Numerical shape optimization using CFD Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in 12’th AeSI CFD Symposium IISc, Bangalore 11–12 August, 2010 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 1 / 49
Outline 1 Approaches to optimization 2 Elements of shape optimization ◮ Shape parameterization ◮ Grid generation/deformation ◮ CFD solution/Adjoint solution ◮ Optimizer 3 Free-form deformation (FFD) 4 Particle swarm method 5 Surrogate models 6 Examples Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 2 / 49
Approaches to optimization Gradient-based Gradient-free ... Finite Adjoint GA PSO Difference • Global optimum possible • Local optimum • “Easy” to implement for engg. • FD accuracy problem problems • Adjoint solver required • Slow convergence: surrogate • Issues with adjoint consistency models and parallelization Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 3 / 49
Elements of shape optimization Shape parameters Surface grid Volume grid CFD solution I Optimizer Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 4 / 49
Shape parameterization Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 5 / 49
Shape parameterization approaches • Aerodynamic DVs: ◮ LE radius, max camber, taper ratio • PARSEC, Kulfan parameterization, etc. • BSplines/NURBS • Need to re-generate surface/volume grid whenever shape is changed • Or, use a free-form approach like RBF-based grid deformation Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 6 / 49
Free Form Deformation • Originated in computer graphics field • Embed the object inside a box and deform the box • Independent of the representation of the object • Deform CFD grid also, independent of grid type Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 7 / 49
Free Form Deformation Consistent parameterization Airplane shape DVs Compact set of DVs Smooth geometry Local control Analytical sensitivity Grid deformation Setup time Existing grids CAD connection (Samareh) Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 8 / 49
Free Form Deformation: Example (R. Duvigneau, INRIA) Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 9 / 49
Free Form Deformation • X 0 ( P ) = coordinate of point P wrt reference shape • Movement of point P under the deformation n j n i n k � � � i ( ξ p ) B n j X ( P ) = X 0 ( P ) + Y ijk B n i j ( η p ) B n k k ( ζ p ) i =0 j =0 k =0 • Bernstein polynomials B n m ( t ) = C n m t m (1 − t ) n − m , t ∈ [0 , 1] , m = 0 , 1 , . . . , n • Design variables { Y ijk } , 0 ≤ i ≤ n i , 0 ≤ j ≤ n j , 0 ≤ k ≤ n k • Cannot change wing planform • Wing twist can be added as additional variables Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 10 / 49
Optimizer Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 11 / 49
Particle swarm optimization • Kennedy and Eberhart (1995) • Modeled on behaviour of animal swarms: ants, bees, birds • Cooperative behaviour of large number of individuals through simple rules • Emergence of swarm intelligence Optimization problem D ⊂ R d min x ∈ D J ( x ) , Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 12 / 49
Particle swarm optimization • Kennedy and Eberhart (1995) • Modeled on behaviour of animal swarms: ants, bees, birds • Cooperative behaviour of large number of individuals through simple rules • Emergence of swarm intelligence Optimization problem D ⊂ R d min x ∈ D J ( x ) , Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 12 / 49
Particle swarm optimization Particles distributed in design space x i ∈ D, i = 1 , ..., N p X2 X1 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 13 / 49
Particle swarm optimization Each particle has a velocity v i ∈ R d , i = 1 , ..., N p X2 X1 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 14 / 49
Particle swarm optimization • Particles have memory ( t = iteration number) Local memory : p t J ( x s i = argmin i ) 0 ≤ s ≤ t Global memory : p t = argmin J ( p t i ) i • Velocity update 2 ⊗ ( p t − x t v t +1 = ωv t i + c 1 r t 1 ⊗ ( p t i − x t + c 2 r t i ) i ) i � �� � � �� � Local Global • Position update x t +1 i + v t +1 = x t i i Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 15 / 49
PSO: embarassingly parallel x n x n x n . . . 1 2 N p ↓ ↓ ↓ J ( x n J ( x n J ( x n 1 ) 2 ) N p ) . . . ↓ ↓ ↓ v n +1 v n +1 v n +1 . . . 1 2 N p ↓ ↓ ↓ x n +1 x n +1 x n +1 . . . 1 2 N p Parallel evaluation of cost functions using MPI Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 16 / 49
Test case: Wing shape optimization M ∞ = 0 . 83, α = 2 o • Minimize drag under lift constraint min C d C l s.t. ≥ 0 . 999 C d 0 C l 0 • FFD parameterization, n = 20 design variables • Particle swarm optimization: (Piaggio Aero. Ind.) 120 particles Grid: 31124 nodes Cost function � � J = C d 0 , 0 . 999 − C l + 10 4 max C d 0 C l 0 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 17 / 49
Test case: Wing shape optimization M ∞ = 0 . 83, α = 2 o • Minimize drag under lift constraint min C d C l s.t. ≥ 0 . 999 C d 0 C l 0 • FFD parameterization, n = 20 design variables • Particle swarm optimization: (Piaggio Aero. Ind.) 120 particles Grid: 31124 nodes Cost function � � J = C d 0 , 0 . 999 − C l + 10 4 max C d 0 C l 0 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 17 / 49
Wing optimization Initial shape Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 18 / 49
Wing optimization Optimized shape Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 19 / 49
PSO computational cost • Slow convergence: O(100)-O(1000) iterations • Require large swarm size: O(100) particles • CFD is expensive: few minutes to hours • Example: Transonic wing optimization (coarse CFD grid) (10 min/CFD) (120 CFD/pso iter) (200 pso iter) = 4000 hours Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 20 / 49
Surrogate Models Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 21 / 49
Metamodels • Expensive PDE-based model Shape parameters Surface grid Volume grid J, C CFD solution • Replace costly model with cheap model: metamodel or surrogate model J, ˜ ˜ Shape parameters Surrogate model C • Approximation of cost function and constraint function(s) ◮ Response surfaces (polynomial model) ◮ Neural networks ◮ Radial basis functions ◮ Kriging/Gaussian Random Process models Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 22 / 49
Kriging I Unknown function f : R d → R Given the data as F N = { f 1 , f 2 , . . . , f N } ⊂ R sampled at X N = { x 1 , x 2 , . . . , x N } ⊂ R d , infer the function value at a new point x N +1 . Treat result of a computer simulation as a fictional gaussian process F N is assumed to be one sample of a multivariate Gaussian process with joint probability density � � − 1 N C − 1 2 F ⊤ p ( F N ) = exp N F N (1) � (2 π ) N det( C N ) where C N is the N × N covariance matrix. Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 23 / 49
Kriging II When adding a new point x N +1 , the resulting vector of function values F N +1 is assumed to be a realization of the ( N + 1)-variable Gaussian process with joint probability density � � − 1 N +1 C − 1 2 F ⊤ p ( F N +1 ) = exp N +1 F N +1 (2) � (2 π ) N +1 det( C N +1 ) Using Baye’s rule we can write the probability density for the unknown function value f N +1 , given the data ( X N , F N ) as � � − ( f N +1 − ˆ f N +1 ) 2 p ( f N +1 | F N ) = p ( F N +1 ) = 1 Z exp 2 σ 2 p ( F N ) f N +1 where ˆ f N +1 = k ⊤ C − 1 σ 2 f N +1 = κ − k ⊤ C − 1 N F N , N k (3) � �� � � �� � Inference Error indicator Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 24 / 49
Kriging III Covariance matrix: Given in terms of a correlation function, C N = [ C mn ], C mn = corr( f m , f n ) = c ( x m , x n ) � � d ( x i − y i ) 2 − 1 � c ( x, y ) = θ 1 exp + θ 2 r i 2 2 i =1 Parameters Θ = ( θ 1 , θ 2 , r 1 , r 2 , . . . , r d ) determined to maximize the likelihood of known data max log( p ( F N )) Θ Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 25 / 49
Kriging: Illustration 12 10 DACE predictor 8 6 12 4 standard error 10 2 of the predictor 8 0 0 2 4 6 8 10 12 6 4 2 0 0 2 4 6 8 10 12 1 1.5 1.5 2 2.5 2.5 3 3.5 3.5 4 4.5 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 Aug 2010 26 / 49
Recommend
More recommend
