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April 2009

The Method of Fundamental Solutions applied to eigenproblems in partial differential equations. Pedro R. S. Antunes - CEMAT (joint work with C. Alves). April 2009. Outline.  Experimental results of resonance  Eigenvalue problem for the Laplacian - some results and questions

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April 2009

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  1. The Method of Fundamental Solutions applied to eigenproblems in partial differential equations Pedro R. S. Antunes - CEMAT (joint work with C. Alves) April 2009

  2. Outline 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

  3. Experimental results of resonance

  4. Chladni figures experimental nodal lines

  5. Eigenvalue problem for the Laplacian

  6.  0 l1 l2 ... ln ...   • 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) General results:  Countable number of eigenvalues  The sequence goes to infinity 1>0 1=0

  7. Numerical methods – eigenfrequency calculation

  8. Numerical methods - eigenfrequency calculation Finite Elements, Finite Differences, Boundary Elements Consider the rigidity matrix Ah(k). (h – discretization parameter, k – frequency) Fixed h, search k : matrix is not invertible (eg: det(Ah(k)) = 0 ) Meshless Methods • 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)

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

  10.   • Theoretical results Given an open set Rd, different points and kC, are linearly independent on . The set is dense in L2(), when  is an admissible curve.  is not an eigenfrequency

  11. xi yi  • Algorithm for the source points (2D) Consider m points x1 ,…, xmcollocation points (almost equally spaced) Define m points y1 ,…, ymsource points

  12. Circle: Plot of Log[g(k)] BesselJ zeros (exact values) • Algorithm for the eigenfrequency calculation Build the matrices Consider g(k)=|det(Am(k))| and look for the minima Due to the ill conditioning of the matrix g(k) is too small Search for local minimum using the Golden Ratio Search

  13.  y0 x0 • Algorithm for the eigenmode calculation Define extra points { The extra point x0is not on a nodal line Given the approximate eigenfrequency k, define To calculate j solve the system { - non null solution, - null at boundary points

  14. Error bounds (Dirichlet case) A posteriori bound (Moler and Payne 1968) 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 .

  15. Numerical Tests (algorithm validation)

  16. Numerical tests (Dirichlet case) – 2D m=dimension of the matrix

  17. Numerical tests (Dirichlet case) – 3D

  18. Numerical tests (on the location of point sources) Point-sources on a boundary of a circular domain Point-sources on an “expansion” of  With the choice proposed Big rounding errors Plot of Log(g(k))

  19. Numerical tests (almost double eigenvalues) Plot of Log(g(k)) with n=60 1+10-8 1 k3-k2≈4.2110-8

  20. Numerical Simulations (non trivial domains)

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

  22. Mixed problemDirichlet - external boundaryNeumann - internal boundary • Numerical Simulations (Dirichlet and Mixed boundary conditions) Dirichlet problem eigenmode nodal lines plot

  23. Numerical simulations – non trivial domains 3D 3D plots of eigenfunctions associated to three resonance frequencies

  24. Domains with corners or cracks Hybrid method

  25. Hybrid method – domains with corners or cracks The classical MFS is not accurate for corner/crack singularities  k(x-yj) is analytic in  ()  u is singular at some corners  If  has an interior angle / (with  irrational), then Lehmann (1959)

  26. u = uReg + uSing 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

  27. 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 xi zi yi MB MI Eigenfrequencies NR NS k • Hybrid method – eigenfrequency calculation (Betcke-Trefethen subspace angle technique)

  28. Hybrid method – Dirichlet problem with cracks 1st eigenfrequency NR=80, MB=180 NS=10, MI=30 2nd eigenfrequency 5th eigenfrequency

  29. Hybrid method – mixed problem NR=200, MB=300 NS=7, MI=30 9th eigenfrequency 1st eigenfrequency 5th eigenfrequency

  30. Shape optimization problems

  31.  Given a quantity depending on some eigenvalues, we want to find a domain which optimizes  Direct setting ... lN 0 l1 l2 Inverse setting • Shape optimization problems

  32. Inverse eigenvalue problems Existence issue:The inverse problem may not have a solution. There are some restrictions to the admissible sets of eigenvalues, eg. Payne & Pölya & Weinberger (1956)  Ashbaugh & Benguria (1991)

  33. Inverse eigenvalue problem Uniqueness issue: No unique solution, in general. 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

  34. MFS  • Shape optimization problem – numerical solution Define the class of star-shaped domains with boundary given by where r is continuous (2)-periodic function  Consider the approximation (M) 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.

  35. Numerical results shape optimization problems

  36. The ball maximizesAshbaugh & Benguria (1991) • 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 L&Y = 3.202... Optimal shape

  37. Can one hear the sound of Riemann Hypothesis? “Harmonic drum” Kane-Shoenauer 1(1995) Kane-Shoenauer 2 (1995) Our approach A drum with the first 12 eigenfrequencies ~ 12 first Im(zeros) of Zeta function k2 / k1 1,5 1,5621 1,4173 1,50041 k3 / k1 1,5 1,6764 1,5229 1,50093 k4 / k1 2 2,0760 1,9327 1,99952 • Numerical results - shape optimization problem Is it possible to build a drum with an almost well defined pitch (fifth and the octave): Optimal shape Is there a drum playing all the non trivial zeros of the Zeta function?(modulo asymptotic behaviour)

  38. Numerical results - shape optimization problems • Optimization problem (Dirichlet) 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 Numerical counterexample(Alves & A., 2005): Elliptical stadium37.9875443 < 38.0021483 (stadium) (3D)

  39. Other PDEs:Bilaplacian

  40.  R2 • Eigenvalue problem for the bilaplacian Search for k (eigenfrequencies) such that there exist non null function u (eigenmodes) : General results:  Countable number of eigenfrequencies  The sequence goes to infinity k1>0

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

  42. Given an open set 2, different points and kC, are linearly independent on . Theorem (density result) If γ is the boundary of a domain which contains , the set is dense in H3/2(). • Theoretical results  is not an eigenfrequency

  43. Eigenfrequency/eigenmode calculation Eigenfrequency calculation Build the matrix M with the four blocks mm di,j=xi-yj Define g(k)=|det(M(k))| and calculate the minima Eigenmode calculation Extra collocation point

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

  45. Numerical results - bilaplacian

  46. Numerical results – location of point sources The proposed algorithm for the source points again presents more stable results Big rounding errors Plot of Log(g(k))

  47. Numerical simulations – equilateral triangle The eigenfunction associated to the first eigenvalue of the plate problem changes the sign “near” each corner

  48. Numerical simulations – non trivial domains 3D plots and nodal domains for the 3rd,7th,10th and 11th resonance frequencies

  49. Other PDEs:Elastodynamics (2D)

  50.  R2 • Eigenvalue problem for elastodynamics Fundamental solution: Kupradze’s tensor MFS: Invertibility of the matrix

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