slide 2a.1 stiff structures, compliant mechanisms, and mems: a short course offered at iisc,...

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 2a.1

Lecture 2aMathematical Preliminaries for Optimal Design

Essential basics of calculus of variations and constrained minimization


Contents• Minimum-time problems

– Fermat’s problem and Snell’s law– Brachistochrone problem

• Constrained minimization– Lagrangian and conventions– Karush-Kuhn-Tucker necessary conditions– Sufficient conditions

• Calculus of variations– Functional and its variation– Fundamental lemma– Euler-Lagrange equations– Extensions to other situations– Constrained variational calculus problems


Fermat’s light-ray problem(Feynman’s “life-guard on the beach” problem)

What is the minimum-time path from A to B?

Can be solved as a constrained minimization problemLeads to Snell’s law of refraction.

Speed of light = c2

Speed of light = c1

Lifeguard’s swimming speed = c2

Lifeguard’s running speed = c1A

B

A

B


Brachistochrone (minimum time) problem

x

Y=f(x)

The bead slides along a wire under the action of gravity.

g

A

B

What shape of the wire (i.e., what f(x)) will lead to the minimum descent time for the bead?

Posed as a challenge by Johann Bernoulli.Solved by Leibnitz, Newton, L’Hospital, and Jacob Bernoulli…

l

dxyhg

dxdydxdyyJ

0

2

f(x)y )(2

)/(1)/,(Minimize

l

h

Functional


Unconstrained minimization

)(Minimize xx

f

Necessary condition: 0x )( *f

Sufficient condition: ))(( *xH f is positive definite

nx

x

x

...2

1

xnf :)(x

nxf

xfxf

f...

)( 2

1

x

2

2

2

2

1

2

2

2

22

2

21

2

1

2

12

2

21

2

)(

nnn

n

n

x

fxxf

xxf

xxf

x

fxxf

xxf

xxf

x

f

f

xH

Gradient Hessian

nT xxxHx 0)( *i.e.,

If is a solution…*x

( equations)n


Equality constrained minimization

0)(

toSubject

)(Minimize

xh

xx

f

Necessary condition: 0Γx ),( **L

Sufficient condition:

xxxHx 0)( *T

If is a solution…),( ** Γx

Define a Lagrangian,L

hΓxx Tm

iii fhfL

1)()(

)(

)(

)(

2

1

x

x

x

h

mh

h

h

( equations)mn

satisfying 0xxh )( *

0Γxhx *** )()(f 0xh )( *( equations)n ( equations)m

and

Lagrange multiplier(s)


General constrained minimization

0)(

0)(

toSubject

)(Minimize

xg

xh

xx

fDefine a Lagrangian,

gΛhΓ

xxx

TT

p

jjj

m

iii

f

ghfL

11

)()()(

)(

)(

)(

2

1

x

x

x

g

pg

g

g

If is a solution…),,( *** ΛΓx

An inequality constraint can be active (= sign) or inactive (<sign).

0ΛxgΓxhx ***** )()()(f

0xg0xh )(,)( **

pjg jj ,...,2,1for0)( * x0 j

Karush-Kuhn-Tucker (KKT)necessary conditions

Complementarity conditions

A Beautiful Mind


Solution to Fermat/Feynman’s minimum-time problem

speed = c2

speed = c1A

B

1

2

l

1d

2d 1x

2x

0

toSubject

),(Minimize

21

2

22

22

1

21

21

21, 21

lxxh

c

xd

c

xdxxf

xx

0Γxhx *** )()(f 021

211

1 xdc

x0

22

222

2 xdc

x&

2

1

2

1

sin

sin

c

c

Snell’s law


Return to brachistochrone problem

ll

dxyhg

dxdydxdxdyyFJ

0

2

0f(x)y )(2

)/(1)/,(Minimize

What is different now? The unknown is a function. The objective is a function of the unknown function and its derivative(s).

)()(lim

0

yJvyJJ

First variation

dxyy

Fy

y

Fy

y

FJ

...

Operationally useful definition:)(xy

)()( xvxy

)(xvx


Fundamental lemma of calculus of variations

b

a

x

x

xvxf 0)()( for any

0)()(satisfying)( ba xvxvxv

then

],[0)( ba xxxxf

If


Now, it is only integration by parts…

Necessary condition for a minimum:

dxyyyFJb

a

x

x

),,(Minimize

f(x)y

0J

Consider

02

2

xb

a

b

a

xb

a

b

a

b

a

x

x

x

x

x

x

x

x

yy

F

dx

dy

y

Fy

y

Fdxy

y

F

dx

dy

y

F

dx

dy

y

F

yy

Fy

y

Fdxy

y

F

dx

dy

y

F

dx

dy

y

F

dxyy

Fy

y

Fy

y

FJ

02

2

y

F

dx

d

y

F

dx

d

y

FBoundary conditions

Euler-Lagrange necessary conditions

By the fundamental lemma


Extensions

• Second variations; sufficient conditions– Refer to any standard text, e.g., Gelfand and Fomin.

• To multiple derivatives of – Simply integrate by parts as many times as

necessary and collect the boundary terms carefully.

• To multiple unknown functions, i.e.,– Straightforward; write the same set of equations for

each.

• To multiple independent variables, i.e., – Need to use the divergence theorem instead of

integrating by parts.

y

,..., 21 yy

,..., 21 xx


Constrained variational calculus problems

],[0),,(

0),,(

toSubject

),,(Minimizef(x)y

ba

x

xG

x

x

xxxyyyh

dxyyyFG

dxyyyFJ

b

a

b

a

Integral (global) constraint

Differential (local) constraint

What is the Lagrangian now?

b

a

x

xJ dxyyyhxGJL ),,()(

Single scalar variableScalar valued function


Main points

0ΛxgΓxhx ***** )()()(f

0xg0xh )(,)( **

pjg jj ,...,2,1for0)( * x0 j

02

2

y

F

dx

d

y

F

dx

d

y

F

KKT necessary conditions for constrained minimization

Euler-Lagrange necessary conditions for a functional

The KKT conditions can be used for variational calculus problems as well.

slide 2a.1 stiff structures, compliant mechanisms, and mems: a short course offered at iisc,...

Documents