System Optimization - II Multi-disciplinary Design Optimization

June-July, 2003 IAT, Pune 1

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)

1n:f);x(f,Minimise

If maximisation is required

ie, Maximise f(x)

then restate it as

Minimise F(x) = -f (x)


Inequality Constraints

mn:f;0)x(gInequality

• 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

X1

X2

g1

g2


Equality Constraints

Ln:f;0)x(hEquality

• 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 s

X1

X2

h(x)=0


Optimization Problem Statement

n - Independant variables

L - Equality constraints;

m - Inequality constraints

0XX;0XX

esinequalitispecial

:g;0)X(ginequality

:h;0)X(hequality

,sintconstratoSubject

:f)X(f,Minimise

LU

mn

Ln

ln

• Side constraints• Boxing design space• Bounding box

f 1n


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


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.


Global & Local Minimum

• 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 }

x1

f

f

f

X*x


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?

Computer Programme

X1f

x1

f


n-D Unconstrained

0x

f..

x

f

x

f

n21

T,gradientSet

• Solve the n equations for x1, x2 , x3 . . . xn

• Check for Positive definiteness of H. All eigenvalues of H should be 0


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/x1x2 2 0

H(f) = =

2f/x2x1 2f/x22 0 2

X* = ( 2, 2); f* = 0; Point of minima, since H +ve Definite (1 =2 = 2)


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)

Computer Programme

X1, X2 , X3h

What if h (x1, x2 , x3) cannot be explicitly inverted?


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


Constrained Optimisation Inequality Constraints

2

2

2

s)x(g)x(f)s,,x(LMinimise

realiss,0s)x(gassameis0)x(g

iablevarslacks

realissWhere,0s)x(g

;0)x(gtoSubject

);x(fMinimise

x1

x2

X*

g1=0

g1= -s2

- Lagrange multiplier; s - slack variable


Constrained Optimisation

conditionskerTucKuhn0)s,,,x(L

m2lnproblemtheofDimension

s)x(g)x(h)x(f)s,,,x(L,Minimise

,assameis

:f;0)x(gInequality

:f;0)x(hEquality

,toSubject

:f);x(f,Minimise

2TT

mn

ln

1n


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 ?


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

X

f


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.


• 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

[email protected]

http://www.casde.iitb.ac.in/

http://www.casde.iitb.ac.in/MDO/

mailto:[email protected]


Thank you


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

• . . . .


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)

• . . . .


Multi-modal Function

System Optimization - II Multi-disciplinary Design Optimization

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