Cyril Imbert

SOME REGULARITY RESULTS FOR ANISOTROPIC MOTION OF FRONTS

Keywords: AMS Subject Classifications: 35A21, 35B65, 35D99, 35J60, 35K55, 35R35. Part

We study the regularity of propagating fronts whose motion is anisotropic. We prove that there is at most one normal direction at each point of the front; as an application, we prove that convex fronts are C 1,1 . These results are by-products of some necessary conditions for viscosity solutions of quasilinear elliptic equations. These conditions are of independent interest; for instance they imply some regularity for viscosity solutions of nondegenerate quasilinear elliptic equations.

Introduction

Following [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF][START_REF] Nochetto | Numerical analysis of geometric motion of fronts[END_REF], we study propagating fronts whose velocity field v Φ is given by the following geometric law:

v Φ = (κ Φ + g)n Φ ,
where n Φ and κ Φ are respectively the inward normal direction and the mean curvature associated with a Finsler metric Φ; g denotes a possible (bounded) driving force.

The main result of this paper states that under appropriate assumptions, there is at most one (outward or inward) "normal direction" at each point of the front.

In order to define the front past singularities, we use the level-set approach initiated by Barles [START_REF] Barles | Remark on a flame propagation model[END_REF] and developed by Osher and Sethian [START_REF] Osher | Fronts moving with curvature dependent speed: Algorithms based on hamilton-jacobi equations[END_REF]. This approach consists in describing the front Γ t at time t as the zero level-set of a (continuous or discontinuous) function u : Γ t = {x : u(x, t) = 0}. Choosing first a continuous function u 0 such that the initial front Γ 0 coincides with {x : u 0 (x) = 0} (consider for instance the signed distance function to Γ 0 ), u turns out to be a solution of the following Hamilton-Jacobi equation: where Du and D 2 u denotes the first and second derivative in x of the function u and Φ • denotes the dual metric associated with Φ. This equation is known as the anisotropic mean curvature equation. It is solved by using viscosity solutions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. The function u depends on the choice of u 0 , but not the front Γ t , even not the two families of sets O t = {x : u(x, t) > 0} and I t = {x : u(x, t) < 0} [START_REF] Evans | Motion of level sets by mean curvature, I[END_REF][START_REF] Gang | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Ishii | Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor[END_REF]. The definition of the front is therefore consistent and the notions of "outside"and "inside" become precise.

The study of the normal directions reduces to the study of the semi-jets of discontinuous semisolutions of (1.1). This latter study is persued by using necessary conditions derived for viscosity solutions of degenerate elliptic and parabolic quasilinear equations. Besides, these conditions are of independent interest. For instance, we derive from them regularity of viscosity solutions of nondegenerate quasilinear elliptic and parabolic equations.

The paper is organized as follows. In Section 2, we first give assumptions and recall definitions that are used in the paper. In particular, the Finsler metric and its dual are introduced and the definition of normal directions and semijets are recalled. In Section 3, we state and prove our main results (Theorem 1 and Corollary 1). Eventually, in Section 4, we present the necessary conditions used in the proof of Theorem 1.

Assumptions and definitions

In this section, we give assumptions and definitions that are used throughout the paper.

2.1. Anisotropic motion. In order to take into account the anisotropy and the inhomogeneity of the environment in which the front propagates, the metric induced by the Euclidian norm is replaced with a so-called Finsler metric. In our context, a Finsler metric Φ is the support function of a given compact set denoted by B Φ • (x) :

Φ(ζ, x) = max{ ζ, ζ * : ζ * ∈ B Φ • (x)}.
The set B Φ • (x) is referred to as the Wulff shape. Here are the assumptions we make concerning Φ and B Φ • (x).

A0. (i) The Wulff shape B Φ • (x) is a compact set that contains the origin in its interior and is symmetric with respect to it;

(ii) Φ ∈ C 2 (R n \{0} × R n ); (iii) for all x ∈ R n , ζ → [Φ(ζ, x)] 2 is strictly convex. For a given x ∈ R n , the dual metric Φ • is defined as the support function of the set B Φ (x) = {ζ ∈ R n : Φ(ζ,
→ Φ • (ζ, x
) is a support function; indeed, a support function is convex and linear along half-lines issued from the origin. Consequently:

D ζζ Φ • (ζ, x) 0 and ζ ∈ KerD ζζ Φ • (ζ, x),
where denotes the usual order associated with S n , the space of n × n symmetric matrices. A second example of motion is the following: consider a (riemannian

) metric Φ(ζ, x) = Φ(ζ) = Gζ, ζ
where G ∈ S n is definite positive. The associated dual metric turns out to be Φ

• (ζ * ) = G -1 ζ * , ζ * .
Finally, let us give a third example in which the inhomogeneity of the environment is taken into account: Φ(ζ, x) = a(x) Gζ, ζ , where a ∈ C 2 (R n ) and a(x) > 0 for all x ∈ R n . The reader can check that in these three examples the kernel of D ζζ Φ • (ζ, x) coincides with Span{ζ}. We next assume that the Finsler metric verifies such a property.

A1. ∀x ∈ R n , ∀ζ ∈ R n \{0}, KerD ζζ Φ • (ζ, x) = Span{ζ}.
We also need the following additional assumption.

A2. There exists L > 0 such that for all x, y ∈ R n and all

ζ * ∈ R n , |Φ • (ζ * , y) -Φ • (ζ * , x)| L |ζ * | |y -x|.
2.2. Semi-jets, P-subgradients and P-normals. We solve (4.1) and (4.2) by using viscosity solutions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. In order to ensure the existence of a solution (using for instance results from [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF][START_REF] Nochetto | Numerical analysis of geometric motion of fronts[END_REF]), we assume throughout the paper that the initial front is bounded. Unboundedness of the domain can be handled with results from [START_REF] Barles | Front propagation and phase field theory[END_REF]. The definition of viscosity solutions is based on the notion of semi-jets. Let Ω be a subset of R n and u be a numerical function defined on Ω and x be a point in Ω. A couple (X, p) ∈ S n × R n is a so-called subjet (resp. a superjet) of the function u at x (with respect to Ω) if for all y ∈ Ω :

1 2 X(y -x), y -x + p, y -x u(y) -u(x) + o(|y -x| 2 ) (2.2) resp. 1 2 X(y -x), y -x + p, y -x u(y) -u(x) + o(|y -x| 2 ) , (2.3)
where o(.) is a function such that o(h)/h → 0 as h → 0 + . The set of all the subjets (resp. superjets) of u at x is denoted by J 2,- Ω u(x) (resp. by J 2,+ Ω u(x)). In order to define viscosity solutions for parabolic equations, one must use so-called parabolic semi-jets P 2,- Ω×[0,T ] u(x, t); see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] for their definition.

A vector p that there exists X ∈ S n such that (X, p) ∈ J 2,- Ω u(x) is a so-called P-subgradient [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] of the function u :

∀y ∈ Ω, p, y -x u(y) -u(x) + O(|y -x| 2 ).
The set of all such vectors is referred to as the proximal subdifferential of the function u and it is denoted by ∂ P u(x). Analogously, a proximal superdifferential (hence P-supergradients) can be defined by ∂ P u(x) = -∂ P (-u)(x). It coincides with the sets of vectors p such that ∃X ∈ S n : (X, p) ∈ J 2,+ Ω u(x). The geometry of a set Ω can be investigated by studying subjets of the function denoted by Zero Ω defined on Ω and that is identically equal to 0. The proximal subdifferential of this function coincides with the proximal normal cone of Ω at x [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]:

N P (Ω, x) = {p ∈ R n : ∀y ∈ Ω, p, y -x O(|y -x| 2 )}.
An element of N P (Ω, x) is referred to as a P-normal. If p is a P-normal of Ω at x and λ is a nonnegative number, then λp is still a P-normal. From the geometrical viewpoint, one can say that N P (Ω, x) is a cone, that is to say it is made of half-lines issued from the origin. Crandall, Ishii and Lions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] proved that for a set with a C 2 boundary:

J 2,- Ω Zero(x) = {(S(x) -Y, λn(x)) : λ ≥ 0, Y 0},
where n(x) denotes the normal vector and S(x) denotes the second fundamental form extended to R n by setting S = 0 along Span{n(x)}. The proximal normal cone is therefore reduced to R + n(x) = {λn(x) : λ 0}. If Ω is a hyperplan and n = 0 denotes a normal vector from H ⊥ , then N P (Ω, x) is the whole line Span{n}.

Main results

In this section, we state and prove our main results, namely Theorem 1 and Corollary 1. The proof of Theorem 1 rely on necessary conditions verified by solutions of possibly degenerate elliptic and parabolic quasilinear equations; these conditions are presented in Section 4.

Theorem 1. Consider a Finsler metric Φ satisfying A0, A1 and A2. Then the associated propagating front Γ t , t > 0, has at most one "outward normal direction" (resp. "inward normal direction"), that is to say the proximal normal cone at any point of I t ∪ Γ t or at any point of I t (resp. O t ∪ Γ t or O t ) is at most a line.

Remarks. 1. Assumptions A0 and A2 ensure the existence and uniqueness of the solution u of (1.1). Assumption A1 can be seen as a regularity assumption on the Franck diagram.

2. Theorem 1 remains valid if the front "fattens" (see [START_REF] Souganidis | Front propagation: Theory and applications[END_REF] for details about the fattening phenomena).

Theorem 1 implies the regularity of convex fronts. See also Theorem 5.5 in [START_REF] Evans | Motion of level sets by mean curvature, III[END_REF]. Corollary 1. Let the metric Φ be independent of the position and such that A0, A1 are satisfied. Assume that the initial front Γ 0 is convex. Then the associated propagating front Γ t is also convex and is C 1,1 ; more precisely, I t ∪ Γ t and I t are convex and their boundary is C 1,1 .

Let us now prove these two results.

Proof of Theorem 1. Assumptions A0 and A2 ensure that the assumptions of Theorem 4.9 in [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF] are satisfied. Then, there exists a unique solution of (1.1). In order to prove Theorem 1, we must prove that for a given point of the boundary of I t , two P-normals p 1 and p 2 colinear. Let us choose λ such that λp 1 + (1 -λ)p 2 = 0. We know [START_REF] Barles | Front propagation and phase field theory[END_REF] that the function Zero It is a supersolution of (1.1). By applying Proposition 1 (see Section 4), we obtain:

p 1 -p 2 ∈ KerD ζζ Φ • (λp 1 + (1 -λ)p 2 ).
Using Assumption A1, we conclude p 1 -p 2 is colinear with λp 1 + (1 -λ)p 2 . We conclude that p 1 and p 2 are colinear.

We proceed analogously with the sets I t , O t ∪ Γ t and O t .

Proof of Corollary 1. The fact that the front is convex for any time t follows from Theorem 3.1 in [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF]. Choosing for u 0 the opposite of the signed distance function to Γ 0 , we ensure that the initial datum is Lipschitz and concave. Therefore, Theorem 2.1 in [START_REF] Nochetto | Numerical analysis of geometric motion of fronts[END_REF] implies that u is Lipschitz; this ensures that u has a sublinear growth. By applying Theorem 3.1 in [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF], we know that x → u(x, t) is concave, hence I t ∪ Γ t and I t are convex sets. The Hahn-Banach theorem ensures the existence of a normal in the sense of convex analysis. Such a normal is also a P-normal [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]. Using the fact that Zero It and Zero It∪Γt are supersolutions of (1.1) (see instance [START_REF] Barles | Front propagation and phase field theory[END_REF]), Theorem 1 implies that there is at most one P-normal. Hence there is exactly one normal in the sense of convex analysis and C 1,1 regularity follows.

Necessary conditions for elliptic and parabolic quasilinear equations

In the present section, we state necessary conditions that are verified by viscosity sub-and supersolutions (hence by solutions) of quasilinear elliptic equations on a domain Ω ⊂ R n :

- n i,j=1 a i,j (Du, u, x) ∂ 2 u ∂x i ∂x j + f (Du, u, x) = 0, ∀x ∈ Ω. (4.1)
These equations may be degenerate and/or singular at Du = 0. We also study the associated parabolic equations on Ω × [0, T ] :

∂u ∂t - n i,j=1 a i,j (Du, u, x, t) ∂ 2 u ∂x i ∂x j + f (Du, u, x, t) = 0, ∀(x, t) ∈ Ω × [0, T ].
(4.2) In the following, the n × n symmetric matrix with entries (a i,j ) is denoted by A. We assume that (4.1) and (4.2) are degenerate elliptic.

(E) For all p, u, x(, t), A(p, u, x(, t)) 0.

In Propositions 1 and 2, we prove that the difference of two P -subgradients (resp. P -supergradients) of a supersolution (resp. of a subsolution) of (4.1) or (4.2) is a degenerate direction, that is to say it lies in the kernel of A.

Proposition 1 (The elliptic case). Consider a supersolution (resp. a subsolution) u of (4.1), a point x ∈ Ω and two subjets (X i , p i ) ∈ J 2,- Ω u(x), i = 1, 2 (resp. two superjets (X i , p i ) ∈ J 2,+ Ω u(x), i = 1, 2). Then for any λ ∈ [0, 1] such that λp 1 + (1 -λ)p 2 = 0, the following holds true:

p 1 -p 2 ∈ KerA(λp 1 + (1 -λ)p 2 , u(x), x).
A straightforward consequence of Proposition 1 is the following result dealing with nondegenerate equations.

Corollary 2. Suppose that the equation (4.1) is nondegenerate, i.e., A(p, u, x)q, q > 0 if q = 0.

Then a solution u : Ω → R of (4.1) has "no corners", that is to say the function u at most one P-subgradient and at most one P-supergradient at any point x ∈ Ω. This corollary applies for instance to the equation associated with the search of minimal surfaces:

div Du 1 + |Du| 2 = 0 ⇔ -∆u + D 2 uDu, Du 1 + |Du| 2 = 0. (4.3)
Before proving Proposition 1, we state its parabolic version.

Proposition 2 (The parabolic case). Consider a supersolution (resp. a subsolution) u of (4.2), a point (x, t) ∈ Ω × [0, T ] and two parabolic subjets (X i , p i , α i ) ∈ P 2,- Ω×[0,T ] u(x, t), i = 1, 2 (resp. two parabolic superjets (X i , p i , α i ) ∈ P 2,+ Ω×[0,T ] u(x, t), i = 1, 2). Then for any λ ∈ [0, 1] such that λp 1 + (1 -λ)p 2 = 0, the following holds true:

p 1 -p 2 ∈ KerA(λp 1 + (1 -λ)p 2 , u(x), x, t).
Corollary 3. Suppose that the equation (4.2) is nondegenerate, i.e., A(p, u, x, t)q, q > 0 if q = 0.

Then a solution u : Ω × [0, T ] → R of (4.1) has "no corners", that is to say the function u has at most one P-subgradient and at most one Psupergradient at any point x ∈ R n .

The Hamilton-Jacobi equation associated with the motion by mean curvature of graphs is an example of nondegenerate quasilinear parabolic equation:

∂u ∂t -∆u + D 2 uDu, Du 1 + |Du| 2 = 0. ( 4.4)
A class of parabolic equations, including (4.4), is studied by a geometrical approach in [START_REF] Barles | Quasilinear parabolic equations, unbounded solutions and geometrical equations I[END_REF]. The proof of Proposition 1 relies on the following technical lemma.

Lemma 1. Consider an arbitrary set Ω and a function u : Ω → R. Let x be a point in Ω and (X i , p i ), i = 1, 2, be two subjets of u at x. Then for any matrix X ∈ S n such that X X i , i = 1, 2, any λ [0, 1] and any M > 0, the following holds true:

(X + M (p 1 -p 2 ) ⊗ (p 1 -p 2 ), λp 1 + (1 -λ)p 2 ) ∈ J 2,- Ω u(x).

Let us show how Lemma 1 implies Proposition 1.

Proof of Proposition 1. Let X ∈ S n be such that X X i for i = 1, 2 and consider any λ ∈ [0, 1] and any M > 0. By applying Lemma 1 to the supersolution u of (4.1) and by denoting p the vector λp 1 + (1 -λ)p 2 and q the vector p 1 -p 2 , we conclude that: (X + M q ⊗ q, p) ∈ J 2,- Ω u(x). As u is a supersolution of (4.1) and p = 0, the following holds true:

-tr [A(p, u(x), x)(X + M q ⊗ q)] + f (p, u(x), x) 0.

Dividing by M and letting M → +∞ yields: 0

A(p, u(x), x)q, q = tr [A(p, u(x), x)q ⊗ q] 0.

The first inequality follows from the ellipticity of (4.1). We conclude that q ∈ KerA(p, u(x), x).

If the function u is a subsolution, apply the lemma to the function -u and use it analogously.

One can easily give a parabolic version of this lemma and use it to prove Proposition 2. We omit these details and we turn to the proof of Lemma 1.

Proof of Lemma 1. By considering v(y) = u(x+y)-u(x), we may assume that x = 0 and u(x) = 0. Let us denote p = λp 1 + (1 -λ)p 2 and q = p 1 -p 2 . A straightforward calculus shows us that for any real number r such that |r| min( 2λ M , 2(1-λ) M ) : 1 2 M r 2 max{(1 -λ)r, -λr}.

Therefore, for any y such that | q, y | min( 2λ M , 2(1-λ) M ), we get: 1 2 M q, y 2 max{(1 -λ) q, y , -λ q, y }.

Finally, for any y in a neighbourhood of the origin such that x + y ∈ Ω, we get:

1 2 (X + M q ⊗ q)y, y + p, y = 1 2 Xy, y + 1 2 M q, y 2 + p, y

  ∂u ∂t -Φ • (Du, x) tr[D ζζ Φ • (Du, x)D 2 u] + D ζ Φ • (Du, x), Du |Du| +tr[D ζx Φ • (Du, x)] + g(Du, x, t) = 0,(1.1)

supported by the TMR "Viscosity solutions and their applications". 1263

max 1 2 X 1 y, y + (1 -λ) q, y + p, y , 1 2 X 2 y, y -λ q, y + p, y

We have therefore proved that (X + M q ⊗ q, p) ∈ J 2,- Ω-x v(0) = J 2,- Ω u(x). Remark. Using Lemma 1, necessary conditions can be derived for any general nonlinear elliptic equation F (D 2 u, Du, u, x) = 0 if (E) is satisfied and if X → F (X, p, u, x) is positively homogenous.


Grigor Nika

Andrei Constantinescu

DESIGN OF MULTI-LAYER MATERIALS USING INVERSE HOMOGENIZATION AND A LEVEL SET METHOD

Keywords: Topology optimization, Level set method, Inverse homogenization, Multi-layer material

This work is concerned with the micro-architecture of multi-layer material that globally exhibits desired mechanical properties, for instance a negative apparent Poisson ratio. We use inverse homogenization, the level set method, and the shape derivative in the sense of Hadamard to identify material regions and track boundary changes within the context of the smoothed interface. The level set method and the shape derivative obtained in the smoothed interface context allows to capture, within the unit cell, the optimal microgeometry. We test the algorithm by computing several multi-layer auxetic micro-structures. The multi-layer approach has the added benefit that contact during movement of adjacent "branches" of the micro-structure can be avoided in order to increase its capacity to withstand larger stresses.

Introduction

The better understanding of the behavior of novel materials with unusual mechanical properties is important in many applications. As it is well known the optimization of the topology and geometry of a structure will greatly impact its performance. Topology optimization, in particular, has found many uses in the aerospace industry, automotive industry, acoustic devices to name a few. As one of the most demanding undertakings in structural design, topology optimization, has undergone a tremendous growth over the last thirty years. Generally speaking, topology optimization of continuum structures has branched out in two directions. One is structural optimization of macroscopic designs, where methods like the Solid Isotropic Method with Penalization (SIMP) [START_REF] Bendsoe | Topology optimization: theory, methods and applications[END_REF] and the homogenization method [START_REF] Allaire | Shape Optimization by the Homogenization Methods[END_REF], [START_REF] Allaire | Shape optimization by the homogenization method[END_REF] where first introduced. The other branch deals with optimization of micro-structures in order to elicit a certain macroscopic response or behavior of the resulting composite structure [START_REF] Bendsoe | Generating optimal topologies in structural design using a homogenization method[END_REF], [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF], [START_REF]Sigmund Materials with prescribed constitutive parameters: An inverse homogenization problem[END_REF], [START_REF] Wang | Level-set method for design of multi-phase elastic and thermoelastic materials[END_REF]. The latter will be the focal point of the current work.

In the context of linear elastic material and small deformation kinematics there is quite a body of work in the design of mechanical meta-materials using inverse homogenization. One of the first works in the aforementioned subject was carried out by [START_REF]Sigmund Materials with prescribed constitutive parameters: An inverse homogenization problem[END_REF]. The author used a modified optimality criteria method that was proposed in [START_REF] Rozvany | Layout and Generalized Shape Optimization by Iterative COC Methods[END_REF] to optimize a periodic micro-structure so that the homogenized coefficients attained certain target values.

On the same wavelength the authors in [START_REF] Wang | Level-set method for design of multi-phase elastic and thermoelastic materials[END_REF] used inverse homogenization and a level set method coupled with the Hadamard boundary variation technique [START_REF] Allaire | Conception optimale de structures[END_REF], [START_REF] Allaire | Structural optimization using sensitivity analysis and a level set method[END_REF] to construct elastic and thermo-elastic periodic micro-structures that exhibited certain prescribed macroscopic behavior for a single material and void. More recent work was also done by [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF], where again inverse homogenization and a level set method coupled with the Hadamard shape derivative was used to extend the class of optimized micro-structures in the context of the smoothed interface approach [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF], [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF]. Namely, for mathematical or physical reasons a smooth, thin transitional layer of size 2 , where is small, replaces the sharp interface between material and void or between two different material. The theory that [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF], [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF] develop in obtaining the shape derivative is based on the differentiability properties of the signed distance function [START_REF] Delfour | Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization, Advances in Design and Control[END_REF] and it is mathematically rigorous.

Topology optimization under finite deformation has not undergone the same rapid development as in the case of small strains elasticity, for obvious reasons. One of the first works of topology optimization in non-linear elasticity appeared as part of the work of [START_REF] Allaire | Structural optimization using sensitivity analysis and a level set method[END_REF] where they considered a non-linear hyper-elastic material of St. Venant-Kirchhoff type in designing a cantilever using a level set method. More recent work was carried out by the authors of [START_REF] Wang | Design of materials with prescribed nonlinear properties[END_REF], where they utilized the SIMP method to design non-linear periodic micro-structures using a modified St. Venant-Kirchhoff model.

The rapid advances of 3D printers have made it possible to print many of these microstructures, that are characterized by complicated geometries, which itself has given way to testing and evaluation of the mechanical properties of such structures. For instance, the authors of [START_REF] Clausen | Topology optimized architectures with programmable Poisson's ratio over large deformations[END_REF], 3D printed and tested a variety of the non-linear micro-structures from the work of [START_REF] Wang | Design of materials with prescribed nonlinear properties[END_REF] and showed that the structures, similar in form as the one in figure 1, exhibited an apparent Poisson ratio between -0.8 and 0 for strains up to 20%. Preliminary experiments by P. Rousseau [START_REF] Rousseau | Design of auxetic metamaterials[END_REF] on the printed structure of figure 1 showed that opposite branches of the structure came into contact with one another at a strain of roughly 25% which matched the values reported in [START_REF] Clausen | Topology optimized architectures with programmable Poisson's ratio over large deformations[END_REF]. To go beyond the 25% strain mark, the author of [START_REF] Rousseau | Design of auxetic metamaterials[END_REF] designed a material where the branches were distributed over different parallel planes (see figure 2). The distribution of the branches on different planes eliminated contact of opposite branches up to a strain of 50%. A question remains whether or not the shape of the unit cell in figure 2 is optimal. We suspect that it is not, however, the novelty of the actual problem lies in its multi-layer character within the optimization framework of a unit cell with respect to two desired apparent elastic tensors. Our goal in this work is to design a multi-layer periodic composite with desired elastic properties. In other words, we need to specify the micro-structure of the material in terms of both the distribution as well as its topology. In section 2 we specify the problem setting, define our objective function that needs to be optimized and describe the notion of a Hadamard shape derivative. In section 3 we introduce the level set that is going to implicitly characterize our domain and give a brief description of the smoothed interface approach. Moreover, we compute the shape derivatives and describe the steps of the numerical algorithm. Furthermore, in Section 4 we compute several examples of multi-layer auxetic material that exhibit negative apparent Poisson ratio in 2D. For full 3D systems the steps are exactly the same, albeit with a bigger computational cost. Notation. Throughout the paper we will be employing the Einstein summation notation for repeated indices. As is the case in linear elasticity, ε ε ε(u u u) will indicate the strain defined by: ε ε ε(u u u) = 1 2 ∇u u u + ∇u u u , the inner product between matrices is denoted by A A A:B B B = tr(A A A B B B) = A ij B ji . Lastly, the mean value of a quantity is defined as M Y (γ) = 1 |Y | Y γ(y y y) dy y y.

Problem setting

We begin with a brief outline of some key results from the theory of homogenization [START_REF] Allaire | Shape Optimization by the Homogenization Methods[END_REF], [START_REF] Bakhvalov | Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials[END_REF], [START_REF] Cioranescu | Introduction to Homogenization[END_REF], [START_REF] Mei | Homogenisation methods for multiscale mechanics[END_REF], [START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF], that will be needed to set up the optimization problem. Consider a linear, elastic, periodic body occupying a bounded domain Ω of R N , N = 2, 3 with period that is assumed to be small in comparison to the size of the domain. Moreover, denote by

Y = - 1 2 , 1 2
N the rescaled periodic unit cell. The material properties in Ω are represented by a periodic fourth order tensor A(y y y) with y y y = x x x/ ∈ Y and x x x ∈ Ω carrying the usual symmetries and it is positive definite:

A ijkl = A jikl = A klij for i, j, k, l ∈ {1, . . . , N } Ω Y Figure 3
. Schematic of the elastic composite material that is governed by eq. (2.1).

Denoting by f f f the body force and enforcing a homogeneous Dirichlet boundary condition the description of the problem is,

-div σ σ σ = f f f in Ω, σ σ σ = A(x x x/ ) ε ε ε(u u u ) in Ω, (2.1) u u u = 0 0 0 on ∂Ω.
We perform an asymptotic analysis of (2.1) as the period approaches 0 by searching for a displacement u u u of the form

u u u (x x x) = +∞ i=0 i u u u i (x x x, x x x/ )
One can show that u u u 0 depends only on x x x and, at order -1 , we can obtain a family of auxiliary periodic boundary value problems posed on the reference cell Y. To begin with, for any m, ∈ {1, . . . , N } we define E E E m = 1 2 (e e e m ⊗e e e +e e e ⊗e e e m ), where (e e e k ) 1≤k≤N is the canonical basis of R N . For each E E E m we have

-div y A(y y y)(E E E m + ε ε ε y (χ χ χ m )) = 0 0 0 in Y, y y y → χ χ χ m (y y y) Y -periodic, M Y (χ χ χ m ) = 0 0 0.
where χ χ χ m is the displacement created by the mean deformation equal to E E E m . In its weak form the above equation looks as follows:

Find χ χ χ m ∈ V such that Y A(y y y) E E E m + ε ε ε(χ χ χ m ) : ε ε ε(w w w) dy y y = 0 for all w ∈ V, (2.2)
where

V = {w w w ∈ W 1,2 per (Y ; R N ) | M Y (w w w) = 0}
. Furthermore, matching asymptotic terms at order 0 we can obtain the homogenized equations for u u u 0 ,

-div x σ σ σ 0 = f f f in Ω, σ σ σ 0 = A H ε ε ε(u u u 0 ) in Ω, (2.3)
u u u 0 = 0 0 0 on ∂Ω.
where A H are the homogenized coefficients and in their symmetric form look as follows,

A H ijm = Y A(y y y)(E E E ij + ε ε ε y (χ χ χ ij )) : (E E E m + ε ε ε y (χ χ χ m ))
J(S) = 1 2 A H -A t 2 η with S = (S 1 , . . . , S d ). (2.5)
where • η is the weighted Euclidean norm, A t , written here component wise, are the specified elastic tensor values, A H are the homogenized counterparts, and η are the weight coefficients carrying the same type of symmetry as the homogenized elastic tensor. We define a set of admissible shapes contained in the working domain Y that have a fixed volume by d . Thus, we can formulate the optimization problem as follows, inf

U ad = S i ⊂ Y is open, bounded, and smooth, such that |S i | = V t i , i = 1, . . . ,
S⊂U ad J(S) χ χ χ m satisfies (2.2) (2.6)
2.2. Shape propagation analysis. In order to apply a gradient descent method to (2.6) we recall the notion of shape derivative. As has become standard in the shape and topology optimization literature we follow Hadamard's variation method for computing the deformation of a shape. The classical shape sensitivity framework of Hadamard provides us with a descent direction. The approach here is due to [START_REF] Murat | Etudes de problmes doptimal design[END_REF] (see also [START_REF] Allaire | Conception optimale de structures[END_REF]). Assume that Ω 0 is a smooth, open, subset of a design domain D. In the classical theory one defines the perturbation of the domain Ω 0 in the direction θ θ θ as

(Id + θ θ θ)(Ω 0 ) := {x x x + θ θ θ(x x x) | x x x ∈ Ω 0 } where θ θ θ ∈ W 1,∞ (R N ; R N )
and it is tangential on the boundary of D. For small enough θ θ θ, (Id + θ θ θ) is a diffeomorphism in R N . Otherwise said, every admissible shape is represented by the vector field θ θ θ. This framework allows us to define the derivative of a functional of a shape as a Fréchet derivative.

Definition Definition Definition 2.2.1. The shape derivative of J(Ω 0 ) at Ω 0 is defined as the Fréchet derivative in W 1,∞ (R N ; R N ) at 0 0 0 of the mapping θ θ θ → J((Id + θ θ θ)(Ω 0 )):

J((Id + θ θ θ)(Ω 0 )) = J(Ω 0 ) + J (Ω 0 )(θ θ θ) + o(θ θ θ) with lim θ θ θ→0 0 0 |o(θ θ θ)| θ θ θ W 1,∞ , and J (Ω 0 )(θ θ θ) a continuous linear form on W 1,∞ (R N ; R N ).
Remark 1. The above definition is not a constructive computation for J (Ω 0 )(θ θ θ). There are more than one ways one can compute the shape derivative of J(Ω 0 ) (see [START_REF] Allaire | Conception optimale de structures[END_REF] for a detailed presentation). In the following section we compute the shape derivative associated to (2.6) using the formal Lagrangian method of J. Cea [START_REF] Céa | Conception optimale ou identification de formes: calcul rapide de la drive directionnelle de la fonction cout[END_REF].

Level set representation of the shape in the unit cell

Following the ideas of [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF], [START_REF] Wang | Level-set method for design of multi-phase elastic and thermoelastic materials[END_REF], the d sub-domains in the cell Y labeled S i , i ∈ {1, . . . , d} can treat up to 2 d distinct phases by considering a partition of the working domain Y denoted by F j , j ∈ {1, . . . , 2 d } and defined the following way, Define for i ∈ {1, . . . , d} the level sets φ i ,

F 1 =S 1 ∩ S 2 ∩ . . . ∩ S d F 2 =S c 1 ∩ S 2 ∩ . . . ∩ S d . . .
φ i (y y y)      = 0 if y y y ∈ ∂S i > 0 if y y y ∈ S c i < 0 if y y y ∈ S i
Moreover, denote by Γ km = Γ mk = F m ∩ F k where k = m, the interface boundary between the m th and the k th partition and let Γ = ∪ 2 d i,j=1i =j Γ ij denote the collective interface to be displaced. The properties of the material that occupy each phase, F j are characterized by an isotropic fourth order tensor

A j = 2 µ j I 4 + κ j - 2 µ j N I 2 ⊗ I 2 , j ∈ {1, . . . , 2 d }
where κ j and µ j are the bulk and shear moduli of phase F j , I 2 is a second order identity matrix, and I 4 is the identity fourth order tensor acting on symmetric matrices.

Remark 2. Expressions of the layer F k , 0 ≤ k ≤ 2 d in terms of the sub-domains S i , 1 ≤ k ≤ d is simply given by the representation of the number k in basis 2. For a number, k its representation in basis 2 is a sequence of d digits, 0 or 1. Replacing in position i the digit 0 with S i and 1 with S c i and can map the expression in basis 2 in the expression of the layer F i . In a similar way, one can express the subsequent formulas in a simple way. However for the sake of simplicity we shall restrain the expressions of the development in the paper to d = 2 and 0 ≥ j ≥ 4.

Remark 3. At the interface boundary between the F j 's there exists a jump on the coefficients that characterize each phase. In the sub-section that follows we will change this sharp interface assumption and allow for a smooth passage from one material to the other as in [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF], [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF].

3.1. The smoothed interface approach. We model the interface as a smooth, transition, thin layer of width 2 > 0 (see [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF], [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF]) rather than a sharp interface. This regularization is carried out in two steps: first by re-initializing each level set, φ i to become a signed distance function, d S i to the interface boundary and then use an interpolation with a Heaviside type of function, h (t), to pass from one material to the next,

φ i → d S i → h (d S i ).
The Heaviside function h (t) is defined as,

h (t) =      0 if t < -, 1 2 1 + t + 1 π sin π t if |t| ≤ , 1 if t > . (3.1)
Remark 4. The choice of the regularizing function above is not unique, it is possible to use other type of regularizing functions (see [START_REF] Wang | Color" level sets: A multiple multi-phase method for structural topology optimization with multiple materials[END_REF]).

The signed distance function to the domain S i , i = 1, 2, denoted by d S i is obtained as the stationary solution of the following problem [START_REF] Osher | Fronts propagating with curvature dependent speed: algorithms based on hamiltonjacobi formulations[END_REF],

∂d S i dt + sign(φ i )(|∇d S i | -1) = 0 in R + × Y, d S i (0, y y y) = φ i (y y y) in Y, (3.2)
where φ i is the initial level set for the subset S i . Hence, the properties of the material occupying the unit cell Y are then defined as a smooth interpolation between the tensors A j 's j ∈ {1, . . . , 2 d },

A (d S ) = (1 -h (d S 1 )) (1 -h (d S 2 )) A 1 + h (d S 1 ) (1 -h (d S 2 )) A 2 + (1 -h (d S 1 )) h (d S 2 ) A 3 + h (d S 1 ) h (d S 2 ) A 4 . (3.3)
where d S = (d S 1 , d S 2 ). Lastly, we remark that the volume of each phase is written as

Y ι k dy y y = V k
where ι k is defined as follows,

         ι 1 = (1 -h (d S 1 )) (1 -h (d S 2 )), ι 2 = h (d S 1 ) (1 -h (d S 2 )), ι 3 = (1 -h (d S 1 )) h (d S 2 ), ι 4 = h (d S 1 ) h (d S 2 ).
(3.4)

Remark 5. Once we have re-initialized the level sets into signed distance functions we can obtain the shape derivatives of the objective functional with respect to each sub-domain S i . In order to do this we require certain differentiability properties of the signed distance function.

Detailed results pertaining to the aforementioned properties can be found in [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF], [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF]. We encourage the reader to consult their work for the details. For our purposes, we will make heavy use of Propositions 2.5 and 2.9 in [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF] as well as certain results therein.

Theorem Theorem Theorem 3.1.1. Assume that S 1 , S 2 are smooth, bounded, open subsets of the working domain Y and θ θ θ 1 , θ θ θ 2 ∈ W 1,∞ (R N ; R N ). The shape derivatives of (2.6) in the directions θ θ θ 1 , θ θ θ 2 respectively are,

∂J ∂S 1 (θ θ θ 1 ) = - Γ θ θ θ 1 • n n n 1 η ijk A H ijk -A t ijk A * mqrs (d S 2 )(E k mq + ε mq (χ χ χ k ))(E ij rs + ε rs (χ χ χ ij )) -h * (d S 2 ) dy y y ∂J ∂S 2 (θ θ θ 2 ) = - Γ θ θ θ 2 • n n n 2 η ijk A H ijk -A t ijk A * mqrs (d S 1 ) (E k mq + ε mq (χ χ χ k )) (E ij rs + ε rs (χ χ χ ij )) -h * (d S 1 ) dy y y
where, for i = 1, 2, A * (d S i ), written component wise above, denotes,

A * (d S i ) = A 2 -A 1 + h (d S i ) A 1 -A 2 -A 3 + A 4 , (3.5) h * (d S i ) = ( 2 -1 + h (d S i )( 1 -2 -3 + 4 )) (3.6)
and j , j ∈ {1, . . . , 4} are the Lagrange multipliers for the weight of each phase.

Proof. For each k, we introduce the following Lagrangian for (u

u u k , v v v, µ µ µ) ∈ V × V × R 2d associated to problem (2.6), L(S S S, u u u k , v v v, µ µ µ) = J(S S S) + Y A (d S S S ) E E E k + ε ε ε(u u u k ) : ε ε ε(v v v) dy y y + µ µ µ • Y ι ι ι dy y y -V V V t , (3.7)
where µ µ µ = (µ 1 , . . . , µ 4 ) is a vector of Lagrange multipliers for the volume constraint, ι ι ι = (ι 1 , . . . , ι 4 ), and V V V t = (V t 1 , . . . , V t 4 ). Remark 6. Each variable of the Lagrangian is independent of one another and independent of the sub-domains S 1 and S 2 .

Direct problem. Differentiating L with respect to v v v in the direction of some test function w w w ∈ V we obtain,

∂L ∂v v v | w w w = Y A ijrs (d S S S ) (E k ij + ε ij (u u u k )) ε rs (w w w) dy y y,
upon setting this equal to zero we obtain the variational formulation in (2.2).

Adjoint problem. Differentiating L with respect to u u u k in the direction w w w ∈ V we obtain,

∂L ∂u u u k | w w w = η ijk A H ijk -A t ijk Y A mqrs (d S S S ) (E k mq + ε mq (u u u k )) ε rs (w w w) dy y y + Y A mqrs (d S S S ) ε mq (w w w) ε rs (v v v) dy y y.
We immediately observe that the integral over Y on the first line is equal to 0 since it is the variational formulation (2.2). Moreover, if we chose w w w = v v v then by the positive definiteness assumption of the tensor A as well as the periodicity of v v v we obtain that adjoint solution is identically zero, v v v ≡ 0.

Shape derivative. Lastly, we need to compute the shape derivative in directions θ θ θ 1 and θ θ θ 2 for each sub-domain S 1 , S 2 respectively. Here we will carry out computations for the shape derivative with respect to the sub-domain S 1 with calculations for the sub-domain S 2 carried out in a similar fashion. We know (see [START_REF] Allaire | Conception optimale de structures[END_REF]) that

∂J ∂S i (S S S) | θ θ θ i = ∂L ∂S i (S S S, χ χ χ k , 0 0 0, λ λ λ) | θ θ θ i for i = 1, 2. (3.8) Hence, ∂L ∂S 1 (θ θ θ 1 ) = η ijk A H ijk -A t ijk Y d S 1 (θ θ θ 1 ) ∂A mqrs ∂S 1 (d S S S )(E k mq + ε mq (u u u k )) (E ij rs + ε rs (u u u ij ))dy y y + Y d S 1 (θ θ θ 1 ) ∂A ijrs ∂d S 1 (d S S S )(E k ij + e yij (u u u k ))ε rs (v v v)dy y y + 1 Y -d S 1 (θ θ θ 1 ) ∂h (d S 1 ) ∂d S 1 (1 -h (d S 2 ))dy y y + 2 Y d S 1 (θ θ θ 1 ) ∂h (d S 1 ) ∂d S 1 (1 -h (d S 2
)) dy y y

+ 3 Y -d S 1 (θ θ θ 1 ) ∂h (d S 1 ) ∂d S 1 h (d S 2 ) dy y y + 4 Y d S 1 (θ θ θ 1 ) ∂h (d S 1 ) ∂d S 1 h (d S 2
) dy y y.

The term on the second line is zero due to the fact that the adjoint solution is identically zero. Moreover, applying Proposition 2.5 and then Proposition 2.9 from [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF] as well as using the fact that we are dealing with thin interfaces we obtain,

∂L ∂S 1 (θ θ θ 1 ) = -η ijk A H ijk -A t ijk Γ θ θ θ 1 • n n n 1 A * mqrs (d S 2 ) (E k mq + ε mq (u u u k )) (E ij rs + ε rs (u u u ij )) dy y y + 1 Γ θ θ θ 1 • n n n 1 (1 -h (d S 2 )) dy y y -2 Γ θ θ θ 1 • n n n 1 (1 -h (d S 2
)) dy y y

+ 3 Γ θ θ θ 1 • n n n 1 h (d S 2 ) dy y y -4 Γ θ θ θ 1 • n n n 1 h (d S 2 ) dy y y
where n n n 1 denotes the outer unit normal to S 1 . Thus, if we let u u u k = χ χ χ k , the solution to the unit cell (2.2) and collect terms the result follows.

Remark 7. The tensor A * in (3.5) as well h * in (3.6) of the shape derivatives in Theorem 3.1.1 depend on the signed distance function in an alternate way which provides an insight into the coupled nature of the problem. We further remark, that in the smooth interface context, the collective boundary Γ to be displaced in Theorem 3.1.1, is not an actual boundary but rather a tubular neighborhood.

3.2.

The numerical algorithm. The result of Theorem 3.1.1 provides us with the shape derivatives in the directions θ θ θ 1 , θ θ θ 2 respectively. If we denote by,

v 1 = ∂J ∂S 1 (S S S), v 2 = ∂J ∂S 2 (S S S),
a descent direction is then found by selecting the vector field θ θ θ 1 = v 1 n n n 1 , θ θ θ 2 = v 2 n n n 2 . To move the shapes S 1 , S 2 in the directions v 1 , v 2 is done by transporting each level set, φ i , i = 1, 2 independently by solving the Hamilton-Jacobi type equation

∂φ i ∂t + v i |∇φ i | = 0, i = 1, 2. (3.9)
Moreover, we extend and regularize the scalar velocity v i , i = 1, 2 to the entire domain Y as in [START_REF] Allaire | Structural optimization using sensitivity analysis and a level set method[END_REF], [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF]. The extension is done by solving the following problem for i = 1, 2,

-α 2 ∆θ θ θ i + θ θ θ i = 0 in Y, ∇θ θ θ i n n n i = v i n n n i on Γ, θ θ θ i Y-periodic,
where α > 0 is small regularization parameter. Hence, using the same algorithm as in [START_REF] Allaire | Structural optimization using sensitivity analysis and a level set method[END_REF], for i = 1, 2 we have:

3.2.1. Algorithm. We initialize S 0 i ⊂ U ad through the level sets φ i 0 defined as the signed distance function of the chosen initial topology, then 1. iterate until convergence for k ≥ 0: a. Calculate the local solutions χ χ χ m k for m, = 1, 2 by solving the linear elasticity problem

(2.2) on O k := S k 1 ∪ S k 2 .
b. Deform the domain O k by solving the Hamilton-Jacobi equations (3.9) for i = 1, 2.

The new shape O k+1 is characterized by the level sets φ k+1 i solutions of (3.9) after a time step ∆t k starting from the initial condition φ k i with velocity v i k computed in terms of the local problems χ χ χ m k for i = 1, 2. The time step ∆t k is chosen so that J(S S S k+1 ) ≤ J(S S S k ). 2. From time to time, for stability reasons, we re-initialize the level set functions φ k i by solving (3.2) for i = 1, 2.

Numerical examples

For all the examples that follow we have used a symmetric 100 × 100 mesh of P 1 elements. We imposed volume equality constraints for each phase. In the smooth interpolation of the material properties in formula (3.3), we set equal to 2∆x where ∆x is the grid size. The parameter is held fixed through out (see [START_REF] Allaire | Multi-phase structural optimization via a level set method[END_REF] and [START_REF] Michailidis | Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method[END_REF]). The Lagrange multipliers were updated at each iteration the following way, n+1 j = n j -β Y ι n j dy y y -V t j , where β is a small parameter. Due to the fact that this type of problem suffers from many local minima that may not result in a shape, instead of putting a stopping criterion in the algorithm we fix, a priori, the number iterations. Furthermore, since we have no knowledge of what volume constraints make sense for a particular shape, we chose not to strictly enforce the volume constraints for the first two examples. However, for examples 3 and 4 we use an augmented Lagrangian to actually enforce the volume constraints,

L(S S S, µ µ µ, β β β) = J(S S S) - 4 i=1 µ i C i (S S S) + 4 i=1 1 2 β i C 2 i (S S S),
here C i (S S S) are the volume constraints and β is a penalty term. The Lagrange multipliers are updated as before, however, this time we update the penalty term, β every 5 iterations. All the calculations were carried out using the software FreeFem++ [START_REF] Hecht | New development in FreeFem++[END_REF].

Remark 8. We remark that for the augmented Lagrangian we need to compute the new shape derivative that would result. The calculations are similar as that of Theorem 3.1.1 and, therefore, we do not detail them here for the sake of brevity.

Example 1.

The first structure to be optimized is multilevel material that attains an apparent Poisson ratio of -1. The Young moduli of the four phases are set to E 1 = 0.91, E 2 = 0.0001, E 3 = 1.82, E 4 = 0.0001. Here phase 2 and phase 4 represent void, phase 2 represents a material that is twice as stiff as the material in phase 3. The Poisson ratio of each phase is set to ν = 0.3 and the volume constraints were set to V t 1 = 30% and V t 3 = 4%.

ijkl 1111 1122 2222 η ijkl 1 30 1 A H ijkl 0.12 -0.09 0.12 A t ijkl 0.1 -0.1 0.1 Table 1. Values of weights, final homogenized coefficients and target coefficients From figure 8 we observe that the volume constraint for the stiffer material is not adhered to the target volume. In this cases the algorithm used roughly 16% of the material with Poisson ratio 1.82 while the volume constraint for the weaker material was more or less adhered to the target constraint. 11 we observe that the volume constraint for the stiffer material is not adhered to the target volume. In this cases the algorithm used roughly 15% of the material with Poisson ratio 1.82 while the volume constraint for the weaker material was more or less adhered to the target constraint. 3. Values of weights, final homogenized coefficients and target coefficients Again just as in the previous two examples we observe that the volume constraint for the stiffer material is not adhered to the target volume, even though for this example a augmented Lagrangian was used. In this cases the algorithm used roughly 20% of the material with Poisson ratio 1.82 while the volume constraint for the weaker material was more or less adhered to the target constraint. The fourth structure to be optimized is multilevel material that attains an apparent Poisson ratio of -0.5. An augmented Lagrangian was used to enforce the volume constraints for this example as well. The Lagrange multiplier was updated the same way as before, as was the penalty parameter β. The Young moduli of the four phases are set to E 1 = 0.91, E 2 = 0.0001, E 3 = 1.82, E 4 = 0.0001. The Poisson ratio of each material is set to ν = 0.3, however, this times we require that the volume constraints be set to V

Conclusions and Discussion

The problem of an optimal multi-layer micro-structure is considered. We use inverse homogenization, the Hadamard shape derivative and a level set method to track boundary changes, within the context of the smooth interface, in the periodic unit cell. We produce several examples of auxetic micro-structures with different volume constraints as well as different ways of enforcing the aforementioned constraints. The multi-layer interpretation suggests a particular way on how to approach the subject of 3D printing the micro-structures. The magenta material is essentially the cyan material layered twice producing a small extrusion with the process repeated several times. This multi-layer approach has the added benefit that some of the contact among the material parts is eliminated, thus allowing the structure to be further compressed than if the material was in the same plane.

The algorithm used does not allow "nucleations" (see [START_REF] Allaire | Structural optimization using sensitivity analysis and a level set method[END_REF], [START_REF] Wang | Level-set method for design of multi-phase elastic and thermoelastic materials[END_REF]). Moreover, due to the non-uniques of the design, the numerical result depend on the initial guess. Furthermore, volume constraints also play a role as to the final form of the design.

The results in this work are in the process of being physically realized and tested both for polymer and metal structures. The additive manufacturing itself introduces further constraints into the design process which need to be accounted for in the algorithm if one wishes to produce composite structures.

Figure 1 .

 1 Figure 1. A 3D printed material with all four branches on the same plane achieving an apparent Poisson ratio of -0.8 with over 20% strain. On subfigure (a) is the uncompressed image and on sub-figure (b) is the image under compression. Used with permission from [18].

Figure 2 .

 2 Figure 2. A 3D printed material with two of the branches on a different plane achieving an apparent Poisson ratio of approximately -1.0 with over 40% strain. Sub-figure (a) is the uncompressed image and sub-figure (b) is the image under compression. Used with permission from [18].

Figure 4 .

 4 Figure 4. Perturbation of a domain in the direction θ θ θ.

F 2 dFigure 5 .

 25 Figure 5. Representation of different material in the unit cell for d = 2.

Figure 6 .

 6 Figure 6. The design process of the material at different iteration steps. Young modulus of 1.82, Young modulus of 0.91, void.

Figure 7 .Figure 8 .

 78 Figure 7. On the left we have the unit cell and on the right we have the macro-structure obtained by periodic assembly of the material with apparent Poisson ratio -1.

Figure 9 .

 9 Figure 9. The design process of the material at different iteration steps. Young modulus of 1.82, Young modulus of 0.91, void.

Figure 10 .Figure 11 .

 1011 Figure 10. On the left we have the unit cell and on the right we have the macro-structure obtained by periodic assembly of the material with apparent Poisson ratio -1.

Figure 12 .

 12 Figure 12. The design process of the material at different iteration steps. Young modulus of 1.82, Young modulus of 0.91, void.

Figure 13 .Figure 14 .

 1314 Figure 13. On the left we have the unit cell and on the right we have the macro-structure obtained by periodic assembly of the material with apparent Poisson ratio -0.5.

Figure 15 .

 15 Figure 15. The design process of the material at different iteration steps. Young modulus of 1.82, Young modulus of 0.91, void.

Figure 16 .Figure 17 .

 1617 Figure 16. On the left we have the unit cell and on the right we have the macro-structure obtained by periodic assembly of the material with apparent Poisson ratio -0.5.

Table 2 .

 2 3, however, this times we require that the volume constraints be set to V t 1 = 33% and V t 3 = 1%. Values of weights, final homogenized coefficients and target coefficients Again, from figure

	ijkl 1111 1122 2222
	η ijkl	1	30	1
	A H ijkl 0.11 -0.09 0.12 A t ijkl 0.1 -0.1 0.1

Table 4 .

 4 Values of weights, final homogenized coefficients and target coefficients

	t 1 = 53%

Acknowledgments

This research was initiated during the sabbatical stay of A.C. in the group of Prof. Chiara Daraio at ETH, under the mobility grant DGA-ERE (2015 60 0009). Funding for this research was provided by the grant "MechNanoTruss", Agence National pour la Recherche, France (ANR-15-CE29-0024-01). The authors would like to thank the group of Prof. Chiara Daraio for the fruitful discussions. The authors are indebted to Grégoire Allaire and Georgios Michailidis for their help and fruitful discussions as well as to Pierre Rousseau who printed and tested the material in figure 1 & figure 2.


Jolita Ralyté
email: jolita.ralyte@unige.ch

Michel Léonard
email: michel.leonard@unige.ch

Evolution Models for Information Systems Evolution Steering

Keywords: Information Systems Evolution, IS evolution steering, IS evolution structure, IS evolution lifecycle, IS evolution impact

Sustainability of enterprise Information Systems (ISs) largely depends on the quality of their evolution process and the ability of the IS evolution steering officers to deal with complex IS evolution situations. Inspired by Olivé [1] who promotes conceptual schema-centric IS development, we argue that conceptual models should also be the centre of IS evolution steering. For this purpose we have developed a conceptual framework for IS evolution steering that contains several interrelated models. In this paper we present a part of this framework dedicated to the operationalization of IS evolution -the evolution metamodel. This metamodel is composed of two interrelated views, namely structural and lifecycle, that allow to define respectively the structure of a particular IS evolution and its behaviour at different levels of granularity.

Introduction

No matter the type and the size of the organization (public or private, big or small), sustainability of its Information Systems (ISs) is of prime importance to ensure its activity and prosperity. Sustainability of ISs largely depends on the quality of their evolution process and the ability of the officers handling it to deal with complex and uncertain IS evolution situations. There are several factors that make these situations complex, such as: proliferation of ISs in the organization and their overlap, independent evolution of each IS, non-existence of tools supporting IS evolution steering, various IS dimensions to be taken into account, etc. Indeed, during an IS evolution not only its information dimension (the structure, availability and integrity of data) is at stake. IS evolution officers have also to pay attention to its activity dimension (the changes in enterprise business activity supported be the IS), the regulatory dimension (the guarantee of IS compliance with enterprise regulation policies), and the technology dimension (the implementation and integration aspects).

In this context, we claim that there is a need for an informational engineering approach supporting IS evolution steering, allowing to obtain all the necessary information for an IS evolution at hand, to define and plan the evolution and to assess its impact on the organization and it ISs. We found the development of such an approach on conceptual modelling by designing a conceptual framework for IS evolution steering. Some parts of this framework were presented in [START_REF] Opprecht | Towards a framework for enterprise information system evolution steering[END_REF] and [START_REF] Ralyté | Defining the responsibility space for the information systems evolution steering[END_REF]. In this paper we pursue our work and present one of its componentsthe operationalization of the IS evolution through the metamodel of IS Evolution.

The rest of the paper is organized as follows: in section 2 we overview the context of our work -the conceptual framework that we are developing to support the IS evolution steering. Then, in section 3, we discuss the role and principles of conceptual modelling in handling IS evolution. In sections 4 and 5 we present our metamodel formalizing the IS evolution, its structural and lifecycle views, and illustrate their usage in section 6. Section 7 concludes the paper.

Context: A Framework for IS Evolution Steering

With our conceptual framework for IS evolution steering we aim to face the following challenges: 1) steering the IS evolution requires a thorough understanding of the underpinning IS domain, 2) the impact of IS evolution is difficult to predict and the simulation could help to take evolution decisions, 3) the complexity of IS evolution is due to the multiple dimensions (i.e. activity, regulation, information, technology) to be taken into account, and 4) the guidance for IS evolution steering is almost non-existent, and therefore needs to be developed. As shown in Fig. 1, the framework contains several components each of them taking into account a particular aspect of IS evolution steering and considering the evolution challenges listed above. Let us briefly introduce these components.

The IS Steering Metamodel (IS-SM ) is the main component of the framework having as a role to represent the IS domain of an enterprise. Concretely, it allows to formalize the way the enterprise ISs are implemented (their structure in terms of classes, operations, integrity rules, etc.), the way they support enterprise business and management activities (the definition of enterprise units, activities, positions, business rules, etc.), and how they comply with regulations governing these activities (the definition of regulatory concepts, rules and roles). Although IS-SM is not the main subject of this paper (it was presented in [START_REF] Opprecht | Towards a framework for enterprise information system evolution steering[END_REF][START_REF] Ralyté | Defining the responsibility space for the information systems evolution steering[END_REF]), it remains the bedrock of the framework, and is necessary to be presented for better understanding other models and illustrations. IS-SM is also the kernel model for implementing an Informational Steering Information System -ISIS. ISIS is a meta-IS for steering enterprise IS (IS upon ISs according to [START_REF] Dinh | Towards a New Infrastructure Supporting Interoperability of Information Systems in Development: the Information System upon Information Systems[END_REF]). While enterprise ISs operate at business level, ISIS performs at the IS steering level. Therefore, we depict IS-SM in Fig. 2 mainly to make this paper self-explanatory, and we invite the reader to look at [5] for further details. The role of the Evolution Metamodel is to specify IS changes, and to assist the IS steering actor responsible for performing these changes. This metamodel comprises two interrelated views: structural and lifecycle. While the former deals with the extent and complexity of the IS evolution, the later supports its planning and execution. The Evolution Metamodel is the main subject of this paper, and is detailed in the following sections.

The Impact Space component provides mechanisms to measure the impact of IS changes on the enterprise IS, on the business activities supported by these ISs, and on the compliance with regulations governing enterprise activities. The impact model of a particular IS evolution is defined as a part of the IS-SM including the IS-SM elements that are directly or indirectly concerned by this evolution. An IS-SM element is directly concerned by the evolution if its instances undergo modifications, i.e. one or more instances of this element are created, enabled, disabled, modified, or deleted. An IS-SM element is indirectly concerned by the evolution if there is no modification on its instances but they have to be known to make appropriate decisions when executing the evolution.

The Responsibility Space (Ispace/Rspace) component [START_REF] Ralyté | Defining the responsibility space for the information systems evolution steering[END_REF] helps to deal with responsibility issues related to a particular IS evolution. Indeed, each IS change usually concerns one or several IS actors (i.e. IS users) by transforming their information and/or regulation spaces (Ispace/Rspace). An IS actor can see her information/regulation space be reduced (e.g. some information is not accessible anymore) or in the contrary increased (e.g. new information is available, new actions has to be performed, new regulations has to be observed). In both cases the responsibility of the IS actor over these spaces is at stake. The Ispace/Rspace model is defined as a part of IS-SM. It allows for each IS evolution to create subsets of information, extracted from ISIS, that inform the IS steering officer how this evolution affects the responsibility of IS users.

Finally, the Evolution Steering Method provides guidelines to use all these aforementioned models when executing an IS evolution.

3 Modelling IS Evolution: Background and Principles

Background

Most of the approaches dealing with IS and software evolution are based on models and metamodels (e.g. [START_REF] Pons | Model evolution and system evolution[END_REF][START_REF] Burger | A change metamodel for the evolution of mof-based metamodels[END_REF][START_REF] Aboulsamh | Towards a model-driven approach to information system evolution[END_REF][START_REF] Kchaou | A mof-based change meta-model[END_REF][START_REF] Ruiz Carmona | TraceME: Traceability-based Method for Conceptual Model Evolution[END_REF]). They mainly address the structural aspects of IS evolution (for example, changing a hierarchy of classes, adding a new class) [START_REF] Pons | Model evolution and system evolution[END_REF], model evolution and transformations [START_REF] Burger | A change metamodel for the evolution of mof-based metamodels[END_REF], and the traceability of changes [START_REF] Kchaou | A mof-based change meta-model[END_REF][START_REF] Ruiz Carmona | TraceME: Traceability-based Method for Conceptual Model Evolution[END_REF]. They aim to support model-driven IS development, the automation of data migration, the evaluation of the impact of metamodel changes on models, the development of forward-, reverse-, and re-engineering techniques, the recording of models history, etc. The importance and impact of model evolution is also studied in [START_REF] Lehman | Software Evolution[END_REF] where the authors stress that understanding and handling IS evolution requires models, model evolution techniques, metrics to measure model changes and guidelines for taking decisions.

In our work, we also claim that the purpose of conceptual modelling in IS evolution steering is manifold, it includes the understanding, building, deciding and realising the intended IS changes. As per [START_REF] Lehman | Evolution as a noun and evolution as a verb[END_REF], the notion of IS evolution has to be considered as a noun and as a verb. As a noun it refers to the question "what" -the understanding of the IS evolution phenomenon and its properties. While as a verb, it refers to the questions "how" -the theories, languages, activities and tools which are required to evolve a software. Our metamodel for IS evolution steering (see Fig. 1) includes two complementary views, namely structural and lifecycle view, and so serves to cope with complex IS artefacts, usually having multiple views.

Models are also known as a good support for taking decisions. In case of IS evolution, usually, there are several possible ways to realise it, each of them having a different impact on enterprise ISs and even on its activities. Taking a decision without any appropriate support can be difficult and very stressful task. Finally, with a set of models, the realisation of IS evolution is assisted in each evolution step and each IS dimension.

Principles of IS Evolution

The focus of the IS evolution is to transform a current IS schema (ASIS-IS) into a new one (TOBE-IS), and to transfer ASIS-IS objects into TOBE-IS objects. We use ISIS (see the definition in section 2), whose conceptual schema is represented by IS-SM (Fig. 2), as a support to handle IS evolution. Indeed, ISIS provides a thorough, substantial information on the IS structure and usage, which, combined with other information outside of ISIS, is crucial to decide the IS evolution to pursue. Furthermore, ISIS is the centre of the management and the execution of the IS evolution processes both at the organizational and informatics levels. So, one main principle of IS evolution is always to consider these two interrelated levels: the ISIS and IS levels with their horizontal effects concerning only one level, and their vertical effects concerning both levels. In the following, to make a clear distinction between the IS and ISIS levels, we use the concepts of "class" and "object" at the IS schema level, and "element" and "instance" at the ISIS schema level.

IS evolution is composition of transformation operations, where the most simple ones are called atomic evolution primitives. Obtaining an initial list of atomic evolution primitives for an IS and its ISIS is simple: we have to consider all the elements of the ISIS schema, and, for each of them, all the primitives usually defined over an element: Search, Create, Read, Update, Delete (SCRUD). In the case of IS-SM as ISIS schema, there are 53 elements and so, 265 atomic evolution primitives. Since the aim of the paper is to present the principles of our framework for IS evolution steering, we simplify this situation by considering only the most difficult primitives Create and Delete. Nevertheless, there are still 106 primitives to be considered. The proposed conceptual framework for IS evolution steering is going to help facing this complexity.

Structural View of IS Evolution

An IS evolution transforms a part of the ASIS-IS ISP into ISP', which is a part of the TOBE-IS, in a way that the TOBE-IS is compliant with:

-the horizontal perspective: the instances of the new ISIS and the objects of TOBE-IS validate all the integrity rules defined respectively over ISIS and TOBE-IS; -the vertical perspective: the TOBE-IS objects are compliant with the instances of the new ISIS.

In a generic way, we consider that an overall IS evolution requires to be decomposed into several IS evolutions, and so the role of the structural IS evolution model view (shown in Fig. 3) consists in establishing the schema of each IS evolution as a schema of composition of evolution primitives defined over IS-SM to pursue the undertaken IS evolution.

An evolution primitive represents a kind of elementary particle of an evolution: we cannot split it into several parts without loosing qualities in terms of manageable changes and effects, robustness, smartness and performances, introduced in a following paragraph. The most basic evolution primitives are the atomic evolution primitives: some of them, like Create, Delete and Update, are classic, since the other, Enable, Disable, Block, Unblock are crucial for the evolution process.

Atomic Evolution Primitives

Since the ISIS schema (i.e. IS-SM) is built only by means of existentially dependencies1 , the starting point of the IS evolution decomposition is very simple -it consists of a list of atomic primitives: create, delete, update, enable, disable, block and unblock an instance of any IS-SM element. We apply the same principle at the IS level, so the IS schema steered by ISIS is also built by using only existential dependencies. Moreover, an instance/object is existentially dependent on its element/class.

These atomic primitives determine a set of possible states that any ISIS instance (as well as IS object) could have, namely created, enabled, blocked, disabled, and deleted. Fig. 4 provides the generic life cycle of an instance/object.

Once an instance is created, it must be prepared to be enabled, and so to be used at the IS level. For example, a created class can be enabled, and so to have objects at the IS level, only if its methods validate all the integrity rules whose contexts contain it. A created instance can be deleted if it belongs to a stopped evolution. Enabled instances are disabled by an evolution when they do not play any role in the targeted TOBE-IS. They are not deleted immediately for two reasons: the first one concerns the fact that data, operations, or rules related to them, which were valid before the evolution, still stay consistent for situations where continuity is mandatory, for instance due to contracts. The second one concerns the evolution itself: if it fails, it is necessary to come back to the ASIS-IS, and so, to enable again the disabled instances.

Enabled instances are blocked during a phase of an evolution process when it is necessary to avoid their possible uses at the IS level through objects and/or execution of operations. At the end of this phase they are unblocked (re-enabled). For instance an activity (an instance of the element Activity) can be blocked temporary because of the introduction of a new business rule. Finally when an instance is deleted, it disappears definitively.

Robust Generic Atomic Evolution Rules

The generic atomic evolution rules must be validated to assure the consistency of the evolution process. Indeed each atomic evolution primitive has effects on other elements than its targeted elements. For example, deleting an integrity rule has effects on the methods of several classes belonging to the context of this integrity rule. Below, we present two kinds of generic atomic evolution rules: the first concerns the horizontal and vertical evolution effects, while the second deals with the dynamic effects.

Evolution Effects Horizontally and Vertically. An evolution primitive is firstly an atomic operation on the ISIS. So, it must verify the integrity rules defined over the IS-SM model to manage the horizontal effects. For example, if an instance cl of the element Class is deleted, then all the instances clo i of the element Class Operation related with cl must be deleted due to the existential dependency between these two elements (see Fig. 2).

An evolution primitive is also an operation on the IS and has to manage the vertical effects of the conformity rules between ISIS instances and IS objects. For example, deleting cl induces also deleting all its objects in the IS.

Then, since the evolution operations on IS are executed from ISIS, they validate the integrity rules defined over IS, which are instances of ISIS.

Generic Dynamic Evolution Rules. The generic evolution rules concern the states of the ISIS elements produced by the use of atomic evolution primitives (Fig. 4): created, enabled, blocked, disabled, deleted, and especially the interactions between instances of different elements in different states. They must be observed only at the ISIS level.

Some generic rules concerning the states "created" and "deleted" are derived directly from the existentially dependencies. Considering the element Einf depending existentially on the element Esup, any instance of Einf may be in the state "created" only if its associated instance esup of Esup is in the state "created", and it must be in the state "deleted" if esup is in the state "deleted".

The generic rules concerning the states "blocked" and "disabled" require to con-sider another relation between the IS-SM elements, called "determined by", defined at the conceptual level and not the instance level. An element Esecond is strictly/weakly determined by the element Efirst if any instance esecond to be exploitable in IS must/can require to be associated to one or several instances efirst.

Then there is the following generic dynamic rule: any instance esecond must be in the state disabled/blocked/deleted if at least one of its efirst is in the state respectively disabled/blocked/deleted.

For instance, the element Operation is strictly determined by the element Class, because any operation to be executed at the IS level must be associated to at least one class (see Fig. 2). Then, if an operation is associated to one class in the state disabled/blocked, it also must be in the state disabled/blocked, even if it is also associated to other enabled classes.

The element Integrity Rule is weakly determined by the element Business Rule because integrity rules are not mandatory associated with a business rule. In the same way, all elements, like Class, associated with the Regulatory Element are weakly determined by it, because their instances are not mandatory associated to an instance of Regulatory Element.

Considering the following elements of the IS-SM models (see Fig. 2): Person, Position, Business Process (BP), Activity, Business Rule (BR), Role, Operation, Class, Integrity Rule (IR), Regulatory Element (RE), here is the list of relations strictly determined by (=>): BP => Activity, BR => Activity, Operation => Class, Operation => IR, IR => Class. The list of the relations weakly determined by (->) (in addition to the aforementioned ones with the Regulatory Element) is: IR -> BR, IR -> RE, Class -> RE, Operation -> RE, Role -> RE, BR -> RE, Activity -> RE, Position -> RE, Event -> RE, BP -> RE.

Robustness. Every aforementioned evolution primitive is robust if it manages all its horizontal and vertical effects and respects all the generic dynamic evolution rules. The use of only existential dependencies at the both levels, IS and ISIS, in our approach, facilitates reaching this quality. Nevertheless, at the IS level, such an approach requires that the whole IS schema (including static, dynamic and integrity rule perspectives) must be easily evolvable, and the IT system supporting the IS (e.g. a database management system) must provide an efficient set of evolution primitives [START_REF] Andany | Management of schema evolution in databases[END_REF].

Composite Evolution Primitives

The composite primitives are built by a composition of the atomic ones (Fig. 3). They are necessary to consider IS evolution at the management level [START_REF] Ralyté | Defining the responsibility space for the information systems evolution steering[END_REF], but also for informational and implementation purposes. For instance, replacing an integrity rule by a new one can be considered logically equivalent to delete it and then to create the new one. But this logic is not pertinent if we consider the managerial, IS exploitation and implementation perspectives. It is much more efficient to build a composite evolution primitive "replace" built from the atomic primitives "create" and "delete".

A composite evolution primitive is robust, if it manages all its horizontal and vertical effects and respects all the generic dynamic evolution rules.

Managerial Effects

The managerial effects consider the effects of the IS evolution at the human level, and so concern the IS-SM elements Role, Activity, Position and Person. The evolution steering officers have to be able to assess whether the proposed evolution has a harmful effect on organization's activities or not, and to decide to continue or not this evolution. The evolution primitives are smart if they alert these levels by establishing a report of changes to all the concerned roles, activities, positions, and persons. To do that, they will use the responsibility space (Fig. 1) with its two sub-spaces: its informational space (Ispace) and its regulatory space (Rspace). This part was presented in [START_REF] Ralyté | Defining the responsibility space for the information systems evolution steering[END_REF]. Below in the paper, all primitives are smart.

Lifecycle View of IS Evolution

Evolution of an information system is generally a delicate process for an enterprise for several reasons. First, it cannot be realized by stopping the whole IS because all the activities supported by IS should be stopped and this situation is unthinkable in most cases. Second, it has impacts, especially on actors and on the organization of activities. It can even induce the need for reorganizing the enterprise. Third, it takes time and often requires to set up a process of adaptation to the changes for all concerned actors to enable them to perform their activities. Moreover, it concerns a large informational space of IS-SM and requires to be decomposed into partial evolutions called sub-evolutions. So, it requires a coordination model to synchronize all processes of these sub-evolutions as well as the process of the main evolution. Furthermore, it is a long process, with an important number of actors who work inside the evolution process or whose activities are changed by the evolution. Finally, most evolutions of ASIS-IS into TOBE-IS are generally nearly irreversible, because it is practically impossible to transform back TOBE-IS into ASIS-IS at least for two main reasons: (1) some evolution primitives, used by the evolution, can be irreversible themselves (e.g. the case of an existing integrity rule relaxed by the evolution), and (2) actors, and even a part of the enterprise, can be completely disoriented to go back to TOBE-IS after all the efforts they have done to adapt to ASIS-IS. So, a decision to perform an evolution must be very well prepared to decrease the risks of failure. For this purpose, we explore a generic lifecycle of an evolution, first at atomic primitive level, then at composite primitive level and finally at evolution level.

An atomic primitive can be performed stand-alone in two steps: (1) preparation and (2) execution or abort. They are defined as follows:

-Preparation: prepares the disabling list of ISIS instances and IS objects to be disabled if success, the creating list of ISIS instances and IS objects to be created if success, the list of reports of changes, and the blocking list of ISIS instances; -Execution: sends the reports of changes, blocks the concerned IS parts, disables/creates effectively contents of the disabling/creating lists, then unblocks the blocked IS parts.

The work done at the preparation step serves to decide whether the primitive should be executed or aborted. Finally, the execution of the primitive can succeed or fail. For example, it fails if it cannot block an IS part. As an example let us consider the deletion of a role: its creating list is empty and its disabling list contains all the assignments of operations to this role. Blocking these assignments signifies these operations cannot be executed by means of this role during the deletion of the role. It can fail if an IS actor is working through this role.

In the case of the atomic evolution primitive "Create an instance Cl of Class", the preparation step defines:

-how to fill the new class with objects, -how to position it in the IS schema by linking Cl to other IS classes by means of existential dependencies; -how to alert the managers of Role, Operation and Integrity Rule about the Cl creation.

Besides, it is important to create together with Cl its methods and attributes, and even the association classes between Cl and other IS classes. For that, we need a more powerful concept, the composite evolution primitive, as presented below.

A composite primitive is composed of other composite or atomic primitives, which builds a hierarchy of primitives. The top composite primitive is at the root of this hierarchy; the atomic primitives are at its leaves. Every composite primitive has a supervision step, which controls the execution of all its subprimitives. Only the top composite primitive has in addition a coordination step, which takes the same decision to enable or abort for all its sub primitives in the hierarchy. The main steps of a composite primitive life cycle are:

-Preparation: creates all direct sub-primitives of the composite primitive; -Supervision: determines the impacts and the managerial effects from the enabling lists and the creating lists established by the sub-primitives; -Coordination: takes the decision of enabling or aborting primitive processing and transmits it to the sub-primitives; -Training: this is a special step for the top primitive; it concerns training of all actors concerned by the whole evolution. This step is performed thanks to the actors' responsibility spaces.

The top composite primitive is successful if all its sub-primitives are successful; it fails if at least one among its sub-primitives fails. The life cycle of the atomic primitives must be adapted by adding the abort decision and by taking into account that enable/abort decisions are made by a higher level primitive. Fig. 5 illustrates the co-ordination between the composite primitive lifecycle and its sub-primitive lifecycle.

Thus, from the atomic evolution primitive "Create Class" we build the composite evolution primitive "C-Create Class" with the following sub-primitives:

-Create Class, which is used to create the intended class Cl and also all the new classes to associate Cl with other classes, as mentioned previously, -Create Class Concept, Create Class Attribute, -if necessary, Create Attribute, and Create Domain, -C-Create Method with its sub-primitives Create Method and Create Attribute Method.

Let us now look at the lifecycle of an entire evolution, which is a composition of primitives. During the processing of an evolution, the preparation step consists in selecting the list of composite primitives, whose processing will realize this Fig. 6. Coordination of the IS evolution lifecycle with the lifecycles of its top composite primitives evolution. Then, from the impacts and the managerial effects determined by the supervision steps of these composite primitives, the supervision step of the evolution determines a plan for processing these primitives. It decides which primitives can/must be executed in parallel and which in sequence. Next, the coordination step launches processing of primitives following the plan. After analyzing their results (success or failure), it decides to launch other primitives and/or to abort some of them. Finally, the evolution is finished and it is time to assess it. Indeed, the evolution processing transforms the enterprise and its ways of working, even if processing of some composite primitives fails. Due to the important complexity, it seems important to place the training step at the evolution level and not at the level of composite primitive. Of course, the training step of a composite primitive must be realized before its execution. But, in this way, it is possible to combine training steps of several composite primitives into one, and to obtain a more efficient training in the context of the evolution. Fig. 6 shows the coordination between the lifecycles of IS evolution and its top composite primitives.

Illustrating Example

To illustrate our approach, we use the example of a hospital. Fig. 7 depicts a small part of the kernel of its IS schema. In this example we will consider:

-one organizational unit: the general medicine department, -two positions: the doctor and the nurse, -two activities of a doctor: a 1 concerning the care of patients (visit, diagnostic, prescription) and a 2 concerning the management of the nurses working in her team. To illustrate an evolution case, let us suppose that now our hospital has to apply new rules for improving patients' safety. To this end, each doctor will be in charge to guarantee that nurses of her team have sufficient competences for administrating the drugs she can prescribe. So, the IS of the hospital must evolve, especially by introducing new classes: Nurse Drug that associates a nurse with a drug for which she is competent according to her doctor, and Doctor Drug that associates a doctor with a drug that she can prescribe. The TOBE-IS schema is shown in Fig. 8. The IS evolution is then composed of 2 top composite primitives, one around Doctor Drug (DD), the other one around Nurse Drug (ND). The first one is built from the composite primitive C-Create Class to create the instance DD of the ISIS element Class. Its preparation step specifies:

-the DD objects will be obtained from the ASIS-IS class Prescription; -DD will be existentially dependent on the IS classes Doctor and Drug, and Prescription will become existentially dependent on DD and no more directly dependent on Drug;

-the alerts for Role, Activities, Positions, Persons about the changes, especially in the creation an object of Prescription, which in TOBE-IS must be related to a DD object; -creation of DD objects, creation or modification of roles for reaching them; -the blocking list for its execution, which includes Doctor, Drug and Prescription.

The second composite primitive is built from the composite evolution primitive C-Create Class to create the instance ND of the ISIS element Class. Its preparation step specifies:

-the ND objects will be obtained from the ASIS-IS class Prescription; -ND will be existentially dependent on the IS classes Nurse and Drug, and Drug Delivery will become existentially dependent on ND; -the alerts for Role, Activities, Positions, Persons about the changes, especially in the creation an object of Drug Delivery, which must be related to a ND object; -creation of ND objects, creation or modification of roles for reaching them; -the blocking list for its execution, which includes Nurse, Drug and Drug Delivery.

In the case of this example, the execution process of the IS evolution after the training of involved actors is simple: to execute the top evolution composite primitives related to Doctor Drug and then to Nurse Drug.

Conclusion

Handling information systems evolution is a complex task that has to be properly defined, planned and assessed before its actual execution. The result of each IS evolution has impact on the sustainability on organization's ISs and also on the efficiency of the organization's activity. So this task is not only complex but also critical.

In this paper, we continue to present our work on a conceptual framework for IS evolution steering that aims to establish the foundation for the development of an Informational Steering Information System (ISIS). In particular, we dedicate this paper to the engineering aspects of the concept of IS evolution, and present its metamodel, which is one of the components in our framework (Fig. 1).

The role of the IS Evolution Metamodel consists in supporting the operationalization of the IS evolution. Therefore, it includes two views: structural and lifecycle. The structural view allows to progressively decompose a complex IS evolution into a set of atomic primitives going trough several granularity levels of composite primitives. The obtained primitives as robust because they follow generic evolution rules and take into account horizontal and vertical effects on ISIS and IS. They are also smart because they pay attention to the managerial effects of IS evolution at the human level. The lifecycle view helps to operate IS evolution at its different levels of granularity by providing a set of models and rules for progressing from one step to another.

To complete our framework for IS evolution steering we still need to define the Impact Space component that will provide mechanisms to measure the impact of IS evolution and to take decisions accordingly. With the IS Evolution Metamodel we have prepared the basis for developing the detailed guidance for IS evolution steering, which will complete our work on this conceptual framework.

Fig. 1 .

 1 Fig. 1. Conceptual Framework for IS Evolution Steering

Fig. 2 .

 2 Fig. 2. Simplified version of IS-SM. The right part (in white) shows the information model generic to any IS implementation, the left part (in red) represents enterprise business activity model, the top part (in grey) represents the regulatory model governing enterprise business and IS implementations. The multi-coloured elements represent pivot elements allowing to interconnect the information, activity and regulation models, and so, to capture how ISs support enterprise activities and comply with regulations.

Fig. 3 .

 3 Fig. 3. Structural view of the IS evolution

Fig. 4 .

 4 Fig. 4. Lifecycle of an instance of any element from IS-SM

Fig. 5 .

 5 Fig. 5. Coordination of the lifecycle of a composite primitive (left) with the lifecycles of its sub-primitives (right); * indicates multiple transitions, dashed lines indicate that the step is under the responsibility of the lower or upper level model.

Fig. 7 .

 7 Fig. 7. A small part of the ASIS-IS schema of the hospital

Fig. 8 .

 8 Fig. 8. A part of the TOBE-IS schema of the hospital

A class C2 is existentially dependent on the class C1, if every object o2 of C2 is permanently associated to exactly one object o1 of C1; o2 is said to be existentially dependent on o1. The existential dependency is a transitive relation. One of its particular cases is the specialization.


Eduardo Dur Án Venegas

Stéphane Le Diz Ès

Christophe Eloy

A coupled model for flexible rotors

Rotors are present in various applications ranging from wind turbines to helicopters and propellers. The rotors are often made of flexible materials which implies that their geometry varies when the operational conditions change. The intrinsic difficulty of rotor modeling lies in the strong coupling between the flow generated by the rotor and the rotor itself that can deform under the action of the flow. In this talk, we propose a model where the strong coupling between the flexible rotor and its wake is taken into account. We are particularly interested in configurations where the general momentum theory [START_REF] Sørensen | General momentum theory for horizontal axis wind turbines[END_REF] cannot be used (for example, for helicopters in descent flight).

The wake is described by a generalized Joukowski model. We assume that it is formed for each blade of a bound vortex on the blade and two free vortices of opposite circulation, same core size a, emitted at the radial locations R i and R e (see figure 1). These parameters are computed from the circulation profile Γ(r) obtained on the blade by applying locally at each radial location r the 2D Kutta-Joukowski formula

Γ(r) = 1 2 C L (α(r))U (r)c(r), (1)
where c(r) is the local chord, C L (α(r)) the lift coefficient of the chosen blade profile, α(r) the angle of attack of the flow, and U (r) the norm of the velocity. The vortex circulation Γ m is the maximum value of Γ(r), and the emission locations R i and R e are the radial distances of the centroid of ∂ r Γ on both sides of the maximum (see figure 1). The wake is computed using a free-vortex method [START_REF] Leishman | Principles of Helicopter Aerodynamics[END_REF]. Each vortex is discretized in small vortex segments for which the induced velocity can be explicitly obtained from the Biot-Savart law [START_REF] Saffman | Vortex Dynamics[END_REF]. We are considering helical wake structures that are stationary in the rotor frame. This frame is rotating at the rotor angular velocity Ω R and translating at a velocity V ∞ corresponding to an external axial wind. For a prescribed rotor of N blades, the wake structure is characterized by five non-dimensional parameters

λ = Ω R R b V ∞ , η = Γ m Ω R R 2 b , R * e = R e R b , R * i = R i R b , ε = a R b , (2)
where R b is the blade length. The aerodynamic forces exerted on the blade are calculated using the blade element theory [START_REF] Leishman | Principles of Helicopter Aerodynamics[END_REF]. From the wake solution are deduced the angle of attack and the velocity amplitude at each radial location on the blade in the rotor plane. Then, the loads are deduced from the lift and drag coefficients C L and C D of the considered blade profile. The blade deformation is obtained using a ribbon model for the blade [START_REF] Dias | meet Kirchhoff: A general and unified description of elastic ribbons and thin rods[END_REF]. This 1D model is a beam model that allows to describe the nonlinear coupling between bending and torsion. In the simplest cases, we assume uniform elastic properties of the blades which are characterized by a Poisson ratio ν and a non-dimensional Young modulus

E * = E/ρ b Ω 2 R 2 b
, where ρ b is the density of the blade.

A typical example with a simple blade geometry is shown in figure 2. In these figures are shown both the case of a rigid rotor and of a flexible rotor for the same operational conditions (same V ∞ and same Ω R ). We do see the effect of blade flexibility. The blades do bend and twist in the presence of the flow. Moreover, this bending and twisting also affect the wake. When the blade bends, the vortices move streamwise and inward, which impacts the expansion of the wake. The vortex circulation is also slightly modified as η changes from 0.0218 to 0.0216 when the blades bend.

Other examples will be presented and compared to available data. The question of the stability will also be addressed. Both flow instabilities and instabilities associated with the blade flexibility will be discussed.

Figure 1 :

 1 Figure 1: Generalized Joukowski model. The parameters (Γ m , R i and R e ) of the model are computed from the circulation profile Γ(r) on the blade as explained in the text.

Figure 2 :

 2 Figure 2: Illustration of the effect of blade flexibility on the wake structure and blade geometry. Dashed lines: wake and blades for the rigid case. solid lines: wake and blades for the flexible case. The undeformed blade is as illustrated in figure 1: it is a flat plate with a constant twist angle θ = -10 • and a linearly decreasing chord from c(r = 0.2R b ) = 0.1R b to c(r = R b ) = 0.07R b . The wake parameters of the rigid rotor are λ = 6.67, η = 0.0218, R * e = 0.99, R * i = 0.24, ε = 0.01. The flexible blades have the characteristics: E * = 10 6 , ν = 0.5. (a) 3D geometry of the rotor and of the wake. Only the deformation and the vortices emitted from a single blade are shown. (b) Locations of the vortices in the plane including a blade and the rotor axis. (c) Twist angle of the blade. (d) Bending of the blade.