Shape Optimization Involving Eigenvalues of Laplace-Beltrami Operator

Chiu-Yen Kao

Joint work with Colin Macdonald, Daryl J. Springer

AMS Sectional Meeting in Tucson, University of Arizona, Oct 26 2012 This work is partially supported by NSF, DMS1216742

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1.  Introduction to shape optimization on eigenvalue problems

2.  Theoretical Results 3.  Previous Numerical Approaches 4.  Rearrangement algorithm: 5.  Closest Point Method and Surface Problem

Outline

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Consider an open bounded , a positive , and satisfies the elliptic eigenvalue problem The eigenvalues are Q1: Shape problem. Q2: Composition problem.

RD→:ρ

⎩⎨⎧

∂∈=

∈=Δ−

.0)(,)()()(DxxuDxxuxxu λρ

nRD⊂

!

(",u)

0 < !1(D,") ! !2 (D,") ! !3(D,") ! ..."#

Shape of the drum

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Theoretical Results

Q1: Shape problem: 1.  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.

2.  Payne, Pólya, and Weinberger conjecture (1955): The disk maximizes the ratio of to has been proved by Ashbaugh and Benguria (1992)

Q2: composition problem: 1.  Krein (1955) provided one dimensional optimal density distribution for

maximal and minimal . 2.  Cox and McLaughling (1993): minimal for higher dimensions.

1λ2λ

nλ

nλDinea ..0))(( =−− βραρ

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Krein Theorem (1955)

The Optimal Distribution

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Composition Problem

Q2: composition problem Let Find àshape optimization problem:

Dinea ..0))(( =−− βραρ

∫ =≤≤→Dn MdxD ρβραρλρ ,),,(

constD =Ω+= ΩΩ ||,\αχβχρ

)(max)(min kk ForF λλΩΩ

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Numerical Approaches 1 Shape derivative: Osher Santosa (drum) 2001, Kao Lou

Yanagida (population dynamics) 2008, Kao Osher Yablonovitch(photonic crystal) 2005

2  Topological derivative: He Kao Osher (drum,

photonic crystal) 2007 3 Multi-level set method: Haber 2004 4 Rearrangement:

Cox 1990, Kao Shu 2010 Hintermuller Kao Laurain 2011

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An Efficient Algorithm:

€

infρλ1 = inf

ρinf

u∈W01,2

|∇u |∫2dx

ρu2dx∫

Idea: 1.  Based on Rayleigh quotient formulation of eigenvalue 2.  Monotone iterative scheme

* Find eigenfunction * Find density such that

€

infu∈W0

1,2

|∇u |∫2dx

ρu2dx∫

€

infρ

|∇u |∫2dx

ρu2dx∫= sup

ρρu2dx∫

!

" fixed :eigenvalueforwardproblem

!

u fixed : rearrangement of r

€

ρnew

€

ρnew = argsupρ

ρu2dx∫

€

Ωmin(λ1)

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Monotone Iterative Scheme Initial: Iteration: * How to find with ? There exists such that

€

ρ,λ1(ρ),u1(ρ)

€

ρnew = argsupρ

ρu12dx∫

€

λ1(ρnew ) = infu∈W0

1,2

|∇u |∫2dx

ρnewu2dx∫

<|∇u1 |∫

2dx

ρnewu12dx∫

<|∇u1 |∫

2dx

ρu12dx∫

= λ1(ρ)

€

ρnew = argsupρ

ρu2dx∫

€

ρ = ρ2χΩ + ρ1χD \Ω, |Ω |= const

€

u*

€

Ω={x | u2(x) > u*2}= const

€

ρnew = ρ2χΩ + ρ1χD \Ω

Discrete Case: Rearrangment Inequality

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n

n

jj

jnjj

and

constwwhere

ψψ

ρρρρρρρ

≤≤

=Ω≈===<=== ∑+=

+

...

||,.....

1

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1211

*

**

∑∑∫ =≈= jjjjjnew uwdxu ψρρρρ

ρ

22suparg

Summary

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Ωmin(λ1) : Level Set Approach

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Ωmin(λ1) : Efficient Approach

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Ωmin(λ2) : Efficient Approach

Symmetry Breaking

S. Chanillo et. al., 2000

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•  Consider the elliptic eigenvalue problem

Let be a domain inside , and Solve the optimization: Subject to the constraint:

Eigenvalue Problem on Surfaces

Ω D

!"su(x) = !m(x)u(x) x # D,$u$n

+"u = 0 x #$D.

%

&'

('

min!!1

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Parametrization Approach

!su(x)

!su(x)

!su(x) =

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Definition (Closest point function):

Definition (Closest point extension): Let S be a smooth surface in Rd. The closest point extension of a surface function u : S → R is a function v : Ω → R, defined in a neighborhood Ω ⊂ Rd of S, as

We will say that v is the closest point extension of u.

Closest Point Approach

cp(x) = x ! d"d

v(x) = u(cp(x))

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Closest Point on Hemisphere

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Laplace-Beltrami Operator

1.  Interpolation Method for closest points on the surfaces (may not on the Cartesian grids): bicubic interpolation for second order scheme

2.  Standard Discretization in the neighboring Cartesian grids near the surface

* Keep center point without evaluating it at closest point.

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Boundary Condition

Boundary Condition: Discretization:

!u!n

+!u = 0

u(x2 )!u(cp(x2 ))2 | cp(x2 )! cp(x2 ) |

+!u(x2 )+u(cp(x2 ))

2"

#$

%

&'= 0

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Closest Point Method – Matrix Implementation (COLIN B. MACDONALD, JEREMY BRANDMAN, STEVEN J. RUUTH)

1.  Interpolation Method for closest points on the surfaces (may not on the Cartesian grids): bicubic interpolation for second order scheme

2.  Standard Discretization in the neighboring Cartesian grids near the surface

Sparsity Matrix

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Minimal Set of Points

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!min(!1D ) = 0.54

Minimization of First Eigenvalue

!min(!2D ) " 2.15

Higher Eigenmodes

!min(!3D ) " 4.91

!min(!1D ) " 2.15

Circular Shape !min(!2P ) " 0.54

!min(!4P ) " 2.16

Ellipse !min(!2P ) " 0.22

!min(!4P ) " 0.91

!min(!1D ) "1.02

!min(!2D ) " 3.05

!min(!3D ) " 6.06

!min(!4D ) " 6.09

!min(!2N ) "1.04

!min(!3R,"=1) "1.65

!min(!3R,"=10 ) " 2.72

!min(!3D ) " 3.04

Ring

Future Directions •  Nonlinear constrain:

•  Manifold problem:

•  Higher order PDEs and Nonlinear Eigenvalue Problem,

e.g. thin plate problem •  General objective function

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!

"Pdx = MD#

!

"2u(x) = #$(x)u(x) x %&,

u(x) ='u'n

= 0 x %'&.

(

) *

+ *

L(x,u,!su(x),"su(x)) = !"(x)u(x) x # D,B(u(x),!su(x)) = 0 x #$D.

%&'

Thank you

The End