April 2009

The Method of Fundamental Solutions applied

to eigenproblems in partial differential equations

Pedro R. S. Antunes - CEMAT

(joint work with C. Alves)

Experimental results of resonance

Eigenvalue problem for the Laplacian

- some results and questions

Numerical solution using the Method of Fundamental Solutions (MFS)

- eigenfrequency calculation

- eigenfunction calculation

- numerical simulations with 2D and 3D domains

Hybrid method for domains with corners or cracks

Shape optimization problem

Extension to the Bilaplacian eigenvalue problem and to the eigenvalue in elastodynamics

Conclusions and future work

• Outline

Experimental results of resonance

experimental nodal linesexperimental nodal lines

• Chladni figures

Eigenvalue problem for the Laplacian

Search for k (eigenfrequencies) such that there exists non null function u (eigenmodes) :

on 0uor 0

2,3d ,in 0

n

2

uRuku d

An application: Calculate the resonance frequencies and eigenmodes associated to a drum (2D) or a room (3D)

n 0

• Eigenvalue problem for the Laplacian

General results: Countable number of eigenvalues The sequence goes to infinity 1>0 1=0

Numerical methods – eigenfrequency calculation

Finite Elements, Finite Differences, Boundary Elements

Meshless Methods

Consider the rigidity matrix Ah(k).(h – discretization parameter, k – frequency)

Fixed h, search k : matrix is not invertible (eg: det(Ah(k)) = 0 )

- particular solutions - angle Green’s functions (Moler&Payne, 1968, Trefethen&Betcke, 2004)

- radial basis functions (JT Chen et al., 2002, 2003)

- method of fundamental solutions (Karageorghis, 2001; JT Chen et al., 2004; Alves&Antunes, 2005)

• Numerical methods - eigenfrequency calculation

~1

m

jjkj yxxuxu

y j

• The Method of Fundamental Solution (MFS)

Fundamental solution:

Consider the approximation

The coefficients j are calculated such that fits the boundary conditionsu~

an admissible curve

Given an open set Rd, different points and kC,

are linearly independent on .

The set is dense in L2(),

when is an admissible curve.

c

myy ,...,

1

yyxSpanxk :

)}(),...,({ 1 mkk yxyx

• Theoretical results

is not an eigenfrequency

Consider m points x1 ,…, xm collocation points (almost equally spaced)

Define m points y1 ,…, ym source points

),(2

1~ ,~/~ 1,,1 jjjjjjjjj xy nnnnn

xi yi

• Algorithm for the source points (2D)

1 2 3 4 5 6

-40

-35

-30

BesselJ zeros (exact values)

Circle:Plot of Log[g(k)]

Search for local minimum using the Golden Ratio Search

Due to the ill conditioning of the matrix g(k) is too small

• Algorithm for the eigenfrequency calculation

)()(

jy

ix

kkAm Build the matrices

Consider g(k)=|det(Am(k))| and look for the minima

m

jjkjm yxxuxu

0

)()()(~

To calculate j solve the system

0)(~, ... ,0)(~,1)(~

1

0

mxuxu

xu

Given the approximate eigenfrequency k, define

- non null solution,- null at boundary points

The extra point x0 is not on a nodal line

• Algorithm for the eigenmode calculation

x0y0

Define extra points { Cy 0

0x

{

• Error bounds (Dirichlet case)

Let be an approximation for the pair (eigenfrequency,eigenfunction) which satisfies the problem

(with small )

Then there exists an eigenfrequency k and eigenfunction u such that

and

where is very small if on .

1

~

1

~k

kk

A posteriori bound (Moler and Payne 1968)

Ωu

ΩΩθ

L

L

2~

Ω, on xu

, in ΩukuΔ

~0~~~ 2

uk ~,~

cuu

L2~

0~ u

Numerical Tests (algorithm validation)

m=dimension of the matrix

m abs. error (k1) m abs. error (k2) m abs. error (k3)

30 5.7210-6 30 1.3610-6 30 1.8110-5

40 8.4210-8 40 1.6710-7 40 2.1710-7

50 7.7610-8 50 1.1110-8 50 6.9410-8

60 1.4610-9 60 1.4410-9 60 3.1710-9

m abs. error (k1) m abs. error (k2) m abs. error (k3)

30 2.3110-6 30 4.9410-6 30 5.2110-6

40 5.9110-8 40 1.2110-8 40 1.2610-7

50 1.6410-9 50 3.0110-10 50 3.2710-9

60 8.2310-11 60 9.3110-12 60 9.3510-11

m abs. error (k5) m abs. error (k5) m abs. error (k5)

20 2.1110-4 30 1.4610-5 40 1.2310-6

50 3.0610-7 60 2.5210-8 70 5.0510-9

80 3.1910-9 90 6.1910-10 100 1.8710-10

• Numerical tests (Dirichlet case) – 2D

m abs. error (k1) m abs. error (k2) m abs. error (k3)

112 1.2510-8 112 9.2110-7 112 8.5710-6

158 8.6110-12 158 1.9710-9 158 6.5310-8

212 2.1810-14 212 1.6110-13 212 9.4610-11

m abs. error (k1) m abs. error (k2) m abs. error (k3)

218 6.1310-10 218 9.2710-7 218 1.5510-6

296 3.1110-10 296 7.3110-8 296 7.0910-8

386 9.1510-12 386 5.2510-9 386 1.9510-10

m abs. error (k5) m abs. error (k5) m abs. error (k5)

226 1.3610-5 304 5.8710-6 374 7.2110-8

• Numerical tests (Dirichlet case) – 3D

Plot of Log(g(k))

Point-sources on a boundary of a

circular domain

Point-sources on an “expansion” of

With the choice proposed

Big rounding errors

• Numerical tests (on the location of point sources)

m abs. error (k2) abs. error (k3)

60 1.1510-10 1.2510-10

70 4.1610-11 6.8310-12

80 3.3310-12 5.0310-12

1

1+10-8

k3-k2≈4.2110-8

Plot of Log(g(k)) with n=60

7.02481 7.02481 7.02481 7.02481 7.02481 7.02481 7.02481-631

-630

-629

-628

-627

-626

• Numerical tests (almost double eigenvalues)

Numerical Simulations (non trivial domains)

Plots of eigenfunctions associated to the 21th,…,24th eigenfrequencies

• Numerical Simulations

(Dirichlet and Mixed boundary conditions)

Dirichlet problem

Mixed problemDirichlet - external boundaryNeumann - internal boundary

nodal lines plot eigenmode

• Numerical Simulations

3D plots of eigenfunctions associated to three resonance frequencies

• Numerical simulations – non trivial domains 3D

Domains with corners or cracks Hybrid method

• Hybrid method – domains with corners or cracks

k(x-yj) is analytic in ( )Cjy

If has an interior angle / (with irrational), then

Lehmann (1959)

sin~ rcu

u is singular at some corners

The classical MFS is not accurate for corner/crack singularities

u = uReg + uSing

MFS approximation

particular solutions

1,2,...j , j β rk J,j β

Sinrj

xyxxu j

N

jj

N

j

SR

1

j1

j

• Hybrid method – domains with corners or cracks

Particular solutions (Dirichlet boundary conditions)

j satisfies the PDE

j satisfies the b.c. on the edges

(Betcke-Trefethen subspace angle technique)

xiyi

zi

Eigenfrequencies

Choose randomly MI points zi

Build the matrices

Calculate A=QR factorization where

Calculate , the smallest singular value of QB(k) Look for the minima

)()(

)()( )(

ijik

ijik

zyz

xyxkA

MI

MB

NR NS

)(

)( )(

kQ

kQkQ

I

B

)(1 k

)(1

k

k

• Hybrid method – eigenfrequency calculation

NR=80, MB=180NS=10, MI=30

1st eigenfrequency

2nd eigenfrequency 5th eigenfrequency

• Hybrid method – Dirichlet problem with cracks

• Hybrid method – mixed problem

1.0 0.5 0.5 1.0

0.5

1.0

1.5

1st eigenfrequency 5th eigenfrequency 9th eigenfrequency

NR=200, MB=300NS=7, MI=30

Shape optimization problems

Given a quantity depending on some eigenvalues, we want to find a domain which optimizes

NQ ,...,1

Q

Direct setting

0

Inverse setting

• Shape optimization problems

Payne & Pölya & Weinberger (1956)

Ashbaugh & Benguria (1991) j

j2

1,0

2

1,1

1

2

There are some restrictions to the admissible sets of eigenvalues, eg.

n1n

3

• Inverse eigenvalue problems

Existence issue: The inverse problem may not have a solution.

Kac’s problem (60’s): can one hear the shape of a drum?Gordon, Webb & Wolpert presented isospectral domains (1992)

In 1994 Buser presented a lot of isospectral domains, e.g

• Inverse eigenvalue problem

Uniqueness issue: No unique solution, in general.

Define the class of star-shaped domains with boundary given by

0,2 t,sin,cos tttrCurveswhere r is continuous (2)-periodic function

M

jjj tjbtjaatrtr

10 sin cos~

Define a non negative function

which depends on the problem to be addressed.To calculate the point of minimum, we

use the Polak-Ribière’s conjugate gradient

method.

),...,(,...,,,...,

,...,,,...,:

110

10

kkQbbaaG

RbbaaG

nMM

MM

bbaa MM ,...,,,..., 10 kkk n,...,, 21

MFS

• Shape optimization problem – numerical solution

Consider the approximation (M)

Numerical resultsshape optimization problems

Which is the shape that maximizes and which is its maximum value?

In 2003 Levitin did a numerical study to find the optimal shape.

2

3

131,

k

kkkQ

1

3

We obtained 3.201999331

3 C

• Numerical results - shape optimization problems

.1

2

The ball maximizes Ashbaugh & Benguria (1991)

Optimal shape

L&Y = 3.202...

Is it possible to build a drum with an almost well defined pitch (fifth and the octave):

2 ;2

3

1

4

1

3

1

2 kk

kk

kk

1,999521,93272,07602k4 / k1

1,500931,52291,67641,5k3 / k1

1,500411,41731,56211,5k2 / k1

Our approach

Kane-Shoenauer 2

(1995)

Kane-Shoenauer 1

(1995)

“Harmonic

drum”

• Numerical results - shape optimization problem

Optimal shape

Can one hear the sound of Riemann Hypothesis?

A drum with the first 12 eigenfrequencies~ 12 first Im(zeros) of Zeta function

Is there a drum playing all the non trivial zeros of the Zeta function?

(modulo asymptotic behaviour)

shapes that minimize the eigenvalues 1 - the circle is the minimizer

2 - two circles minimize, … but the convex minimizer is unknown1973- Troesch - conjectured the stadium2002- Henrot&Oudet - reffuted - no circular parts

• Optimization problem (Dirichlet)

Numerical counterexample (Alves & A., 2005): Elliptical stadium

37.9875443 < 38.0021483 (stadium)

(3D)

• Numerical results - shape optimization problems

Other PDEs:Bilaplacian

Search for k (eigenfrequencies) such that there exist non null function u (eigenmodes) :

on 0

on 0

in 0

42

u

u

uku

n

R2

• Eigenvalue problem for the bilaplacian

General results: Countable number of eigenfrequencies The sequence goes to infinity k1>0

~11

m

jjj

m

jjj yxnyxxuxu

y j

xHxiHi

x

2

01

028)(

• The MFS application to the bilaplacian problem

y j

Consider the approximation (m)

an admissible curve

Fundamental solution:

is analytic in satisfies the PDE

u~ u~

Given an open set 2, different points and kC,

are linearly independent on .

c

myy ,...,

1

)}(),...,(),(),...,({ 111

mm yxnyxnyxyxymy

• Theoretical results

If γ is the boundary of a domain which contains , the set

is dense in H3/2().

: :

yyxnyyxSpanxyx

Theorem (density result)

is not an eigenfrequency

DC

BAM Build the matrix M

di,j=xi-yj

,dA ji

,dnC jii

,dnB jij

,dnnD jiji

with the four blocks mm

• Eigenfrequency/eigenmode calculation

Eigenfrequency calculation

Define g(k)=|det(M(k))| and calculate the minima

Eigenmode calculation

Extra collocation point

12

22~

~

k

kk

Let and be an approximate eigenfrequency

and eigenfunction which satisfies the problem

Then there exists an eigenfrequency k such that

Ω, on xu

Ω, on xεu

, in ΩukuΔ

n ~

~0~~~ 42

.21

ΩΩεc

LL

k~ u~

• Error bounds

Theorem (a posteriori bound)

where

Generalizes Moler-Payne’s result for the bilaplacian

m abs. error (k1) m abs. error (k2) m abs. error (k3)

20 4.2310-6 20 7.88 10-5 20 5.54 10-3

25 4.17 10-8 25 8.80 10-7 25 7.58 10-5

30 3.66 10-10 30 3.85 10-8 30 3.57 10-6

40 1.96 10-11 40 7.90 10-10 40 1.23 10-7

• Numerical results - bilaplacian

Plot of Log(g(k))Big rounding errors

• Numerical results – location of point sources

The proposed algorithm for the source points again presents more stable results

The eigenfunction associated to the first eigenvalue of the plate problem changes the sign “near” each corner

• Numerical simulations – equilateral triangle

3D plots and nodal domains for the 3rd,7th,10th and 11th resonance frequencies

• Numerical simulations – non trivial domains

Other PDEs:Elastodynamics (2D)

on 0

in 0)(

2

u

uu

R2

• Eigenvalue problem for elastodynamics

Fundamental solution: Kupradze’s tensor

MFS: Invertibility of the matrix

• Eigenvalue problem for elastodynamics - numerics

Test for the disk (6th eigenfrequency) with Poisson ratio =3/8

The choice of source points- same conclusions -

• Eigenvalue problem for elastodynamics - numerics

Other shapes - simulations for the 1st, 2nd, 3rd eigenfrequencies, with Poisson ratio =1/2

1st component 2nd component

(i) The algorithm proposed for the location of source points produces stable results.

(ii) With that algorithm, the MFS solves accurately Laplace eigenvalue problems for quite general 2D and 3D regular shapes.

(iii) An hybrid method was proposed for domains with corners/cracks, that clearly improves MFS results.

(iv) Eigenvalues shape optimization problems were solved using the MFS and allowed to obtain better results than in previous studies by other authors (harmonic drum, stadium…).

(v) This MFS approach was extended to eigenvalue problems with other PDE’s, such as the Bilaplacian eigenvalue problem.

Some Conclusions

Future work(i) Extension of the enrichment technique for: 3D domains; exterior problems; Bilaplacian eigenvalue problem.

(ii) Further analysis of the eigenvalue problem in elastodynamics.

(iii) Shape optimization problems in polygonal domains