Numerical shape optimization for compressible flows (Minimization of expensive cost functions) Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in Mechanical Engineering Seminar Dept. of Mechanical Engg., IISc, Bangalore 11 March, 2011 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 1 / 72
Collaborators 1 Regis Duvigneau Project OPALE, INRIA, Sophia Antipolis 2 Biju Uthup ADA, Bangalore Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 2 / 72
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 March 2011 3 / 72
Example of optimization: RAE2822 Initial shape Solver: euler2d Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 4 / 72
Example of optimization: RAE2822 Optimized shape Optimizer: Torczon Simplex, 20 Hicks-Henne parameters Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 5 / 72
Example of optimization: RAE2822 Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 6 / 72
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 March 2011 7 / 72
Elements of shape optimization Shape parameters Surface grid Volume grid CFD solution I Optimizer Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 8 / 72
Shape parameterization Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 9 / 72
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 March 2011 10 / 72
Free Form Deformation • Originated in computer graphics field (Sederberg and Parry) • 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 March 2011 11 / 72
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 March 2011 12 / 72
Free Form Deformation: Example (R. Duvigneau, INRIA) Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 13 / 72
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 March 2011 14 / 72
Optimizer Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 15 / 72
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 March 2011 16 / 72
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 March 2011 16 / 72
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 March 2011 17 / 72
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 March 2011 18 / 72
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 March 2011 19 / 72
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 March 2011 20 / 72
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 March 2011 21 / 72
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 March 2011 21 / 72
Wing optimization Initial shape Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 22 / 72
Wing optimization Optimized shape Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 23 / 72
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 March 2011 24 / 72
Surrogate Models Praveen. C (TIFR-CAM) Shape optimization IISc, 11 March 2011 25 / 72
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 Shape parameters Surrogate model J, ˜ ˜ 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 March 2011 26 / 72
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 March 2011 27 / 72
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 March 2011 28 / 72
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 March 2011 29 / 72
Recommend
More recommend
