Pontryagin Approximations for Optimal Design of Elastic Structures

Transkript

1 Pontryagin Approximations for Optimal Design of Elastic Structures Jesper Carlsson NADA, KTH Collaborators: Anders Szepessy, Mattias Sandberg October 5, 2005

2 A typical optimal design problem Optimal Design inf l(u), a ρ(u,v) = l(v), v V, ρ: {0,1} ρdx = C, with compliance and bilinear energy functional An alternative formulation inf ρ: {0,1} l(u) a ρ (u,v) f b udx + Γ N f s uds, ρε i j (u)e i jkl ε kl (v)dx. ( ) l(u) + η ρdx, a ρ (u,v) = l(v), v V. Problem Optimal design problems are typically ill-posed i.e. no existence of solution. Cure To relax the admissible set of controls A, ρ A.

3 A harder inverse problem Optimal design, continued inf ρ Γ (u u given ) 2 ds a ρ (u,v) = l(v), v V. Ill posed due to non-continuous dependence of error in measured data.

4 Hamilton-Jacobi-Bellman and the Pontryagin Principle Value function T inf {g(x T) + α A 0 u(x,t) = solution to the HJB-equation h(x t,α t )dt}, X t = f (X t,α t ), X(0) = X 0 T inf α A,X(t)=x t h(x s,α s )ds + g(x T ) u t + H(u x,x) {}}{ min α A {u x f (x,α) + h(x,α)} = 0, u(x,t ) = g(x) Differentiation along optimal paths α t, X t, λ t u x (X t,t) gives Pontryagin s Principle Also λ t = λ t f X (X t,α t ) + h X (X t,α t ), λ(t ) = g (X T ) α t = argmin a A {λ t f (X t,a) + h(x t,a)}

5 Assume H, f,h,g differentiable, λ C 1. Lagrange principle X t = H λ (λ t,x t ) λ t = H X (λ t,x t ) with control determined by Pontryagin principle α t = argmin a A {λ t f (X t,a) + h(x t,a)} Two reasons for non-smooth control: 1. Only Lipschitz continuous Hamiltonian 2. Backward characteristic paths X(t) may collide Remedy for 1: construct a C 2 concave approximation H δ of the Hamiltonian H. Error estimate [Sandberg, Szepessy 2005]: ū u L (R d R + ) = O(δ)

6 Concave Maximization in Electric Conduction Minimize power-loss in conductive medium [Pironneau 1984] Lagrangian Hamiltonian min ( qϕds + η σdx) div σ ( σ ϕ ) = 0, x σ ϕ = q n H = min σ σvdx + σ (η ϕ λ) dx + }{{} v q(ϕ + λ)ds = q(ϕ + λ)ds min σv dx + σ }{{} s(v) q(ϕ + λ)ds s v

7 Concave regularization H δ = s δ (η ϕ λ)dx + q(ϕ + λ)ds, By symmetry λ = ϕ the Hamiltonian system reduces to s δ (η ϕ 2 ) ϕ wdx = qwds or div ( s δ (η ϕ 2 ) ϕ(x) ) = 0, s ϕ δ = q, n x Concave maximization problem: ϕ V is the unique maximizer of H δ (ϕ) = s δ (η ϕ(x) 2 )dx + 2 qϕds.

8 Numerical Examples of Electric Conduction

9 Optimal Design of an Elastic Structure Lagrangian for compliance minimization l(u) + l(λ) + ρ ( ) η ε i j (u)e i jkl ε kl (λ) dx, }{{} v Hamiltonian H = l(u) + l(λ) + minρvdx. ρ }{{} s(v) Regularization gives Hamiltonian system which by symmetry u = λ reduces to s δ( η εi j (u)e i jkl ε kl (u) ) ε i j (u)e i jkl ε kl (v)dx = l(v) Concave maximization problem: u is the unique maximizer of H δ = 2l(u) + s δ ( η εi j (u)e i jkl ε kl (u) ) dx.

10 Figure 1: Plot of s δ as approximation of ρ [0.001,1]. Compliance =

11 Figure 2: Plot of the density ρ {0.001,1} after 1, 5 and 200 iterations.

12 Figure 3: 10 iterations when only removing material. Compliance = 0.51.

13 References [1] Allaire G., Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, 146. Springer-Verlag, New York, [2] Bendsø M. P. and Sigmund O., Topology optimization. Theory, methods and applications. Springer-Verlag, Berlin, [3] Pironneau O., Optimal Shape Design for Elliptic Systems. Springer Series in Computational Physics. Springer-Verlag, New-York, [4] Sandberg M. and Szepessy A., Convergence rates of Symplectic Pontryagin Approximations in Optimal Control Theory, to appear in M2AN Mathematical Modelling and Numerical Analysis. Preprint available at szepessy.