System Optimization II Multidisciplinary Design Optimization K Sudhakar

System Optimization - II Multi-disciplinary Design Optimization K Sudhakar PM Mujumdar Centre for Aerospace Systems Design & Engineering Indian Institute of Technology, Mumbai June-July, 2003 IAT, Pune 1

Objective Function(s) If maximisation is required ie, Maximise f(x) then restate it as Minimise F(x) = -f (x) June-July, 2003 IAT, Pune 2

Inequality Constraints • Each inequality constraint reduces design space X 2 • No restriction on number of inequality constraints • If an inequality is required to be gi(x) 0 then restate it as g’i(x) = -gi(x) 0 g 1 g 2 June-July, 2003 X 1 IAT, Pune 3

Equality Constraints • Each equality constraint reduces dimensionality of design space by one, eg h 1(x 1, x 2, . . . , xn) = 0; xn = y 1(x 1, x 2, . . xn-1) f {x 1, x 2, . . xn} = f{x 1, x 2, . . xn-1, y 1(x 1, x 2, . . xn-1)} = F {x 1, x 2, . . xn-1} h(x)=0 s • L<n X 2 X 1 June-July, 2003 IAT, Pune 4

Optimization Problem Statement n 1 • Side constraints • Boxing design space • Bounding box n - Independant variables L - Equality constraints; m - Inequality constraints June-July, 2003 f IAT, Pune 5

Engineering Design • Parameterize - identify variables, which if given values, design is realizable. {b, C_r, S, , , a/f, VH, VV. . } • Identify analysis to determine feasibility compliance with customer requirements. ie. Evaluate all constraints • Identify goodness criteria • Optimize June-July, 2003 IAT, Pune 6

Constraints • Customer needs are constraints g = {Range, Vmax, ROC, . , OEI, . , noise, . . }; g m These are usually inequality constraints, gi 0 • Laws of nature that cannot be violated are constraints h = {Newton’s laws, conservation laws, . }; h L These are usually equality constraints, h = 0. June-July, 2003 IAT, Pune 7

Global & Local Minimum f • Global Minimum : A function f(x) is said to have a global minimum at x*, if X* f x f(x*) f(x) for all x S • Local Minimum : A function f(x) is said to have a local minimum at x*, if f(x*) f(x) for all x N S where N= { x S ¦ x-x* ; > 0 } f x 1 June-July, 2003 IAT, Pune 8

1 -D Unconstrained f Minimise f(X) = (x 1 - 2)2 f/ x 1 = 2 (x 1 - 2) ; 2 f/ x 12 = 2 x 1* = 2, f* = 0 ; x 1 Point of minima Some points for consideration! • f - Simple function, easy to analytically differentiate. • What if f is complex? Numerically differentiate? X 1 June-July, 2003 IAT, Pune Computer Programme f 9

n-D Unconstrained • Solve the n equations for x 1, x 2 , x 3. . . xn • Check for Positive definiteness of H. All eigenvalues of H should be 0 June-July, 2003 IAT, Pune 10

Two Variable Problem Minimise f(X) = (x 1 - 2)2 + (x 2 - 2)2 Tf = [ f/ x 1 ; f/ x 2 ] = [ 2(x 1 - 2) ; 2(x 2 - 2) ] 2 f/ x 12 2 f/ x 1 x 2 H(f) = 2 0 0 2 = 2 f/ x 2 x 1 X* = ( 2, 2); 2 f/ x 22 f* = 0; Point of minima, since H +ve Definite ( 1 = 2) June-July, 2003 IAT, Pune 11

Equality Constrained Minima Minimise Subject to f (x 1, x 2, x 3) h (x 1, x 2 , x 3) = 0 Solve for x 1 = (x 2 , x 3) Minimise f ( (x 2 , x 3), x 2 , x 3) = F (x 2 , x 3) What if h (x 1, x 2 , x 3) cannot be explicitly inverted? X 1 , X 2 , X 3 June-July, 2003 IAT, Pune Computer Programme h 12

Equality Constrained Minima Minimise f(x 1, x 2); Subject to h(x 1, x 2) = 0 Is equivalent to the un-constraint problem Minimise L(x 1, x 2, ) = f(x 1, x 2) + h(x 1, x 2) L/ x 1 = f/ x 1 + h/ x 1 = 0 L/ x 2 = f/ x 2 + h/ x 2 = 0 L/ = h = 0 - Lagrange multiplier June-July, 2003 IAT, Pune 13

Constrained Optimisation Inequality Constraints g 1=0 x 2 g 1= -s 2 X* x 1 - Lagrange multiplier; June-July, 2003 IAT, Pune s - slack variable 14

Constrained Optimisation June-July, 2003 IAT, Pune 15

Issues in Posing the Problem • Of all variables that influence the design which to pick as design variables? XD X • Of all functions that determine system behaviour which one to choose as objectives? f F • How to confirm that all constraints are specified ? • How to evaluate f, g, h ? June-July, 2003 IAT, Pune 16

Issues in Optimisation • Which optimisation algorithm to use? – Gradient based? How to generate gradients? – Evolutionary? Too many function evaluations? • Evaluation of gradients? Requirements on convergence will be more severe than that required for engineering analysis. • Noisy functions f X June-July, 2003 IAT, Pune 17

Issues in Optimisation • Special issues in large scale problems? Experience of others? • Issues when strong inter-disciplinary interactions exist - especially when disciplinary analysis is complex. • Intensive, complex legacy codes for analysis • Are there environments that make problem posing and problem solving easy. June-July, 2003 IAT, Pune 18

• To know more about CASDE http: //www. casde. iitb. ac. in • To know more about MDO work at CASDE http: //www. casde. iitb. ac. in/MDO/ • To contact me sudhakar@aero. iitb. ac. in June-July, 2003 IAT, Pune 19

Thank you June-July, 2003 IAT, Pune 20

Customer Requirements • • Mach no. M 0. 8 ( -M -0. 8) ROC 100 m/s (-ROC -100 m/s) Take off distance, TOD 800 m Noise level 90 EPNDB • . . June-July, 2003 IAT, Pune 21

Conservation Principles, etc • Mass is conserved, Menrty - Mexit = 0 • Linear momentum is conserved, T - D - m d. V/dt = 0 (for accelerated flight) L-W = 0 (for 1 -g level flight) • . . June-July, 2003 IAT, Pune 22

Multi-modal Function June-July, 2003 IAT, Pune 23