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Fatih Ecevit Max Planck Institute for Mathematics in the Sciences

New Galerkin Methods for High-frequency Scattering Simulations. Fatih Ecevit Max Planck Institute for Mathematics in the Sciences. Collaborations. Universidad Pública de Navarra University of Bath. V í ctor Dom í nguez Ivan Graham. I. Electromagnetic & acoustic scattering problems. II.

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Fatih Ecevit Max Planck Institute for Mathematics in the Sciences

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  1. New Galerkin Methods forHigh-frequency Scattering Simulations Fatih Ecevit Max Planck Institute for Mathematics in the Sciences Collaborations Universidad Pública de Navarra University of Bath Víctor Domínguez Ivan Graham

  2. I. Electromagnetic & acoustic scattering problems II. High-frequency integral equation methods • Main principles (BGMR 2004) • A robust Galerkin scheme (DGS 2006) • Required improvements III. New Galerkin methods for high-frequency scattering simulations • Two new algorithms New Galerkin methods for high-frequency scattering simulations Outline

  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. High-frequency Integral Equation Methods Integral Equation Formulations Boundary Condition: Radiation Condition:

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

  8. II. High-frequency Integral Equation Methods Integral Equation Formulations Boundary Condition: Radiation Condition: Single layer potential: 1st kind 2nd kind Double layer potential: 2nd kind

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

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

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

  12. II. current High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density: Bruno, Geuzaine, Monro, Reitich (2004) Double layer density: is non-physical

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

  14. II. High-frequency Integral Equation Methods Single Convex Obstacle: Ansatz Single layer density: BGMR (2004)

  15. II. Highly oscillatory! High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach

  16. II. Highly oscillatory! High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach Localized Integration: for all n BGMR (2004)

  17. II. (Melrose & Taylor, 1985) High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach

  18. II. (Melrose & Taylor, 1985) High-frequency Integral Equation Methods Single Convex Obstacle A Convergent High-frequency Approach Change of Variables: BGMR (2004)

  19. II. High-frequency Integral Equation Methods Single Smooth Convex Obstacle • Bruno, Geuzaine, Monro, Reitich … 2004 … • Bruno, Geuzaine (3D)……………. 2006 …

  20. II. High-frequency Integral Equation Methods Single Smooth Convex Obstacle • Bruno, Geuzaine, Monro, Reitich … 2004 … • Bruno, Geuzaine (3D)……………. 2006 … • Huybrechs, Vandewalle …….…… 2006 …

  21. II. High-frequency Integral Equation Methods Single Smooth Convex Obstacle • Bruno, Geuzaine, Monro, Reitich … 2004 … • Bruno, Geuzaine (3D)……………. 2006 … • Huybrechs, Vandewalle …….…… 2006 … • Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.)

  22. II. High-frequency Integral Equation Methods Single Smooth Convex Obstacle • Bruno, Geuzaine, Monro, Reitich … 2004 … • Bruno, Geuzaine (3D)……………. 2006 … • Huybrechs, Vandewalle …….…… 2006 … • Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.) Single Convex Polygon • Chandler-Wilde, Langdon ….…….. 2006 .. • Langdon, Melenk …………..……… 2006 ..

  23. II. High-frequency Integral Equation Methods Single Smooth Convex Obstacle • Bruno, Geuzaine, Monro, Reitich … 2004 … • Bruno, Geuzaine (3D)……………. 2006 … • Huybrechs, Vandewalle …….…… 2006 … • Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.) • Domínguez, E., Graham, ………… 2007 … (circler bd.) Single Convex Polygon • Chandler-Wilde, Langdon ….…….. 2006 .. • Langdon, Melenk …………..……… 2006 ..

  24. II. A High-frequency Galerkin Method DGS (2006) The Combined Field Operator

  25. II. A High-frequency Galerkin Method DGS (2006) The Combined Field Operator Continuity: Giebermann (1997) circler domains …………… DGS (2006) general smooth domains …

  26. II. A High-frequency Galerkin Method DGS (2006) The Combined Field Operator Continuity: Giebermann (1997) circler domains …………… DGS (2006) general smooth domains … Coercivity: DGS (2006) circler domains …………… general smooth domains … open problem

  27. II. A High-frequency Galerkin Method DGS (2006) Plane-wave Scattering Problem

  28. II. is an explicitly defined entire function with known asymptotics are smooth periodic functions is not explicitly known but behaves like: A High-frequency Galerkin Method DGS (2006) Plane-wave Scattering Problem

  29. II. is an explicitly defined entire function with known asymptotics are smooth periodic functions is not explicitly known but behaves like: A High-frequency Galerkin Method DGS (2006) Plane-wave Scattering Problem Melrose, Taylor (1985) DGS (2006)

  30. II. A High-frequency Galerkin Method DGS (2006) Plane-wave Scattering Problem

  31. II. A High-frequency Galerkin Method DGS (2006) Plane-wave Scattering Problem for some on the “deep” shadow

  32. II. A High-frequency Galerkin Method DGS (2006) Plane-wave Scattering Problem for some on the “deep” shadow DGS (2006)

  33. II. A High-frequency Galerkin Method DGS (2006) Polynomial Approximation Deep Shadow Shadow Boundaries Illuminated Region

  34. II. A High-frequency Galerkin Method DGS (2006) Polynomial Approximation Deep Shadow Shadow Boundaries Illuminated Region … gluing together

  35. II. A High-frequency Galerkin Method DGS (2006) Polynomial Approximation Deep Shadow Shadow Boundaries Illuminated Region … gluing together

  36. II. A High-frequency Galerkin Method DGS (2006) Polynomial Approximation Deep Shadow Shadow Boundaries Illuminated Region … gluing together … approximation by zero

  37. II. A High-frequency Galerkin Method DGS (2006) Polynomial Approximation Deep Shadow Shadow Boundaries Illuminated Region … gluing together is the optimal choice

  38. II. A High-frequency Galerkin Method DGS (2006) Galerkin Method Deep Shadow Shadow Boundaries Illuminated Region … gluing together Discrete space

  39. II. A High-frequency Galerkin Method DGS (2006) Galerkin Method Deep Shadow Shadow Boundaries Illuminated Region … gluing together Final Estimate

  40. II. A High-frequency Galerkin Method DGS (2006) Galerkin Method Deep Shadow Shadow Boundaries Illuminated Region … gluing together Final Estimate QuestionCan one obtain a robust Galerkin method that works for higher frequencies as well as low frequencies?

  41. II. A High-frequency Galerkin Method DGS (2006) Galerkin Method Deep Shadow Shadow Boundaries Illuminated Region … gluing together Final Estimate In other words higher frequencies: low frequencies: do an approximation on the deep shadow region??

  42. II. A High-frequency Galerkin Method DGS (2006) Galerkin Method Deep Shadow Shadow Boundaries Illuminated Region … gluing together Final Estimate

  43. II. A High-frequency Galerkin Method DGS (2006) Galerkin Method Deep Shadow Shadow Boundaries Illuminated Region … gluing together Final Estimate In other words higher frequencies: low frequencies: do an approximation on the deep shadow region??

  44. III. A straightforward extension of the Galerkin approximation in DGS (2006) applies to deep shadow region New Galerkin methods for high-frequency scattering simulations New Galerkin Methods Deep Shadow Shadow Boundaries Illuminated Region … new Galerkin methods … gluing together Treat these four transition regionsseparately

  45. III. A straightforward extension of the Galerkin approximation in DGS (2006) applies to deep shadow region New Galerkin methods for high-frequency scattering simulations New Galerkin Methods Deep Shadow Shadow Boundaries Illuminated Region … new Galerkin methods … gluing together Treat these four transition regionsseparately The highly oscillatory integrals arising in the Galerkin matrices can be efficiently evaluated as the stationary phase points are apriory known

  46. III. New Galerkin methods for high-frequency scattering simulations New Galerkin Methods

  47. III. New Galerkin methods for high-frequency scattering simulations New Galerkin Methods

  48. III. New Galerkin methods for high-frequency scattering simulations New Galerkin Methods

  49. III. New Galerkin methods for high-frequency scattering simulations New Galerkin Methods

  50. III. New Galerkin methods for high-frequency scattering simulations New Galerkin Methods

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