TA5 Test Case Praveen. C 1 R. Duvigneau 2 1 Tata Institute of - PowerPoint PPT Presentation

TA5 Test Case Praveen. C 1 R. Duvigneau 2 1 Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 2 Projet Opale, INRIA Sophia Antipolis Integrated Multiphysics Simulation & Design Optimization Database Workshop for multiphysics optimization software validation Presentation of the Academic Test Case Results Agora, Jyv¨ asky¨ a, Finland December 3-4, 2009 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 1 / 25

TA5 test case • Optimize RAE5243 airfoil to reduce drag under lift constraint Mach Re C l Flow condition 0.68 19 million 0.82 Fully turbulent • Modify shape of upper airfoil surface by adding a bump X cr ∆ Y h X br X bl Airfoil chord is taken to be unity P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 2 / 25

Bound constraints for bump parameters X cr ∆ Y h X br X bl 0 < X cr < 1 0 < X br < X bl 0 < X bl < 0 . 4 0 < ∆ Y h < 0 . 05 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 3 / 25

Modification of bound constraints • If X br and/or X bl − X br is too small, then CFD grid will not be able to resolve the bump. X bl − X br > L min X br > L min , • Also, we restrict X cr and ∆ Y h 0 . 4 < X cr < 0 . 8 L min < X br < X bl 2 L min < X bl < 0 . 4 0 < ∆ Y h < 0 . 01 In the computations, we use L min = 0 . 05 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 4 / 25

NUWTUN flow solver Based on the ISAAC code of Joseph Morrison http://isaac-cfd.sourceforge.net • Finite volume scheme • Structured, multi-block grids • Roe flux • MUSCL reconstruction • Implicit scheme, grid sequencing, multigrid • Wilcox k − ω turbulence model Source code of NUWTUN available online http://nuwtun.berlios.de P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 5 / 25

Grid for CFD C-grid of size 353 × 97, 270 points on airfoil, y + < 1 . 5, outer boundary at 20 chords 0.4 0.2 0 -0.2 -0.4 -0.5 0 0.5 1 1.5 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 6 / 25

Validation with experiments M = 0 . 68, Re = 19 million, α = 0 . 77 deg. NUWTUN 1 Experiment 0.5 0 -Cp -0.5 -1 -1.5 0 0.2 0.4 0.6 0.8 1 x/c P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 7 / 25

Global metamodel-based optimization • Global models: provide global trends in objective function ◮ Faster convergence towards global optimum • Metamodels are approximate, inaccurate • Not possible to construct accurate metamodel in one-shot • Difficult to construct uniformly accurate model in high dimensions ◮ Curse of dimensionality • Model must be accurate in regions of optima • But need to sufficiently explore the design space • Balance between exploration and exploitation P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 8 / 25

Gaussian process models • Treat results of a computer code as a stochastic process !!! • Provides an estimate of the variance in predicted value 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 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 9 / 25

Merit functions • Statistical lower bound f M ( x ) = ˜ J ( x ) − κ ˜ s ( x ) • Probability of improvement � � T − ˜ J ( x ) PoI( x ) = Φ ˜ s ( x ) • Expected improvement u ( x ) = J min − ˜ J ( x ) EI( x ) = ˜ s ( x )[ u Φ( u ) + φ ( u )] , ˜ s ( x ) P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 10 / 25

Minimization of 2-D Branin function: Initial database 15 10 5 0 −5 0 5 10 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 11 / 25

Minimization of 2-D Branin function: after 20 iter 15 10 5 0 −5 0 5 10 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 12 / 25

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 | Deformed grid j =1 where r = ( x, y ) � • Results in smooth grids P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 13 / 25

Reference solution and design variables Result α deg. C l C d C d p C d v Present 2.5 0.8244 0.01627 0.01052 0.005757 Qin et al. - 0.82 0.01622 0.01063 0.005586 α range for optimization: 2 < α < 3 0 . 4 < < 0 . 8 X cr L min < X br < X bl 2 L min < < 0 . 4 X bl 0 < ∆ Y h < 0 . 01 2 < < 3 α P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 14 / 25

Reference solution: Pressure α = 2 . 5 deg. P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 15 / 25

Transformed design variables • Transformations x 1 = X cr X cr = x 1 X br − L min x 2 = X bl − 2 L min X br = x 2 x 3 + L min x 3 = X bl − 2 L min = x 3 + 2 L min X bl = ∆ Y h x 4 ∆ Y h = x 4 x 5 = α − 2 . 5 = x 5 + 2 . 5 α • Bounds 0 . 4 < < 0 . 8 x 1 0 < x 2 < 1 . 0 0 < x 3 < 0 . 4 − 2 L min 0 < x 4 < 0 . 01 − 0 . 5 < x 5 < 0 . 5 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 16 / 25

Bump function y (0) , y (0) u : lower and upper l curves of RAE5243 airfoil y (0) y u ( x ) = u ( x ) + y b ( x ) X cr y (0) ∆ Y h y l ( x ) = ( x ) X br l X bl Bump function  0 x ≤ X cr − r 1 or x ≥ X cr + r 2   y b ( x ) = ∆ Y h · Cubic ( x ) X cr − r 1 < x ≤ X cr  ∆ Y h · Cubic ( x ) X cr < x < X cr + r 2  r 1 = X br , r 2 = X bl − X br P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 17 / 25

Optimization test Constrained problem min C d subject to C l = C l 0 We replace equality constraint with inequality constraint C l ≥ C l 0 Constraint is enforced using penalty approach Unconstrained problem � � min C d 0 , 1 − C l + 10 4 max C d 0 C l 0 At convergence, we recover C l ≈ C l 0 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 18 / 25

Optimization test • 5 design variables Annotation = Number of CFD evaluations 8 2 6 0 4 • Initial database of 48 4 5 5 6 6 using LHS 0.86 8 2 6 7 6 0 7 8 4 8 2 • 4 merit functions based 8 8 9 0.84 Objective function on statistical lower 6 9 0 4 8 2 6 0 bound with 0 0 0 1 1 2 1 1 1 1 1 1 0.82 κ = 0 , 1 , 2 , 3 • Gaussian process 0.8 4 8 2 2 2 3 models 1 1 1 6 3 1 0 4 8 2 6 0 4 8 4 4 4 5 5 6 6 6 1 1 1 1 1 1 1 1 0.78 • Merit functions 0 5 10 15 20 25 30 Number of iterations minimized using PSO P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 19 / 25

Shape parameters ∆ Y h × 10 − 3 Case X cr X bl X br Present 0.688 0.399 0.257 8.578 Qin et al. 0.597 0.313 0.206 5.900 0.05 RAE5243 Optimized 0 -0.05 0 0.2 0.4 0.6 0.8 1 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 20 / 25

Force and Pressure coefficient Case C d ∆ C d C d p C d v C l AOA Present 0.01266 -22.2% 0.00680 0.00586 0.8204 2.19 Qin et al. 0.01326 -18.2% 0.00756 0.00570 0.82 - 1.5 RAE5243 Optimized 1 0.5 -Cp 0 -0.5 -1 0 0.2 0.4 0.6 0.8 x/c P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 21 / 25

Pressure contours P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 22 / 25

Lift curve and drag polar 0.02 0.85 0.018 0.8 RAE5243 Lift coefficient Drag coefficient RAE5243 0.016 Optimized Optimized 0.75 0.014 0.7 0.012 0.65 0.6 0.01 1 1.5 2 2.5 3 0.6 0.65 0.7 0.75 0.8 0.85 Angle of attack Lift coefficient P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 23 / 25

Close-up view of grids Initial 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0 0.2 0.4 0.6 0.8 1 Optimized 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0 0.2 0.4 0.6 0.8 1 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 24 / 25

Close-up view of grids Initial 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.4 0.5 0.6 0.7 0.8 0.9 Optimized 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.4 0.5 0.6 0.7 0.8 0.9 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 25 / 25