MIest Taylor Series Expansion 16888 E77 →(x),weg"R Taylor Series Expansion of objective Function (x)=J(xo)+VJ(x )(x-x)+3x-xyH(x)x-x)+HOT first order term second order term Tangential Effect of curvature hyperplane (2nd derivative) at xo at xo Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
11 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Taylor Series Expansion Taylor Series Expansion , kk n x x 6 \ 6\ J where Taylor Series Expansion of Objective Function Tangential hyperplane at xo Effect of curvature (2nd derivative) at xo 0 00 0 0 1 ( ) ( ) ( ) ( ) ( ) ( )( ) H.O.T. 2 T T JJ J ª º ¬ ¼ 0 x x x x x x x Hx x x first order term second order term
MlesdJacobian Matrix-multiple objectives E5. If there is more than one objective function ie if we have a gradient vector for each J;, arrange them columnwise and get Jacobian matrix aa, a 2 VJ=Ox, a o aJ Zx 1 Ox a n nxT Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
12 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Jacobian Matrix Jacobian Matrix – multiple objectives multiple objectives If there is more than one objective function, i.e. if we have a gradient vector for each Ji, arrange them columnwise and get Jacobian matrix: 1 2 11 1 1 2 22 2 1 2 z z z nn n J J J xx x J J J xx x J J J xx x ª º ww w « » ww w « » « » ww w « » ww w « » « » « » ww w « » « » ww w ¬ ¼ J "" # #%# " 12z JJJª º « » « » « » « » ¬ ¼ J # n x z z x 1
Mest Normalization 16888 ESD.J7 In order to compare sensitivities from different design variables in terms of their relative sensitivity it is necessary to normalize raw-unnormalized sensitivity= partial derivative evaluated at point xi. 0 Normalized sensitivity captures relative sensitivity Ax/xJ(x”)ax| change in objective per change in design variable L Important for comparing effect between design variables Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Normalization Normalization In order to compare sensitivities from different design variables in terms of their relative sensitivity it is necessary to normalize: i J x w w o x “raw” - unnormalized sensitivity = partial derivative evaluated at point xi,o , ( ) i o ii i JJ J x xx J x ' w ' w o o x x Normalized sensitivity captures relative sensitivity ~ % change in objective per % change in design variable Important for comparing effect between design variables
M| esd Example: Dairy Farm Problem影男 Dairy Farm sample problem L-Length =100 [m] N-it of cows 10 R-Radius= 50 With respect to which COW design variable is the COW objective most sensitive? fence Parameters A=2LR+R2C=f·F+n,N f=100$m n=2000/c0W F=2L+2R m=2S/liter M=100√ANP=I-C Assume that we are not at the optimal point x*! Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
14 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Example: Dairy Farm Problem Example: Dairy Farm Problem With respect to which design variable is the objective most sensitive? “Dairy Farm” sample problem L R N L – Length = 100 [m] N - # of cows = 10 R – Radius = 50 [m] fence 2 2 2 2 100 / A LR R FL R M A N S S C f F nN I NMm P IC Parameters: f=100$/m n=2000$/cow m=2$/liter xo Assume that we are not at the optimal point x* ! COW COW COW
M Dairy Farm Sensitivity 16888 E77 aP Compute objective at Xo J(x)=13092 OL[ 36.6 Then compute raw sensitivities vi op 22254 5884 aP · Normalize 100.366 13092 0.28 LOR 10 VJ 22254 13092 2.25 5884 13092 Dairy Farm Normalized sensitivities R Show graphically (optional)s 00.511.522.5 Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
15 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Dairy Farm Sensitivity Dairy Farm Sensitivity • Compute objective at x o • Then compute raw sensitivities • Normalize • Show graphically (optional) 36.6 2225.4 588.4 P L P J N P R ª º w« » w« » ª º « » w « » « » « » w« » « » ¬ ¼ w« » « » ¬ ¼ w 100 36.6 13092 0.28 10 2225.4 1.7 ( ) 13092 2.25 50 588.4 13092 o o J J J ª º « » « » ª º « » « » « » « » « » « » ¬ ¼ « » « » ¬ ¼ x x ( ) 13092 o J x Dairy Farm Normalized Sensitivities 0 0.5 1 1.5 2 2.5 L N R Design Variable