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System Optimization - II Multi-disciplinary Design Optimization

System Optimization - II Multi-disciplinary Design Optimization. K Sudhakar PM Mujumdar Centre for Aerospace Systems Design & Engineering Indian Institute of Technology, Mumbai. Objective Function(s). If maximisation is required ie, Maximise f(x) then restate it as

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System Optimization - II Multi-disciplinary Design Optimization

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

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

  3. X2 g1 g2 X1 Inequality Constraints • Each inequality constraint reduces design space • 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 IAT, Pune

  4. h(x)=0 s X2 X1 Equality Constraints • Each equality constraint reduces dimensionality • of design space by one, eg • h1(x1, x2, . . ., xn) = 0;  xn = y1(x1, x2, . . xn-1) • f {x1, x2,. . xn} = f{x1, x2,. . xn-1, y1(x1, x2, . .xn-1)} • = F {x1, x2,. . xn-1} • L < n IAT, Pune

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

  6. 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 IAT, Pune

  7. 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. IAT, Pune

  8. x X* f f x1 Global & Local Minimum f • Global Minimum : A function f(x) is said to have • a global minimum at x*, if • 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 } IAT, Pune

  9. f x1 f Computer Programme X1 1-D Unconstrained Minimise f(X) = (x1 - 2)2 f/x1 = 2 (x1- 2) ; 2f/x12 = 2 x1* = 2, f* = 0 ; Point of minima • Some points for consideration! • f - Simple function, easy to analytically differentiate. • What if f is complex? Numerically differentiate? IAT, Pune

  10. n-D Unconstrained • Solve the n equations for x1, x2 , x3 . . . xn • Check for Positive definiteness of H. • All eigenvalues of H should be  0 IAT, Pune

  11. Two Variable Problem Minimise f(X) = (x1 - 2)2 + (x2 - 2)2 Tf = [f/x1 ; f/x2 ] = [ 2(x1- 2) ; 2(x2- 2) ] 2f/x12 2f/x1x2 2 0 H(f) = = 2f/x2x1 2f/x22 0 2 X* = ( 2, 2); f* = 0; Point of minima, since H +ve Definite (1 =2 = 2) IAT, Pune

  12. h Computer Programme X1, X2 , X3 Equality Constrained Minima Minimise f (x1, x2, x3) Subject to h (x1, x2 , x3) = 0 Solve for x1 =  (x2 , x3) Minimise f ( (x2 , x3), x2 , x3) = F (x2 , x3) What if h (x1, x2 , x3) cannot be explicitly inverted? IAT, Pune

  13. Equality Constrained Minima Minimise f(x1, x2); Subject to h(x1, x2) = 0 Is equivalent to the un-constraint problem Minimise L(x1, x2, ) = f(x1, x2) +  h(x1, x2) L/x1 = f/x1 +  h/x1 = 0 L/x2 = f/x2 +  h/x2 = 0 L/ = h= 0  - Lagrange multiplier IAT, Pune

  14. x2 g1=0 g1= -s2 X* x1 Constrained Optimisation Inequality Constraints  - Lagrange multiplier; s - slack variable IAT, Pune

  15. Constrained Optimisation IAT, Pune

  16. 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 ? IAT, Pune

  17. f X 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 IAT, Pune

  18. 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. IAT, Pune

  19. 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 IAT, Pune

  20. Thank you IAT, Pune

  21. 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 • . . . . IAT, Pune

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

  23. Multi-modal Function IAT, Pune

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