Airfoil shape optimization using adjoint method and automatic differentiation

Transcription

1 Airfoil shape optimization using adjoint method and automatic differentiation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore AeSI CFD Symposium August, 29 Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 1 / 25

2 Objectives and controls Objective function I(β) = I(β, Q) mathematical representation of system performance Control variables β Parametric controls β R n Infinite dimensional controls β : X Y Shape β set of admissible shapes State variable Q: solution of an ODE or PDE R(β, Q) = = Q = Q(β) Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 2 / 25

3 Mathematical formulation Constrained minimization problem min β I(β, Q) subject to R(β, Q) = Find δβ such that δi < (to first order) δi = I [ I I δβ + β Q δq = β + I ] Q δβ Q β }{{} G Steepest descent δβ = ɛg, ɛ > δi = ɛgg = ɛ G 2 < How to compute gradient G cheaply and accurately? Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 3 / 25

4 Elements of shape optimization 1 Shape parameterization 2 Surface grid generation/deformation 3 Domain grid generation/deformation 4 Flow solution (Euler/Navier-Stokes solver) 5 Adjoint flow solution 6 Optimization method Shape parameters β Surface grid X s Volume grid X CFD solution Q I di dβ = di dx dx s dx dx s dβ Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 4 / 25

5 Adjoint approach For shape optimization: I = I(X, Q) di dx = I X + I Q Q X Flow sensitivity Q X ; costly to evaluate Differentiate state equation R(X, Q) = R X + R Q Q X = Introducing an adjoint variable Ψ, we can write di dx = I X + I Q Q X + Ψ [ R X + R Q ] Q X Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 5 / 25

6 Adjoint approach Collect terms involving the flow sensitivity di dx = I R + Ψ X X + Choose Ψ so that flow sensitivity vanishes [ I R + Ψ Q Q ] Q X I R + Ψ Q Q = or ( ) R Ψ + Q ( ) I = Q Gradient di dx = I R + Ψ X X Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 6 / 25

7 Optimization steps β = X s = X Solve the flow equations to steady-state dq dt + R(X, Q) = = Q, I(X, Q) Solve adjoint equations to steady-state dψ dt + ( ) R Ψ + Q Compute gradient wrt grid X ( ) I = = Ψ Q di dx = I R + Ψ X X di dβ = di dx dx s dx dx s dβ = β β ɛ di dβ Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 7 / 25

8 Continuous and discrete approaches Continuous approach (differentiate and discretize) PDE Adjoint PDE Discrete adjoint Discrete approach (discretize and differentiate) PDE Discrete PDE Discrete adjoint We use the discrete approach R(X, Q) = represent the finite volume equations which are algebraic equations Use ordinary calculus to differentiate Need to compute I Q, I X, ( ) R Ψ, Q ( ) R Ψ X Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 8 / 25

9 Automatic differentiation Computer code available to compute I(X, Q), R(X, Q) Code is made of composition of elementary functions T = X, T r = F r (T r 1 ) Y = F (X) = F p F p 1... F 1 (T ) Use differentiation by parts formula Ẏ = F (X)Ẋ = F p(t p 1 )F p 1(T p 2 )... F 1(T )Ẋ Automated using AD tools Computer code P Automatic Differentiation New code Ṗ Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 9 / 25

10 Reverse differentiation Reverse mode computes transpose: (X, Ȳ ) X X = [F (X)] Ȳ = [F 1(T )] [F 2(T 1 )]... [F p(t p 1 )] Ȳ Forward sweep and then reverse sweep T Func: T F 1 T 1 2 T p 1 F2... Grad: T p 1, Ȳ [F p] T Tp 2 [F p 1 ]T Tp 3... T F p [F 1 ] T Forward variables T j required in reverse order: store or recompute Reverse mode useful to compute ( ) I, Q ( ) I, X ( ) R Ψ, Q ( ) R Ψ X Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 1 / 25

11 Differentiation: Example A simple example f = (xy + sin x + 4)(3y 2 + 6) Computer code, f = t 1 t 1 = x t 2 = y t 3 = t 1 t 2 t 4 = sin t 1 t 5 = t 3 + t 4 t 6 = t t 7 = t 2 2 t 8 = 3t 7 t 9 = t t 1 = t 6 t 9 Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

12 F77 code: costfunc.f s ubroutine c o s t f u n c ( x, y, f ) t1 = x t2 = y t3 = t1 t2 t4 = s i n ( t1 ) t5 = t3 + t4 t6 = t5 + 4 t7 = t2 2 t8 = 3. t7 t9 = t t1 = t6 t9 f = t1 end Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

13 Automatic Differentiation: Reverse mode SUBROUTINE COSTFUNC_B(x, xb, y, yb, f, fb) t1 = x t2 = y t3 = t1*t2 t4 = SIN(t1) t5 = t3 + t4 t6 = t5 + 4 t7 = t2**2 t8 = 3.*t7 t9 = t t1b = fb t6b = t9*t1b t9b = t6*t1b t8b = t9b t7b = 3.*t8b t5b = t6b t3b = t5b t2b = t1*t3b + 2*t2*t7b t4b = t5b t1b = t2*t3b + COS(t1)*t4b yb = t2b xb = t1b fb =. END Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

14 Implementation of AD CFD code written with many subroutines Subroutines differentiated individually Then assembled together to form adjoint solver Only non-linear portions differentiated with AD numerical flux (Roe) Limiters Linear portions differentiated manually Leads to an efficient code with less memory requirements Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

15 Shape parameterization Parameterize the deformations [ ] [ ] xs x () s = y s y s () + h(ξ) = [ nx n y m β k B k (ξ) k=1 ] h(ξ) Hicks-Henne bump functions n A B ξ h(ξ) B k (ξ) = sin p (πξ q k ), q k = log(.5) log(ξ k ) Move points along normal to reference line AB ξ Exact derivatives dxs dβ computed can be Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

16 Grid deformation Interpolate displacement of surface points to interior points using RBF f(x, y) = a + a 1 x + a 2 y + N b j r r j 2 log r r j j=1 where r = (x, y) Initial grid Deformed grid Results in smooth grids Exact derivatives dx dx s can be computed Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

17 NUWTUN flow solver Based on the ISAAC code of Joseph Morrison Finite volume scheme Structured, multi-block grids Roe flux MUSCL reconstruction with Koren limiter Implicit scheme Source code of NUWTUN available online Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

18 Convergence tests Residue Adjoint residual.1 1x1-6 1x1-7 1x1-8 1x1-9 Flow residual 1x1-1 1x1-11 1x1-12 1x1-13 1x1-14 1x Number of iterations Cl Number of iterations Cl Cd Cd Convergence characteristics for the flow and adjoint solutions, and convergence of lift and drag coefficients, for RAE2822 airfoil at M =.73 Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

19 Validation of adjoint gradients 8 AD FD 6 4 Dot-product test Limiters can cause non-differentiability Koren limiter: dependance on parameter Check adjoint derivatives against finite difference Gradient Gradient Hicks-Henne parameter AD FD Hicks-Henne parameter Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

20 Test cases NACA12: M =.8, α = 1.25 o I = C d C l C l C d RAE2822: M =.729, α = 2.31 o Penalty approach Constrained minimization I = C d + C d 1 C l C l min I = C d C d s.t. C l = C l Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug 29 2 / 25

21 NACA12: Maximize L/D Initial -Cp -.5 Initial conmin_frcg optpp_q_newton steep conmin_frcg optpp_q_newton steep x/c Method I 1C d C l N fun N grad Initial conmin frcg optpp q newton steep Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

22 RAE2822: Drag minimization, penalty approach Cp.5 y.2 Initial conmin_frcg optpp_q_newton steep Initial conmin_frcg optpp_q_newton steep x x Method I 1C d C l N fun N grad Initial conmin frcg optpp q newton steep Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

23 RAE2822: Lift-constrained drag minimization Cp.5.2 Initial conmin_mfd fsqp ipopt Initial conmin_mfd fsqp ipopt x/c Method I 1C d C l N fun N grad Initial conmin mfd fsqp ipopt Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

24 RAE2822: Lift-constrained drag minimization Cp.5.2 ipopt Initial -.5 ipopt Initial x/c Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25

25 Sensitivity to perturbations.4 Initial Optimized 2 Initial Optimized.3 15 Drag coefficient.2 Lift/Drag Mach number Mach number (a) (b) Variation of (a) drag coefficient and (b) L/D with Mach number for RAE2822 airfoil and optimized airfoil Need for robust aerodynamic optimization Praveen. C (TIFR-CAM) Shape Optimization AeSI, Aug / 25