Numerical shape optimization and adjoint equations Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in TIFR-CAM 18 August, 2009 Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 1 / 37
Effect of shape on flow http://www.aerospaceweb.org Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 2 / 37
Airfoils http://www.centennialofflight.gov Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 3 / 37
Compressible flow and shocks speed of air Mach number, M = speed of sound Range, R = M a C L log W i c T C D W f Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 4 / 37
Golf ball http://www.aerospaceweb.org Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 5 / 37
Effect of shape on flow http://www.aerospaceweb.org Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 6 / 37
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 ) = 0 = ⇒ Q = Q ( β ) Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 7 / 37
Mathematical formulation • Constrained minimization problem min I ( β, Q ) subject to R ( β, Q ) = 0 β • Find δβ such that δ I < 0 (to first order) � ∂ I � δ I = ∂ I ∂β δβ + ∂ I ∂β + ∂ I ∂Q ∂QδQ = δβ ∂Q ∂β � �� � G • Steepest descent δβ = − ǫG ⊤ , ǫ > 0 δ I = − ǫGG ⊤ = − ǫ � G � 2 < 0 How to compute gradient G cheaply and accurately ? Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 8 / 37
Elements of shape optimization Shape parameters Surface grid Volume grid CFD solution I Q β X s X d β = d I d I d X d X s d X d X s d β Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 9 / 37
Adjoint approach • For shape optimization: I = I ( X, Q ) d X = ∂ I d I ∂X + ∂ I ∂Q ∂Q ∂X • Flow sensitivity ∂Q ∂X ; costly to evaluate • Differentiate state equation R ( X, Q ) = 0 ∂X + ∂R ∂R ∂Q ∂X = 0 ∂Q • Introducing an adjoint variable Ψ, we can write � ∂R � d X = ∂ I d I ∂X + ∂ I ∂Q ∂X + ∂R ∂Q ∂X + Ψ ⊤ ∂Q ∂Q ∂X Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 10 / 37
Adjoint approach • Collect terms involving the flow sensitivity � ∂ I � ∂Q d X = ∂ I d I ∂X + Ψ ⊤ ∂R ∂Q + Ψ ⊤ ∂R ∂X + ∂Q ∂X • Choose Ψ so that flow sensitivity vanishes � ∂ I � ∂R � ⊤ � ⊤ ∂Q + Ψ ⊤ ∂R ∂ I ∂Q = 0 or Ψ + = 0 ∂Q ∂Q • Gradient d X = ∂ I d I ∂X + Ψ ⊤ ∂R ∂X Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 11 / 37
Optimization steps • β = ⇒ X s = ⇒ X • Solve the flow equations to steady-state d Q d t + R ( X, Q ) = 0 = ⇒ Q, I ( X, Q ) • Solve adjoint equations to steady-state � ∂ I � ∂R � ⊤ � ⊤ dΨ d t + Ψ + = 0 = ⇒ Ψ ∂Q ∂Q • Compute gradient wrt grid X d X = ∂ I d I ∂X + Ψ ⊤ ∂R ∂X d β = d I d I d X d X s − β − ǫ d I = ⇒ β ← d X d X s d β d β Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 12 / 37
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 ) = 0 represent the finite volume equations which are ◮ algebraic equations ◮ Use ordinary calculus to differentiate ◮ Need to compute � ∂R � ∂R � ⊤ � ⊤ ∂ I ∂ I ∂Q, ∂X , Ψ , Ψ ∂Q ∂X Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 13 / 37
Automatic differentiation • Computer code available to compute I ( X, Q ) , R ( X, Q ) • Code is made of composition of elementary functions T 0 = X r ′ th line of program: T r = F r ( T r − 1 ) Y = F ( X ) = F p ◦ F p − 1 ◦ . . . ◦ F 1 ( T 0 ) ◮ Use differentiation by parts formula Y = F ′ ( X ) ˙ ˙ 1 ( T 0 ) ˙ X = F ′ p ( T p − 1 ) F ′ p − 1 ( T p − 2 ) . . . F ′ X ◮ Automated using AD tools: Algorithmic Differentiation Computer code New code Automatic ˙ Differentiation P P Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 14 / 37
Reverse differentiation • Reverse mode computes transpose: ( X, ¯ → ¯ Y ) − X X = [ F ′ ( X )] ⊤ ¯ 2 ( T 1 )] ⊤ . . . [ F ′ p ( T p − 1 )] ⊤ ¯ ¯ Y = [ F ′ 1 ( T 0 )] ⊤ [ F ′ Y • Forward sweep and then reverse sweep T p − 1 T 1 T 2 Func: T 0 − → F 1 − → F 2 − → . . . − → F p ¯ ¯ ¯ T p − 2 T p − 3 T 0 T p − 1 , ¯ [ F ′ p ] T [ F ′ p − 1 ] T [ F ′ 1 ] T Grad: Y − → − → − → . . . − → Forward variables T j required in reverse order: store or recompute • Reverse mode useful to compute � ∂ I � ∂ I � ∂R � ⊤ � ⊤ � ∂R � ⊤ � ⊤ , , Ψ , Ψ ∂Q ∂X ∂Q ∂X Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 15 / 37
Shape parameterization • Parameterize the � n deformations � � � x s � � n x � x (0) s = + h ( ξ ) A B y (0) ξ y s n y 1 s 0.9 0.8 m � 0.7 h ( ξ ) = β k B k ( ξ ) 0.6 k =1 h( ξ ) 0.5 0.4 • Hicks-Henne bump functions 0.3 0.2 q k = log(0 . 5) B k ( ξ ) = sin p ( πξ q k ) , 0.1 log( ξ k ) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ Exact derivatives d X s d β can be • Move points along normal to computed reference line AB Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 16 / 37
Grid deformation Initial grid • Interpolate displacement of surface points to interior points using RBF ˜ f ( x, y ) = a 0 + a 1 x + a 2 y + N � r j | 2 log | � b j | � r − � r − � r j | j =1 Deformed grid where � r = ( x, y ) • Results in smooth grids d X • Exact derivatives d X s can be computed Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 17 / 37
Test cases: airfoil shape optimization RAE2822: M ∞ = 0 . 729, α = 2 . 31 o min I = C d s.t. C l = C l 0 C d 0 Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 18 / 37
RAE2822: Lift-constrained drag minimization 1.5 0.06 1 0.04 0.5 0.02 ipopt Initial -Cp 0 0 ipopt -0.02 Initial -0.5 -0.04 -1 -0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x/c Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 19 / 37
Sensitivity to perturbations 0.04 Initial 200 Initial Optimized Optimized 0.03 150 Drag coefficient Lift/Drag 0.02 100 0.01 50 0 0 0.7 0.71 0.72 0.73 0.74 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.75 0.76 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 TIFR, 18 Aug 2009 20 / 37
Wing optimization Initial shape Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 21 / 37
Wing optimization Optimized shape Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 22 / 37
ATR Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 23 / 37
Planform optimization for propeller aircraft (Rakshith, JNCASR) Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 24 / 37
Unmanned air vehicle Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 25 / 37
Adjoint-based aposteriori error estimation • u h = numerical solution on grid h • Numerical analysis � u − u h � = O ( h p ) , p > 0 as h → 0+ How to choose the grid size ? • Functional of interest J ( u ) J ( u ) − J ( u h ) = O ( h p ) Choose grid so that error in J is small Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 26 / 37
Cartesian Points NACA-0012 - coarse M ∞ = 1 . 2, α = 0 o 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Points Mach number (Mohan Varma) Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 27 / 37
Cartesian Points NACA-0012 - coarse M ∞ = 1 . 2, α = 0 o 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Points Mach number (Mohan Varma) Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 28 / 37
PDE with homogeneous BC • Linear PDE Lu = f in D • Adjoint operator ( v, Lu ) = ( L ∗ v, u ) • Functional of interest J = ( g, u ) • Adjoint PDE L ∗ v = g • Primal-dual equivalence J = ( g, u ) ( L ∗ v, u ) = = ( v, Lu ) = ( v, f ) Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 29 / 37
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