Transcript Lecture 1a Role of Structures and Mechanisms in MEMS

Lecture 2a
Mathematical Preliminaries for
Optimal Design
Essential basics of calculus of variations and
constrained minimization
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.1
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
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.2
Fermat’s light-ray problem
(Feynman’s “life-guard on the beach” problem)
What is the minimum-time path from A to B?
Speed of light = c2
B
A
Speed of light = c1
Lifeguard’s swimming speed
= c2
B
A
Lifeguard’s running speed
= c1
Can be solved as a constrained minimization problem
Leads to Snell’s law of refraction.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.3
Brachistochrone (minimum time)
problem
The bead slides along a wire under the action of gravity.
h A
g
Functional
1  (dy / dx) 2
Minimize J ( y, dy / dx)  
dx
y f(x)
2 g (h  y )
0
l
Y=f(x)
B
l
x
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…
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.4
Unconstrained minimization
Minimize f ( x)
x
 x1 
x 
 2
x 
 ... 
 xn 
f (x) : n  
*
x
If
is a solution…
( n equations)
Necessary condition:
 f (x* )  0
Sufficient condition:
H( f (x* )) is positive definite
i.e., xT H(x* )x  0 x  n
f

Gradient x1 
 f 
 x2 
 f ( x)   
 ... 
 xf 
 n
 2
x1

Hessian
2 f
 x x
H f ( x)   1 2

2

 x xf
 1n
2 f
2 f
x2 x1
2 f
x22

2 f
x2 xn


2
 xn xf 2 

  
2 f 
 x 2
n 

2 f
xn x1
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.5
Equality constrained minimization
Minimize f (x)
x
Subject t o
h( x)  0
 h1 ( x) 
 h ( x) 
 2 
h

  
hm ( x) 
If (x , Γ ) is a solution…
*
*
m
L  f (x)   i hi (x)  f  ΓT h
i 1
Lagrange multiplier(s)
 L(x* , Γ* )  0 ( n  m equations)
Necessary condition:

Define a Lagrangian, L
( n equations)
(mequations)
*
h
(
x
)0
 f (x )   h(x )Γ  0 and
*
*
*
Sufficient condition:
xT H(x* )x  0 x
satisfying h(x* )x  0
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.6
General constrained minimization
A Beautiful Mind
Minimize f (x)
x
Subject t o
h ( x)  0
g ( x)  0
 g1 ( x)  Define a Lagrangian,
p
 g (x) 
m
 2  L  f (x)   i hi (x)    j g j (x)
g

i 1
j 1



T
T

f

Γ
h

Λ
g
 g p ( x) 
An inequality constraint can be active (= sign) or inactive (<sign).
*
*
*
If (x , Γ , Λ ) is a solution…
 f (x* )   h(x* )Γ*   g(x* )Λ*  0
h(x* )  0, g(x* )  0
g j (x* ) j  0 for j  1,2,..., p
j  0
Karush-Kuhn-Tucker
(KKT)
necessary conditions
Complementarity
conditions
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.7
Solution to Fermat/Feynman’s
minimum-time problem
speed
= c2
2
d 2 x1
B
x2
1
d1
d12  x12
d 22  x22
Minimize f ( x1 , x2 ) 

x1 , x2
c1
c2
Subject to
speed
A
= c1
h  x1  x2  l  0
l
 f (x )   h(x )Γ  0
*
*
*
x1
c1 d12  x12
   0&
x2
c2 d 22  x22
  0
sin 1 c1 Snell’s law

sin  2 c2
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.8
Return to brachistochrone problem
1  (dy / dx) 2
Minimize J   F ( y, dy / dx) dx  
dx
y f(x)
2 g (h  y )
0
0
l
l
What is different now?
The unknown is a function.
The objective is a function of the unknown function and its derivative(s).
First variation

J ( y  v)  J ( y) 
 J  lim



 0

x
y (x) v(x)
y ( x)  v ( x)
Operationally useful definition:
 F

F
F
 J    y  y 
y  ... dx
y
y
 y

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.9
Fundamental lemma of calculus of
variations
xb
If
 f ( x )v ( x )  0
for any
xa
v( x) satisfyingv( xa )  v( xb )  0
then
f ( x)  0 x [ xa , xb ]
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.10
Now, it is only integration by parts…
xb
Consider Minimize J   F ( y, y, y) dx
y f(x)
xa
Necessary condition for a minimum:  J  0
 F

F
F
y 
y 
y dx
y
y 
xa  y
xb
 J  
 F
 F

d  F 
d  F  
F
y  
y dx  
   y  
y 
y 
dx  y 
dx  y  
y  x
xa  y
 y
a
xb
xb
 F
 F
d  F 
d 2  F  
F
d  F  
y  2 
y dx  
y   0
   y  
y 
y  
dx  y 
dx  y  
y
dx  y  
xa  y
 y
xa
xb
xb
By the fundamental lemma
F d  F  d 2  F 
  2 
  0
 
y dx  y  dx  y 
Boundary conditions
Euler-Lagrange necessary conditions
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.11
Extensions
• Second variations; sufficient conditions
– Refer to any standard text, e.g., Gelfand and
Fomin.
• To multiple derivatives of y
– Simply integrate by parts as many times as
necessary and collect the boundary terms
carefully.
• To multiple unknown functions, i.e., y1 , y2 ,...
– Straightforward; write the same set of
equations for each.
• To multiple independent variables, i.e., x1 , x2 ,...
– Need to use the divergence theorem instead of
integrating by parts.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.12
Constrained variational calculus problems
xb
Minimize J   F ( y, y, y) dx
y  f(x)
xa
Subject t o
xb
G   FG ( y, y, y) dx  0
Integral (global) constraint
xa
h( y, y, y)  0 x  [ xa , xb ] Differential (local) constraint
What is the Lagrangian now?
xb
LJ  J  G    ( x) h( y, y, y)dx
xa
Single scalar variable
Scalar valued function
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.13
Main points
KKT necessary conditions for constrained minimization
 f (x* )   h(x* )Γ*   g(x* )Λ*  0
h(x* )  0, g(x* )  0
g j (x* ) j  0 for j  1,2,..., p
j  0
Euler-Lagrange necessary conditions for a functional
F d  F  d 2  F 
  2 
  0
 
y dx  y  dx  y 
The KKT conditions can be used for variational calculus
problems as well.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 2a.14