2 T 3 T 4 U 2 Figure 4. Geometry.
Since in triangles of the form T m i , i ∈ {1, 2, 3, 4}, where the material is strong,

det ∇ψ = 1 =⇒ f 2 (det ∇ψ) = 0, one has F η (ψ, Q 2 ) = ηF 1 (ψ, Q 2 ). Hence, since ψ -ϕ λI ∈ S per (Q 2 , R 2 ), we have W η hom (λI) ≤ F η (ψ, Q 2 ) ≤ ηF 1 (ψ, Q 2 ).
3.2. An example from solid-solid phase transformations. To build the following counterexample, we introduce energy densities on Q such that a phenomenon similar to the one on Figure 2 may occur at the continuous level. The rigidity now relies on the set of matrices we introduce hereafter.

• Matrices in M 2 A 1 := diag(1, 1), A 2 := diag(4, 3), B 1 := diag(1, 3), B 2 := diag(4, 1),

C := 1 2 diag(5, 4), R := 1 √ 2 1 1 -1 1 . • Compact sets in M 2 K 1 := SO 2 A 1 ∪ SO 2 A 2 , K 2 := SO 2 B 1 ∪ SO 2 B 2 , H 1 := K 1 R, H 2 := (K 2 R) pc .
• Geometry (see Figure 4)

U 1 := 4 i=1 T i , U 2 := Q \ U 1 ,
where T 1 , . . . , T 4 are defined as in (3.1).

The counterexample is as follows.

Theorem 5. Let W 1 , W 2 ∈ W(a, p) be two quasiconvex functions (to be built later) such that

W -1 1 (0) = H 1 and W -1 2 (0) = H 2 . (3.3)
Consider the energy density W : R 2 × M 2 → [0, +∞) defined by

W (x, Λ) := χ(x)W 1 (Λ) + (1 -χ(x))W 2 (Λ),
where χ is given by χ := χ U 1 in Q and extended by periodicity to the whole R 2 . The following properties hold: 1) the cell integrand W cell related to W is bounded from below by a constant c > 0;

2) CR belongs to the zero levelset of the homogenized integrand W hom related to W . Therefore QW cell (CR) ≥ c > W hom (CR).

We will make use of the following facts in the proof.

i) The compact set K 1 is polyconvex and rigid, i.e., if U ⊆ R 2 is an open connected set and ψ : U → R 2 is a Lipschitz function such that ∇ψ(x) ∈ K 1 for a.e. x ∈ U, then ψ is affine. We refer to [START_REF] Šverák | On the problem of two wells[END_REF]Theorem 2] and [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF]Theorem 4.11] for the proofs. Since R is a rotation, the same properties hold for

H 1 . ii) H 1 ∩ H 2 = Ø, because by Lemma 4 H 2 ⊆ Λ ∈ M 2 : det Λ ∈ [3, 4] .
iii) A 1 is rank-one connected to B 1 and B 2 , and A 2 to B 1 and B 2 also. More precisely, denoted by {e 1 , e 2 } the canonical basis in R 2 ,

A 1 -B 1 = -2e 2 ⊗ e 2 A 1 -B 2 = -3e 1 ⊗ e 1 A 2 -B 1 = 3e 1 ⊗ e 1 A 2 -B 2 = 2e 2 ⊗ e 2 .
Proof of Theorem 5. Property 1). Since W cell grows superlineary at infinity and is continuous by Lemma 6, it is enough to prove that W cell (Λ) = 0 for any Λ ∈ M 2 . We procede by contradiction and assume there exists Λ ∈ M 2 such that W cell (Λ) = 0. By Lemma 8, there exists a

Q-periodic Lipschitz function φ : R 2 → R 2 such that Λ + ∇φ(x) ∈ H 1 for a.e. x ∈ U 1 H 2 for a.e. x ∈ U 2 . (3.4)
Due to the rigidity, we infer that there exists D i ∈ H 1 such that Λ+ ∇φ(x) = D i for a.e. x ∈ T i . Again by the rigidity, the periodicity condition implies that there exists D ∈ H 1 such that D i = D for all i ∈ {1, 2, 3, 4}.

By observing that ψ(x) := (Λ -D) • x + φ(x) belongs to W 1,∞ 0 (U 2 ) (up to a translation), from the definition of quasiconvexity we get

W 2 (D) ≤ - U 2 W 2 D + ∇ψ(x) dx = - U 2 W 2 Λ + ∇φ(x) dx = 0 and so D ∈ H 2 , which contradicts H 1 ∩ H 2 = Ø. Property 2). It is sufficient to find φ ∈ W 1,p per (Q 2 , R 2 ) such that Q 2 W x, CR + ∇φ(x) dx = 0.
This can be accomplished by using the following function

ψ : (-1/ √ 2, 1/ √ 2) 2 → R 2 , ψ (1) (x) :=      x 1 if x ∈ (-1/ √ 2, - √ 2/4) × (-1/ √ 2, 1/ √ 2) 4x 1 -3 √ 2/4 if x ∈ [- √ 2/4, √ 2/4] × (-1/ √ 2, 1/ √ 2) x 1 + 3 √ 2/2 if x ∈ ( √ 2/4, 1/ √ 2) × (-1/ √ 2, 1/ √ 2)
;

ψ (2) (x) :=      x 2 if x ∈ (-1/ √ 2, 1/ √ 2) × (-1/ √ 2, - √ 2/4) 3x 2 - √ 2/2 if x ∈ (-1/ √ 2, 1/ √ 2) × [- √ 2/4, √ 2/4] x 2 + √ 2 if x ∈ (-1/ √ 2, 1/ √ 2) × ( √ 2/4, 1/ √ 2)
.

Let ϕ be the (-

1/ √ 2, 1/ √ 2) 2 -periodic extension of ϕ : x → ψ(x) -C • x. Then φ : x → ϕ(R • x)
does the job. Actually, as illustrated on Figure 5,

CR + ∇φ(x + m) ∈ H 1 for (x, m) ∈ U 1 × Z 2 and CR + ∇φ(x + m) ∈ H 2 for (x, m) ∈ U 2 × Z 2 . A 1 R B 2 R A 1 R B 1 R A 2 R B 1 R A 1 R B 2 R A 1 R Figure 5. The values of CR + ∇φ in R -1 (-1/ √ 2, 1/ √ 2) 2 .
On the left the axis are oriented in the directions R -1 e 1 and R -1 e 2 .

To complete the counterexample we need to build two quasiconvex functions W 1 , W 2 satisfying (3.3). Lemma 9. Let H be a compact, polyconvex and frame-invariant subset of M 2 . Given p ∈ (1, +∞), for a suitable a > 0 there exists a quasiconvex frame-invariant function W ∈ W(a, p) such that W -1 (0) = H.

If p ≥ 2, then W can be chosen polyconvex.

Proof. Let V (Λ) := dist(Λ, H) p and set W (Λ) := QV (Λ). By Lemma 5, W -1 (0) = H. Since the Frobenius norm is frame-invariant, the same holds for V , and therefore for W since for all Λ ∈ M 2 and all R ∈ SO 2 ,

W (RΛ) = inf Q V (RΛ + ∇φ)dx : φ ∈ W 1,p 0 (Q, R 2 ) = inf Q V (RΛ + R∇R -1 φ)dx : φ ∈ W 1,p 0 (Q, R 2 ) = inf Q V R(Λ + ∇ϕ) dx : ϕ ∈ W 1,p 0 (Q, R 2 ) = inf Q V (Λ + ∇ϕ)dx : ϕ ∈ W 1,p 0 (Q, R 2 ) = W (Λ).
A different construction allows us to consider a polyconvex energy density in the case p ∈ [2, +∞). We define the functions

V 1 (Λ) := dist(Λ, H co ) p and V 2 (Λ) := dist M(Λ), L p 2 ,
where L := {M(Λ) : Λ ∈ H} co . Both are polyconvex and with p-growth, moreover V 1 is p-coercive and, by Lemma 4, the zero levelset of V 2 is H. The function W := max{V 1 , V 2 } does the job. In addition, it is easy to verify that also in this case W is frame-invariant.

Remark 4. The previous lemma is optimal, because a polyconvex function with subquadratic growth is convex (see [9, Corollary 5.9]).

Comparison of boths examples.

In the discrete example, the zero levelset of the energy density of the strong phase is J = {Λ ∈ M 2 , det Λ = 1}, the space of isochoric deformations, which is not rigid. The rigidity comes from the structure of Q-periodic discrete gradients on T 1 .

In the continuous example, we replace S 1 per (Q, R 2 ) by W 1,p per (Q, R 2 ), hence adding much more flexibility to the periodic gradients. In order to keep the required rigidity, we then replace J by H 1 in the strong phase.

The rigidity of the discrete example lies in (3.2), whereas the rigidity of the continuous example lies in (3.4).

Compared to Stefan Müller's example, the repartition of the strong phase in Q allows to take full advantage of the constraint of periodicity in both examples of this section. On the contrary, in Section 2.3, the periodicity constraint is lost in the x 1 -direction, as shown by Proposition 1.

Counterexample on perforated domains

We obtain the counterexample using the same strategy as for the case of mixtures: we choose a suitable polyconvex function W : M d → [0, +∞) by focusing on its zero levelset K ⊆ M d first. Since Q ∩ P is connected and we need a certain flexibility to construct a Q 2 -periodic competitor, the set K shall not be too rigid.

Let us begin by describing the geometry of the subset P , sketched on Figure 6.

Definition 11. The set P is the complement in R 2 of the set m∈Z 2 O + m, where Let us introduce some sets in the space M 2 , that we will use to describe the energy density.

O := x ∈ Q : x 1 ∈ [1/8, 7/8] and 3x 1 ≤ 4x 2 ≤ 3x 1 + 1 .
Definition 12. Let l 1 := 3/4, l 2 := 9/16 and l 3 := 1/4. We consider the following sets (see Figure 7):

• K 1 := diag(s, 0) : s ∈ (0, 1] and K 2 := diag(0, t) : t ∈ (0, 1] ;

• K := K 1 ∪ K 2 ∪ {diag(0, 0)};
• H := diag(s, t) : s, t ∈ [0, 1] and t ≤ 1s ;

• L := diag(s, t) : 0 < s ≤ l 1 l 3 /l 2 and 0 < t ≤ l 3l 2 s/l 1 ;

• M := diag(s, t) : 0 < s ≤ l 1 /2 and 0 < t ≤ l 3 .

l 3 l 1 l 3 /l 2 1 1 l 3 l 1 /2 1 1 Figure 7. A representation of K ∪ L and K ∪ M in R 2 ,
identified with the set of the diagonal matrices.

The counterexample is as follows.

Theorem 6. Let p ∈ [2, +∞) and let W : M 2 → [0, +∞) be given by

W (Λ) := dist(Λ, H) p + |det(Λ)| p 2
, where H is as in Definition 12. Then, W is polyconvex and belongs to W(a, p) for a suitable a > 0. Moreover, with the notation of Definitions 11 and 12, there holds 1) the zero levelset of the cell integrand W cell related to (W, P ) coincides with K ∪ L;

2) the zero levelset of the homogenized integrand W hom related to (W, P ) contains

K ∪ M ; 3) K ∪ L is quasiconvex. Therefore, for all Λ ∈ (K ∪ M ) \ (K ∪ L) = Ø, QW cell (Λ) > W hom (Λ) = 0. Proof. The set H being convex, Λ → dist(Λ, H) p is a convex function. Since Λ → | det Λ| p 2
is polyconvex, W is polyconvex. It is easy to verify that W satisfies (2.1) for a suitable a > 0. The strict inequality QW cell (Λ) > W hom (Λ) is a direct consequence of 1)-3) using Lemma 5. Let us split the proof of 1)-3) into three steps.

Step 1. Since W -1 (0) = K, by using φ ≡ 0 as a test function in (2.2), we obtain

K ⊆ W -1 cell (0). Let us check that L ⊆ W -1 cell (0). Given s ∈ (0, l 1 l 3 /l 2 ) and t ∈ (0, l 3 -l 2 s/l 1 ), we define ψ : Q → R 2 by ψ (1) (x) :=      0 if x ∈ (0, 1/2) × (0, 1) x 1 -1/2 if x ∈ [1/2, 1/2 + s] × (0, 1) s if x ∈ (1/2 + s, 1) × (0 , 1)
;

ψ (2) (x) :=      0 if x ∈ (0, 1) × (0, 5/8 -t) x 2 -5/8 + t if x ∈ (0, 1) × [5/8 -t, 5/8] t if x ∈ (0, 1) × (5/8, 1) . (4.1)
We have that φ(x)

:= ψ(x) -diag(s, t) • x is Q-periodic and that ∇ψ(x) ∈ K if x ∈ P . More precisely, ∇ψ(x) / ∈ K only if x belongs to [1/2, 1/2 + s] × [5/8 -t, 5/8] ⊆ O.
There, ∇ψ ≡ diag(1, 1) (see Figures 8 and9).

It remains to procede with the delicate part of the argument: the opposite inclusion

W -1 cell (0) ⊆ K ∪ L. Let C = (c ij ) ∈ W -1 cell (0)
. By Lemma 7, there exists a Lipschitz function

ψ : Q → R 2 such that ∇ψ(x) ∈ K for L 2 a.e. x ∈ Q ∩ P and φ(x) := ψ(x) -C • x is Q-periodic.
We will show that ψ is substantially a laminate as in (4.1). Let us point out that if Λ ∈ K, then either Λ 11 = 0 or Λ 22 = 0.

We use the following notation:

L s := r ∈ (0, 1) : (s, r) ∈ ({s} × (0, 1)) ∩ P ;

L s := r ∈ (0, 1) : (r, s) ∈ ((0, 1) × {s}) ∩ P . Since ∂ 2 ψ (1) (x) = 0 for L 2 a.e. x ∈ Q ∩ P , ψ (1) (s, •) is constant along any connected component of L s for all s ∈ (0, 1). In particular for s ∈ (0, 1/8) ∪ (7/8, 1), ψ (1) (s, •) is constant and therefore the (0, 1)-periodicity of φ(s, •) imposes that c 12 = 0. For s ∈ 1/2 Figure 8. The first component of ψ is flat on Q ∩ P with the exception of the grey zone, where the gradient is equal to (1, 0).

Notice that L s = (0, 1) if s ∈ (0,

5/8

Figure 9. The second component of ψ is flat on Q ∩ P with the exception of the grey zone, where the gradient is equal to (0, 1).

[1/8, 7/8], ψ (1) (s, •) is constant on each of the two connected components of L s . Hence, by periodicity, ψ (1) (s, •) is constant on the whole L s .

From the fact that ψ (1) (s, •) is constant along L s for any s ∈ (0, 1), we can deduce that if x ∈ Q ∩ P is a differentiability point for ψ (1) , then ψ (1) is differentiable in all {x 1 } × L x 1 and ∇ψ (1)

(x) = ∇ψ (1) (x) ∀x ∈ {x 1 } × L x 1 . (4.2)
Similarly, one can show that c 21 = 0 and that if x ∈ Q ∩ P is a differentiability point for ψ (2) , then ψ (2) is differentiable in all L x 2 × {x 2 } and ∇ψ (2)

(x) = ∇ψ (2) (x) ∀x ∈ L x 2 × {x 2 }. (4.3)
Let X 1 , X 2 be two L 1 -negligible subsets of the interval (0, 1) such that ψ is differentiable in P := (Q ∩ P ) \ (X 1 × X 2 ) and ∇ψ(x) ∈ K for all x ∈ P . Let us show that, if for some x ∈ P there holds ∇ψ(x) ∈ K 1 , then ∇ψ(x) ∈ K 1 ∪ {diag(0, 0)} ∀x ∈ P ∩ (0, 1) × L x 1 .

(4.4)

In fact, since ∇ψ (1) (x) = (0, 0), we have ∇ψ (1) = (0, 0) in {x 1 } × L x 1 due to (4.2) and therefore ∇ψ ∈ K 1 in {x 1 } × L x 1 \ X 2 . As now ∇ψ (2) ≡ (0, 0) in {x 1 } × L x 1 \ X 2 , (4.3) implies that ∇ψ (2) ≡ (0, 0) in P ∩ (0, 1) × L x 1 . We are in position to conclude the first step. Given s ∈ (0, 1/8) \ X 1 and t ∈ (0, 3/32) \ X 2 , we define the following two sets.

S : = s ∈ (0, 1) \ X 1 : ∂ 1 ψ (1) (s, t) > 0 ; T : = t ∈ (0, 1) \ X 2 : ∂ 2 ψ (2) (s, t) > 0 .
Since ∇ψ ∈ K, ∂ 1 ψ (1) (s, t) ≤ 1 for all s ∈ S, and we infer from (1) (s, t)ds, that 0 ≤ c 11 ≤ L 1 (S). Similarly, 0 ≤ c 22 ≤ L 1 (T ). In particular this shows that

c 11 = 1 0 c 11 + ∂ 1 φ (1) (s, t)ds = 1 0 ∂ 1 ψ (1) (s, t)ds = S ∂ 1 ψ
c 11 , c 22 ∈ [0, 1]. If c 11 > 0, for any ε > 0 there exist s 1 , s 2 ∈ S such that s 2 -s 1 ≥ c 11 -ε. Recalling that if ∇ψ(x) ∈ K and ∂ 1 ψ (1) (x) > 0 then ∇ψ(x) ∈ K 1 , from (4.4), we obtain ∇ψ(x) ∈ K 1 ∪ {diag(0, 0)} ∀x ∈ P ∩ (0, 1) × (L s 1 ∪ L s 2 ) .
Since T ⊆ (0, 1) \ (L s 1 ∪ L s 2 ), we have the estimate

c 22 ≤ L 1 (T ) ≤ max 0, l 2 l 1 (s 1 -s 2 ) + l 3 ≤ max 0, l 2 l 1 (-c 11 + ε) + l 3 .
The arbitrariness of ε > 0 completes the proof of the step.

Step 2. Since W hom is rank-one convex and W -1 hom (0) ⊇ K, it is sufficient to prove that

C := diag(l 1 /2, l 3 ) ∈ W -1 hom (0). Let us construct a Lipschitz function ψ : Q 2 → R 2 such that ∇ψ ∈ K a.e. in Q 2 ∩ P and φ(x) := ψ(x) -C • x is Q 2 -periodic.
In this way we get

W hom (C) ≤ 1 4 Q 2 ∩P W C + ∇φ(x) dx = 0.
Despite the complexity of the following description, the function ψ is very simple. Consider the following sets (see Figure 10):

• U := {x ∈ [1/8, 1] × [3/32, 3/32 + l 2 ] : 4x 2 < 3x 1 }; • U 1 := [0, 1] × [0, 3/32 + l 2 ] \ (U ∪ O); • U 2 := {x ∈ [1/8, 1] × [3/32, 3/32 + l 3 ] : 4x 2 < 3x 1 }; • U 3 := {x ∈ [0, 1] 2 : (1, 1) -x ∈ U 2 }; • U 4 := {x ∈ [0, 1] 2 : (1, 1) -x ∈ U 1 }; • U 5 := U ∪ [0, 1] × [0, 1/8) ; • U 6 := [0, 1/8) × [3/32, 3/32 + l 3 ); • U 7 := {x ∈ [0, 1] 2 : (1, 1) -x ∈ U 6 }; • U 8 := {x ∈ [0, 1] 2 : (1, 1) -x ∈ U 5 }. U 1 U 2 U 3 U 4 3 32 l 2 + 3 32 l 3 + 3 32 U 5 U 6 U 7 U 8

Figure 10

To simplify the exposition, we introduce two auxiliary Lipschitz functions ϕ 1 , ϕ 2 : [0, 1] 2 \ O → R:

ϕ 1 (x) :=          0 if x ∈ U 1 x 2 -3/32 if x ∈ U 2 x 2 -l 2 -3/32 if x ∈ U 3 h 3 if x ∈ U 4 ; ϕ 2 (x) :=          0 if x ∈ U 5 x 2 -3/32 if x ∈ U 6 x 2 -l 2 -3/32 if x ∈ U 7 h 3 if x ∈ U 8 .
Both ϕ 1 and ϕ 2 can be extended to Lipschitz maps in all [0, 1] 2 .

We are now in position to define the desired function ψ : Figures 11,12 and 13).

Q 2 → R 2 (see
ψ (1) (x) :=      0 if x ∈ (0, 9/8] × (0, 2) x 1 -9/8 if x ∈ (9/8, 9/8 + l 1 ] × (0, 2) l 1 if x ∈ (9/8 + l 1 , 2) × (0, 2) ; ψ (2) (x) :=          ϕ 1 (x) if x ∈ (0, 1] × (0, 1] ϕ 2 (x -(1, 0)) if x ∈ (1, 2) × (0, 1] ϕ 1 (x -(0, 1)) + l 3 if x ∈ (0, 1] × (1, 2) ϕ 2 (x -(1, 1)) + l 3 if x ∈ (1, 2) × (1, 2) . (4.5)
Step 3. The set H is convex and so the inclusion (K ∪ L) qc ⊆ H is immediate.

As a direct consequence of [15, Theorem 1], for any C ∈ M 2 sym the set we can conclude that

N C := D ∈ M 2 sym : D -C is not positive definite is quasiconvex. Since for any D ∈ H \ (K ∪ L) there exists a C ∈ H \ (K ∪ L) such that D / ∈ N C , whereas K ∪ L ⊆ N C , 9
(K ∪ L) qc ⊆ H ∩ C∈H\(K∪L) N C = K ∪ L and then K ∪ L is quasiconvex.
Remark 5. The proof of Theorem 6 does not take advantage of the particular structure of W but it is based only on the fact that W -1 (0) = K. Therefore, instead of W we can consider the function V : M 2 → [0, +∞) defined as the quasiconvex envelope of dist(•, K) p , p ∈ (1, +∞). Indeed, since K is polyconvex (because zero levelset of the polyconvex function W ), by Lemma 5 it follows that V -1 (0) = K. Note that in this way our counterexample covers also the case of energy densities with growth p ∈ (1, 2).

Remark 6. Let N be a convex and compact subset of M 2 sufficiently large so that ∇ψ ∈ N a.e. in Q 2 , where ψ is defined as in (4.5). Consider now the function V : R 2 ×M 2 → [0, +∞) defined by V (x, Λ) := χ P (x)W (Λ) + (1χ P (x))dist(Λ, N ) p .

Since W cell ≤ V cell , we have the inclusion V -1 cell (0) ⊆ K ∪ L. We also have the inclusion

K ∪ M ⊆ V -1 hom (0): in fact K ⊆ V -1 hom (0) (because K ⊆ N ) and diag(h 1 /2, h 3 ) ∈ V -1 hom (
0) (by using again ψ). In this way we can conclude that also by mixing a polyconvex function and a convex function, the inequality QV cell > V hom can occur.

Appendix: Stefan Müller's example in dimension three

To show that Proposition 1 is not peculiar to dimension two, let us consider the corresponding energy for a three-dimensional soft material reinforced by a strong plate. The energy is now given by W 0 :

M 3 ∋ Λ → |Λ| 4 + f (det Λ) where f (z) :=      12(1 + a) 2 z + a -12(1 + a) -9 if z > 0 12(1 + a) 2 a -12(1 + a) -9 - 12(1 + a) 2 a 2 z if z ≤ 0
for a ∈ (0, 1/2). The energy density under consideration is still of the form W η : R 3 ×M 3 → [0, +∞), (y, Λ) → χ η (y)W 0 (Λ) where χ η is the Q-periodic extension on R 3 of

χ η (y) := 1 if y 1 ∈ (0, 1/2) η if y 1 ∈ [1/2, 1) ,
with Q = (0, 1) 3 ∋ y = (y 1 , y 2 , y 3 ) and η > 0. Such an energy density is nonnegative, polyconvex, frame-invariant and its zero-levelset is SO 3 .

Proposition 2. For all λ 1 , λ 2 ∈ (0, 1), there exists c > 0 independent of η such that

QW η cell (Λ) ≤ η c, where Λ := diag(1, λ 1 , λ 2 ).
Proof. Essentially, one has to construct a Lipschitz domain

U of R 3 , a function φ ∈ W 1,∞ per (U, R 3 ) and a function ϕ ∈ L ∞ (U, W 1,∞ per (Q, R 3
)) such that Λ + ∇φ(x) + ∇ y ϕ(x, y) ∈ SO 3 for all x ∈ U and y ∈ (0, 1/2) × (0, 1) 2 (the strong phase of the material).

Notation. The canonical basis of R 3 is denoted by {e 1 , e 2 , e 3 }.

We will use the angles θ and γ defined by

cos θ = λ 1 , sin θ = 1 -λ 2 1 , cos γ = λ 2 , sin γ = 1 -λ 2 2 .
We set ρ := sin 2 θ + cos 2 θ sin 2 γ and define the unit vectors e Definition of U . In order to describe U , we need the following quantity τ := cos θ sin γ sin θ .

We set U := U 1 ∪ U 2 ∪ U 3 ∪ U 4 , where U 1 := (x 1 , x 2 , x 3 ) : 0 < x 1 < 1; 0 < x 3 ≤ 1/2; 1/2τ x 3 ≤ x 2 < 1τ x 3 , U 2 := (x 1 , x 2 , x 3 ) : 0 < x 1 < 1; 0 < x 3 ≤ 1/2; -τ x 3 < x 2 ≤ 1/2τ x 3 , U 3 := (x 1 , x 2 , x 3 ) : 0 < x 1 < 1; 1/2 ≤ x 3 < 1; 1/2τ (1x 3 ) ≤ x 2 < 1τ (1x 3 ) , U 4 := (x 1 , x 2 , x 3 ) : 0 < x 1 < 1; 1/2 ≤ x 3 < 1; -τ (1x 3 ) < x 2 ≤ 1/2τ (1x 3 ) .

The domains U 1 , U 2 , U 3 , U 4 are sketched on Figure 14. Note that the interface between U 1 and U 2 (resp. U 3 and U 4 ) is perpendicular to e 4 (resp. e 5 ).

Definition of φ. We consider the piecewise constant function G ∈ L ∞ (U, M 3 ) of the form

G =   0 × × 0 × × 0 0 0   ,
where the non zero two-dimensional submatrix × × × × is one the following four possible submatrices: Let us check that G is a gradient field: = 2ρ e 1 ⊗ e 5 .

These couples of matrices being rank-one connected, G is actually a gradient field ∇φ on U . This gradient can be extended by periodicity (due to the boundary conditions).

Definition of ϕ. We consider the following specific rotations in 3D, which are compositions of rotations around e .

We construct a function ϕ of the form ϕ(x, y) := 4 i=1 χ U i (x)ϕ i (y), where ϕ 1 , . . . , ϕ 4 are defined as follows. We then choose ϕ 2 ∈ W 1,∞ per (Q, R 3 ) such that

2 .

 2 Homogenization of multiple integrals and the cell formula 2.1. Continuous and discrete homogenization of nonconvex functionals 2.2. Short summary of convexity properties 2.3. Stefan Müller's counterexample 2.4. Counterexample by comparison of the zero levelsets 3. Counterexamples from composite materials 3.1. Discrete example 3.2. An example from solid-solid phase transformations 3.3. Comparison of boths examples 4. Counterexample on perforated domains Appendix: Stefan Müller's example in dimension three References 1. Introduction For the homogenization of periodic integral functionals of the type

Since W - 1

 1 (0) is compact and Q ∩ P has a Lipschitz boundary, the function φ |Q∩P has a Lipschitz representative. The conclusion follows by taking a Lipschitz Q-periodic extension of φ |Q∩P on R d .

Figure 1 .

 1 Figure 1. Compression of one periodic cell and buckling of several periodic cells

Remark 3 . 2 . 4 . 2 O

 3242 As shown in the Appendix, Proposition 1 holds in dimension three in a weaker form, by substituting the rank-one convex hull with the quasiconvex hull of W η cell . Counterexample by comparison of the zero levelsets. Let us consider the following matrices of M := diag(0, 0), I := diag(1, 1), A := diag(-1, 1), B := diag(0, 1), and C := diag(0, 1/2), and two quasiconvex functions W 1 , W 2 ∈ W(a, p) such that

3. 1 .

 1 Discrete example. Let us first describe the geometry of the model.

4 Figure 2 .

 42 Figure 2. Geometry.

l 3 l 2 l 1 Figure 6 .

 216 Figure 6. In grey the set Q ∩ P .

/ 8 Figure 11 .

 811 Figure 11. In [(0, 2) × (0, 1)] ∩ P the first component of ψ is flat with the exception of the grey zone, where the gradient is equal to (1, 0).

Figure 12 .

 12 Figure[START_REF] Iwaniec | The Failure of Rank-One Connections[END_REF]. In [(0, 2) × (0, 1)] ∩ P the second component of ψ is flat with the exception of the grey zone, where the gradient is equal to (0, 1).

Figure 13 .

 13 Figure 13. The graph of the second component of ψ in [(0, 2) × (0, 1)] ∩ P .

4 and e 5

 5 by e 4 := 1 ρ (sin θ e 2 + cos θ sin γ e 3 ) , e 5 := 1 ρ (sin θ e 2cos θ sin γ e 3 ) .

A 1 := sin θ cos θ sin γ 0 - 3 Zone U 2 : 2 x 3 Figure 14 .

 0322314 Figure 14. Domain U (in the plane generated by {e 2 , e 3 }).

A 1 -A 3 = 2 [cos θ sin γ e 1 -sin θ sin γ e 2 ] ⊗ e 3 A 2 -A 4 = 2 [-cos θ sin γ e 1 -sin θ sin γ e 2 ] ⊗ e 3 A 1 -

 1321232421231 A 2 = 2e 1 ⊗ [sin θ e 2 + cos θ sin γ e 3 ] = 2ρ e 1 ⊗ e 4 A 3 -A 4 = 2e 1 ⊗ [sin θ e 2cos θ sin γ e 3 ]

2 and e 3 :cos β cos α -sin α cos α sin β cos β sin α cos α sin α sin β -sin β 0

 30 where α and β are the two angles. We also denote by χ the Q-periodic extension on R 3 ofχ(y) := 1 if y 1 ∈ (0, 1/2) -1 if y 1 ∈ [1/2, 1)

Zone 1 .We then choose ϕ 1 ∈ 2 -Zone 2 . 1 -

 11221 For x ∈ U 1 , Λ + ∇φ(x) =   1 sin θ cos θ sin γ 0 cos θsin θ sin γ W 1,∞ per (Q, R 3 ) such that ∇ϕ 1 (y) = χ(y)   cos γ cos θ -1 0 0 cos γ sin θ 0 0 -In the strong phase ( χ(y) = 1), Λ + ∇φ(x) + ∇ϕ 1 (y) =   cos γ cos θ sin θ cos θ sin γ cos γ sin θ cos θsin θ sin γ sin γ 0 cos γ   = R( θ, γ)is a rotation, with θ = -θ. In the soft phase ( χ(y) = -1)Λ + ∇φ(x) + ∇ϕ 1 (y) = cos γ cos θsin θ cos θ sin γ cos γ sin θ cos θ sin θ sin γ sin γ For x ∈ U 2 , Λ + ∇φ(x) =  sin θcos θ sin γ 0 cos θsin θ sin γ

∇ϕ 2 ( 2 - 4 .

 224 In the strong phase ( χ(y) = 1),Λ + ∇φ(x) + ∇ϕ 2 (y) =   cos γ cos θsin θcos θ sin γ cos γ sin θ cos θsin θ sin γ sin γ 0 cos γ   = R(θ, γ)is a rotation, with γ = -γ. In the soft phase ( χ(y) = -1),Λ + ∇φ(x) + ∇ϕ 2 (y) = cos γ cos θsin θ cos θ sin γ cos γ sin θ cos θ sin θ sin γ sin γ Proceeding as above, one may find ϕ3 , ϕ 4 ∈ W 1,∞ per (Q, R 3 ) such that for all x ∈ U 3 and y ∈ (0, 1/2) × (0, 1) 2 , Λ + ∇φ(x) + ∇ϕ 3 (y) =   cos γ cos θ sin θcos θ sin γ cos γ sin θ cos θ sin θ sin γ sin γ 0 cos γ   = R( θ, γ)is a rotation, with θ = -θ, γ = -γ; and for all x ∈ U 4 and y ∈ (0, 1/2) × (0, 1)2 , Λ + ∇φ(x) + ∇ϕ 4 (y) =   cos γ cos θsin θ cos θ sin γ cos γ sin θ cos θ sin θ sin γ sin γ 0 cos γ   = R(θ, γ)is a rotation.Finally, we are in position to conclude the proof. By using φ and ϕ defined above as test-functions, one obtainsQW η cell (Λ) ≤ U Q W η y, Λ + ∇φ(x) + ∇ y ϕ(x, y) dy dx ≤ η c,where c = max(W (C i ))/2 and {C i } i is the finite set of values taken by Λ + ∇φ(x) + ∇ y ϕ(x, y) in the soft phase.

  8, Theorem 19.1 and Remark 19.2]).

Theorem 1. Assume that W satisfies Hypothesis 1 and that P satisfies Hypothesis 2. Then the homogenized integrand W hom related to (W, P ) is a quasiconvex function belonging to W(b, Ω) for a suitable b = b(a, Ω), and for any ε h ց 0 + the sequence I ε h Γ-converges to the functional I hom : L p (Ω, R d ) → [0, +∞] defined by

  Due to the periodicity constraint on Q, we have that 1 4 (det ∇ψ |T 1 + det ∇ψ |T 2 + det ∇ψ |T 3 + det ∇ψ |T 4 ) = det Λ, 1 4 (det ∇ψ |T 5 + det ∇ψ |T 6 + det ∇ψ |T 7 + det ∇ψ |T 8 ) = det Λ.

	(3.2)

  ) .

	Thus, as expected,
	1 4	(det ∇ψ |T

1 + det ∇ψ |T 2 + det ∇ψ |T 3 + det ∇ψ |T 4 ) = 1

Acknowledgments M. Barchiesi wishes to thank Andrea Braides, Giovanni Leoni and Enzo Nesi for their useful comments and suggestions. A. Gloria gratefully thanks Andrea Braides and Stefan Müller for stimulating discussions. The research of M. Barchiesi was supported by the Center for Nonlinear Analysis (NSF Grants No. DMS-0405343 and DMS-0635983). The research of A. Gloria was supported by the Hausdorff Center for Mathematics. He also acknowledges the support from the Marie Curie Research Training Network MRTN-CT-2004-505226 'Multi-scale modelling and characterisation for phase transformations in advanced materials' (MULTIMAT).


Mathieu Acher

Patrick Heymans

Philippe Collet
email: collet@i3s.unice.fr

Philippe Lahire
email: lahire@i3s.unice.fr

Next-Generation Model-based Variability Management: Languages and Tools

Keywords: Feature Models, Model Management, Variability Modelling, Automated Reasoning, Language, Model Composition and Differences

Variability modelling and management is a key activity in a growing number of software engineering contexts, from software product lines to dynamic adaptive systems. Feature models are the defacto standard to formally represent and reason about commonality and variability of a software system. This tutorial aims at presenting next generation of feature modelling languages and tools, directly applicable to a wide range of model-based variability problems and application domains. Participants (being practitioners or academics, beginners or advanced) will learn the principles and foundations of languages and tool-supported techniques dedicated to the model-based management of variability.

The estimated length of the tutorial is 3 hours. Specific details about the topic, goal, schedule and supporting material of the tutorial are given in Section 1. Target audience and any pre-requisite background required by attendees are described in Section 2. A brief biography of presenters is given in Section 3.

Description of the Tutorial 1.Topic

In many domains, systems heavily rely on software and have to be efficiently extended, changed, customized or configured for use in a particular context (e.g., to respond to the specific expectations of a customer) [START_REF] Svahnberg | A taxonomy of variability realization techniques: Research articles[END_REF][START_REF] Pohl | Software Product Line Engineering: Foundations, Principles and Techniques[END_REF]. The challenge for software practitioners is to develop and use the right models, languages, and tool-supported techniques to produce and maintain multiple similar software products (variants), exploiting what they have in common and managing what varies among them. A lack of flexibility in the reusable artifacts or a scope that is too large may have severe consequences on the model-based engineering process. Therefore the modelling and management of variability in software intensive systems, for example Software Product Lines (SPLs), is a critical task [START_REF] Chen | Variability management in software product lines: a systematic review[END_REF].

Feature Models (FMs) are the defacto standard for modelling and reasoning about commonality and variability of a software system (e.g., an SPL). FMs are widely used to compactly define the valid combinations of features supported by an SPL, each valid combination of features (called configuration) corresponding to a given product. Numerous model-based SPL approaches rely on FMs for various purposes: generation of product configurators, customisation of other artifacts (e.g., code, models) to derive members of an SPL [START_REF] Czarnecki | Mapping features to models: A template approach based on superimposed variants[END_REF][START_REF] Czarnecki | Verifying feature-based model templates against wellformedness ocl constraints[END_REF], reasoning about properties of an SPL [START_REF] Classen | Model checking lots of systems: efficient verification of temporal properties in software product lines[END_REF][START_REF] Classen | Symbolic model checking of software product lines[END_REF], etc. FMs have been intensively studied by academics during the last two decades. The research output is notably a formal semantics, the development of various automated reasoning operations and benchmarks, tools and languages [START_REF] Benavides | Automated analysis of feature models 20 years later: a literature review[END_REF]. Industrial tools (pure::variants, Gears) have also been developed and support FMs to fulfill the growing interest of practitioners. FMs will also be part of the Common Variability Language (CVL) standard 4 .

The effort still continues since FMs are becoming increasingly complex (e.g., 6000+ features for Linux), handled by several stakeholders or organizations, and used to describe features at various levels of abstraction. Yet, barriers hindering a wider adoption of FMs, for example in industrial contexts, remain and have been reported (e.g., scalability [START_REF] Benavides | Automated analysis of feature models 20 years later: a literature review[END_REF], inadequate mechanisms to achieve separation of concerns [START_REF] Acher | Separation of Concerns in Feature Modeling: Support and Applications[END_REF], lack of relationships among languages [START_REF] Hubaux | A preliminary review on the application of feature diagrams in practice[END_REF]). To overcome current limitations, we have developed a set of complementary languages (TVL and FAMILIAR) and tool-supported techniques.

Goal

During the tutorial, participants will learn solutions applicable to a wide range of modelbased variability problems and application domains while being supported by concrete tools and languages. We will notably expose new capabilities offered to modellers and show that, without the advances made, some model-based analysis would not be made possible in several case studies. The languages and tools have already been applied in practical contexts (industrial and academic), in different application domains (printers, semi-conductors, medical imaging, video surveillance, etc.) and for various purposes (extraction of FMs from legacy software artifacts [START_REF] Acher | Reverse engineering architectural feature models[END_REF], SPL evolution [START_REF] Acher | Feature Model Differences[END_REF], management of multiple models [START_REF] Acher | Separation of Concerns in Feature Modeling: Support and Applications[END_REF][START_REF] Acher | Composing multiple variability artifacts to assemble coherent workflows[END_REF], model-based validation of SPLs [START_REF] Acher | Modeling variability from requirements to runtime[END_REF], large scale configuration of FMs [START_REF] Hubaux | Supporting multiple perspectives in feature-based configuration[END_REF], etc.)

The tutorial aims at introducing new feature modelling languages, tools and their underlying environments, describing their theoretical foundations and showing how their combined use can practically handle the whole process from variability modelling to product configuration. Firstly, we give the participants an overview of TVL, a textbased feature modelling language. We emphasize its concise, natural, yet expressive syntax that subsumes most existing FM dialects [16]. Secondly, we present FAMIL-IAR (for FeAture Model scrIpt Language for manIpulation and Automatic Reasoning) a language dedicated to the management of multiple FMs [START_REF] Acher | A domain-specific language for managing feature models[END_REF]. We introduce practical techniques to import, export, compose, decompose, edit, configure, reason about several FMs and combine these operations to realise complex model-based variability management tasks. As effort has been made to support interoperability between the two languages, their combine use can be achieved. At the end of the tutorial, participants of the tutorial will be able to go further with the languages and modelling techniques, and to reuse them in practical or academic contexts.

Plan

The tutorial is based on slides and supporting demonstrations. An archive will be provided to participants, including tutorial slides, related articles, TVL models, FAMILIAR scripts and packaged tools to interactively play with the models during the tutorial. A concrete, end-to-end demonstration is used for modeling FMs with TVL and performing management operations of FMs with FAMILIAR. The demonstration is based on the reengineering of a car configurator, that includes the modelling of configuration options using TVL and the management of the TVL model(s) using FAMILIAR. The estimated length of the tutorial is typically 3 hours:

1. (1 hour) The tutorial will start with a short explanation of variability modelling and FMs. We will present TVL, exposing its textual nature and formal semantics. Using the illustrative example, we will explain the main constructions (feature, group cardinalities, attributes) and modularity capabilities of the language. The resulting TVL models will then be analysed by FAMILIAR. 2. (1,5 hours) In the second part, we will present some relevant analyses and operations for the illustrative example and show how to perform them using FAMILIAR. These analyses will include usual satisfiability checking, detection of dead features and wrong cardinalities, but also more advanced operations (aggregation, differencing, slicing and merging of FMs). 3. (0,5 hour) The session will end up with a short summary of what have been done, a broader overview of the applications already realised by our tool suite and a report on our experience with the design and development of the languages. It should strengthen discussions with the audience (tools information, applicability of the techniques, etc.)

Material and Experience

Numerous presentations of the languages and tools have already taken place in the past.

An empirical evaluation of TVL on four real industrial projects has been published and presented at SLE '10 [18]. We plan to reuse introductory slides of TVL used in industrial contexts. TVL has also been presented in VaMoS'10. FAMILIAR has been presented in numerous venues: VaMoS'11 (tool demonstration), SAC'11 (formal presentation), french SPL day (tool demonstration), and ASE'11 (tool demonstration track). It is also currently being used in the french national project YourCast (http://yourcast. unice.fr, started in 2012) which aims at building a complex industrial strength SPL of model-based information broadcast systems. Furthermore, the languages and tools are extensively used during an SPL course given to Masters' students at University of Namur. As a running project, students have to re-engineer car configurators using TVL, FAMILIAR and modelling techniques.

The web page http://www.info.fundp.ac.be/spltutorials/ gathers some available material we plan to reuse for the tutorial. It includes the slides and screencasts used during the aforementioned demonstrations or the teaching material and illustrative example used in the SPL course.

Audience

Target audience. The tutorial targets software practitioners -professionals working in industrial contexts: requirements engineers, domain analysists, software architects, or product line managers -that look for languages and tools to model and manage variability. For example, TVL has already been applied in four industrial case studies [START_REF] Hubaux | Evaluating a textual feature modelling language: Four industrial case studies[END_REF] while FAMILIAR can tackle numerous variability problems reported in the industry [START_REF] Acher | Separation of Concerns in Feature Modeling: Support and Applications[END_REF]. The tutorial also targets an academic audience. For example, the recent research results as well as state-of-the-art existing techniques (e.g., reasoning operations reported in [START_REF] Benavides | Automated analysis of feature models 20 years later: a literature review[END_REF]) are integrated in FAMILIAR. During the tutorial, academics will learn the design and developement of novel model-based operations (e.g., merging, slicing, differencing techniques for FMs) [START_REF] Acher | A domain-specific language for managing feature models[END_REF][START_REF] Acher | Separation of Concerns in Feature Modeling: Support and Applications[END_REF].

Pre-requisite background. Attendees, being industrial practitioners or academics, may have basic knowledge about software product lines and variability, but it is not required. The tutorial targets both beginners and advances. Target participants of MODELS'12. More specifically, PhD students or modellers unaware of variability modelling techniques will have the opportunity to learn a widely used modelling formalism (FM) as well as existing languages and tools they can directly use. Furthermore, advanced modellers will learn novel techniques that may be reused in their academical or industrial contexts. We particularly target people involved or interested in the development of model management tools. Our experience in several case studies has shown that syntactical mechanisms are likely to be insufficient for operations like the computation of differences of FMs [START_REF] Acher | Feature Model Differences[END_REF] or the composition/decomposition of FMs [START_REF] Acher | Comparing approaches to implement feature model composition[END_REF][START_REF] Acher | Separation of Concerns in Feature Modeling: Support and Applications[END_REF]. We have rather developed a set of semantical operations. We will present their foundations and report on our experience, which can benefit for the attendees.

Presenters

Dr. Mathieu Acher is a post-doctoral researcher at University of Namur, Belgium. His research focuses on the modelling and management of variability in software intensive systems. He is the main developer of FAMILIAR for which he has designed and implemented novel automated operations. He is teaching a comprehensive SPL course at University of Namur, relying extensively on TVL, FAMILIAR and models in general. He has authored more than 20 peer-reviewed papers and has presented them in numerous international venues. He has performed demonstrations of FAMILIAR at VaMoS'11, SAC'11 and ASE'11. Dr. Patrick Heymans is full professor of software engineering at University of Namur, Belgium and visiting researcher at INRIA/University of Lille/CNRS. He is founding member and co-director of the PReCISE research centre (50 researchers) where he leads the requirements engineering and software product line efforts. He has supervised 9 PhD theses and authored 85 peer-reviewed papers. He is a regular referee for top SE journals and conferences, and associate editor of IEEE TSE. Patrick was recently the program chair of RE'11. He is principal investigator on various international software engineering research projects and regularly acts as an advisor and trainer for IT companies. Dr. Philippe Lahire is a full professor at the Université Nice Sophia Antipolis, France. He is involved in research on model-driven engineering, aspect-oriented modeling and software product line engineering. He has supervised 6 PhD theses and authored more than 30 peer-reviewed papers. Philippe regularly acts as member of program committees (TOOLS'07-'09, ECMFA'11, MODELS '12, etc.). He has been involved in several european and french research projects and also led Tempus collaborative projects with eastern countries. Dr. Philippe Collet is an associate professor (HDR) at the Université Nice Sophia Antipolis, France. He is involved in research on applying model-driven techniques to large scale software architectures with focus on self-adaptive systems and software product lines. He has supervised 4 PhD theses and authored more than 40 peer-reviewed papers. In 2012, Philippe is program co-chair of the main french software engineering conference (CIEL). From 2002, he has been principal investigator on various industrial and national software engineering research projects.

http://www.omgwiki.org/variability/doku.php