Variant of Optimality Criteria Method for Multiple State Optimal Design Problems

Transcription

1 Variant of Optimality Criteria Method for Multiple State Optimal Design Problems Ivana Crnjac J. J. STROSSMAYER UNIVERSITY OF OSIJEK DEPARTMENT OF MATHEMATICS Trg Ljudevita Gaja Osijek, Hrvatska icrnjac@mathos.hr Joint work with K. Burazin and M. Vrdoljak [PDE, OPTIMAL DESIGN AND NUMERICS]

2 Stationary diffusion equation Stationary diffusion equation Let Ω R d be open and bounded, A L (Ω; Sym d ) satisfying Aξ ξ α ξ 2, A ξ ξ β ξ 2, ξ R d and f H (Ω). Stationary diffusion equation with homogenous Dirichlet boundary condition: { div (A u) = f u H 0(Ω) Ω - mixture of two isotropic materials with conductivities 0 < α < β: where χ L (Ω; {0, }). A = χαi + ( χ)βi, Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 2/25

3 Stationary diffusion equation Stationary diffusion equation Let Ω R d be open and bounded, A L (Ω; Sym d ) satisfying Aξ ξ α ξ 2, A ξ ξ β ξ 2, ξ R d and f H (Ω). Stationary diffusion equation with homogenous Dirichlet boundary condition: { div (A u) = f u H 0(Ω) Ω - mixture of two isotropic materials with conductivities 0 < α < β: where χ L (Ω; {0, }). A = χαi + ( χ)βi, Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 2/25

4 Multiple state optimal design problem J(χ) = [χ(x)g α (x, u(x)) + ( χ(x))g β (x, u(x))] dx min, Ω χ L (Ω; {0, }), χ dx = q α, Ω 0 < q α < Ω, and g α, g β Caratheodory functions which satisfy growth condition g j (x, u) a u s + b(x), j = α, β, for some a > 0, b L (Ω) and s < 2d d 2, and u = (u,..., u m ), m 2, where u i is the solution of { div (A ui ) = f i u i H 0 (Ω), i =,..., m where f i H (Ω), A = χαi + ( χ)βi, 0 < α < β. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 3/25

5 Definition (Composite material) If a sequence of characteristic functions χ n L (Ω; {0, }) and conductivities A n (x) = χ n(x)αi + ( χ n(x))βi satisfy χ n θ A n H A, then it is said that A is homogenised tensor of two-phase composite material with proportions θ of first material and microstructure defined by the sequence (χ n). Definition (H-convergence) A sequence of matrix functions A n is said to H-converge to A if for every f the sequence of solutions of { div (A n u n) = f u n H 0(Ω) satisfies u n u in H 0(Ω), A n u n A u in L 2 (Ω; R d ), where u is the solution of the homogenised equation { div (A u) = f u H 0(Ω). Example simple laminates: if χ ε depend only on x, then where A = diag(λ θ, λ+ θ, λ+ θ,..., λ+ θ ), λ + θ = θα + ( θ)β, λ θ = θ α + θ β. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 4/25

6 Set of all composites: A := {(θ, A) L (Ω; [0, ] Sym d ) : A K(θ) a.e. } G closure problem: for given θ find all possible homogenised (effective) tensors A K(θ) is given in terms of eigenvalues (Murat & Tartar; Lurie & Cherkaev): 2D: λ 2 λ + θ λ θ d j= d j= λ θ λ j λ + θ j =,..., d 3D: λ j α λ θ α + d λ + θ α β λ j β λ θ + d β λ + θ, λ3 λ θ λ + θ λ λ2 λ Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 5/25

7 Original problem: J(χ) = [χ(x)g α (x, u) + ( χ(x))g β (x, u)] dx min, Ω χ L (Ω; {0, }), χ dx = q α. Ω Generalized objective function is J(θ, A) = [θ(x)g α (x, u(x)) + ( θ)(x))g β (x, u(x))] dx Ω where u = (u,..., u m ) and u i is the solution of the state equation { div (A ui ) = f i u i H, i =,..., m. 0(Ω) Relaxed problem: J(θ, A) min, (θ, A) A, θ dx = q α. Ω Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 6/25

8 Optimality condition Let (θ, A ) be optimal point of the relaxed problem and ε (θ ε, A ε ) be a smooth path with (θ 0, A 0 ) = (θ, A ). If ε J(θ ε, A ε ) is smooth, then necessary condition of optimality is δj(θ, A ) := d dɛ J(θε, A ε ) ε=0 0 Let us introduce adjoint states p,..., p m as solutions of div (A p i ) = θ g α (, u) + ( θ) g β (, u) u i u i i =,..., m. p i H 0 (Ω) Then δj(θ, A ) = Ω δθ(x)[g α(x, u (x)) g β (x, u (x))+l] dx Ω ( ) dθ ε dε, daε. dε ε=0 for any admissible variation (δθ, δa) := m δa u i p i dx, Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 7/25 i=

9 Optimality conditions How to choose a smooth path?. Let us fix θ. a) Then set K(θ ) is convex and we can choose segment A ε = A + ε(a A ), A K(θ ). [L.Tartar, G. Allaire, M. Vrdoljak] b) We choose segment in terms of inverse matrices (A ε ) = (A ) + ε(a (A ) ), A K(θ ) Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 8/25

10 Optimality conditions How to choose a smooth path?. Let us fix θ. a) Then set K(θ ) is convex and we can choose segment A ε = A + ε(a A ), A K(θ ). [L.Tartar, G. Allaire, M. Vrdoljak] b) We choose segment in terms of inverse matrices (A ε ) = (A ) + ε(a (A ) ), A K(θ ) Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 8/25

11 Optimality conditions How to choose a smooth path?. Let us fix θ. a) Then set K(θ ) is convex and we can choose segment A ε = A + ε(a A ), A K(θ ). [L.Tartar, G. Allaire, M. Vrdoljak] b) We choose segment in terms of inverse matrices (A ε ) = (A ) + ε(a (A ) ), A K(θ ) Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 8/25

12 Sets K(θ) and K(θ) One can show equvalence of the following sets K(θ) - Set of all symmetric matrices A with eigenvalues λ,..., λ d which satisfy inequalities d j= d j= λ θ λ j λ + θ, j =,..., d λ j α λ θ α + d λ + θ α β λ j β λ θ + d β λ + θ K(θ) - Set of all symmetric matrices A with eigenvalues ν,..., ν d which satisfy inequalities d j= d j= ν + θ ν j ν θ, j =,..., d α ν j ν j β α ν θ + d α ν + θ ν θ + d β ν + θ β A K(θ) A K(θ) Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 9/25

13 Optimality conditions From necessary condition of optimality Ω then follows (for a.e. in Ω) where N = Sym i =,..., m. m δa u i p i dx 0 i= (A ) : N = min A K(θ ) A : N, m σi τi, σ i = A u i and τ i = A p i, i= Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 0/25

14 Theorem (von Neumann) For symmetric matrices A and M following inequality is valid A : M λ(a) λ(m), where λ(a) and λ(m) are vectors of eigenvalues of A and M in nondecreasing order. Equality holds if and only if A and M are diagonalizable in same basis. d j= d j= d ν iη i min, i= ν + θ νj ν θ, j =,..., d α ν j α ν θ + d α ν + θ ν j β ν θ + d β ν + θ β where ν i are increasing, while η i decreasing for i =..d Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems /25

15 Optimality conditions 2. Let us now consider variation of θ. We take smooth path (θ ε, A ε ) such that a.e. x Ω (A ε ) (x) : N(x) = min A K(θ ε ) (A (x) : N(x)) Again, from neccesary condition of optimality it follows (a.e. x Ω) ( δθ(x) g α (x, u (x)) g β (x, u (x)) + l + g ) θ (θ (x), N(x)) 0. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 2/25

16 Theorem Let (θ, A ) be minimizer of objective functional J(θ, A). We introduce m symmetric matrix N = Sym σi τi, for σi = A u i, τ i = A p i and define function g(θ, N) = i= min A K(θ) (A : N). Then (A ) (x) : N(x) = g(θ (x), N(x)), a.e x Ω. Moreover, if we define function R (x) = g α (x, u) g β (x, u) + l + g θ (θ (x), N (x)) the optimal θ satisfies (a. e. on Ω) θ (x) = 0 if R (x) > 0 θ (x) = if R (x) < 0 0 θ (x) if R (x) = 0. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 3/25

17 Algorithm Take some initial θ 0 and A 0. For k from 0 to N: Calculate u k i, i =,..., m, the solution of { div (A k u i ) = f i u i H 0 (Ω). 2 Calculate p k i, i =,..., m, the solution of div (A k p i ) = θ k g α (, u k ) + ( θ k ) g β (, u k ) u i u i p i H 0 (Ω), uk = (u k,..., uk m) and define σ k i := Ak u k i, τ k i 3 For x Ω, let θ k+ (x) be the zero of function m := A k p k i and Nk := Sym (σi k τi k ). i= θ g α (x, u k (x)) g α (x, u k (x)) + l + g θ (θ, Nk (x)), and if a zero doesn t exist, take 0 (or ) in case when this function is positive (or negative) on 0,. 4 Let (A k+ ) (x) be the minimizer in the definition of g(θ k+ (x), N k (x)). Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 4/25

18 Theorem (d=2) If dimension d = 2, then for given θ [0, ] and matrix N with eigenvalues η η 2 we have ( ) A. If η 2 > 0 and θ A η := α β η2 α β, then ( g β 2 α 2) ( η + ) η 2 2, θ < θ A (θ, N) = β (θ(α β) + ( β + α θ (β α) η (θ(α β) + β) 2 + η 2 α )., θ θ A β ( ) B. If η < 0 and θ B η β := η2 α β, then g ( β 2 α 2) ( η + ) η 2 2, θ > θ B (θ, N) = α θ(α β) ( + 2β θ (β α) η (θ(α β) + β) 2 + η 2 α )., θ θ B β C. If η 0 and η 2 0, then ( g θ (θ, N) = (β α) η (θ(α β) + β) 2 + η 2 α ). β Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 5/25

19 Theorem (d=3) For d = 3, given θ [0, ] and matrix N with eigenvalues η η 2 η 3 we have A. If η 3 = 0 then g β α (θ, N) = (η + η2). θ (θα + ( θ)β) 2 B. If η 3 > 0 and additionally η 2 + η 3 η > 0, it holds g (θ, N) = θ ( β (β 2 α 2 η2 + η 3 ) θ(α β) + α + β ( β η + η 2 + η 3 (β α)(α + 2β) ) 2θ(α β) + α + 2β 2 η + (β α) where θ B = α( η2+ η3 2 η) (α β)( η2+ η3 η) and θb 2 = α( η3 η2) (α β). η3 If η 2 + η 3 η 0 then we omit the first case in the above formula. C. If η 3 < 0 then, if η 2 and η are negative as well then g (θ, N) = θ ) 2, θ < θ B, (θα + ( θ)β) 2, θb θ < θb 2, η 3(α β β α ) + (θα + ( θ)β) 2 (η + η2), θ θb 2, ( α η + η 2 + ) η 2 3 (β α)(2α + β), θ > θ C 2θ(α β) + 3β, ( α (β 2 α 2 η2 + ) η 2 3 β α ) + η θ(α β) + 2β (θα + ( θ)β) 2, θc 2 < θ θc, (α β β α )η 3 + (θα + ( θ)β) 2 (η + η2), θ θc 2, where θ C = β( η2+ η3 2 η) (β α)( η2+ η3 η) and θc 2 = β( η3 η2) (β α). η3 If η 2 < 0 and η 0 then θ C g is not defined and we can express θ (θ, N) by the second and third term in the formula above, omitting the assumption θ θ C in the second case. If η 2 0 then both θ C and θc g 2 are not defined and θ is given by formula in the third case above, for any θ [0, ]. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 6/25

20 Ball Cube Example. We consider two dimensional problem of weighted energy minimization J(θ, A) = 2 f u dx + f 2 u 2 dx min, Ω Ω where Ω R 2 is a ball B(0, 2), α =, β = 2, while u and u 2 are state functions for { div (A ui ) = f i u i H 0 (Ω), i =, 2, where we take f = χ B(0, ) and f 2 for right-hand sides. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 7/25

21 Ball Cube Figure: Optimal distribution of materials in circle with volume constraint 25% of the first material. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 8/25

22 Ball Cube (a) Cost functional J. (b) θ k θ k+ L 2 in terms of the iteration number k. Figure: Convergence history for energy minimization in circle. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 9/25

23 Introduction Ball Cube Comparison between numerical and exact solution numerical θ exact θ numerical θ exact θ (a) Mesh refinement (b) Mesh refinement 6 Figure: Values of exact and numerical θ. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 20/25

24 Ball Cube 0 η=0.5 η=0.25 η=0.7 η= Figure: Dependence of L error between numerical and exact solution with respect to mesh refinement. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 2/25

25 Ball Cube Example 2. Second example is three dimensional energy minimization problem J(θ, A) = (f u + f 2 u 2 ) dx min, Ω with α =, β = 2 and two state equations { div (A ui ) = f i u i H 0 (Ω), i =, 2. We take cube Ω = [, ] 3 as domain and set function f to be zero on the upper half (z > 0) and 0 on the lower half of the cube, while function f 2 to be zero on the left half (y < 0) and 0 on the right half of the cube. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 22/25

26 Ball Cube (a) Optimal distribution of materials in a cube. (b) Intersection of the cube with x=0 plane. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 23/25

27 Ball Cube (c) Cost functional J. (d) θ k θ k+ L 2 in terms of the iteration number k. Figure: Convergence history for energy minimization. Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 24/25

28 Ball Cube Thank you for your attention! Ivana Crnjac Variant of Optimality Criteria Method for Multiple State Optimal Design Problems 25/25