Constrained Single-Step One-Shot Method with Applications in Aerodynamics

Transcription

1 Constrained Single-Step One-Shot Method with Applications in Aerodnamics icolas R. ager 2 erman Aerospace Center DLR Branschweig Institte of Aerodnamics and Flow echnolog merical Methods Branch C 2 A 2 S 2 E 2 Hmboldt Universit Berlin Department of Mathematics rier Jne ico ager Slide

2 Collaborators HU Berlin: A. riewan A. Hamdi E. Özaa A. Ploce DLR: C. Ilic Uni Paderborn: A. Walther Uni rier: V. Schlz S. Schmidt rier Jne ico ager Slide 2

3 oal: min s.t. where and are the state and design variables. iven fixed point iteration e.g. psedo-time stepping to solve PDE f Problem Statement c 0 + c 0. Assmptions: c alwas invertible. IF given! s.t. f C 2. contractive: ρ < c 0. rier Jne ico ager Slide 3

4 Slide 4 rier Jne ico ager B OS f L + One-Shot approach L L L primal pdate dal pdate design pdate One-step one-shot step +: Stationar point: 4243 Aims: Choose B sch that: Convergence of OS. Bonded retardation. shifted Lagragian

5 Slide 5 rier Jne ico ager Jacobian of the extended iteration: Whenever we can define sch that B I B B J * 0 * * * B const J < ˆ ρ ρ Bonded retardation we have bonded retardation.

6 ecessar condition for contractivit J * Eigenvales of are the zeros of the eqation det λ B + H λ 0 where H λ λi λi I. I ecessar bt not sfficient condition for contractivit: B B f 0 and B f 2 H. [riewan 2006] rier Jne ico ager Slide 6

7 Exact penalt fnction: L a Remar: Deriving sfficient conditions on B for J * to have a spectral radis smaller than has proven difficlt. Instead we loo for descent on the agmented Lagrangian L a α : primal residal where α > 0 and β >0. + β dal residal Lagrangian rier Jne ico ager Slide 7

8 Slide 8 rier Jne ico ager Ms L L L a a a B s he fll gradient of L a is given b where B I I I I M β α β β α 0 0 and. Correspondence condition

9 Correspondence condition Conseqence Correspondence condition: here is a - correspondence between the stationar points of L a and the roots of s if det[ αβ I I I β ] 0 for which it is sfficient that 2 αβ ρ > + β. [Hamdi riewan 2008] rier Jne ico ager Slide 9

10 Descent condition heorem Descent condition: s is a descent direction for all large positive B if and onl if β αβ I + > I + I + I β which is implied b αβ ρ > c Satisfied for β α with c 2. c ρ heorem: A sitable B is given b: B α + β +. β 2 [Hamdi riewan 2008] rier Jne ico ager Slide 0

11 One-step one-shot Aerodnamic shape design 2 2c Descent for β α with c. 2 c ρ In practice choose c β 2 α >>. A sitable B is given b B α + β +. Instead BFS pdates for the Hessian 2 L a + β + + α + β B * 0 * 0 a L α + β + α 4 2 he gradient is evalated b Algorithmic Differentiation AD.. rier Jne ico ager Slide

12 ransonic case: RAE 2822 at Ma 0.73 with α 2 Cost fnction: drag cd One-step one-shot Aerodnamic shape design AUij 2D Eler + mesh deformation + parameterization First and second derivatives b AD tool ADOL-C eometric constraint: constant thicness Camberline/hicness decomposition 20 Hics-Henne coefficients define camberline rier Jne ico ager Slide 2

13 Atomatic Differentiation of Entire Design Chain x new dx m design vector P defgeo difgeo meshdefo flow solver C D srface grid grid Adjoint version of entire design chain b ADOL-C AUij 2D Eler + mesh deformation + parameterization dc dp D C m D m dx dx x new x P new and dx xnew xold x x new new Id AUij_AD meshdefo_ad defgeo_ad rier Jne ico ager Slide 3

14 Drag redction RAE 2822 M 0.73 α 2.0 inviscid flow mesh 6x33 cells 20 design variables Hics-Henne One-step one-shot One-step one-shot Flow Solver: AUij Compressible Eler Explicit RK-4 Mltigrid Implicit residal smoothing rier Jne ico ager Slide 4

15 Primal compared to copled iteration Retardation-Factor 4 [Özaa ager 2008] rier Jne ico ager Slide 5

16 min C reatment of lift constraint b penalt mltiplier method s. t. C C Penalt fnction for lift: h C C h 0 Redefine objective fnction: min D C D + λh ; h Update the penalt parameter in each one-shot step : λ + λ + ch c > h > 0 λ L 0 0 L t arget and! L t arget L f CD + λh h < 0 λ A good starting vale is: λ 0 rier Jne ico ager C h D Slide 6

17 Constrained One-Step One-Shot Drag redction b constant lift RAE 2822 M 0.73 α 2.0 inviscid flow mesh 6x33 cells 40 design variables Hics-Henne One-step one-shot Flow Solver: AUij Compressible Eler Explicit RK-4 Mltigrid Implicit residal smoothing rier Jne ico ager Slide 7

18 Primal compared to copled iteration Retardation-Factor 6 [ager Ploce 2008] rier Jne ico ager Slide 8

19 Histor of Penalt Mltiplier [ager Ploce 2008] rier Jne ico ager Slide 9

20 Extension to avier-stoes ELA Code Flow Solver: ELA U Berlin 3D avier-stoes RAS incompressible with pressre correction mltibloc -ω Wilcox trblence model and others Fortran lines AD ool: APEADE IRIA sorce to sorce reverse for first derivatives tangent on reverse for second derivatives rier Jne ico ager Slide 20

21 Extension to avier-stoes ELA Code Drag redction with lift constraint ACA 442 Re α5. RAS -ω Wilcox trblence model 300 srface mesh points Approaches for Optimization one-shot method entire design chain differentiated gradient smoothing penalt mltiplier method rier Jne ico ager Slide 2

22 Extension to avier-stoes ELA Code Drag redction with lift constraint ACA 442 Re α5. RAS -ω Wilcox trblence model 300 srface mesh points Approaches for Optimization one-shot method entire design chain differentiated gradient smoothing penalt mltiplier method rier Jne ico ager Slide 22

23 Extension to avier-stoes ELA Code Drag redction with lift constraint ACA 442 Re α5. RAS -ω Wilcox trblence model 300 srface mesh points 5% drag redction Approaches for Optimization one-shot method entire design chain differentiated gradient smoothing penalt mltiplier method [Özaa ager 2009] rier Jne ico ager Slide 23

24 hans for or attention! rier Jne ico ager Slide 24