Multiple state optimal design problems with explicit solution

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1 Multiple state optimal design problems with explicit solution Marko Vrdoljak Department of Mathematics Faculty of Science University of Zagreb, Croatia Krešimir Burazin Department of Mathematics University of Osijek, Croatia 88th GAMM Annual Meeting, Weimar, March 2017 Marko Vrdoljak Optimal design problems with explicit solution 1

2 Multiple state problem optimizing energy Fill R d with two isotropic materials with conductivity 0 < α < β, quantity q α of the first material is given: State equations A = χαi + (1 χ)βi, χ L (; {0, 1}) χ dx = q α { div (A ui ) = f i u i H 1 0 () i = 1,..., m, Goal functional is a conic sum of energies (µ i > 0) I (χ) = µ i f i (x)u i (x)dx min / max Relaxation via homogenization theory: classical design relaxed design χ L (; {0, 1}) θ L (; [0, 1]) A K(θ) a.e. on I (χ) J(θ, A) given by the same formula Marko Vrdoljak Optimal design problems with explicit solution 2

3 Example maximization := B(0, 1) R 2, q α := 0.8 I (χ) = 2 f i(x)u i (x)dx max f 1 = χ A + εχ B, where f 2 = χ A εχ B A := B ( c, ( ) 0, 2) 1 B := B 0, 1 5 ±ε B(0, 1) 1 Numerical solution, ε = 0.01 Numerical solution, ε = 0 Marko Vrdoljak Optimal design problems with explicit solution 3

4 Representation by a concave maximization problem The space of admissible local fractions { T := θ L (; [0, 1]) : Admissible (relaxed) designs θ dx = q α } A = {(θ, A) T L (; Sym) : A K(θ) (a.e. on )} λ 2 λ + θ K(θ) λ 2 λ + θ B(θ) λ + θ 1 λ θ = θα + (1 θ)β = θ α + 1 θ β λ θ λ θ λ θ λ + θ λ 1 λ θ λ + θ λ 1 B = {(θ, A) T L (; Sym) : A B(θ) (a.e. on )} Marko Vrdoljak Optimal design problems with explicit solution 4

5 Maximization over B Murat and Tartar (1985), Casado-Díaz (2015) one state equation. Lemma There exists a unique σ S = {σ L 2 (; R d ) m : div σ i = f i, i = 1..m} such that max J(θ, A) = max µ i A 1 σi σi dx. (1) (θ,a) B (θ,a) B Moreover, if (θ, A ) is an optimal design for problem max B J and u the corresponding state function, then A u i = σ i, i = 1,..., m. Above maximization problems is easily solved: Design (θ, A ) is optimal if and only if (almost everywhere in ) A σ i = λ θ σ i, i = 1..m. If u is the corresponding state function, we have σ i = λ θ u i or equivalently A u i = λ θ u i, i = 1..m. Marko Vrdoljak Optimal design problems with explicit solution 5

6 Simpler relaxation problem Theorem Let (θ, A ) be an optimal design for the problem max B J. Then θ solves I (θ) = µ i f i u i dx max θ T and u determined uniquely by div (λ θ u i) = f i u i H 1 0 () i = 1,..., m, (2) Conversely, if θ is a solution of optimal design problem (2), and ũ is the corresponding state function, then for any measurable à B( θ) such that à ũ i = λ ( θ) ũ i, e.g. for à = λ ( θ)i, ( θ, Ã) is an optimal design for the problem max B J. Marko Vrdoljak Optimal design problems with explicit solution 6

7 Necessary and sufficient optimality conditions Similar to Lemma above, one can rephrase the simpler relaxation problem (2): there exists a unique σ S = {σ L 2 (; R d ) m : div σ i = f i, i = 1..m} such that max I = max T θ T µ i Moreover, σ is the same as for max B J. Lemma β α αβ θ σ i 2 dx. The necessary and sufficient condition of optimality for solution θ T of optimal design problem (2) simplifies to the existence of a Lagrange multiplier c 0 such that µ i σi 2 > c θ = 1, µ i σi 2 < c θ = 0. Marko Vrdoljak Optimal design problems with explicit solution 7

8 Spherically symmetric case Let R d be spherically symmetric (ball or annulus), and let the right-hand sides be radial functions: f i = f i (r). Since σ is unique, it must be radial: σ i = σ i (r)e r. Theorem For any maximizer θ for max T I, the radial function θ(r) = θ ds is also a maximizer. B(0,r) If θ is a maximizer of I over T, then for a simple laminate à K( θ) with layers orthogonal to e r, ( θ, Ã) is a maximizer of J over A. For any maximizer (θ, A ) of J over A, θ is a maximizer of I over T. For problems on a ball, σ is a unique (radial) solution od div σ i = f i, i = 1..m, and so conditions of optimality easily determine optimal θ. Marko Vrdoljak Optimal design problems with explicit solution 8

9 Energy minimization A. Single state equation: [Murat & Tartar, 1985] I (θ) = fu dx min θ T, and u determined uniquely by div (λ + θ u) = f u H 1 0 () B. Multiple state equations: I (θ) = µ i f i u i dx min θ T, and u i determined uniquely by div (λ + θ u i) = f i u i H 1 0 () i = 1,..., m A: Holds always! min A J min I T B: Holds in spherically symmetric case or when m < d. Marko Vrdoljak Optimal design problems with explicit solution 9

10 m < d Theorem If m < d then min A J = min T I and: There is unique u H 1 0 (; Rm ) which is the state for every solution of min A J and min T I. If (θ, A ) is an optimal design for the problem min A J, then θ is optimal design for min T I. Conversely, if θ is a solution of optimal design problem min T I, then any (θ, A ) A satisfying A ui = λ + θ u i, i = 1,..., m (e.g. simple laminates) is an optimal design for the problem min A J. Marko Vrdoljak Optimal design problems with explicit solution 10

11 Spherical symmetry R d is spherically symmetric and right-hand sides f i = f i (r), i = 1,..., m are radial functions. Theorem There is a unique radial u which is the state for any solution of min A J and min T I. Moreover, If (θ, A ) A is a solution of the relaxed problem min A J then θ is optimal for min T I, and A ui = λ + θ u i, i = 1,..., m. There exists a radial minimizer θ of I over T and for any radial minimizer θ of I over T there exists a simple laminate A K(θ ) such that (θ, A ) is an optimal design for min A J. Marko Vrdoljak Optimal design problems with explicit solution 11

12 Optimality conditions for min T I min θ T I (θ) = max θ T min v H 1 0 (;Rm ) µ i Saddle points exist... share the same v (aka u ). min I (θ) = max θ T θ T µ i λ + θ v i 2 2f i v i dx λ + θ u i 2 2f i u i dx Lemma θ T is a solution min T I if and only if there exists a Lagrange multiplier c 0 such that m µ i u i 2 > c θ = 0, µ i ui 2 < c θ = 1. Marko Vrdoljak Optimal design problems with explicit solution 12

13 Example energy minimization = B(0, 2), f 1 = χ B(0,1), f 2 1, { div (λ + θ u i) = f i u i H 1 0 () i = 1, 2 µ f 1 u 1 dx + f 2 u 2 dx min Solving state equation in polar coordinates u i (r) = σ i (r) θ(r)α + (1 θ(r))β, i = 1, 2, with r σ 1 (r) = 2, 0 r < 1, r 1 and σ 2 (r) = 2r, 1 r 2, 2. Define ψ := µσ σ2 2, g α := ψ α 2, g β := ψ β 2. Marko Vrdoljak Optimal design problems with explicit solution 13

14 Geometric interpretation of optimality conditions c c g g α α g β g β µ 2 r A: 0 < µ 1 B: 1 < µ 4 2 r c c g α g α g β 0 4 µ 2 r C: 4 < µ 16 g β 0 2 r D: 16 < µ Marko Vrdoljak Optimal design problems with explicit solution 14

15 Optimal θ for case B Optimal state u is unknown but m µ i ui 2 = µ u1 2 + u2 2 [g β, g α ]. By necessary conditions of optimality, on a set where c > g α we have θ = 1, on a set where c < g β we have θ = 0, and if g β < c < g α we have θ 0, 1, and θ is uniquely determined from ψ λ +(θ ) = c. 2 c g α θ 1 c α β β g β c 0 p c 1 q c 1 q c 2 q c 3 g α g β 0 4 µ 2 r B: 1 < µ 4 2 r 0 p c 1 q c 1 q c 2 q c 3 All possible optimal configurations (for various q α ): α α mix α mix α mix α mix β α mix β mix β Marko Vrdoljak Optimal design problems with explicit solution 15 2 r