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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
12
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
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