In this paper we consider an optimal design problem in wave propagation proposed in [1] SIGMUND O. and JENSEN J.S. Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization, The Royal Society of London, Philosophical Transactions, Series A, Vol. 361, pp. 1001–1019, 2003 in the one-dimensional situation: Given two materials at our disposal with different Young’s modulus and different density, the problem consists of finding the best distributions of the two initial materials in a rod in order to minimize the vibration energy in the structure under periodic loading of driving frequency Ω. We comment on relaxation and optimality conditions, and perform numerical simulations of the optimal configurations. We also prove the existence of classical solutions in certain
cases

Bellido Guerrero, José Carlos

Donoso Bellón, Alberto

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–8)An optimal design problem in wave propagationJ.C. Bellido1, A. Donoso11 Dpto. Matema´ticas (ETSII), Universidad de Castilla la Mancha, Edificio Polite´cnico, Avda. CamiloJose´ Cela s/n, 13071, Ciudad Real. E-mails: JoseCarlos.Bellido@uclm.es, Alberto.Donoso@uclm.es.Palabras clave: Optimal design, wave propagation, classical solutionsResumenIn this paper we consider an optimal design problem in wave propagation proposedin [1] in the one-dimensional situation: Given two materials at our disposal with dif-ferent Young’s modulus and different density, the problem consists of finding the bestdistributions of the two initial materials in a rod in order to minimize the vibrationenergy in the structure under periodic loading of driving frequency Ω. We commenton relaxation and optimality conditions, and perform numerical simulations of theoptimal configurations. We also prove the existence of classical solutions in certaincases.1. IntroductionIn [1] a structural model was proposed for the design of finite structures made up oftwo given materials in the context of wave propagation. The model is an optimal designproblem in which the distribution of two given materials is optimized so as to minimizea cost functional related to the vibration or propagation of waves along the medium. Inpractice, this model may be useful for the systematic design of wave filters, damping ofwaves, or wave guides. The authors develop a numerical method for the optimization ofthose structures based on topology optimization (see [2]), but there is no mathematicalanalysis of the model. In the numerical examples, they also observe the surprising factthat no microstructure appears between the two materials when one tries to design a filteror to minimize the vibration energy. Our aim here is to analyze mathematically the modelproposed in [1] in the one-dimensional situation for longitudinal propagation.We consider the one-dimensional wave equationρwtt = (Ewx)x,1J.C. Bellido, A. Donosowhere t is the time variable, x the space variable, w is the displacement, and E and ρare the Young’s modulus and material density respectively. Notice we do not considerany damping term in our equation. We are interested in time-harmonic solutions underperiodic loading of the previous equation, i.e. in solutions of the formw(x, t) = u(x) exp(iΩt),where Ω is the driving frequency and u the amplitude function. Then the amplitude, u,verifies the equation (E(x)u′)′ +Ω2ρ(x)u = 0.Now, let us assume that we have two materials at our disposal, and consequently theelastic coefficient, E(x), and the density, ρ(x), take two values on, E1, E2 (0 < E1, E2)and ρ1, ρ2 (0 < ρ1, ρ2) respectively. We make the technical assumptionE1 ≤ E2, ρ1 ≤ ρ2,and that at least one of the inequalities is strict (otherwise we would just have one mate-rial). If we have a rod of length l, that we represent by the interval [0, l], and we distributethe two materials in the rod, the elastic coefficient function and the density function arerespectivelyE(x) = E1χ(x) + E2(1− χ(x)), ρ(x) = ρ1χ(x) + ρ2(1− χ(x)),where χ is the characteristic function of the (measurable) subset of the rod where we placethe first material. Now it is natural to consider the following optimal design problem,(P) MinAad J(χ),where the admissible set of designs isAad = {χ : characteristic function of a measurable subset of [0, l]} ,the cost functional is the vibration energyJ(χ) =∫ l0[Ω2(ρ1χ(x) + ρ2(1− χ(x)))|u(x)|2 + (E1χ(x) + E2(1− χ(x)))|u′(x)|2]dx,and u is computed from χ through the state equation[(E1χ(x) + E2(1− χ(x)))u′]′ +Ω2(ρ1χ(x) + ρ2(1− χ(x)))u = 0,(E1χ(0) + E2(1− χ(0)))u′(0) = γ, (E1χ(l) + E2(1− χ(l)))u′(l) = 0,γ is a fixed given nonzero number. Here we consider the case of boundary conditionscorresponding to one external load on the left extreme of the rod, and homogeneousNeumann boundary conditions on the other extreme. Roughly speaking, the optimal designproblem consists of determining the best distribution (i.e., the best χ) of the two givenmaterials in the rod in order to minimize the vibration energy in the structure, so that weminimize the vibration amplitude along the rod. Notice that our optimal design problemindeed is an optimal design problem for an ODE, as we have simplified the wave equationby considering time harmonic solutions.2Optimal design in wave propagationWe do not consider any volume constraint in this problem, and this is in accordancewith the physical nature of the problem since waves propagate better through homogeneousmaterials than through mixtures of two materials. However the problems makes perfectsense if we impose a restriction on the amount of the materials that we may use, as forinstance the constraint ∫ l0χ(x) dx ≤ δ, or∫ l0χ(x) dx = δ,with δ ∈ (0, l) a fixed constant.The solutions of the state equations are understood as weak solutions. That equationis obviously nonelliptic; however by the Fredholm’s alternative, we can claim the existenceof a unique solution for any boundary data if and only if Ω2 is not an eigenvalue for theproblem− [(E1χ(x) + E2(1− χ(x)))u′]′ = λ(ρ1χ(x) + ρ2(1− χ(x)))u, u ∈ H1(0, l), (1a)(E1χ(0) + E2(1− χ(0)))u′(0) = 0, (1b)(E1χ(l) + E2(1− χ(l)))u′(l) = 0. (1c)Recall that, as the operator involved is compact and self-adjoint, we know that there existsan increasing (maybe not strictly) sequence of eigenvalues. For the sake of simplicity, andin order to formulate the optimal design problem, we will assume that Ω2 is far away ofany eigenvalue of the problem (1a-1c) for any characteristic function χ. This hypothesismakes physical sense (see [1]).It is well-known that in principle we cannot hope the previous optimal design problemto admit optimal solutions, and in general we need to study relaxation for it (see [4] andthe references therein).The plan of the paper is the following. Section 2 is devoted to state a relaxation resultfor our optimal design problem and to give necessary conditions of optimality for therelaxed formulation of the problem. In Section 3 we obtain qualitative properties of theoptimal solutions of the relaxed problem, being able to prove, in certain cases, that thereare optimal solutions in the form of characteristic functions, what implies the existence ofminimizers for the original optimal design problem (P). Finally, in Section 4 we performnumerical simulations by using a gradient method based on the sensitivities computed inSection 2 and analyze numerically examples of practical interest.2. RelaxationRelaxation for optimal control or optimal design problems for ODE’s is a classicalsubject. In this sense our optimal design problem fits into this framework and relaxationfor it is straightforward, and so to prove the relaxation result stated below is standard(basic references on the subject are [4, 5, 6]).Theorem 2.1 The optimal design problem(P˜) MinA?ad J˜(E(θ), ρ(θ)),3J.C. Bellido, A. Donosowhere the admissibility set isA?ad = {θ ∈ L∞(0, l) : 0 ≤ θ ≤ 1} ,E(θ) and ρ(θ) are given byE = E(θ) = E1E2/(E1 + θ(E2 − E1)), ρ = ρ(θ) = ρ1θ + ρ2(1− θ),the cost functional isJ˜(E, ρ) =∫ l0[Ω2ρ(x)|u(x)|2 + E(x)|u′(x)|2] dx,and u is computed from the pair (E, ρ) through the state equation[Eu′]′ +Ω2ρu = 0, (2a)E(0)u′(0) = γ, E(l)u′(l) = 0 (2b)is a relaxation of the optimal design problem (P).We will need for the numerical simulations the gradient of the cost functional J˜ in therelaxed problem (P˜). Its computation is direct by using the classical adjoint method.Theorem 2.2 The functional J˜ in problem (P˜) is Gaˆteaux differentiable on the set A?ad,and the Gaˆteaux derivative at θ¯ in the direction of the admissible variation θ (admissiblevariations means that θ ∈ L∞(0, l) such that there exists h0 > 0 such that for everyh ≤ h0, θ¯ + hθ ∈ A?ad) is given byJ˜ ′(θ¯; θ) =∫ l0{[E′(θ¯(x))(u¯′(x)p¯′(x) + |u¯′(x)|2)+ Ω2(ρ2 − ρ1)(u¯(x)p¯(x)− |u¯(x)|2)]θ(x)− (E′(θ¯(x))u¯(x)p¯(x)θ(x))′} dx,where u¯ is the state associated to θ¯ (i.e. the solution of (2a-2b) for the pair (E(θ¯)ρ(θ¯))),and p¯ is the adjoint state, being in this case the unique solution of[E(θ¯)p¯′]′ +Ω2ρ(θ¯)p¯ = 2(Ω2ρ(θ¯)u¯− [E(θ¯)u¯′]′), (3a)E(θ¯(0))p¯′(0) = −2γ, E(θ¯(l))p¯′(l) = 0. (3b)Using the Taylor expansion of the cost functional we obtain, as a direct consequenceof the previous result, the following maximum principle.Corollary 2.1 If θ¯ (with θ¯ an element of the interior of A?ad) is an optimal solution forproblem (P˜) thenJ˜ ′(θ¯; θ − θ¯) ≥ 0for any θ such that θ − θ¯ is an admissible variation.4Optimal design in wave propagation3. Existence of Solutions for Problem (P)In this section we explore the consequences of the optimality conditions given in theform of a maximum principle in Corollary 2.1 on the optimal solutions. Our objective isto check whether it is possible to prove existence results for the original optimal designproblem (P). In the case that E1 = E2 (and we assume then that E1 = E2 = 1), thedesign variable does not act on the principle part of the state equation and the problemis mathematically much simpler, we are able to prove that indeed any optimal solutionfor problem (P˜) is a characteristic function, so that the original optimal design problemadmits minimizers. We use the optimality conditions in this particular case to show theresult, which is given in Theorem 3.1. The general situation is more difficult and doesnot seem possible to use the same argument, however we give a condition implying theexistence of at least one optimal solution in the form of a characteristic function. We dividethis section into two subsections corresponding to the two cases.3.1. Case E1 = E2We assume that E1 = E2 = 1 and our problem now, that we denote as (P˜1), is thefollowingMinA?ad J˜(ρ(θ)),where the admissibility set isA?ad = {θ ∈ L∞(0, l) : 0 ≤ θ ≤ 1} ,ρ(θ) is given byρ(θ) = ρ1θ + ρ2(1− θ),the cost functional isJ˜(θ) =∫ l0[Ω2ρ(θ(x))|u(x)|2 + |u′(x)|2] dx,and u is computed from θ through the state equation[u′]′ +Ω2ρ(θ(x))u = 0,u′(0) = γ, u′(l) = 0Following Theorem 2.1, this problem is in the relaxed form and therefore it admits atleast one optimal solution. The following result establishes that any optimal solution isactually a classical one (classical solution means a solution in the form of a characteristicfunction).Theorem 3.1 Any optimal solution, θ¯, for problem (P˜1) takes on only the values {0, 1},i.e. any optimal solution for (P˜1) is a characteristic function.5J.C. Bellido, A. Donoso3.2. General caseThe existence of classical solutions for the general situation of (P˜) is not clear in generalas far as the authors are able to claim, although something may be said about this. Wedo not state a formal result in this section, but show using new ideas that the problemadmits optimal solutions in certain cases. All the technical details are included in [3], butwe give numerical evidence of this in the next section.4. Numerical Approach and ExamplesAfter the analysis carried out in the previous sections, this section is devoted to showseveral numerical simulations for our problem. The approach that we follow in the pre-sent work is to approximate directly the optimal solutions for the relaxed optimal designproblem (P˜) by a gradient method. Of course this implies approximation of the statesand adjoint states, i.e. solutions of the homogenized state equation (see Theorem 2.1) andthe corresponding adjoint equation (see Theorem 2.2). We believe that this approach willcertainly give good approximations of the optimal designs for problem (P˜).The numerical algorithm used in this work is a variable step-size descent method givenbyθk+1 = θk − J˜ ′(θk)sk,where sk is a small positive step such that J˜(θk+1) < J˜(θk), J˜ ′(θk) is the gradient of thecost functional respect to the control, computed in Section 2, and the step sk is lookedfor as the step that minimizes J˜(θk+1(sk)) (we use a descent method for computing sk).On the other hand, since the control must verify that 0 ≤ θk+1 ≤ 1, then the updatingformula can be easily compacted byθk+1 = ma´x(0,mı´n(1, θk − J˜ ′(θk)sk)),being the same that we obtained by using a projected gradient method. Concerning thestopping criteria, it has been checked that the cost functional slightly changes its valueand therefore, the L∞−norm of the difference J˜(θk+1) − J˜(θk) reduces to zero, as weapproximate to the minimum. All numerical examples belonging to this work have beenobtained under the stopping criteria, tol = 10−5, requiring for convergence about 15iterations of the algorithm, on average. It has also been tested that very similar resultsare obtained when we run the same numerical examples with different mesh sizes.For the sake of briefness, we illustrate our approach through two numerical examples.In both of them, the design domain is an elastic rod of unit length subjected to timeharmonic longitudinal excitation of amplitude γ in the left end. Material properties areE2 = 200 GPa, ρ2 = 7,800 kg/m3, and E1 = 70 GPa, ρ1 = 2700 kg/m3 corresponding tosteel and aluminum, respectively. In previous sections it has been possible to prove thatoptimized solutions are classical when the Young’s modulus E keeps constant along therod and to show that there are cases with E varying its value in which still there areclassical solutions. In general, numerical simulations indicate that optimized layouts areclassical, and therefore, there is no microstructure among the two phases we are mixing.In the first example we have used the frequency value Ω = 50 kHz. Even though we letthe control θ vary continuously between 0 and 1 during the optimization process, we can6Optimal design in wave propagationsee in Figure 1(a) that the optimized solution obtained is classical at the end of the process.Notice that this configuration make the wave damp itself in an optimized way, as it travelsalong the rod (see Figure 1(b)). Similar conclusions as before can be obtained in the secondexample for Ω = 200 kHz (see Figure 1(c) and (d)). We can repeat the same numericalexamples but keeping constant the Young’s modulus, for instance, E2 = E1 = 70 GPa.The numerical results for Ω = 50 kHz and 200 kHz (Figure 2) corroborate the existenceof classical solutions such as it was analytically proved.0 0.25 0.5 0.75 100.51Axial positionDesign variable, θ(a) Optimized design0 0.25 0.5 0.75 1−50510 x 10−6Axial positionAmplitude, u(b) Wave amplitude0 0.25 0.5 0.75 100.51Axial positionDesign variable, θ(c) Optimized design0 0.25 0.5 0.75 1−1012 x 10−6Axial positionAmplitude, u(d) Wave amplitudeFigura 1:It is important to notice that (for this particular case E2 = E1) we initially tried witha numerical approach based on optimality conditions. Although in some situations suchalgorithm gave us the same results as the ones obtained by the gradient method, the factis that it did not enjoy good convergence properties in general, but when it did we gotsimilar results.In view of the numerical results we can state that, even though the design problemtreated is this paper is rather different to the band-gap one, as we commented in theIntroduction, the optimized finite structures (almost periodic in some cases) obtained (forminimum vibration energy) for a particular value of Ω, not avoid (like in the band-gapsituation) but make rather difficult the wave can propagate itself along the rod only atthat corresponding value of Ω.To conclude, it is worth emphasizing that first, all this analysis makes sense for highfrequency values (typically greater than 10 kHz), otherwise the optimal designs are “bo-ring” because they would correspond to take only the material with higher ratio (Ei/ρi),for i = 1, 2, and second, the same layouts we present here are obtained whenever theproduct Ω l keeps constant.7J.C. Bellido, A. Donoso0 0.25 0.5 0.75 100.51Axial positionDesign variable, θ(a) Optimized design0 0.25 0.5 0.75 1−101 x 10−5Axial positionAmplitude, u(b) Wave amplitude0 0.25 0.5 0.75 100.51Axial positionDesign variable, θ(c) Optimized design0 0.25 0.5 0.75 1−2024 x 10−6Axial positionAmplitude, u(d) Wave amplitudeFigura 2:AcknowledgementThe authors acknowledge many stimulating discussions on the subject of this paperwith Martin Bendsøe, Jacob Jensen and Ole Sigmund. Financial support was providedby MEC (Spain) Grant MTM2004-07114, and by Junta de Comunidades de Castilla-LaMancha (Spain) Grant PAI-05-027.Referencias[1] SIGMUND O. and JENSEN J.S. Systematic Design of Phononic Band-Gap Materials and Structuresby Topology Optimization, The Royal Society of London, Philosophical Transactions, Series A, Vol.361, pp. 1001–1019, 2003.[2] BENDSØE M.P. and SIGMUND O. Topology Optimization: Theory, Methods and Applications,Springer-Verlag, Berlin, Germany, 2003.[3] BELLIDO J.C. and DONOSO A. An optimal design problem in wave propagation, to appear inJournal of Optimization Theory and Applications.[4] LURIE K.A. Applied Optimal Control Theory of Distributed Systems, Plenum Press, New York,New York, 1993.[5] WARGA J. Optimal Control of Differential and Functional Equations, Academic Press, New York,New York, 1972.[6] YOUNG L.C. Lectures on the Calculus of Variations and Optimal Control Theory, W. B. SaundersCoorporation, Philadelphia, Pennsylvania, 1969.8

An optimal design problem in wave propagation

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An optimal design problem in wave propagation

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idUS. Depósito de Investigación Universidad de Sevilla