Theoretical Background for Aerodynamic Shape Optimization John C. Vassberg Antony Jameson Boeing Technical Fellow T. V. Jones Professor of Engineering Advanced Concepts Design Center Dept. Aeronautics & Astronautics Boeing Commercial Airplanes Stanford University Long Beach, CA 90846, USA Stanford, CA 94305-3030, USA Von Karman Institute Brussels, Belgium 7 April, 2014 Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 1
LECTURE OUTLINE • INTRODUCTION • THEORETICAL BACKGROUND – SPIDER & FLY – BRACHISTOCHRONE • SAMPLE APPLICATIONS – MARS AIRCRAFT – RENO RACER – GENERIC 747 WING/BODY • DESIGN-SPACE INFLUENCE Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 2
THE SPIDER & THE FLY • PROBLEM STATEMENT • PROBLEM SET-UP – COST FUNCTION – DESIGN SPACE – GRADIENT & HESSIAN • SEARCH METHODS – STEEPEST DESCENT – NEWTON ITERATION – NASH EQUILIBRIUM • EXACT SOLUTION Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 3
THE SPIDER & THE FLY Block Size 4" x 4" x 12" Path Length 16.00" SPIDER FLY PATH Obvious Local-Minimum Path between Spider and Fly. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 4
THE SPIDER & THE FLY Block Size 4" x 4" x 12" Path Length Sqrt(250.0)" ~ 15.81" SPIDER FLY PATH Non-Obvious Global-Minimum Path between Spider and Fly. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 5
SPIDER-FLY DESIGN SPACE Path type to optimize is partitioned into four segments. Path described as the piecewise linear curve that connects: (2 , 0 , 3) , ( X, 0 , 4) , (4 , Y, 4) , (4 , 12 , Z ) , (2 , 12 , 1) . Three design variables ( X, Y, Z ), constrained by: 0 4 , ≤ X ≤ 0 12 , ≤ Y ≤ 0 4 . ≤ Z ≤ Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 6
SPIDER-FLY COST FUNCTION Segment Lengths: 1 + ( X − 2) 2 � 1 2 , � = S 1 ( X − 4) 2 + Y 2 � 1 2 , � = S 2 ( Y − 12) 2 + ( Z − 4) 2 � 1 2 , � = S 3 � 1 ( Z − 1) 2 + 4 2 . � = S 4 Total Path Length: I ≡ S = S 1 + S 2 + S 3 + S 4 . Minimize I Subject to Constraints. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 7
SPIDER-FLY GRADIENT First Variation of Cost Function: δI = I X δX + I Y δY + I Z δZ ≡ G δ X ( X − 2) + ( X − 4) = I X S 1 S 2 G ≡ Gradient V ector S 2 + ( Y − 12) Y = I Y S 3 X ≡ Design Space V ector ( Z − 4) + ( Z − 1) = I Z S 3 S 4 Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 8
SPIDER-FLY HESSIAN MATRIX + Y 2 1 = I XX S 3 S 3 1 2 I Y X = (4 − X ) Y = I XY I XX I Y X I ZX S 3 2 = I ZX = 0 I XZ ( X − 4) 2 + ( Z − 4) 2 A = I XY I Y Y I ZY , = I Y Y S 3 S 3 2 3 = ( Y − 12)(4 − Z ) I XZ I Y Z I ZZ = I Y Z I ZY S 3 3 ( Y − 12) 2 + 4 I ZZ = S 3 S 3 3 4 Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 9
FINITE-DIFFERENCE APPROXIMATION Consider the Taylor series expansion of a function f . f ( x + ∆ x ) = f ( x )+ ∆ x f x ( x )+ ∆ x 2 f xx ( x )+ . . . + ∆ x n n ! f n ( x )+ . . . 2 A first-order accurate approximation of f x ( x ) can be determined with the forward differencing formula f x ( x ) ≃ f ( x + ∆ x ) − f ( x ) . ∆ x Here ∆ x is a small perturbation of the X coordinate. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 10
FINITE-DIFFERENCE APPROXIMATION In the case of the spider-fly, let’s approximate I X . I X ≃ I ( X + h, Y, Z ) − I ( X, Y, Z ) h For example, using h = 10 − 3 at ( X, Y, Z ) = (2 , 6 , 2) gives: I X ≃ − 0 . 31565661, an error of about 0.1%. 2 The exact value of I X at this location is − 40 ≃ − 0 . 31622777. √ Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 11
COMPLEX-VARIABLE APPROXIMATION Consider the Taylor series expansion of a complex function f . f ( x + ∆ x ) = f ( x )+ ∆ x f x ( x )+ ∆ x 2 f xx ( x )+ . . . + ∆ x n n ! f n ( x )+ . . . 2 A second-order accurate approximation of f x ( x ) can be found with the complex-variable formula f x ( x ) ≃ Im [ f ( x + ih )] . h Here ∆ x = ih is an imaginary perturbation of X . Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 12
COMPLEX-VARIABLE APPROXIMATION In the case of the spider-fly, let’s approximate I X . I X ≃ Im [ I ( X + ih, Y, Z )] h For all h ≤ 10 − 3 at ( X, Y, Z ) = (2 , 6 , 2), we get: I X ≃ − 0 . 31622777. This is identical to the exact value to 8 significant digits. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 13
GRADIENT APPROXIMATION log 10 ( Error I X ) log 10 ( h ) Finite Difference Complex Variable -1 -1.244 -4.449 -2 -2.243 -6.449 -3 -3.243 -8.449 -4 -4.243 -10.449 -5 -5.243 -12.449 -6 -6.244 -14.449 -7 -7.192 -16.256 -8 -6.778 -16.256 -9 -5.977 -16.256 -10 -4.768 -16.256 Stability of Finite-Difference and Complex-Variable Methods Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 14
GRADIENT APPROXIMATION Finite Difference vs Complex Variables 0 -2 -4 -6 log10 ( Error[Ix] ) -8 -10 -12 -14 Finite Difference -16 Complex Variables -18 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 log10 ( h ) Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 15
SPIDER-FLY SEARCH METHODS Trajectory: X n +1 = X n + δ X n Steepest Descent: δ X n = − λG, λ > 0 δI n = G δ X n = − λG 2 ≤ 0 Newton Iteration: δ X n = − A − 1 G = − HG Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 16
SPIDER-FLY SEARCH METHODS Rank-1 quasi-Newton: H n +1 = H n + ( P n )( P n ) T ( P n ) T δG n , where δG n = G n +1 − G n and P n = δ X n − H n δG n . Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 17
SPIDER-FLY SEARCH METHODS Nash Equilibrium: minimize I ( X ⋆ , Y n , Z n ) I x ( X ⋆ , Y n , Z n ) = 0 X ⋆ , = > = > minimize I ( X n , Y ⋆ , Z n ) I x ( X n , Y ⋆ , Z n ) = 0 Y ⋆ , = > = > minimize I ( X n , Y n , Z ⋆ ) I x ( X n , Y n , Z ⋆ ) = 0 Z ⋆ . = > = > These reduce to: X ⋆ = 2(2 + Y n ) 12(4 − X n ) Z ⋆ = 4 − 3(12 − Y n ) Y ⋆ = (1 + Y n ) , (8 − X n − Z n ) , (14 − Y n ) . Update design vector: [ X n +1 , Y n +1 , Z n +1 ] T = [ X ⋆ , Y ⋆ , Z ⋆ ] T Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 18
SPIDER-FLY INITIAL PATH − 2 √ 2 − 0 . 31623 40 X 0 = G 0 = 6 0 . 0 ≈ 0 . 0 , , ( − 3 1 2 − 0 . 02713 √ 40 + √ 5 ) 1 . 14230 0 . 04743 0 . 0 A 0 ≈ 0 . 04743 0 . 03162 − 0 . 04743 , 0 . 0 − 0 . 04743 0 . 50007 √ √ I 0 = (1 + 2 40 + 5) ≈ 15 . 88518 Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 19
SPIDER-FLY INITIAL PATH Initial Path between Spider and Fly. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 20
SPIDER-FLY STEEPEST DESCENT 0 Step: 1.885 -1 -2 -3 LOG_10 ( GRMS ) -4 -5 -6 -7 -8 -9 0 0 50 100 150 200 250 300 Iteration Convergence of Gradient for Steepest Descent. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 21
SPIDER-FLY STEEPEST DESCENT 6.1 6.1 Top View Side View Baseline Baseline 6.0 6.0 5.9 5.9 5.8 5.8 5.7 5.7 5.6 5.6 5.5 Y 5.5 Y 5.4 5.4 5.3 5.3 5.2 5.2 5.1 5.1 5.0 5.0 Optimum Optimum 4.9 4.9 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.1 2.0 1.9 1.8 1.7 1.6 X Z Steepest-Descent Trajectory through Design Space. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 22
SPIDER-FLY NEWTON ITERATION 0 -1 -2 -3 LOG_10 ( GRMS ) -4 -5 -6 -7 -8 -9 0 0 1 2 3 4 Iteration Convergence of Gradient for Newton Iteration. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 23
SPIDER-FLY NEWTON ITERATION 6.1 6.1 Top View Side View Baseline Baseline 6.0 6.0 5.9 5.9 5.8 5.8 5.7 5.7 5.6 5.6 5.5 Y 5.5 Y 5.4 5.4 5.3 5.3 5.2 5.2 5.1 5.1 Optimum 5.0 5.0 Optimum 4.9 4.9 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.1 2.0 1.9 1.8 1.7 1.6 X Z Newton-Iteration Trajectory through Design Space. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 24
SPIDER-FLY NEWTON ITERATION X n Y n Z n I n n 0 2.000000 6.000000 2.000000 15.88518 1 2.319023 4.984009 1.641696 15.81167 2 2.333268 4.999744 1.666556 15.81139 3 2.333333 5.000000 1.666667 15.81139 Convergence of Newton Iteration on the Spider-Fly Problem. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 25
SPIDER-FLY RANK-1 QUASI-NEWTON 0 -1 -2 -3 LOG_10 ( GRMS ) -4 -5 -6 -7 -8 -9 0 0 1 2 3 4 5 6 7 8 9 10 Iteration Convergence of Gradient for Rank-1 quasi-Newton Iteration. Vassberg & Jameson, VKI Lecture-I, Brussels, 7 April, 2014 26
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