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Shape Optimization for Elliptic Eigenvalue Problems

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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

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

Goal: Minimize a certain design objective

such that is the eigenvalue of

subjects to boundary condition on and is the elliptic differential operator.

Consider an open bounded , a positive ,

and satisfies the elliptic eigenvalue problem

The eigenvalues are

Q1:Shape problem:

Q2:Composition problem:

Q1:Shape problem:

- 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 to has been proved by Ashbaugh and Benguria (1992)
Q2:composition problem:

- Krein (1955) provided one dimensional optimal density distribution for maximal and minimal .
- Cox and McLaughling (1993): minimal for higher dimensions.

Q2:composition problem:

Let

Find

shape optimization problem:

Shape Optimization of a drum head with a fixed domain

Let be a domain inside , and

Solve the optimization:

Subject to the constraint:

The set is defined by

The vector field is the displacement of .

Framework of Murat-Simon:

Let be a reference domain. Consider its variations

with

Definition: the shape derivative of at is the Frechet differential of at .

Compute shape derivatives

subject to

use Lagrange multiplier method:

the necessary condition for a minimizer is

together with the constraint

allows us, in principle, to find and .

Shape derivative

Gradient descent algorithm for the shape

The normal advection velocity of the shape is .

We solve the level set equation:

Because of , we need to have .

Then .

Shape Optimization of a photonic crystal with a fixed domain

Let be a domain inside , and

We begin with a shape with

and we want to maximize

Motivation II : Photonic Crystal

Dielectric: 1 : 11.4

Initial Gap: none Final Gap: 0.1415

Maximize Band Gap

Dielectric: 1 : 11.4

Initial Gap: none Final Gap: 0.0989

Maximize Band Gap

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/

The displacement satisfies: (1)

For small damping , the solution is

The total energy is and

where the period .

The quality factor is defined as :

- Finite potential well:

- One-Dimensional Case
- Higher-Dimensional Case
- Goal: minimize the quality factor

- Solve
- by finite element method. Apply the test function
- By incorporating the boundary condition

Thus

The equation can be written as

It is a nonlinear Eigenvalue Problem !!

- The equation can be written as

- Boundary Integral Method
- It is a nonlinear Eigenvalue Problem !!

- Nonlinear Eigenvalue Problem: (Newton’s method) inverse iteration

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.

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

Suppose there is a small perturbation s.t.

We keep only the first order term

Premultiplying by the corresponding eigenvector leads to

Thus

Consider the elliptic eigenvalue problem with indefinite weight

Let be a domain inside , and

Solve the optimization:

Subject to the constraint:

where represents the density of a species and is the growth rate.

1. If , uniformly as

2. If , uniformly as

The effect of dispersal and spatial heterogeneity in population dynamics.

Theorem: When is an interval, then there are exactly two global minimizers of . 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.

In general domain: open question

Suppose

If then is not minimal. In particular, the strip at the end with much longer edge can’t be the optimal favorable region.

Thank you