Transcript otws2 7790

Shape Optimization for
Elliptic Eigenvalue Problems
Chiu-Yen Kao
Collaborators:
He Lin, Yuan Lou, Stanley Osher,
Fadil Santosa, Eli Yabolnovich, Eiji Yanagida
IPAM, Numerics and Dynamics for Optimal Transport
April 17, 2008
Motivation
• Control the resonance frequencies of devices:
– Maximize or minimize certain frequencies
– Maximize the gap between adjacent frequencies
– Maximize the ratio between real part and imaginary part of
eigenvalues (quality factor optimization)
– Minimize the principle eigenvalue
• Application:
–
–
–
–
Vibration system control
Photonic crystal design
Optical Resonator
Population biology
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Shape Optimization for
Elliptic Eigenvalue Problem
• Goal: Minimize a certain design objective
min F ( )

such that
 is the eigenvalue of
L(u,  ( x),  )  0 , 1   ( x)  2
subjects to boundary condition on
elliptic differential operator.
D
and L is the
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Motivation I : Shape of the drum
Consider an open bounded D  R n , a positive  : D  R ,
and ( , u ) satisfies the elliptic eigenvalue problem
 u ( x)   ( x)u ( x)

 u ( x)  0
x  D,
x  D.
The eigenvalues are
0  1 ( D,  )  2 ( D,  )  3 ( D,  )  ...  
Q1: Shape problem: D  n ( D,  ),   1, D  A
Q2: Composition problem:   n ( D,  ), 1    2 , D dx  M
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Theoretical Results
Q1: Shape problem:
1.
2.
Rayleigh (1877) conjectured, and Faber (1923) and Krahn (1925) proved,
that if you fix the area of a drum, the lowest eigenvalue is minimized
uniquely by the disk.
Payne, Pólya, and Weinberger conjecture (1955): The disk maximizes the
ratio of 2 to 1 has been proved by Ashbaugh and Benguria (1992)
Q2: composition problem:
1.
2.
Krein (1955) provided one dimensional optimal density distribution for
maximal and minimal n .
Cox and McLaughling (1993): minimal n for higher dimensions.
(   1 )(   2 )  0 a.e. in D
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Numerical Approach
Q2: composition problem:
  n ( D,  ), 1    2 ,  dx  M
D
(   1 )(   2 )  0 a.e. in D
Let    2    1 D \ , | D | const
F (k ) or max F (k )
Find min


shape optimization problem:
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Eigenvalue Separation for Drums
• Shape Optimization of a drum head with a fixed domain D
 u   ( x)u x  D
{
u0
x  D
• Let  be a domain inside D
1 for x  
, and  ( x)  {
 2 for x  
• Solve the optimization:
•
max (k  k 1 )  max F () for a given fixed k


• Subject to the constraint:
||  || const  G ()  0
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Shape Mapping
• The set   ( I   )
is defined by
  {x   ( x) | x  }
• The vector field  ( x ) is the displacement of  .
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Shape Derivative
• Framework of Murat-Simon:
• Let  be a reference domain. Consider its variations
•   ( I   ) with  W 1, ( R N ; R N )
• Definition: the shape derivative of F () at  is the
Frechet differential of   F (( I   )) at 0 .
F (( I  h ))  F ()
d s F ()( )  lim
h 0
h
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Shape Derivatives of
Eigenvalues and Area
Compute shape derivatives
1. F ()  (k  k 1 )
d s F ()( ) 
k (  2  1 ) 2
k 1 (  2  1 ) 2
uk   nds 
uk 1  nds


2
2
  ( x)uk dx 
  ( x)uk 1dx 
D
D
2.G ()   1dx  const

d s G ()( )     ndx

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Lagrange Multiplier Method
• max F () subject to G ()  0

use Lagrange multiplier method:
L ( )  F (  )   G (  )
the necessary condition for a minimizer is
d S L()( )  d S F ()( ) d S G()( )  0
together with the constraint
G ()  0
allows us, in principle, to find  and  .
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Gradient Ascent
• Shape derivative
L' ()( )   ( f (uk , uk 1 , k , k 1 ,  ) g ) ( x)  nds

Gradient descent algorithm for the shape
 new  ( I  ( f (uk , uk 1 , k , k 1 ,  ) g )n old ) old
The normal advection velocity of the shape is f g .
We solve the level set equation:
t  ( f g ) |  | 0
Because of G ()  0, we need to have  ( f g )ds  0 .

Then    f ds
. s
gd




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max (

2
 1 )
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max (

3
 2 )
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max (

3
 2 )
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max (

5
 4 )
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Photonic Crystals
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Motivation II : Photonic Crystal
• Shape Optimization of a photonic crystal with a fixed domain 
1
2
   i     i  En  2 En , En ( x  a )  En

c
1
2
   i     i  Hn  2 Hn , Hn ( x  a )  Hn

c
1 for x  S
Let S   be a domain inside  , and
 ( x)  
We begin with a shape with
 2 for x  S
Ej ( 0 ,  )  02 / c 2  Ej 1 ( 0 ,  )
E
E
Hj ( 0 ,  )  02 / c 2  Hj 1 ( 0 ,  )
H
H
  K
and we want to maximize
J1 ( )  inf  EjE 1 ( ,  )  sup  EjE ( ,  ) J 2 ( )  inf  HjH 1 ( ,  )  sup  HjH ( ,  )






J 3 ( )  inf inf  HjH 1 ( ,  ),inf  EjE 1 ( ,  )  sup sup  HjH ( ,  ),sup  EjE ( ,  )


 
18 

Maximize 
Dielectric: 1 : 11.4
Initial Gap: none Final Gap: 0.1415
1 2
TM
Band Gap
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Maximize 
1 2
TE
Band Gap
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Motivation III: Quality Factor Optimization
• Design the material to have lower loss of energy
– Mechanics Systems
Ex: damped mass spring
– Electrical Systems
Ex: RLC circuit, quartz crystal
– Optical Systems
Ex: photonic crystal
Pictures:
http://en.wikipedia.org/wiki/Quartz_clock
http://minty.stanford.edu/PBG/
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The Mass Spring System (1)
The displacement satisfies:
For small damping
(1)
, the solution is
The total energy is
where the period
and
.
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The Mass Spring System (2)
The quality factor is defined as :
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1-D Schrödinger’s Equation
• Finite potential well:
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Optical Resonator
• One-Dimensional Case
• Higher-Dimensional Case
•Goal: minimize the quality factor
Re( )
sup
 2 Im( )
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1-D Forward Eigenvalue Solver (1)
Solve
by finite element method. Apply the test function
By incorporating the boundary condition
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1-D Forward Eigenvalue Solver (2)
Thus
The equation can be written as
It is a nonlinear Eigenvalue Problem !!
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1-D Forward Eigenvalue Solver (3)
The equation can be written as
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2-D Forward Eigenvalue Solver (1)
• Boundary Integral Method
It is a nonlinear Eigenvalue Problem !!
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2-D Forward Eigenvalue Solver (2)
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2-D Forward Eigenvalue Solver (3)
• Nonlinear Eigenvalue Problem: (Newton’s method) inverse iteration
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Gradient Flow
In terms of a single mode damped oscillator, the quality factor is proportional
to the ratio between the real part and the imaginary part of the eigenvalue.
Q
real ( )
 2imag ( )
Our goal here is to maximize the quality factor subject to the wave equation
we discussed previously which can be written in the general eigenvalue problem
p( )VR  0
Suppose there is a small perturbation s.t.
We keep only the first order term
Premultiplying by the corresponding eigenvector leads to
Thus
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Algorithm
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1D Numerical Results (1)
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1D Numerical Results (2)
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2D Numerical Results (1)
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2D Numerical Results (2)
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IV: Eigenvalue with Indefinite Weight
• Consider the elliptic eigenvalue problem with indefinite
weight
 u  m( x)u x  D
{
u / n  0
x  D
Let  be a domain inside D , and
Solve the optimization:
m1  0 for x  
m( x )  {
m2  0 for x  
min 1

Subject to the constraint:
 mdx  M  0
D
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Diffusive Logistic Equation
t    [m( x)   ]


0

n

  ( x,0)  0,  ( x,0)  0
where

represents the density of a species and
1. If   1 (m) ,
2. If 
 ( x, t )  0
uniformly as
 1 (m),  ( x, t )   * ( x)
xD
x  D
xD

is the growth rate.
t 
uniformly as
t 
The effect of dispersal and spatial heterogeneity in population dynamics.
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Eigenvalue and Eigenfunction
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Minimizers in 1D
• Theorem: When D  ( a, b) is an interval, then there are exactly
two global minimizers of  (m) . For the logistic model, this
means that a single favorable region at one of the two ends of
the whole habitat provides the best opportunity for the species
to survive.
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Minimizers in High Dimensions
• In general domain: open question
m  0 for x  (c,1)  (0, b)
• Suppose D  (0,1)  (0, b). m( x)  { 1
m2  0 for x  (0, c)  (0, b)
If b   / c , then  is not minimal. In particular, the strip
at the end with much longer edge can’t be the optimal favorable
region.
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Square domains
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Square domains
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Rectangular domains
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Ellipse domain
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More general domains
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The End
Thank you
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