Mest 16888 ESD.J7 Multidisciplinary System Design Optimization(MSDO) Sensitivity Analysis Lecture 8 1 March 2004 Olivier de weck Karen willcox Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) Design Optimization (MSDO) Sensitivity Analysis Lecture 8 1 March 2004 Olivier de Weck Karen Willcox
Mest Today's Topics 16888 ESD.J7 Sensitivity Analysis effect of changing design variables effect of changing parameters effect of changing constraints Gradient calculation methods Analytical and Symbolic Finite difference Adjoint methods Automatic differentiation Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Today’s Topics Today’s Topics • Sensitivity Analysis – effect of changing design variables – effect of changing parameters – effect of changing constraints • Gradient calculation methods – Analytical and Symbolic – Finite difference – Adjoint methods – Automatic differentiation
Mesd Standard Problem Definition 16888 ESD.J7 min J(x) st、9(X)≤0j=1 h2(x)=0k=1m2 X'<x<X. i=1.n For now, we consider a single objective function, J(x) There are n design variables, and a total of m constraints(m=m, +m2) The bounds are known as side constraints Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Standard Problem Definition Standard Problem Definition 1 2 min ( ) s.t. ( ) 0 1,.., ( ) 0 1,.., 1,.., j k u i ii J g jm h km x x xi n d dd x x x A For now, we consider a single objective function, J(x). There are n design variables, and a total of m constraints (m=m1+m2). The bounds are known as side constraints
M Sensitivity Analysis 16888 ESD.J7 Sensitivity analysis is a key capability aside from the optimization algorithms we discussed Sensitivity analysis is key to understanding which design variables, constraints, and parameters are important drivers for the optimum solution X The process is not finished once a solution X* has been found. a sensitivity analysis is part of post- processIng Sensitivity /Gradient information is also needed by gradient search algorithms isoperformance/goal programming robust design o Massachusets Institute of Technology - Prof. de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Sensitivity Analysis Sensitivity Analysis • Sensitivity analysis is a key capability aside from the optimization algorithms we discussed. • Sensitivity analysis is key to understanding which design variables, constraints, and parameters are important drivers for the optimum solution x*. • The process is NOT finished once a solution x* has been found. A sensitivity analysis is part of postprocessing. • Sensitivity/Gradient information is also needed by: – gradient search algorithms – isoperformance/goal programming – robust design
Mest Sensitivity Analysis 16888 ESD.J7 How sensitive is the "optimal solution to changes or perturbations of the design variables x*? How sensitive is the“° optimal solution x*to changes in the constraints g(x), h(x)and fixed parameters p? Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and astronautics
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Sensitivity Analysis Sensitivity Analysis • How sensitive is the “optimal” solution J* to changes or perturbations of the design variables x*? • How sensitive is the “optimal” solution x* to changes in the constraints g(x), h(x) and fixed parameters p ?
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
