Lecture 4b
Highlights of some solution methods
Aspects of optimization algorithms used in topology
design.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.1
Contents
•
•
•
•
•
Components of optimal synthesis
Sensitivity analysis
Mathematical programming algorithms
An optimality criteria method
Convex approximation methods
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.2
Optimal synthesis: what it entails
Objective
function
Design variables
Constraints
Equations
governing the
device behavior
Design
needs
A study to
ensure the wellposedness of
the optimization
problem
Sensitivity
analysis
Function
evaluation
Optimization
algorithm
Solution
No
Satisfactory?
Yes
OPTIMAL SYNTHESIS
Stop
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.3
Sensitivity analysis
Determining the gradients of the objective and constraint
functions with respect to the design variables.
(the body force
is assumed to be
absent.)
T
Minimize
J

f
u dA

t
Consider
ρ

Subject to
T
T
 ε(u) D (ρ)ε( v) dV   ft vdA  0


g   S (ρ) dV  V *  0

u and v satisfy essential (Dirichlet) boundary conditions.
S (  x ) = smoothened state (exists or not) of a point x
Need to compute:  ρ J ,  ρ g
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.4
Sensitivity analysis in discrete
modeling: direct method
1
Minimize J  F T U
ρ
2
Subject to
KU  F
0
dJ
J  J  U  J  U





di i  U  i  U  i
T
T
N
g   S ( i )  V *  0
i 1
KU  F
(differentiate w.r.t.  i )
K
U
UK
 0 (assuming that F does not depend on  i )
 i
 i
U
U
K
K

U (needs to be solved for each variable to get
)


 i
 i
i
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.5
Sensitivity analysis in discrete
modeling: adjoint method
0
dJ
J  J  U  J  U





di i  U  i  U  i
T
T
U
K
K

U
 i
 i
dJ
 J  1 K
 
U
 K
di
i
 U 
T K
 λ
U
i
T
where
 J 
Kλ  
 Needs to be solved for λ only once!
 U 
Adjoint equation
(if there are constraints dependent on ρ ,
then λ needs to be solved for those as well).
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.6
Sequential linear programming
(SLP)
(k )
(k )
(k )
Minimize
J


J

ρ
0
(k )
T
Minimize J
ρ
Linearize
Subject to
h0
g 0
ρ
Subject to
(k )
0
h
g
(k )
0
 h
( k )T
 g
ρ ( k )  0
( k )T
ρ(k )  0
Solve the LP problem and
repeat until convergence.
Works reasonably well, even “black-box” usage of standard
packages once the problem if well formulated and understood.
Especially suitable when multiple and constraints exist.
Somewhat slower rates of convergence.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.7
Sequential quadratic programming
(SQP)
Minimize J
ρ
Minimize
J
(k )
ρ
(k )
0
 J
( k )T
“Quadratize” Subject to
Subject to
ρ
(k )
1 ( k )T ˆ ( k ) ( k )
 ρ H ρ
2
h0
h0( k )  h ( k ) ρ ( k )  0
g 0
g 0( k )  g ( k ) ρ(k )  0
T
T
Solve the QP problem and
repeat until convergence.
Works quite well in conjunction with trust-region method
(Matlab’s optimization toolbox has a routine: constr( )
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.8
Return to continuum model:
sensitivity analysis
Minimize J   f tT u dA
ρ

Subject to
T
T
ε
(
u
)
D
(
ρ
)
ε
(
v
)
dV

f

 t v dA  0


g   S (ρ) dV  V *  0





L   ftT u dA     ε(u)T D (ρ)ε( v) dV   ftT v dA     S (ρ) dV  V * 








 S

T D
 i L     ε(u)
ε( v )i dV    
i dV   0
 i


   i

D
S
   ε(u)T
ε( v)  
0
 i
 i

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.9
Adjoint sensitivity analysis for the
continuum model



T
T
*
L   f u dA     ε(u) D (ρ)ε( v) dV   ft v dA     S (ρ) dV  V 






T
t




 u L   ftT u dA     ε(u)T D ε( v) dV   0 Adjoint equation



 v L     ε(u)T D ε(v) dV    ftT v dA  0 Equation of equilibrium



recovered from the
weak form
v  u
(Same conclusion that we saw in slide # 2b.9 in the
context of bars)
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.10
Optimality criteria method


T
 f u dA     ε(u) D ε( v) dV   0 Adjoint. Equn.


u 


T
   ε(u) D ε(v) dV    ftT v dA  0 Equilib. Equn.
v 
 
S
T D
  ε(u)
ε( v) dV  
0
Design Equn.
i
i

T
t
A  B  0
Optimality criterion
Ai   ε(u)T

Turned out to be the
same here but not
always true.
D
S
ε( v) dV ; Bi 
0
i
i
 i( k 1)   i( k )   Ai( k )  ( k ) Bi( k ) 

or

 i( k 1)

Ai( k )  ( k )
    ( k ) ( k )   i
  Bi 
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.11
Optimality criteria method:
evaluating the Lagrange multiplier
 i( k 1)   i( k )   Ai( k )  ( k ) Bi( k ) 

or

 i( k 1)
Check if any  i ' s
exceeded their
upper or lower
limits; if yes, limit
them to the bounds.
Inner loop
at kth
iteration

Ai( k )  ( k )
    ( k ) ( k )   i
  Bi 
*
S
(
ρ
)
dV

V
0
Use: 

( k 1)
Repeat until  i ' s do not
change anymore.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.12
Convex approximation methods
• Linearization
• Reciprocal linearization
Replace xi with
yi 
1
xi
• Convex linearization
Replace xi
with yi 
1
only if the partial derivative with
xi
respect to that variable is positive.
Advantage: leads to convex, separable problems that can be easily
solved using the more efficient dual methods (Lagrange multipliers
become the variables).
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.13
Method of moving asymptotes (MMA)
2 f
 0, i  j
xi x j
 pi
qi 

f  r   

xi  Li 
i 1  U i  xi
N
 pi
qi 

r  f 0   

xi 0  Li 
i 1  U i  xi 0
f

2 f
if
 U i  xi 0  x
xi
i
pi  
f
0
if

xi
f

if
 0
xi
qi  
f
f
 xi 0  Li 2
if

xi
xi
N
0
0
0
0
  f

2

  xi 
f
if
0

2

x
 f  U i  xi 0
i



xi2  2 f
 x 
i
 
if f
0

x

i
xi 0  Li
Adjustable bounds to get a
conservative or accurate convex
approximation of the objective
and constraint expressions as
necessary.
K. Svanberg, “The Method of Moving Asymptotes—A New Method for Structural Optimization,” Int.
J. for Num. Meth. In Engineering, Vol. 24, 1987, pp. 359-373.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.14
Main points
• Function evaluation and sensitivity analysis
• Optimality criteria method
• Standard mathematical programming
techniques will do (SLP, SQP)
• Or use convex linearization algorithms such
as MMA
• Posing the problem correctly is crucial;
most algorithms would work for properly
posed problems
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4b.15