Optimal design under the one-dimensional wave equation

OPTIMAL DESIGN UNDER THE ONE-DIMENSIONAL WAVE EQUATION Faustino Maestre ∗ , Arnaud M¨unch †and Pablo Pedregal‡ September 25, 2007 Abstract An optimal design problem governed by the wave equation is examined in detail. Specifically, we seek the time-dependent optimal layout of two isotropic materials on a 1-d domain by minimizing a functional depending quadratically on the gradient of the state with coefficients that may depend on space, time and design. As it is typical in this kind of problems, they are ill-posed in the sense that there is not an optimal design. We therefore examine relaxation by using the representation of two-dimensional ((x, t)∈IR2) divergence free vector fields as rotated gradients. By means of gradient Young measures, we transform the original optimal design problem into a non-convex vector variational problem, for which we can compute an explicit form of the “constrained quasiconvexification ” of the cost density. Moreover, this quasiconvexification is recovered by first or second-order laminates which give us the optimal microstructure at every point. Finally, we analyze the relaxed problem and some numerical experiments are performed. The perspective is similar to the one developed in previous papers for linear elliptic state equations. The novelty here lies in the state equation (the wave equation), and our contribution consists in understanding the differences with respect to elliptic cases. 1 Introduction Optimal design problems in conductivity and elasticity have been extensively studied from various perspectives. For the homogenization viewpoint, see [1]. For more simulation-oriented approaches, see [4, 9]. For treatments based on variational reformulations, see [23]. In many of these examples, the state equation is assumed to ∗Departamento de Matem´aticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, SPAIN. Email address: F[email protected]. †Laboratoire de Math´ematiques de Besan¸con, Universit´e de Franche-Comt´e - 16, route de Gray 25030 Besan¸con, FRANCE. Email address: [email protected] ‡Departamento de Matem´aticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, SPAIN. Email address: pablo.p[email protected] . 1 be isotropic. There has also been attempts to understand non-isotropic situations ([20] and references therein). Suppose we choose two diagonal, non-isotropic, 2 ×2 matrices of the form Aα=1 0 0α, Aβ=1 0 0β, and consider the state equation div ([χ(x)Aα+ (1 −χ(x))Aβ]∇u) = 0 in Ω where Ω ⊂IR2is a bounded, regular, simply-connected domain. It is easy to see that we can also write div 1 0 0χ(x)α+ (1 −χ(x))β∇u= 0 in Ω, and even more so div (ux1,[χ(x)α+ (1 −χ(x))β]ux2) = 0. In the case where Ω = (0, T )×(0,1), and we take both αand βnegative, we see that we have a 1-d wave equation as state equation, and 2-d optimal designs can be interpreted as 1-d time-dependent optimal designs. For this reason, we change the notation and write x(= x2) for the spatial variable, t(= x1) for the time variable, and replace α, β by −α, −β, respectively, so that we focus on such a wave equation. We also change accordingly the domain, and consider initial and boundary condition as is usual in hyperbolic problems. Yet notice that the non-isotropic elliptic example is contained in our analysis too. Optimal control problems in the coefficients are rather well-known in the elliptic case. Homogenization has been the main tool to deal with these when cost functionals do not depend on derivatives of the state. As indicated earlier [1] is still, as far as we can tell, an up-to-date reference for the use of homogenization in optimal design problems. Our interest in understanding optimal design problems for cost functionals depending on gradients of the state led us to explore the use of gradient Young measures in this kind of problems ([23], [24]). See [27] for a pioneering situation that is quite instructive to better understand how different optimal design problems with quadratic cost functionals in the gradient can be. Other references that have treated similar situations from the perspective of homogenization are [11], [14]. Our perspective does not require a full understanding of the G-closure problem (as in homogenization) as the emphasis is n ot placed on the tensors that can be obtained by mixtures (the G-closure), but rather on the set of pairs of vectors that can be related through some tensor of the G-closure. Hence, although intimately connected to homogenization, our approach focuses directly on pairs of fields that can occur in relaxed state equations. In this way, we can treat cost functionals depending on the gradient of the state directly without further ingredients. 2 Having succeeded, to some degree, in understanding the elliptic situation in the conductivity setting (even producing numerical simulations as in [9] of optimal microstructures), a next natural step is to examine the same optimal design problems with quadratic cost functionals on the gradient of the state under a hyperbolic state equation, so as to better understand the differences introduced on the analysis and on the numerics because of this hyperbolic nature. One issue here is the phenomenon of concentration of cost (energy). It is well-understood that Young measures cannot capture concentration effects. We resolve easily this issue by demanding some extra regularity on initial data. Notice that if designs were not allowed to vary with time, the situation would be much simpler as it would not require “homogenization” or the formation of microstructure on this time variable. Many of the standard homogenization facts could be used and exploited. Indeed, one of our main goals is to grasp this dependence of designs on time. Except for the works of Lurie ([15], [16], [17]), this sort of problems have not been addressed in the literature. In particular, he has been investigating over the years the analogue of the G-closure for dynamic designs. Being motivated by realistic applications, one main concerned in his work is to understand the relationship and interaction between the dynamic nature of the problem and the “dynamics” of microstructure. In a sense, even in a static situation, microstructure (laminates) is something dynamic as it is a never-ending refinement process. When this process interacts with real time, some funny situations may occur (including the formation of shocks). This is well-documented for instance in [17], where restrictions between the velocity of formation of microstructure and the dynamics of the state equation are explicitly given so that undesirable behavior is ruled out. We have avoided altogether this issue as we model dynamic microstructure through families of probability measures (Young measures) depending both on space and time so that there is only one dynamic process associated with time. Even so, it is interesting to stress that the laminates we get in our numerical simulations (Section 7) are of the kind that satisfy these requirements in the best way possible as they are oriented parallel to the time axis. This is the most favorable situation in Lurie’s work. See also the comments after Conjecture 1. Another main difference of our work with that of Lurie is that we are interested from the beginning on a cost functional which is quadratic in the gradient of the state. Our methods allow to treat directly this sort of problems without understanding first the G-closure set. A main concern in the work of Lurie is to better understand the G-closure set corresponding to a dynamical situation, and, in particular, to discover the differences with respect to its elliptic counterpart. Other works dealing with optimal control problems under the wave equation in greater dimensions can be found in [6], where the control is a time dependent coefficient, and under other constraints on modes where there is vibration. In this sense another work in which the authors examine time-harmonic solutions of the wave equation is [3], where they prove a relaxation result for this problem and very interesting results of existence of classical solutions for some particular cases. In the wave equation literature, we can find a huge family of optimal control problems 3 where the design variable is not in the highest derivative term. When the control term acts on the first order derivative in time, the term is known as a ”damping” term. These problems are of a different nature physically as well as mathematically. Some relevant references in this topic are [5, 10, 12]. 1.1 Problem Statement We will thus study the following optimal design problem. We consider a design domain Ω = (0,1) ⊂IR, a positive time T > 0, and a maximum amount of one material at our disposal Vα∈(0,1). The optimal design problem consists in deciding, for each time 0 < t < T , the best distribution in Ω of the two materials in order to minimize the time-dependent cost functional depending on the square of the gradient (with respect to both variables (t, x)) of the underlying state. More precisely, let us denote by (P) the problem that consists in minimizing (P)I(χ) = ZT 0ZΩu2 t(t, x) + a(t, x, χ)u2 x(t, x)dx dt where uis the unique solution of utt −div([αχ +β(1 −χ)]ux) = 0 in (0, T )×(0,1), u(0, x) = u0(x), ut(0, x) = u1(x) in Ω,(1) u(t, 0) = 0, u(t, 1) = 0 in [0, T ],(2) and the functions a, u0and u1are known. We have put vanishing boundary conditions on both end-points and source term for simplicity. Functions of time f(t) and g(t) on both points could be considered as well. The function χ∈L∞([0, T ]× Ω; {0,1}) is the design variable, and it indicates where we place the α-material for each time t. Since χis a binary variable a(t, x, χ)∈ {a(t, x, 0), a(t, x, 1)}, we can write a(t, x, χ) = χ(t, x)aα(t, x) + (1 −χ(t, x))aβ(t, x), where aα(t, x) = a(t, x, 1), aβ(t, x) = a(t, x, 0). In addition, we make the assumption 0 < α < β, and aα(t, x) + α≥0, aβ(t, x) + β≥0. The amount of the α-material is given, and therefore we have to enforce the volume constraint ZΩ χ(t, x)dx ≤Vα|Ω|,∀t∈[0, T]. The lack of classical solutions for this sort of problems is well understood (see. Theorem 11, [22]). In this sense we propose and analyze a relaxation of the problem. 4 Our approach is based on an equivalent variational reformulation of the original optimal design problem as a non-convex vector variational problem. As in other situations examined under this perspective [2, 23], we change a scalar problem with differential constraints by a vector variational problem with integral constraints (where the state equation is implicit in the new cost function). It is well-known that the non-existence of optimal solution for vector variational problem is intimately associated with the lack of quasiconvexity of the cost functional, and in this sense we propose to analyze the “constrained quasiconvexification” for this last problem by using gradient Young measures as generalized solutions of variational problems. We compute an explicit relaxation of the original optimal design problem in the form of a relaxed (quasiconvexified) variational problem. It is elementary to check (this is done with some detail in Section 2), the equivalence of our dynamic optimal design problem with the following non-convex, vector variational problem (V P) min U ˆ I(U) = ZT 0ZΩ W(t, x, ∇U(t, x))dxdt subject to U= (U(1), U(2))∈H1([0, T ]×Ω)2, U(1)(0, x) = u0(x), U(1) t(0, x) = u1(x) in Ω, U(1)(t, 0) = f(t), U(1)(t, 1) = g(t) in [0, T ], ZΩ V(t, x, ∇U(t, x))dx ≤Vα|Ω| ∀t∈[0, T]. The two integrands involved are W(t, x, A) =    a2 11 +aα(t, x)a2 12,if A∈Λα, a2 11 +aβ(t, x)a2 12,if A∈Λβ\Λα, +∞,else, V(t, x, A) =    1,if A∈Λα, 0,if A∈Λβ\Λα, +∞,else. Here Λγ={A∈M2×2:M−γA(1) −RA(2) = 0}, γ =α, β, (3) where A(i)is the i-th row of the matrix A=a11 a12 a21 a22, and M−γ=1 0 0−γ, R =0−1 1 0 . 5 1.2 Results Statement To write down an explicit relaxation, put h(t, x) = βaα(t, x)−αaβ(t, x) and for F=F11 F12 F21 F22, s ∈IR, set ψ(F, s) = F12F21 +α s(β−α)2(βF12 +F21)2+β (1 −s)(β−α)2(αF12 +F21)2. Consider the variational problem, (RP ) min U,s ZT 0ZΩ ϕ(t, x, ∇U(t, x), s(t, x))dxdt subject to U∈H1([0, T]×Ω)2, tr(∇U(t, x)) = 0, U(1)(0, x) = u0(x), U(1) t(0, x) = u1(x) in Ω, U(1)(t, 1) = f(t), U(1)(t, 0) = g(t) in [0, T], 0≤s(t, x)≤1,ZΩ s(t, x)dx ≤Vα|Ω| ∀t∈[0, T], where ϕ(t, x, F, s) is explicitly given by the surprising expression                                    h sβ(β−α)2(β2|F12|2+|F21|2+ 2βF12F21) + |F11|2−aβ βF12F21 if h(x, t)≥0, ψ(F, s)≤0, −h (1 −s)α(β−α)2(α2|F12|2+|F21|2+ 2αF12F21) + |F11|2−aα αF12F21, if h(x, t)≤0, ψ(F, s)≤0, −detF +1 s(1 −s)(β−α)2(1 −s)β2(α+aα) + sα2(β+aβ)|F12|2 +(1 −s)(α+aα) + s(β+aβ)|F21|2+ 2(α+aα)β−shF12F21 if ψ(F, s)≥0. tr stands above for the trace of a matrix. All that matters is that this integrand ϕis known in closed form. Theorem 1 Suppose that the initial data u0and u1have the regularity u0∈H2(0,1) ∩H1 0(0,1), u1∈H1 0(0,1). 6 Then the variational problem (RP)is a relaxation of the initial optimization problem (P)in the sense that a) the infima of both problems coincide; b) there are optimal solutions for the relaxed problem (RP); c) these solutions codify (in the sense of the Young measures) the optimal microstructures of the original optimal design problem. For the interpretation of Young measure solutions in this statement, we refer the reader to the already-mentioned contributions in the elliptic case. It is closely related to relaxation in vector, non-convex variational problems ([8]). These optimal Young measures carry the information of optimal microstructures, both on the local distribution of materials, and on the geometry of optimal microarrangements. See more on this interpretation in Section 7. In addition, we can provide explicitly optimal microstructures. Theorem 2 Optimal, dynamic microarrangements of the two materials leading to the relaxed formulation are always laminates which can be given in a completely explicit form. The formulae for all of these laminates are given later at the end of Section 4, where we compute these optimal microstructures corresponding to first and second order laminates. The main new contribution here is therefore to understand the character of the hyperbolic state law, and the differences it introduces with respect to the better known elliptic case. Some of these differences are related to the fact that the manifolds Λγare two 2-dimensional subspaces whose intersection is another 1-dimensional manifold. Moreover there are rank-one connections within those manifolds. An interesting consequence is that the relaxed integrand is finite everywhere (except for the condition involving the trace) in contrast with the elliptic case where the relaxed integrand is finite only in a certain (quasi)convex subset. An important issue is that optimal Young measures gives us the necessary information about the behavior of minimizing sequences of the original optimal design problem. A subsequent important step is to explore the relaxed problem (RP ) in some particular cases, like the ones described in Section 5, with the objective of producing numerical simulations of optimal time-dependent structures [18]. For some particular situations in the (static) elliptic case, it has been shown that a simple relaxation consists in replacing the original discrete design variable χ∈L∞(Ω,{0,1}) by its convex envelop s∈L∞(Ω,[0,1]). For the (dynamic) hyperbolic case with aα=aβ= 1, some numerical experiments (see Section 7) suggest that the above assertion is true. In this regard, we establish the following conjecture (examined briefly in Section 6). 7 Conjecture 1 Suppose the coefficients aα= 1,aβ= 1. The optimization problem (g RP ) min s ˜ I(s) = ZT 0ZΩ u2 t(t, x) + u2 x(t, x)dx dt where uis the unique solution of utt −div[αs +β(1 −s)]ux= 0 in (0, T)×(0,1), u(0, x) = u0(x), ut(0, x) = u1(x)in Ω, u(t, 0) = f(t), u(t, 1) = g(t)in [0, T ], ZΩ s(t, x)dx ≤Vα|Ω|,∀t∈[0, T], 0≤s(t, x)≤1. is equivalent to the original optimal design problem (P)in the sense that a) the infima of both problems coincide, i.e., inf(g RP ) = inf(P); b) the above optimal design problem (g RP )admits optimal solutions; c) these solutions ( in the sense of Young measures) show that optimal microstructures are first order laminates with normal n= (0,1) and volume fraction s. One can pass from (RP) to (g RP ) simply by minimizing the general relaxed integrand ϕ(t, x, F, s) on the auxiliary variable F(2), the second row of F, keeping all other variables fixed (and taking aα= 1, aβ= 1). This is an elementary calculus exercise (Section 6). This vector, the second row of F, was introduced as an auxiliary field to go from the original formulation (P) to its variational form (V P ). After relaxation, in which this auxiliary vector plays an important role, we eliminate it by minimizing over it, so that we are back to a state law which is the result of this minimization process. More importantly, first-order laminates involved in this process (the passage from (RP ) to (g RP )) always correspond to normal direction n= (0,1), i.e., the optimal laminates have to be arranged in the perpendicular direction to the space-axis with volume fraction s(t, x). These are, in particular, laminates of the class considered by Lurie in the previously cited works. Our conjecture establishes that this procedure should capture the optimal relaxed state law. Notice how this process cannot produce a general relaxation theorem (this is in fact our previous theorem for (RP)), as it is tailored and computed for the particular choice of the coefficients aα= 1, aβ= 1. It asserts that among the many relaxed wave-like equations that can be produced by mixing dynamically the two materials, the one providing optimal microstructures for the particular choice of the coefficients aα= 1, aβ= 1, is precisely the one obtained by replacing χ∈ {0,1}by s∈[0,1] (as in the situation in [27]). For other choices of the coefficients aαand aβ, the optimal relaxed equation would eventually be different. One can see this phenomenon for the elliptic situation in [9]. The importance of having this more “economic” relaxation 8 (compared Conjecture 1 with Theorem 1) is that simulations can be performed for these, while it is out of the question to use directly (RP ). We have written this in the form of a conjecture because, even though the passage from (RP ) to (g RP ) is elementary, its formal rigorous proof requires a careful analysis. It has been shown to be correct in a number of situations in the elliptic case ([26]). For our situation here, showing the validity of the conjecture is in progress ([18]). The paper is organized as follows. In Section 2, we describe in more detail the equivalent variational reformulation as well as a general relaxation result when integrands are not continuous and may take on infinite values abruptly. As there is nothing new here compared to other previous works in the elliptic case, our description is rather a remainder included here for the sake of completeness. Sections 3 and 4 are technical in nature but interesting, as we first compute a lower bound of the constrained quasiconvexification (Section 3), by using in a fundamental way the weak continuity of the determinant. Section 4 is concerned with the search for laminates furnishing the precise value of the lower bound in an attempt to show equality of the three convex hulls (poly-, quasi- and rank-one convex hulls), as it is standard in this kind of calculation. In Section 5, we show some particular examples of this relaxation for different and interesting choices of the coefficients aα, aβ. Finally, in Section 6 we analyze the relaxed problem and propose a simpler relaxation, while in Section 7 we numerically solve it by using a gradient descent method. 2 Reformulation and relaxation The lack of classical solution of the original optimal design problem is well-established. We propose to reformulate the problem as a vector variational problem to which we apply suitable tools to study its relaxation. We follow a similar approach to the one in [2, 23]. Under the hypothesis of simple-connectedness of Ω (an interval), there exists a potential v∈H1((0, T)×Ω) such that the state equation can be recast as −div(ut(t, x),−[αχ(t, x) + β(1 −χ(t, x))]ux(t, x)) = 0 in [0, T]×Ω where the div operator is consider now with respect to the variables tand x. The state equation is equivalent to the pointwise constraint ut(t, x) −[αχ(t, x) + β(1 −χ(t, x))]ux(t, x)=R∇v(t, x)a.e. (t, x)∈[0, T ]×Ω where Ris the counterclockwise π/2-rotation in the space-time plane. If we let Λ−γ be as in (3), this constraint reads ∇u(t, x) ∇v(t, x)∈Λ−α∪Λ−βa.e. (t, x)∈[0, T]×Ω.(4) 9 given by CPW(F, s) =                                              h sβ(β−α)2(β2y2+z2+ 2βyz) + x2−aβ βyz if h(x, t)≥0, ψ(s, F)≤0, tr(F) = 0, −h (1 −s)α(β−α)2(α2y2+z2+ 2αyz) + x2−aα αyz, if h(x, t)≤0, ψ(s, F)≤0, tr(F) = 0, 1 s(1 −s)(β−α)2(1 −s)β2(α+aα) + sα2(β+aβ)y2 +(1 −s)(α+aα) + s(β+aβ)z2 +2(α+aα)β−sβyz−det F if ψ(s, F)≥0, tr(F) = 0, +∞if tr(F)6= 0 We claim that in fact this is an exact value. This amounts to finding laminates which yield this same optimal value. 4 Optimal microstructures: laminates We have the lower bound given by the polyconvexification, and we will show that this bound is in fact attained. To this end, we seek an optimal microstructure (a laminate) whose second moments recover the value of the bound. We try to find ν=sνα+ (1 −s)νβ, a laminate with supp(νγ)⊂Λγ,γ=α, β, s∈(0,1), and first moment F. We have different optimal conditions depending of the sign of ψand h, and we analyze different cases accordingly. 4.1 Case ψ≥0 We start with the case when ψ(F, s)≥0 holds. In this case the optimal conditions (12) tell us that Sα,2=y2 α, Sβ,2=y2 β and therefore, by the strict convexity of the square function, we can deduce that ν(2) γ=δyγ, γ =α, β. Hence Fα= (λ, yα,−αyα), Fβ=x−sλ 1−s, yβ,−βyβ,(13) with λ∈IR arbitrary. This means that for every λ∈IR we can decompose Fas a convex combination of two matrices in Λα,Λβrespectively, and satisfying the volume constraint, see Figure 2. 16 La Lb Fa Fb F Figure 2: Infinite decompositions of F. The next step is to check that there exist some λ∈IR such that rank(Fα−Fβ) = 1. After some algebra, we can write rank(Fα−Fβ) = 1 ⇔CF,s(λ) = 0 where CF,s(λ) = −detF −s(λ2−αy2 α)−(1 −s)(F11 −sλ 1−s)2−βy2 β) is a second degree polynomial on λ. It is easy to check that the discriminant of CF,s is ψ(F, s), and so that their roots are λi=x+ (−1)ir1−s sψ(F, s)i= 1,2. Therefore for all pair (F, s) such that ψ(F, s)≥0, there exist two first order laminate ν=sδFα,i + (1 −s)δFβ,i i= 1,2 where Fα,i =λiFα,12 −αFα,12 −λi, Fβ,i = x−sλi 1−sFβ,12 −βFβ,12 −x−sλi 1−s! and they provide the optimal value of the polyconvexification. Thanks to the spatial identification F= (x,y,z), we can observe the above computations from a geometric point of view (see Figure 3). For any matrix F= (x,y,z) the determinant is det F=−(x2+yz), this means that for any matrix F there exist a cone {x2+yz = 0}of rank one directions through this matrix. From 17 La Lb FaFb F Fa1 2 Fb1 2 Figure 3: Two first order laminates optimality conditions we obtain an explicit identification of Fγγ=α, β, up to the first component (13), which let us a degree of freedom in the search of the optimal decomposition. Geometrically, we notice that the intersection between the manifolds Λγand the rank one cone are ellipses, whose intersection with the admissible Fγare two points Fγ,i, γ =α, β and i= 1,2. 4.2 Case ψ≤0 We study now the other case, ψ(F, s)<0. In this situation, we have two different optimal conditions depending of the sign of h. We treat the case h≥0. The other case is similar. From the optimal condition for this case (10), we have Sα,2=y2 α, S1=x2 and by using similar arguments as above, we can deduce ν(2) α=δyα, ν(1) =δx where 1. να=δFαwith Fα= (x, yα,−αyα),(14) 2. by using that Fis the first moment of ν, there exists a unique decomposition F=sFα+ (1 −s)Fβ 18 with Fγ∈Λγ, γ =α, β where Fαis of the form just indicated, and Fβ= (x, yβ,−βyβ).(15) Consider a pair (F, s) such that ψ(F, s)<0. After an elementary manipulation, we get ψ(F, s)≤0⇐⇒ −(β−α)2yzs2+αβ(α−β)y2+ (β−α)z2+ (β−α)2yzs +αβ2y2+αz2+ 2αβyz≤0. Let PF(s) be this second degree polynomial in sfor fixed F. The set where ψ(F, s)≤ 0 is the set where PFhas solutions in [0,1], and slies between those two solutions. There exist real solutions if the discriminant is non-negative, and, in addition, it is easy to check that PF(0), PF(1) are positive1if F /∈Λα∪Λβ. Therefore there are positive solutions if PFis decreasing at 0. After some algebra the discriminant is g(F) = α2β2y4+z4+ (α2+ 4βα +β2)yz +2αβy3z+ 2(α+β)z3y≥0, and the decreasing condition h(F) = (α+β)yz +αβy2+z2≤0. Therefore the set of pairs (F, s) where ψ(F, s)≤0 can be described as {(F, s)∈M2×2×IR : g(F)≥0, h(F)≤0, s ∈(r1, r2)} where ri=1 2−1 2(β−α)yz αβy2−z2+ (−1)ipg(F)i= 1,2. We thus have a characterization of the set ψ(F, s)≤0. We now look for rank-one connections between both manifolds. We would like to write F=rBα+ (1 −r)Bβ with r∈(0,1), Bγ∈Λγ,(Bγ)1=x, γ =α, β, and rank(Bα−Bβ) = 1. On the one hand, Bγ∈Λγ (Bγ)1=x⇒Bγ= (x, yγ,−γyγ)γ=α, β. 1PF(0) = α|βy+z|2, PF(1) = β|αy+z|2 19 La Lb Ma Fa1 Fb1 Fb,2 Fa,2 Fb,2 Fb1 F Figure 4: Second order laminates. The constraint on the vanishing determinant can be rewritten, after some manipulation, as PF(r) = 0, whose roots are ri. We can therefore guarantee that there exist two rank-one directions between Λαand Λβwith barycenter F. We are now in a position to find an optimal second order laminate which recovers the lower bound given by the polyconvexification. We take να=δF α and νβas a convex combination of two Dirac masses supported in the βmanifold (see Figure 4). Put Fβ,i = (x, yβ,i,−βyβ,i) with yβ,i =−1 (1 −ri)(β−α)(αy+z), i = 1,2. Since r1≤s≤r2, it is clear that yβis a convex combination of Fβ,i, i = 1,2. If we consider ¯ Fβ,i =Fα+li(Fβ,i −Fα,i) with lisuch that ¯ Fβ,i ∈Λβ, and take li=ri s, ρi,j =(1 −rj)(ri−s) ri−rj , τi,j =(rj−s)(ri−1) rj(1 + ri) + s(1 −rj),(16) we can define the second-order laminate with support on Λα∪Λβ, barycenter F, and mass in Λαequal to s, by putting νi,j =τi,jδFβ,i + (1 −τi,j)(ρi,jδ¯ Fβ,j + (1 −ρi,j)δFα) 20 with i, j ∈ {1,2},i6=jwhere, det( ¯ Fβ,j −Fα) = 0 and det(Fβ,i −(ρi,j ¯ Fβ,j + (1 −ρi,j)Fα)) = 0. Again, using the spatial identification we can interpret geometrically the above analytical computations. We lost the degree of freedom of the first component of the matrices Fαand Fβ, since these matrices are explicitly determined by (14) and (15), and their first component is xin both cases. This fact lets us simplify the spatial situation to a 2-d case in the plane determined by the first component equal to x. The intersection between the manifolds Λγthe cone of rank one directions - and that reduces to two matrices in each manifold, which we noted by Fγ,i. From these matrices connected by rank one directions we can obtain a second order laminate with volume fraction son Λαand (1 −s) on Λβ. This construction is shown in the Figure 4, where the spatial situation is reduced to the plane of the first component equal to x. A similar result holds for the other point where the optimal value is attained (h(x, s)≤0). We summarize all of these computations of optimal laminates leading to the relaxed integrand ϕ. When ψ(F, s)≥0 there exist two optimal first-order laminates leading to the value of the relaxed integrand ϕ ν=sδFα,i + (1 −s)δFβ,i i= 1,2 (17) where, Fα,i =λiFα,12 −αFα,12 −λi, Fβ,i = F11−sλi 1−sFβ,12 −βFβ,12 −F11−sλi 1−s! with λi=F11 + (−1)ir1−s sψ(F, s)i= 1,2, Fα,12 =1 s(β−α)(βF12 +F21), Fβ,12 =−1 (1 −s)(β−α)(αF12 +F21). When ψ(F, s)≤0 and h(x, t)≥0, there exist two optimal second-order laminates νi,j =τi,jδFβ,i + (1 −τi,j)(ρi,jδ¯ Fβ,j + (1 −ρi,j)δFα) (18) with i, j ∈ {1,2},i6=jwhere the scalars are ρi,j =(1−rj)(ri−s) ri−rj, τi,j =(rj−s)(ri−1) rj(1+ri)+s(1−rj) 21 and the matrices are Fα=F11 Fα,12 −αFα,12 −F11 , Fβ,i =F11 Fβ,12,i −βFβ,12,i −F11  with Fβ,12,i =−1 (1 −ri)(β−α)(αF12 +F21), i = 1,2 ri 1 2−1 2(β−α)F12F21 αβ|F12|2− |F21|2+ (−1)ipg(F)i= 1,2 ¯ Fβ,i =Fα+li(Fβ,i −Fα,i), li=ri s. Similarly, when ψ(F, s)≤0 and h(x, t)≤0, the optimal microstructure is another second-order laminate given by νi,j =τi,jδFα,i + (1 −τi,j)(ρi,jδ¯ Fα,j + (1 −ρi,j)δFβ) with i, j ∈ {1,2},i6=jwhere the scalars are ρi,j =rj(ri−s) ri−rj, τi,j =(s−rj)ri ri(rj−1)+rj(1−s), and the matrices involved are Fβ=F11 Fβ,12 −βFβ,12 −F11 , Fα,i =F11 Fα,12,i −βFα,12,i −F11  with Fα,12,i =1 ri(β−α)(βF12 +F21), i = 1,2 ¯ Fα,i =Fβ−li(Fβ,i −Fα,i), li=1−ri 1−s. 5 Some particular examples In this section we would like to emphasize some particular examples where, by using Theorem 1, we can compute explicitly the relaxed cost functional. Example 1 - An interesting and academic example is the corresponding to aα(t, x) = α,aβ(t, x) = βso that h≡0, the cost functional can be written as ZT 0ZΩu2 t(t, x) + (αχ +β(1 −χ))u2 x(t, x)dxdt, 22 and the constrained quasiconvexification is ϕ(F, s)                F2 11 −F12F21 if ψ(s, F )≤0, −det F+1 s(1 −s)(β−α)22αβ(sα + (1 −s)β)|F12|2 +2(1 −s)α+sβ|F21|2+ 4αβF12F21 if ψ(s, F)≥0. Example 2 - Another interesting case occurs when we take a(t, x, χ) = 1, the most simple quadratic cost function but very interesting from the mathematical point of view. In this case the relaxed cost functional is ZT 0ZΩu2 t(t, x) + u2 x(t, x)dxdt, and therefore aα(t, x) = aβ(t, x) = 1. Hence h(t, x) = β−α, and the constrained quasiconvexification simplifies to ϕ(F, s) =                        1 sβ(β−α)(sβ(β−α)|F11|2+β2|F12|2+|F21|2+ (sα +β(2 −s))F12F21) if ψ(s, F)≤0, −det F+1 s(1 −s)(β−α)2(1 −s)β2(α+ 1) + sα2(β+ 1)|F12|2 +(1 −s)α+sβ + 1|F21|2+ 2β(1 −s) + α(s+β)F12F21 if ψ(s, F)≥0. (19) Example 3 - The last case lies in the border line for our computations to be valid. We take aα(t, x) = −αand aβ(t, x) = −βso that hidentically vanishes. The cost functional is ZT 0ZΩu2 t(t, x)−(αχ +β(1 −χ))u2 x(t, x)dxdt, and for this case the relaxed integrand surprisingly is -det (recall the restriction on the trace) ϕ(F, s) = F2 11 +F12F21 =−det F. Note that depending on the choice of the coefficients aα,aβwe obtain different cost densities for the relaxed problem, yet this choice of the coefficients is independent of the state equation. It is interesting to remark that for all these examples the optimal laminates correspond to the ones computed in the last section (17) when ψ≥0 and (18) when ψ≤0, which are independents of aα,aβ. 23 6 Analysis of (RP ) in the quadratic case In this section we would like to analyze the quadratic case which is Example 2 in the preceding section, and thus focus on (RP) where the cost density is given by (19). From the previous sections we know that this problem admits optimal solutions, and moreover we know that such optimal solutions are first or second-order laminates depending on the sign of the function ψ. An interesting fact is that all functions involved are quadratic in the vector gradient variable and therefore regular, yet it is the presence of gradients and the pointwise constraint that make the problem difficult to examine. One first attempt would lead us to look at optimality conditions introducing several multipliers to keep track of the restrictions. This makes the problem more difficult in the sense that we have to solve a system of partial differential equations. Instead we follow a similar strategy as in [9]. The pointwise constraint given by ψdepends only on the variables F12, F21 , therefore we try to find the “optimal” relationship between these two variables. The next lemma is completely elementary. Lemma 1 For fixed s, the optimal solution of the quadratic, mathematical programming problem Minimize in F(21) :ϕ(F, s) occurs when (αs +β(1 −s))F12 +F21 = 0. In addition, the associated optimal microstructures are first-order laminates with volume fraction s for the α-material and orientation of layers always vertical (along the time axis): s(t, x)δFα+ (1 −s(t, x))δFβ with normal direction of lamination n= (0,1). Having in mind the trace condition F11 +F22 = 0 the optimal value of the cost function simplifies to F2 11 +F2 12.(20) The idea is then to replace the complicated cost function ϕby the expression (20) and then minimize under the constraints (αs +β(1 −s))F12(t, x) + F21(t, x) = 0, F11(t, x) + F22(t, x) = 0 i.e. F11(t, x) −[αs(t, x) + β(1 −s(t, x))]F12(t, x)=TF(2)(x, t), a.e.(t, x)∈[0, T]×Ω, 24 which is equivalent to div F11(t, x) −[αs(t, x) + β(1 −s(t, x))]F12(t, x)= 0 Therefore we can write the minimization problem in terms of the original variable U(1) =uleading to the new relaxed problem (stated in Conjecture 1): (g RP ) min sZT 0ZΩ u2 t(t, x) + u2 x(t, x)dx dt where uis the unique solution of utt −div([αs +β(1 −s)]ux) = 0 in (0, T )×(0,1), u(0, x) = u0(x), ut(0, x) = u1(x) in Ω, u(t, 0) = 0, u(t, 1) = 0 in [0, T ],(21) This new problem may be seen as the continuous version of the original design problem in which the function χ(x, t) is replaced by the continuous function s(x, t). We cannot prove directly that the above problem admits optimal solutions, though we claim, by our conjecture, that it indeed does because of the particular form of the problem and not as a consequence of general results. A deeper and exhaustive analysis of this problem is still in progress (see [18]). Hopefully, the existence of solutions of these problems will be proved. We support numerically our conclusion in the next section. All we can say at this point is contained in the following assertion. Lemma 2 The equalities inf(P) = inf(g RP ) = min(RP ) hold. Proof. It is easy to see that inf(P)≥inf(g RP ) and inf(g RP )≥min(RP ) and using the relaxation Theorem 3 inf(P) = min(RP ) holds, therefore we have all equalities.  25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 t Figure 8: Case 1 -T= 2, (α, β) = (1,2) - E(t) vs. t- Left : Vα= 0.3 - e I(slim)≈ 7.9567 - Right : Vα= 0.5 - e I(slim)≈6.1439. follows: we first decompose the cylinder (0, T )×Ω into M×Ncells such that (0, T)×Ω = ∪i=1,M [ti, ti+1]× ∪j=1,N [xj, xj+1]. Then, we associate with each cell the mean value mi,j ∈[0,1] defined by mi,j =1 (ti+1 −ti)(xj+1 −xj)Zti+1 tiZxj+1 xj slim(t, x)dx dt. (28) At last, we define the function spen M,N in L∞([0, T]×Ω) by spen M,N (t, x) = M X i=1 N X j=1 χ[ti,(1−√mi,j )ti+√mi,j ti+1]×[xj,(1−√mi,j )xj+√mi,j xj+1](t, x).(29) We easily check that ||spen M,N ||L1((0,T )×Ω) =||slim||L1((0,T )×Ω), for all M, N > 0. Thus, the bi-valued function spen M,N takes advantage of the information codified in the density slim. In order to illustrate this, we consider the case 1 with T= 1, (α, β) = (1,2) and Vα= 0.5. Figure 14 depicts the corresponding optimal density slim and the associated function spen M,N for M=N= 30. We obtain e I(spen 30,30) = 5.62 and e I(slim) = 4.7584 respectively. By letting Mand Ngo to infinity, we expect to converge to the value e I(slim) and then construct a minimizing sequence of domains ωM,N such that χω∞,∞be the infimum for I(see Table 1). We refer to [19] for more example. M=N10 20 30 40 50 e I(spen M,N ) 7.45 6.21 5.62 5.09 4.93 Table 1: Case 1 - T= 1 - (α, β) = (1,3) - Vα= 0.5 - Values of the cost function e I(spen M,N ) 32 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 t x 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 t x Figure 9: Case 1 with the volume constraint (27) - T= 2, Vα= 0.5 - Iso-value of the limit density - Top : (α, β) = (1,1.1) e I(slim)≈9.2147 - Bottom : (α, β) = (1,6) -e I(slim)≈4.3109. Acknowledgements The first and third author are partially supported by project MTM2004-07114 from Ministerio de Educaci´on y Ciencia (Spain), by project PAI05-029 from JCCM (Castilla-La Mancha) and the first author by PhD grant 03/034 of JCCM. We would like to thank various referees for helpful comments and constructive criticism which led to an improved version of the manuscript. 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