A LINEARIZED APPROACH TO WORST-CASE DESIGN IN SHAPE OPTIMIZATION

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1 1 A LINEARIZED APPROACH TO WORST-CASE DESIGN IN SHAPE OPTIMIZATION Grégoire ALLAIRE CMAP, Ecole Polytechnique Results obtained in collaboration with Ch. Dapogny (Rutgers University, formerly LJLL UPMC and Renault). Reims, 4 novembre 2014.

2 2 CONTENTS Ecole Polytechnique, RODIN project UPMC, INRIA, Renault, EADS, ESI group, etc. 1. Introduction and model problems. 2. Abstract setting for linearized worst-case design. 3. Applications in thickness optimization. 4. Applications in geometric optimization.

3 3 -I- INTRODUCTION Shape optimization : minimize an objective function over a set of admissibles shapes Ω (including possible constraints) inf J(Ω) Ω U ad The objective function is evaluated through a partial differential equation (state equation) J(Ω) = j(u Ω )dx where u Ω is the solution of Ω PDE(u Ω ) = 0 in Ω Thickness optimization : the shape is parametrized by its thickness h (a coefficient in the p.d.e.). Geometric optimization : the boundary of Ω is varying.

4 4 Thickness optimization Mid-plane Ω R d with boundary Ω = Γ N Γ D. Thickness of the plate h(x) : Ω [h min,h max ] with h max > h min > 0.

5 5 Thickness optimization (Ctd.) For given applied loads g : Γ N R d, f : Ω R d, the displacement u : Ω R d is the solution of div(hae(u)) = f in Ω u = 0 on Γ D ( ) hae(u) n = g on ΓN with the strain tensor e(u) = 1 2 ( u+ t u), the stress tensor σ = hae(u), and A an homogeneous isotropic elasticity tensor. Typical objective function: the compliance J(h) = f udx+ Ω Γ N g udx,

6 6 Geometric optimization Shape Ω R d with boundary Ω = Γ Γ N Γ D, where Γ D and Γ N are fixed. Only Γ is optimized (free boundary).

7 7 Geometric optimization (Ctd.) For given applied loads g : Γ N R d, f : Ω R d, the displacement u : Ω R d is the solution of div(ae(u)) = f in Ω u = 0 on Γ D ( ) Ae(u) n = g on ΓN ( Ae(u) ) n = 0 on Γ Typical objective function: the compliance J(Ω) = f udx+ Ω Γ N g udx,

8 8 Uncertainties location, magnitude and orientation of the body forces or surface loads elastic material s properties geometry of the shape (thickness or boundary) Crucial issue: optimal structures are so optimal for a given set of loads that they cannot sustain a different load!

9 9 Example: minimal weight and minimal compliance Allowed perturbations Source term f Itération 201

10 10 Optimal design with load uncertainties

11 State of the art: many works! Probabilistic approach (Ben-Tal et al. 97, Choi et al. 2007, Frangopol-Maute 2003, Kalsi et al ) Monte-Carlo methods Polynomial chaos, Karhunen-Loève expansions... First-Order Reliability-based Methods (FORM) Various objectives or goals: Minimization of expected value or mean Worst case desing Minimal failure probability 11 Special cases with simplifications: Robust compliance: Cherkaev-Cherkaeva (1999, 2003), de Gournay-Allaire-Jouve (2008). Mean expectation of compliance: Alvarez-Carrasco 2005, Dunning-Kim

12 12 Present work: two main ideas worst case optimization (min-max problem), linearization for small uncertainties (similar idea in Babuska-Nobile-Tempone 2005).

13 13 Worst case design Example in the case of force uncertainties. The force is the sum f +ξ where f is known and ξ is unknown. The only information is the location of ξ and its maximal magnitude m > 0 such that ξ m. We replace the standard objective function J(Ω,f +ξ) by its worst case version J(Ω,f). Worst case design optimization problem: min Ω J(Ω,f) = min Ω max J(Ω,f +ξ) ξ m

14 14 -II- ABSTRACT (AND FORMAL) SETTING Designs h H State equation A(h)u(h) = b with a linear operator A(h) Perturbations δ P in a Banach space P Assume for simplicity that only b (not A) depends on δ Perturbed state equation A(h)u(h, δ) = b(δ) Worst case objective function J(h) = sup δ P δ P m J(u(h,δ)) Goal inf J(h) h H

15 15 Linearization Assume that the perturbations are small, i.e., m << 1. Unperturbed case δ = 0, u(h) = u(h,0) Derivative of the state equation A(h) u db (h,0) = δ dδ (0) Linearization of the worst-case objective function J(h) J(h) = sup δ P δ P m ( J(u(h))+ dj ) du (u(h)) u δ (h,0)(δ) Since the right hand side is linear in δ we deduce J(h) = J(u(h))+m dj du (u(h)) u δ (h,0) P

16 16 Adjoint approach The previous formula for J(h) is not fully explicit: J(h) = J(u(h))+m dj du (u(h)) u δ (h,0) Introduce an adjoint state from which we deduce A(h) T p(h) u δ Conclusion: A(h) T p(h) = dj du (u(h)), dj (h,0) = A(h) u(h,0) p(h) = δ (u(h)) u du δ J(h) = J(u(h))+m db dδ (0) p(h) P P (h,0) = db dδ (0) p(h)

17 17 Linearized worst-case design We add to the usual objective function a perturbation term which is proportional to m and to the standard adjoint p: J(h) = J(u(h))+m db dδ (0) p(h) Classical sensitivity approach can be applied to J(h) The appearance of the adjoint is not a surprise: it is known to measure the sensitivity of the optimal value with respect to the constraint level (or right hand side in the state equation). The entire argument needs to be made rigorous in each specific case. We don t say anything about the existence of optimal designs. We don t prove that optimal designs for J(h) are close to those of J(h). P

18 18 What remains to be done (in this talk) Linearized worst-case design optimization: { inf J(h) = J(u(h))+m db h H dδ (0) p(h) where P A(h)u(h) = b(0) and A(h) T p(h) = dj du (u(h)), We compute a derivative of J(h): it requires two additional adjoints! We build a gradient-based algorithm. We test it on various objective functions. }

19 19 -III- THICKNESS OPTIMIZATION First case: loading uncertainties. Given load f L 2 (Ω) d. Unknown load ξ L 2 (Ω) d with small norm ξ L2 (Ω) d m. Solution u h,ξ of div(hae(u h,ξ )) = f +ξ in Ω u h,ξ = 0 on Γ D ( hae(uh,ξ ) ) n = g on Γ N Many variants are possible (ξ may be localized, or parallel to a fixed vector, or on Γ N, etc.)

20 20 Given a smooth (+ growth conditions) integrand j, consider J(h,ξ) = j(ξ,u h,ξ )dx Worst case design objective function: Ω J(h) = sup J(h,ξ) ξ L 2 (Ω) d ξ L 2 (Ω) d m Linearized worst case design objective function: J(h) = sup ξ L 2 (Ω) d ξ L 2 (Ω) d m ( J(h,0)+ J ) f (h,0)(ξ)

21 21 Theorem. J(h) = Ω j(0,u h )dx+m f j(0,u h ) p h L2 (Ω) d, where p h is the first adjoint state, defined by div(hae(p h )) = u j(0,u h ) in Ω, p h = 0 on Γ D, hae(p h )n = 0 on Γ N. If f j(0,u h ) p h 0 in L 2 (Ω) d, then J is Fréchet differentiable J (h)(s) = D(u h,p h,q h,z h )sdx, with two additional adjoints q h,z h and Ω D(u h,p h,q h,z h ) := Ae(u h ) : e(p h )+m Ae(u h) : e(z h )+Ae(p h ) : e(q h ) 2 f j(0,u h ) p h L2 (Ω) d

22 22 The second and third adjoint states q h,z h are defined by div(hae(q h )) = 2(p h f j(0,u h )) in Ω, q h = 0 on Γ D, hae(q h )n = 0 on Γ N, div(hae(z h )) = 2 f u j(u h ) T ( f j(u h ) p h ) 2 uj(u h )q h in Ω, z h = 0 on Γ D, hae(z h )n = 0 on Γ N.

23 23 Second case: thickness uncertainties. Given thickness h L (Ω). Uncertainty s L (Ω) with s L (Ω) m. div((h+s)ae(u h+s )) = f in Ω u h+s = 0 on Γ D ( (h+s)ae(uh+s ) ) n = g on Γ N Worst case design objective function: J(h) = sup s L (Ω) s L (Ω) m { J(h+s) = Ω } j(u h+s )dx Linearized worst case design objective function: J(h) = sup s L (Ω) s L (Ω) m ( J(h)+ J ) h (h)(s)

24 24 Theorem. J(h) = Ω j(u h )dx+m Ae(u h ) : e(p h ) L1 (Ω), where p h is the first adjoint state, defined by div(hae(p h )) = u j(u h ) in Ω p h = 0 on Γ D. hae(p h )n = 0 on Γ N If E h := {x Ω, Ae(u h ) : e(p h ) = 0} has zero Lebesgue measure, then J is differentiable ( ( )) J (h)(s) = s Ae(u h ) : e(p h )+m Ae(p h ) : e(q h )+Ae(u h ) : e(z h ) dx, Ω with two additional adjoint states q h,z h.

25 25 NUMERICAL ALGORITHM 1. Initialization of the thickness h Iteration until convergence for k 1: (a) Computation of u k and the 3 adjoints p k,q k,z k by solving linearized elasticity problem with the thickness h k. Evaluation of the gradient J (h k ) (b) Update of the thickness h k+1 by a projected gradient step (to satisfy bounds and volume constraint). All computations are made with FreeFem++.

26 26 Load uncertainties in thickness optimization Compliance minimization J(h,ξ) = with a fixed volume constraint Ω Vol(h) := (f +ξ) u h,ξ dx Ω hdx = 0.7 Rectangular 2 1 domain. Bounds h min = 0.1 and h max = 1. Material properties E = 1, ν = 0.3. We compute optimal designs for increasing values of m.

27 27 Load uncertainties in thickness optimization

28 28 -IV- GEOMETRIC OPTIMIZATION First case: loading uncertainties. Given load f L 2 (R d ) d. Unknown load ξ L 2 (R d ) d with small norm ξ L2 (R d ) m. Solution u d Ω,ξ of div(ae(u Ω,ξ )) = f +ξ in Ω u Ω,ξ = 0 on Γ D ( Ae(uΩ,ξ ) ) n = g on Γ N ( Ae(uΩ,ξ ) ) n = 0 on Γ Many variants are possible (ξ may be localized, or parallel to a fixed vector, or on Γ N, etc.)

29 29 Theorem. J(Ω) = Ω j(0,u Ω )dx+m f j(0,u Ω ) p Ω L 2 (Ω) d, where p Ω is the first adjoint state, defined by div(ae(p Ω )) = u j(0,u Ω ) in Ω, p Ω = 0 on Γ D, Ae(p Ω )n = 0 on Γ Γ N. If f j(0,u Ω ) p Ω 0 in L 2 (Ω) d, then J is shape differentiable (with two additional adjoint states).

30 30 Second case: geometric uncertainties. Perturbed shapes (I +χv)(ω), V W 1, (R d,r d ), V L (R d ) d m. χ is a smooth localizing function such that χ 0 on Γ D Γ N.

31 31 Theorem. The linearized worst-case design objective function is J(Ω) = j(u Ω )dx+m χ j(u Ω )+Ae(u Ω ) : e(p Ω ) f p Ω ds, Ω where p Ω is the (previous) adjoint state. Γ If E Γ := {x Γ, (j(u Ω )+Ae(u Ω ) : e(p Ω ) f p Ω )(x) = 0} has zero Lebesgue measure, then it admits a (hugly) shape derivative J (Ω)(θ) involving two (new) additional adjoints q Ω,z Ω.

32 32 Load uncertainties in geometric optimization (compliance)

33 33 Load uncertainties in geometric optimization (compliance)

34 34 Geometric uncertainties in geometric optimization

35 35 Geometric uncertainties (stress minimization)