Mlesd Sensitivity Analysis: Aircraft 16888 E77 Questions for aircraft design How does my solution change if I change the cruise altitude? change the cruise speed? change the range? change material properties? relax the constraint on payload? Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Sensitivity Analysis: Aircraft Sensitivity Analysis: Aircraft Questions for aircraft design: How does my solution change if I • change the cruise altitude? • change the cruise speed? • change the range? • change material properties? • relax the constraint on payload? •
Mest Sensitivity Analysis 16888 E77 Questions for spacecraft design How does my solution change if change the orbital altitude? change the transmission frequency ? change the specific impulse of the propellant? change launch vehicle? Change desired mission lifetime? Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Sensitivity Analysis Sensitivity Analysis Questions for spacecraft design: How does my solution change if I • change the orbital altitude? • change the transmission frequency? • change the specific impulse of the propellant? • change launch vehicle? • Change desired mission lifetime? •
M esd Gradient Vector- single objective E50.77 How does the objective function J value change as we change elements of the design vector x? a/ Compute partial derivatives oJ VJ=a of with respect to X VJ aJ Ox Gradient vector points normal to the tangent hyperplane of (x) 8 Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Gradient Vector Gradient Vector – single objective single objective “How does the objective function J value change as we change elements of the design vector x?” 1 2 n J x J x J x ª º w « » w« » « » w « » w« » « » « » w « » « » w¬ ¼ J # Compute partial derivatives of J with respect to xi i Jxww J Gradient vector points normal to the tangent hyperplane of J(x) 1 x 2 x 3 x
M esd Geometry of Gradient vector(2D)E50. EXample function (x1,x2)=x1 +x t Contour plo 1.8 1.6 14 1.2 0.8 x Xx. Gradient normal to contours Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Geometry of Gradient vector (2D) Geometry of Gradient vector (2D) 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x1 x 2 Contour plot 3.1 3 .1 3.1 3.25 3.25 3 .25 3.25 3.25 3.5 3.5 3.5 3.5 4 4 4 4 5 5 Example function: 12 1 2 1 2 1 Jxx x x , x x 2 1 12 2 2 12 1 1 1 1 J x xx J J x xx ª ºª º w « »« » w « »« » « »« » w « »« » w¬ ¼¬ ¼ Gradient normal to contours
M esd Geometry of Gradient vector( 3D)E50. Example J x Increasing values of j VJ=22 2x2 Tangent plane 2x+2x2+2x3-6=0 J=3 xI Gradient vector points to larger values of Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Geometry of Gradient vector (3D) Geometry of Gradient vector (3D) 222 1 23 Jx x x 1 2 3 2 2 2 x J x x ª º « » « » « » ¬ ¼ increasing values of J 1 x 2 x 3 x Tangent plane 123 2 2 2 60 xxx >111@ o T x o >222@T J x J=3 Example Gradient vector points to larger values of J