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  1. Asymptotics for high-frequencymultiple scattering Fatih Ecevit Max Planck Institute for Mathematics in the Sciences Collaborations Caltech University of Minnesota Max Planck Institute for MIS Max Planck Institute for MIS University of Minnesota Akash Anand Yassine Boubendir Wolfgang Hackbusch Ronald Kriemann Fernando Reitich

  2. I. Electromagnetic & acoustic scattering problems II. High-frequency integral equation methods • Single convex obstacle • Generalization to multiple scattering configurations • Interpretation of the series and rearrangement into periodic orbit sums III. Asymptotic expansions of iterated currents • Asymptotic expansion on arbitrary orbits • Rate of convergence formulas on periodic orbits High-frequency scattering by a collection of convex bodies Outline IV. Numerical examples & acceleration of convergence

  3. I. Electromagnetic & Acoustic Scattering Simulations Governing Equations Maxwell Eqns. Helmholtz Eqn. (TE, TM, Acoustic)

  4. I. Electromagnetic & Acoustic Scattering Simulations Scattering Simulations Basic Challenges: Fields oscillate on the order of wavelength • Computational cost • Memory requirement Numerical Methods: Convergent (error-controllable) • Variational methods (MoM, FEM, FVM,…) • Differential Eqn. methods (FDTD,…) • Integral Eqn. methods(FMM, H-matrices,…) • Asymptotic methods(GO, GTD,…) Demand resolution of wavelength Discretization independent of frequency Non-convergent (error )

  5. I. Combine… Electromagnetic & Acoustic Scattering Simulations Scattering Simulations Basic Challenges: Fields oscillate on the order of wavelength • Computational cost • Memory requirement Numerical Methods: Convergent (error-controllable) • Variational methods (MoM, FEM, FVM,…) • Differential Eqn. methods (FDTD,…) • Integral Eqn. methods(FMM, H-matrices,…) • Asymptotic methods(GO, GTD,…) Demand resolution of wavelength Discretization independent of frequency Non-convergent (error )

  6. II. Radiation Condition: High-frequency Integral Equation Methods Integral Equation Formulations

  7. II. Radiation Condition: Single layer potential: High-frequency Integral Equation Methods Integral Equation Formulations

  8. II. Radiation Condition: Single layer potential: current Single layer density: High-frequency Integral Equation Methods Integral Equation Formulations

  9. II. High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density:

  10. II. High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density:

  11. II. High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density:

  12. II. High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density:

  13. II. High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density:

  14. II. Highly oscillatory! High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach Localized Integration:

  15. II. Highly oscillatory! for all n High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach Localized Integration:

  16. II. Highly oscillatory! for all n Boundary Layers: (Melrose & Taylor, 1985) High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach Localized Integration:

  17. II. Highly oscillatory! for all n Boundary Layers: (Bruno & Reitich, 2004) (Melrose & Taylor, 1985) High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach Localized Integration:

  18. II. • Bruno, Geuzaine, Reitich ………….. 2004 … holy grail !! • Bruno, Geuzaine (3D)…………….. 2006 … • Huybrechs, Vandewalle …….…… 2006 … • Domínguez, E., Graham, ………… 2007 … (circle) Single Convex Polygon(2D) • Chandler-Wilde, Langdon ………… 2006 … • Langdon, Melenk …………..……… 2006 … High-frequency Integral Equation Methods Single Smooth Convex Obstacle • Domínguez, Graham, Smyshlyaev … 2006 … (circle)

  19. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  20. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  21. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  22. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  23. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  24. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  25. II. High-frequency Integral Equation Methods Multiple Scattering Configurations

  26. II. High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  27. II. Disjoint Scatterers: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  28. II. Disjoint Scatterers: Component form: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  29. II. Disjoint Scatterers: Component form: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  30. II. Disjoint Scatterers: Component form: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  31. II. Disjoint Scatterers: Component form: Multiply with the inverse of the diagonal operator High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Invert the diagonal:

  32. II. Disjoint Scatterers: Component form: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Invert the diagonal:

  33. II. Disjoint Scatterers: Component form: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Invert the diagonal:

  34. II. Disjoint Scatterers: Component form: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Invert the diagonal:

  35. II. Disjoint Scatterers: … Operator equation of the 2nd kind High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  36. II. Disjoint Scatterers: … Operator equation of the 2nd kind … Neumann series High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  37. II. Disjoint Scatterers: … Operator equation of the 2nd kind … Neumann series is the superposition over all infinite paths of the solutions of the integral equations High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  38. II. Disjoint Scatterers: … Operator equation of the 2nd kind … Neumann series is the superposition over all infinite paths of the solutions of the integral equations twice the normal derivative (evaluated on ) of the field scattered from High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind:

  39. II. Disjoint Scatterers: High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Reduction to the Interaction of Two-substructures:

  40. II. Disjoint Scatterers: Generalized Phase Extraction:(for a collection of convex obstacles) High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Reduction to the Interaction of Two-substructures:

  41. II. Disjoint Scatterers: Generalized Phase Extraction:(for a collection of convex obstacles) … given by GO High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Reduction to the Interaction of Two-substructures:

  42. II. Disjoint Scatterers: Rearrangement into Sums over Periodic Orbits: can be represented as the superposition of the solution of the above integral equations overprimitive periodic orbits High-frequency Integral Equation Methods Multiple Scattering Formulation Integral Equation of the 2nd Kind: Reduction to the Interaction of Two-substructures:

  43. II. High-frequency Integral Equation Methods A Convergent High-frequency Approach

  44. II. High-frequency Integral Equation Methods A Convergent High-frequency Approach

  45. II. High-frequency Integral Equation Methods A Convergent High-frequency Approach

  46. II. High-frequency Integral Equation Methods A Convergent High-frequency Approach

  47. II. Iteration 1 Iteration 2 Iteration 3 Iteration 10 High-frequency Integral Equation Methods A Convergent High-frequency Approach Iterated Currents:

  48. II. 3rd reflections 2nd reflections High-frequency Integral Equation Methods A Convergent High-frequency Approach Iterated Phase Functions: 1st reflections

  49. II. No reflections 1st reflections 3rd reflections 2nd reflections High-frequency Integral Equation Methods A Convergent High-frequency Approach Iterated Phases on Patches

  50. II. High-frequency Integral Equation Methods A Convergent High-frequency Approach Shadow Boundaries: