Risk Averse Shape Optimization

Transcription

1 Risk Averse Shape Optimization Sergio Conti 2 Martin Pach 1 Martin Rumpf 2 Rüdiger Schultz 1 1 Department of Mathematics University Duisburg-Essen 2 Rheinische Friedrich-Wilhelms-Universität Bonn Workshop on PDE Constrained Optimization of Certain and Uncertain Processes 2009 Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 1 / 28

2 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 2 / 28

3 Conceptual sketch D Γ0 O ΓD ΓN Figure: General setting in 2D Optimization Task min J(O) O ad = {O D : O Lipschitz-continuous } O O ad Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 3 / 28

4 Linear elasticity model The displacement u is given by the equation system PDE div(ae(u)) = f in O, u = 0 on Γ D, (Ae(u))n = g on Γ N Elastic body O R 3 O = Γ N Γ D, Γ D Volume forces f in O Neumann forces g on Γ N where e(u) = 1 2 ( u + ut ) is the linearized strain tensor and Hooke s law Aξ = 2µξ + λ(trξ)id, for any symmetric matrix ξ Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 4 / 28

5 Shape optimization problem Compliance J(O) = O f u dx + g u ds Γ N Least square error compared to target displacement ( J(O) = u u 0 2 dx O ) 1 2 Shape optimization problem min J(O) + lv(o) with l R, l > 0 O O ad O ad = {O D : O Lipschitz-continuous } Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 5 / 28

6 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 6 / 28

7 Level set formulation Implicit description of the domain O via a level set function φ φ(x) = 0 <=> x O φ(x) < 0 <=> x O φ(x) > 0 <=> x O Figure: Levelset description in 2D Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 7 / 28

8 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 8 / 28

9 Shape gradient We consider variations O t = (Id + t V)(O), t > 0 of a smooth elastic domain O for a smooth vector field V defined on the working domain D. The shape derivative of J(O) at O in direction V is defined as the Fréchet derivative of the mapping t J(O t ), i.e. J J(O t ) = J(O) + O, V + o( V ) Ot = (Id + t V) (O) Ot O cf. [Sokolowski, Zolesio 92], [Delfour, Zolesio ] Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 9 / 28

10 Shape gradient As a classical result of the shape sensitivity analysis the shape derivative takes the form < J ( [ ] ) (g u) O, V > = 2 + hg u + f u Aɛ(u) : ɛ(u) V n dν n Γ N + (Aɛ(u) : ɛ(u)) V n dν Γ D Here h denotes the mean curvature of O and n the outer normal. Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 10 / 28

11 Shape gradient in level set formulation When the domain O is implicitly deformed by varying the level set function φ φ t = φ + tψ the level set equation t φ + φ v n = 0 n = φ φ allows to define < J J, ψ >:=< φ O, ψ n φ > cf. [Osher, Sethian 88], [Burger, Osher 04] Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 11 / 28

12 Shape gradient in level set formulation We take into account a regularized gradient descent, based on the metric G(θ, ζ) = which is related to a Gaussian filter with width σ. The shape gradient is the solution of equation D θζ + σ2 θ ζ dx 2 G(grad φ J, θ) = < J, θ > θ H1,2 0 O (D) Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 12 / 28

13 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 13 / 28

14 Topological Derivative Asymptotic behavier for infinitesimal small hole T (x) = lim ρ 0 J(O \ B ρ (x)) J(O) B ρ (x) Toppological derivative for the compliance D topo J(x) = π (λ + 2µ) 2µ (λ + µ) {4µAe(u i) : e(u i ) + (λ µ)tr Ae(u i )tr e(u i )} Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 14 / 28

15 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 15 / 28

16 Optimization algorithm time continuous regularized gradient descent: with time discrete relaxation : φ(t) = grad φ J(φ) G(φ k+1 φ k, θ) = τ < J, θ > θ H1,2 0 O (D) additional ingrediens of the algorithm : multigrid method for the primal and the dual problem (d = 3) preconditioned CG (d = 2) cascadic optimization (from coarse to fine grid resolution) morphological smoothing when switching the grid resolution (σ = 2.5h or 4.5h) topological changes are performed every 10 steps Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 16 / 28

17 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 17 / 28

18 Structure of Random Forces Assume that ω follows a discrete distribution with scenarios ω σ and probabilities π σ with S σ=1 π σ = 1 and basis loads (f k, g m ) spanning the load space: by linearity : f (ω) = K α k f k, g(ω) = k=1 ū(o, ω) = K α k u k f + k=1 M β m f m m=1 M β m u m g m=1 solves A(O, ū(o, ω σ ), ϕ) = l(o, ϕ, ω σ ) Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 18 / 28

19 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 19 / 28

20 Expected Excess Considering the following problem { S } min π σ max{j(o, ω σ ) η, 0} O U ad σ=1 and the smooth approximation of the maximun function a2 + a a2 + ɛ + a max{a, 0} = =: Max(a) ɛ > we get the differentiable Expected Excess functional. expected excess S min{ee(o) := π σ Max(J(O, ω σ )) : O U} σ=1 Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 20 / 28

21 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 21 / 28

22 Excess Probability Now consider min{p(j(o, ω) > η) : O U ad } The random stage ω follows a discrete distribution with finitely many scenarios ω σ and probabilities π σ according to S σ=1 π σ = 1 and we get min{p(j(o, ω σ ) > η) = S π σ H(J(O, ω σ ) η) where H(x) is supposed to be the Heaviside function. Again we use a smooth approximation H(x) tanh(kx) = 1 1+e 2k excess probability min{ep(o) := σ=1 S π σ H(J(O, ω σ )) : O U} σ=1 Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 22 / 28

23 Outline 1 Introduction 2 Level set method Level set formulation Shape gradient Topological Derivative Optimization Algorithm 3 Risk Averse Functionals Uncertainty Expected Excess Excess Probability 4 Numerical Results Examples Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 23 / 28

24 Test Setting % 95% Figure: Nonsymmetric stochastic loading Figure: The results for EV, EE and EP; the threshold η is set to 0.4 Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 24 / 28

25 Expected Exccess Figure: A sequence of results for the optimization with respect to the expected excess for η = 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0., 1.5. Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 25 / 28

26 Cantilever load 1 load expected value expected value Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 26 / 28

27 Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 27 / 28

28 Thank you! Martin Pach (University Duisburg-Essen) Risk Averse Shape Optimization Trier09 28 / 28