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 lines Chladni figures

Eigenvalue problem for the Laplacian

 Search for k (eigenfrequencies) such that there exists non null function u (eigenmodes) :   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 A h (k). (h – discretization parameter, k – frequency)  Fixed h, search k : matrix is not invertible (eg: det(A h (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

   The Method of Fundamental Solution (MFS) Fundamental solution:  Consider the approximation  The coefficients  j are calculated such that fits the boundary conditions  an admissible curve

 Given an open set  R d, different points and k  C, are linearly independent on .  The set is dense in L 2 (  ), when  is an admissible curve. Theoretical results     is not an eigenfrequency

 Consider m points x 1,…, x m  collocation points (almost equally spaced)  Define m points y 1,…, y m source points  xixi yiyi  Algorithm for the source points (2D)

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  Build the matrices  Consider g(k)=|det(A m (k))| and look for the minima

 To calculate  j solve the system  Given the approximate eigenfrequency k, define - non null solution, - null at boundary points The extra point x 0 is not on a nodal line Algorithm for the eigenmode calculation   x0x0 y0y0 Define extra points { {

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 . A posteriori bound (Moler and Payne 1968)

Numerical Tests (algorithm validation)

m=dimension of the matrix mabs. error (k 1 )mabs. error (k 2 )mabs. error (k 3 )             mabs. error (k 1 )mabs. error (k 2 )mabs. error (k 3 )             mabs. error (k 5 )m m          Numerical tests (Dirichlet case) – 2D

m abs. error (k 1 ) mabs. error (k 2 )mabs. error (k 3 )          mabs. error (k 1 )mabs. error (k 2 )mabs. error (k 3 )          mabs. error (k 5 )m m    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)

mabs. error (k 2 )abs. error (k 3 )       k 3 -k 2 ≈4.21  Plot of Log(g(k)) with n=60 Numerical tests (almost double eigenvalues)

Numerical Simulations (non trivial domains)

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

(Dirichlet and Mixed boundary conditions) Dirichlet problem Mixed problem Dirichlet - external boundary Neumann - 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-y j ) is analytic in  ( )  If  has an interior angle  /  (with  irrational), then Lehmann (1959)  u is singular at some corners The classical MFS is not accurate for corner/crack singularities

u = u Reg + u Sing MFS approximation particular solutions 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) xixi yiyi zizi Eigenfrequencies  Choose randomly M I points z i   Build the matrices  Calculate A=QR factorization where  Calculate, the smallest singular value of Q B (k)  Look for the minima MIMI MBMB NRNR NSNS k Hybrid method – eigenfrequency calculation

N R =80, M B =180 N S =10, M I =30 1 st eigenfrequency 2 nd eigenfrequency5 th eigenfrequency Hybrid method – Dirichlet problem with cracks

Hybrid method – mixed problem 1 st eigenfrequency5 th eigenfrequency 9 th eigenfrequency N R =200, M B =300 N S =7, M I =30

Shape optimization problems

Given a quantity depending on some eigenvalues, we want to find a domain which optimizes Direct setting     0   Inverse setting Shape optimization problems

 Payne & Pölya & Weinberger (1956)  Ashbaugh & Benguria (1991) There are some restrictions to the admissible sets of eigenvalues, eg. 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 where r is continuous (2  )-periodic function  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.   MFS Shape optimization problem – numerical solution  Consider the approximation (M   )

Numerical results shape 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.  We obtained Numerical results - shape optimization problems  The ball maximizes Ashbaugh & Benguria (1991) Optimal shape L&Y =

Is it possible to build a drum with an almost well defined pitch (fifth and the octave) : 1,999521,93272,07602 k 4 / k 1 1,500931,52291,67641,5 k 3 / k 1 1,500411,41731,56211,5 k 2 / k 1 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 unknown Troesch - conjectured the stadium Henrot&Oudet - reffuted - no circular parts Optimization problem (Dirichlet) Numerical counterexample (Alves & A., 2005): Elliptical stadium < (stadium) (3D) Numerical results - shape optimization problems

Other PDEs: Bilaplacian

Search for k (eigenfrequencies) such that there exist non null function u (eigenmodes) :   R 2 Eigenvalue problem for the bilaplacian General results:  Countable number of eigenfrequencies  The sequence goes to infinity  k 1 >0

   The MFS application to the bilaplacian problem  Consider the approximation (m   )  an admissible curve Fundamental solution:  is analytic in   satisfies the PDE

 Given an open set  2, different points and k  C, are linearly independent on . Theoretical results If γ is the boundary of a domain which contains , the set is dense in H 3/2 (  ). Theorem (density result)  is not an eigenfrequency

 Build the matrix M d i,j =x i -y j with the four blocks m  m Eigenfrequency/eigenmode calculation Eigenfrequency calculation  Define g(k)=|det(M(k))| and calculate the minima Eigenmode calculation  Extra collocation point

Let and be an approximate eigenfrequency and eigenfunction which satisfies the problem Then there exists an eigenfrequency k such that Error bounds Theorem (a posteriori bound) where Generalizes Moler-Payne’s result for the bilaplacian

mabs. error (k 1 )mabs. error (k 2 )mabs. error (k 3 )             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 3 rd,7 th,10 th and 11 th resonance frequencies Numerical simulations – non trivial domains

Other PDEs: Elastodynamics (2D)

  R 2 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 component2nd 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