TOWARD THE SOLUTION OF KKT SYSTEMS IN AERODYNAMICAL SHAPE OPTIMISATION

Transcription

1 CIRM, Marseille, 7-10 juillet TOWARD THE SOLUTION OF KKT SYSTEMS IN AERODYNAMICAL SHAPE OPTIMISATION Alain Dervieux, François Courty, INRIA, Sophia-Antipolis alain.dervieux@sophia.inria.fr 1

2 CIRM, Marseille, 7-10 juillet OVERVIEW Automated Differentiation One Shot SQP Multilevel preconditioner Application to sonic boom 2

3 CIRM, Marseille, 7-10 juillet The minimisation problem Find x opt = (Y opt,u opt ) in IR N Y IR N u Shape such that J(x opt ) = min Ψ(x) = 0 J(x) Lift, Boom j (u) = ( ) J u (Y,u) < Ψ(Y,u) = 0 ( ) (State: Euler) ( ) Ψ J Y (Y,u) Π = Y (Y,u) ( ) Ψ u (Y,u) Π, 1 > = 0 (Adjoint state) (Stationarity) 3

4 CIRM, Marseille, 7-10 juillet I. Progress in Automated Differentiation TROPICS: An INRIA project (team+research program) focalised on AD and applications : - a new generation tool: TAPENADE - the analysis of its application to optimal control Maurizio Araya, François Courty, Laurent Hascoet, Valérie Pascual, Alain Dervieux 4

5 CIRM, Marseille, 7-10 juillet Progress in inverse mode AD f = f p f p 1 f 1 f (x)ẋ = (f p f p 1 f p 2 f 1 (x)). (f p 1 f p 2 f 1 (x))..... (f 2 f 1 (x)). (f 1(x))ẋ. f t (x).y = (f 1 t (x)). (f 2 t f 1 (x))..... (f p 1 t f p 2 f 1 (x)). (f p t f p 1 f p 2 f 1 (x)). y. 5

6 CIRM, Marseille, 7-10 juillet DO i=1,n... A <- B... ENDDO Case of a loop 6

7 CIRM, Marseille, 7-10 juillet DO i=1,n... A <- B SAVE A(i)=A... ENDDO DO i=n,1,-1... RESTORE A(i) B <- function(a(i),ā(i))... ENDDO Case of a loop, end d 7

8 CIRM, Marseille, 7-10 juillet Inverse differentiation of a fixed point algorithm do iter = 1, update state variable if(residual.lt.eps)stop loop enddo 8

9 CIRM, Marseille, 7-10 juillet Inverse differentiation of a fixed point algorithm do iter = 1, update ST AT E ST ORE ST AT E(iter) if(residual.lt.eps)stop loop enddo do enddo iter = last iter,1 READ ST AT E(iter) update ADJOINT ST AT E - exact but subject to discontinuities - subject to instabilities, if update is nonlinear - very large storage requirement. 9

10 CIRM, Marseille, 7-10 juillet Inverse differentiation of a fixed point algorithm - Keep the differentiated algorithm, but replace the inverse-order read data by the direct-order ones : piggy back method of Griewank-Faure, - Keep the differentiated algorithm, but replace the inverse-order read data by the converged state: Hovland-Mohammadi-Bishof, etc. - Take the converged state, apply an adapted algorithm (preconditioned fixed point). Griewank-Faure, Piggyback Differentiation for Optimization, to be published in Proceedings of the Workshop on PDE Constrained Optimization in Santa FE, July 2001, Springer Lecture Notes on Computational Science and Engineering, Hovland-Mohammadi-Bishof, Automatic Differentiation of Navier-Stokes computations, Argonne National Laboratory MCS-P , 1997 Courty-Koobus-Dervieux-Hascoet 10

11 CIRM, Marseille, 7-10 juillet DO i=1,nelement... A <- B... ENDDO Inverse differentiation of an assembly loop An assembly loop can be executed in any order of the index. 11

12 CIRM, Marseille, 7-10 juillet Inverse differentiation of an assembly loop DO i=1,nnelement... A <- B SAVE A(i)=A... ENDDO Enormous memory storage DO i=nnelement,1,-1... RESTORE A(i) B <- function(a(i),ā(i))... ENDDO 12

13 CIRM, Marseille, 7-10 juillet DO i=1,n... A <- B SAVE A=A... Inverse differentiation by iterationwise adjoining... RESTORE A B <- function(a,ā)... ENDDO Hascoët, The Data-Dependence Graph of Adjoint Programs, INRIA-RR Courty-Koobus-Dervieux-Hascoet 13

14 CIRM, Marseille, 7-10 juillet Return to minimisation problem Find x opt = (Y opt,u opt ) in IR N Y IR N u Shape such that J(x opt ) = min Ψ(x) = 0 J(x) Lift, Bang j (u) = J u (Y,u) < ( Ψ u (Y,u) ( Ψ Y (Y,u) Ψ(Y,u) = 0 ) (State: Euler) Π = J Y (Y,u) (Adjoint state) ) Π, 1 > = 0 (Stationarity) 14

15 CIRM, Marseille, 7-10 juillet II. SQP / One shot optimisation algorithms A classical Sequential Quadratic Programming algorithm: For k=1,... In k : -Compute state update v (Newton). -Compute adjoint Π and gradient. -Compute Hessian 2 xl(x,π) (BFGS). -Compute step h for optimisation subproblem. Let d = v + h. Linear search in k : d ρd Trust region heuristics: Compare decrements of Merit function and quadratic model. Update Trust Region ball k. Next k. Byrd, Third SIAM Conference on Optimization, Houston, TX,1987, Omojokun, PhD Thesis, 1991 Byrd-Gilbert-Nocedal, Math. Programming, 2000 Nocedal-Wright, Numerical Optimization, Springer, 1999 N.B.: Still Three linearised state system at each iteration. 15

16 CIRM, Marseille, 7-10 juillet One Shot Methods S : a fixed (small) number of iterations of a given algorithm Iteration on state: Y k = Y k 1 SΨ( Ψ Iteration on adjoint Π k = Π k 1 S Π Y Iteration on control: u = u ρg(y k,u,π k ) ( J Y )) Ta asan, One Shot methods for optimal control of distributed parameter systems I: finite dimensional control, Report 91-2, Nasa report , ICASE, Marco-Beux, Multilevel optimization : application to shape optimum design with a One Shot method INRIA research report, no 2068,

17 CIRM, Marseille, 7-10 juillet Orientation In the seventies, gradient methods : exact solution of state and co-state, convergence control based on descent direction, Eighties: SQP, one Newton step for state, convergence control based on descent direction. Next step: One-Shot + descent direction 17

18 CIRM, Marseille, 7-10 juillet Assumption 1 : A few assumptions... Ψ(Y SΨ(Y,u),u) 0.99 Ψ(Y,u) Ψ (Π Sr Π ) J Y Y 0.99 Ψ Π J Y Y Assumption 2 : After k 0 applications of S, the gradient candidate, g(y k,u,π k ) is close enough to gradient for the satisfaction of Wolfe condition: k 0, k k 0, ρ k / 0 < ρ k ρ max, d k = M 1 g(y k,u,π k ), { J(Y k,u + ρ k d k ) J(Y k,u) 0.95ρ k < g(y k,u,π k ), d k > where Y k = Y (u + ρ k d k ) with k applications of S 18

19 CIRM, Marseille, 7-10 juillet SQP-like One-Shot algorithm Pour k=0,... *State partial restoration Y = Y + δy ( = Y SΨ Ψ ( )) J Adjoint partial restoration Π = Π + S Π Y Y Compute pseudo gradient g(y k,u,π k ) Linear search for descending under Wolfe conditions (restore state). If not successful, go to * (Optional) Trust Region heuristics Next k. 19

20 CIRM, Marseille, 7-10 juillet Application to a model problem Min Y (u) Y target 2, tel que Y = u on [0,1], Y (0) = Y (1) = 0 P 1 Finite Elements. Dimension of Y, u and adjoint Π is N = 500. Classical SQP algorithm One Shot SQP 20

21 CIRM, Marseille, 7-10 juillet Comparison of the efficiency of the two algorithms Algo Opt. iterations Conj. Grad. iterations Byrd-Omojokun, N= One shot, N= Byrd-Omojokun, N= One shot, N= Tab. 1 Efficiency of the two algorithms from the total number of conjugate gradient iterations for setting all residuals to F. Courty, A. Dervieux, A SQP-like one shot algorithm for Optimal Control problems, submitted to Mathematical Programming 21

22 CIRM, Marseille, 7-10 juillet Application to sonic boom reduction j(γ) = α 1 (C D C target D ) 2 + α 2 (C L C target L ) 2 + α 3 Ω B ( p) 2 dv α 1 = 1.0 α 2 = 10.0 α 3 = Fig. 1 Aircraft shape and gradient 22

23 CIRM, Marseille, 7-10 juillet Application to sonic boom reduction (cont d) Pressure in a plan under the aircraft (10 iterations) Pressure in two cuts, parallel to symmetry axis 23

24 CIRM, Marseille, 7-10 juillet Wing optimisation ONERA M6. initial and optimised wing profiles. ONERA M6. Pressure in a plane under wing. 24

25 CIRM, Marseille, 7-10 juillet Efficiency of the one-shot approach N Y = = N u = 780 one-shot SQP optimisation is equivalent to 10 complete state+adjoint solutions. 200 iterations: 14h 40 min 25

26 CIRM, Marseille, 7-10 juillet Efficiency of the one-shot approach: finer meshes N Y = = N u = iterations cost 6h CPU instead of 44h, - the one-shot optimisation CPU is equivalent to 15 state+adjoint - of course, with the finer mesh, convergence is slower. 26

27 CIRM, Marseille, 7-10 juillet III. Which preconditioners for Shape Optimal Design? u n+1 = u n ρg n 27

28 CIRM, Marseille, 7-10 juillet III. Which preconditioners for Shape Optimal Design? u n+1 = u n ρ C g n

29 CIRM, Marseille, 7-10 juillet III. Which preconditioners for Shape Optimal Design? u n+1 = u n ρ C g n Damp high frequencies: Reuther-Jameson, Mohammadi-Pironneau: - Laplace-Beltrami C = (Id µ ) 1 Identify the degree of the Fourier transform of Hessian: Arian-Ta asan - If degree = 2: Laplace-Beltrami C = (Id µ ) 1 - If degree = +1: C =Neumann-Dirichlet Pseudo-differential operator Reuther-Jameson, Aerodynamic Shape Optimization of Wing and Wing-Body Configurations Using Control Theory, AIAA Paper , 1995 Mohammadi-Pironneau, Applied shape optimization for fluids, Clarendon Press - Oxford, 2001 Arian-Ta asan, Analysis of the Hessian for aerodynamic optimization: Inviscid flow., Comput. Fluids, 28, 7, p ,

30 CIRM, Marseille, 7-10 juillet Shape Optimal Design for the Dirichlet problem Find γ, the optimal displacement of boundary in normal direction: γ = ArgMin J(Z); Z = 1 in Ω γ ; Z Ωγ = 0 Hadamard s variational formula: Unbounded operator in C k. γ γ ρ Z Π n n Regularity loss is 1. Palmerio-Dervieux, CRAS,A,280, , Murat-Simon, Sur le contrôle d un domaine géométrique. Publication 76015, Lab. d Analyse Numérique, Univ. Paris 6, Pironneau, Optimal shape design for elliptic systems, Springer-Verlag,

31 CIRM, Marseille, 7-10 juillet Hadamard formula for the Euler system (formal) j (γ 0,θ) = (F (W ) Π + G(W ) Π + H(W ) Π Ω γ0 x y z )( n V )θ d Ω γ0 + ( p Π + p Π)( n V )θ d Ω γ0 Ω γ0 + (Π 5 p + W 1 Π 1 + W 2 Π 2 + W 3 Π 3 + W 4 Π 4 + W 5 φ 5 ) q Ω γ0 γ δγ d Ω γ 0 - spatial derivatives of pressure p - spatial derivatives of boundary parameter γ Formally, one degree of smoothness is lost Beux-Dervieux, Exact-gradient shape optimization of a 2D Euler flow, Finite Elements in Analysis and Design, 12, p ,

32 CIRM, Marseille, 7-10 juillet Additive multilevel preconditioner (Au,v) = (f,v) v V, f given in V. V 1 V k V Let Q k be the projector: u V, v V k, (Q k u,v) = (u,v). Cg = + k ( ) k 1 2 a (Q k g Q k 1)g Property: C is continuous from H s 0(Ω) to H s+a 0 (Ω) (s, a in adhoc intervals) Bramble, Pasciak, and Xu, Parallel multilevel preconditioners, Math. Comput., 1990, 55, 191,p 1-22 Kunoth, PhD thesis, University of Berlin, Courty, Dervieux, Multilevel Functional Preconditioning for Shape Optimization, submitted to Numerische Mathematik. 31

33 CIRM, Marseille, 7-10 juillet Application to a model problem u opt = ArgMin 1 2 ( u 2 fu) dv

34 CIRM, Marseille, 7-10 juillet Application to a model problem u opt = ArgMin 1 2 ( u 2 fu) dv Verification of the optimal value of a, mesh independant convergence 32

35 CIRM, Marseille, 7-10 juillet Extension to unstructured : agglomeration 4 cells grouped in a single macro-cell Lallemand, Steve, Dervieux, Unstructured multiggridding by volume agglomeration : current status, Computer Fluids, 21, 3, , 1992 Marco, Dervieux, Multilevel parametrization for aerodynamical optimization of 3D shapes, Finite Elements in Analysis and Design, 26, p ,

36 CIRM, Marseille, 7-10 juillet Smooth transfer operators P : prolongation, from coarse to fine. P : restriction, from fine to coarse. L : A smoothing operator, L its transpose: (Lu) i = (1 θ)u i + θ Meas(j) u j j V(i) {i} j V(i) {i} Meas(j) where j belongs to the set V(i) of neighboring cells of i, Meas(j) is cell j measure. 34

37 CIRM, Marseille, 7-10 juillet Building the preconditioner in the unstructured case 1. Build the different levels by agglomeration, 2. Define operators L m,p m,pm,l m, 3. Define: Q k = L m P m PmL m 4.Compute: Cg = n k 1 m k ( ) k 1 2 a (Q k g Q k 1)g 35

38 CIRM, Marseille, 7-10 juillet Application to sonic boom (coarse) Optimal shape design by SQP/conjugate gradient: Gradient convergence Experimental evaluation of the optimal parameter a opt 36

39 CIRM, Marseille, 7-10 juillet Application to sonic boom (FINE) Optimal shape design by SQP/conjugate gradient: Gradient convergence Experimental evaluation of the optimal parameter a opt 37

40 CIRM, Marseille, 7-10 juillet To conclude... - The previous results are recent. It remains to combine One-Shot and Multi-Level. - The Multi-Level approach is being extended to non-symmetric problems. The whole optimality system would then be preconditioned by this mean. - Optimality systems can also be constructed for optimal meshing, and we shall try to solve them with the above methods. 38