fatih ecevit max planck institute for mathematics in the sciences

Fatih EcevitMax Planck Institute for Mathematics in the Sciences

Akash AnandYassine Boubendir

Wolfgang HackbuschRonald KriemannFernando Reitich

High-frequency scattering bya collection of convex bodies

CaltechUniversity of MinnesotaMax Planck Institute for MISMax Planck Institute for MISUniversity of Minnesota

Collaborations

Outline

High-frequency integral equation methods Single convex obstacle Generalization to multiple scattering configurations Interpretation of the series and rearrangement into

periodic orbit sums

Numerical examples & acceleration of convergenceIV.

Asymptotic expansions of iterated currentsIII. Asymptotic expansion on arbitrary orbits Rate of convergence formulas on periodic orbits

Electromagnetic & acoustic scattering problemsI.

High-frequency scattering by a collection of convex bodies

Governing Equations

(TE, TM, Acoustic)

Maxwell Eqns. Helmholtz Eqn.

Electromagnetic & Acoustic Scattering SimulationsI.

Scattering Simulations

Basic Challenges:Fields oscillate on the order of wavelength Computational cost Memory requirement

Variational methods (MoM, FEM, FVM,…) Differential Eqn. methods (FDTD,…) Integral Eqn. methods (FMM, H-matrices,…)

Asymptotic methods (GO, GTD,…)

Numerical Methods:Convergent (error-controllable) Demand resolutionof wavelength

Non-convergent (error )

Discretization independentof frequency

Scattering Simulations

Basic Challenges:Fields oscillate on the order of wavelength Computational cost Memory requirement

Variational methods (MoM, FEM, FVM,…) Differential Eqn. methods (FDTD,…) Integral Eqn. methods (FMM, H-matrices,…)

Asymptotic methods (GO, GTD,…)

Numerical Methods:Convergent (error-controllable) Demand resolutionof wavelength

Non-convergent (error )

Discretization independentof frequency

Combine…

Integral Equation Formulations

Radiation Condition:

High-frequency Integral Equation MethodsII.

Single layer potential:

current

Single layer density:

Single Convex Obstacle: AnsatzSingle layer density:

Single Convex Obstacle

A Convergent High-frequency Approach

LocalizedIntegration:

Highly oscillatory!

for all n

Highly oscillatory!

for all n

Highly oscillatory!

BoundaryLayers:

(Melrose & Taylor, 1985)

for all n

Highly oscillatory!

(Bruno & Reitich, 2004)

BoundaryLayers:

(Melrose & Taylor, 1985)

Single Smooth Convex Obstacle

Bruno, Geuzeine, Reitich ………….. 2004 …

Bruno, Geuzeine (3D) …………….. 2006 …

Bruno, Geuzeine (3D) …………….. 2006 … holy grail !!

Domínguez, Graham, Smyshlyaev … 2006 … (circle/sphere)

Huybrechs, Vandewalle …….…… 2006 …

Bruno, Geuzaine, Reitich ………….. 2004 …

Bruno, Geuzaine (3D) …………….. 2006 …

Chandler-Wilde, Langdon ………… 2006 …

Langdon, Melenk …………..……… 2006 …

Single Convex Polygon (2D)

holy grail !!

Huybrechs, Vandewalle …….…… 2006 …

Multiple Scattering Configurations

Multiple Scattering FormulationIntegral Equation of the 2nd Kind:

Disjoint Scatterers:

Disjoint Scatterers:Component form:

Multiply with theinverse of thediagonal operator

Invert the diagonal:

… Operator equation of the 2nd kind

… Neumann series

is the superposition over all infinite pathsof the solutions of the integral equations

… Neumann series

twice the normal derivative (evaluated on )

of the field scattered from

is the superposition over all infinite pathsof the solutions of the integral equations

Disjoint Scatterers:Reduction to the Interaction of Two-substructures:

Generalized Phase Extraction: (for a collection of convex obstacles)

… given by GO

Rearrangement into Sums over Periodic Orbits:can be represented as the superposition of the solution of the above

integral equations over primitive periodic orbits

Iteration 1 Iteration 2 Iteration 3

Iteration 10

A Convergent High-frequency ApproachIterated Currents:

1st reflections 2nd reflections 3rd reflections

Iterated Phase Functions:

Noreflections

3rdreflections

1streflections

2ndreflections

Iterated Phases on Patches

A Convergent High-frequency ApproachShadow Boundaries:

Asymptotic Expansions in 2DOn Illuminated Regions: (E., Reitich, 2006)

and are defined recursively as

and for

Asymptotic Expansions of Multiple Scattering IterationsIII.

Asymptotic Expansions in 3DOn Illuminated Regions: (Anand, Boubendir, E., Reitich, 2006)

are defined recursively asHere

and for

Asymptotic Expansions in 3DOn Illuminated Regions: (Anand, Boubendir, E., Reitich, 2006)

are defined recursively asHere

and for

Asymptotic expansions ofthe surface current for thevector electromagnetic case(E., Hackbusch, Kriemann, 2006)

Asymptotic Expansions in 2D

Intuition … Fermat’s principle

Thanks to Daan Huybrechs for teaching mehow to make movies in Matlab

Rate of Convergence on Periodic Orbits Periodic Phase on:

Periodic Phase Minimizer:

Periodic Phase Differences:

Periodic Ratios:

Rate of Convergence on Periodic Orbits Rate of Convergence:

Solutions of explicit quadratic equations

2-Dimensions:

curvatures

2-Dimensions:

curvatures 3-Dimensions:

principal curvatures matrix

rotation

Concerning Approximate Currents:

Concerning Exact Currents:

Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:

Extension of Rate of Convergence over the Entire Boundaries:

Rate of Convergence on Periodic Orbits In Summary:

depend only on the geometry and the direction of incidence.The constants involved in the order terms, and

Numerically for a fixed periodic orbit:

Displayed in Numerical Examples:

IV. Numerical Examples & Acceleration of Convergence

2 Periodic Example:

PlanewaveIllumination

Numerical Examples in 2D

2 Periodic Example:

Numerical Examples in 2DIV. Numerical Examples & Acceleration of Convergence

2 Periodic Example:

Point SourceIllumination

3 Periodic Example:

Point SourceIllumination

Acceleration of ConvergenceIV. Numerical Examples & Acceleration of Convergence

2 Periodic Example:

0.07240.07400.07850.0718

Iteration 1 Iteration 2 Iteration 3

Iteration 10

Numerical Examples in 3D

Two ellipsoids and with radii and centers and

IV. Numerical Examples & Acceleration of Convergence

Maxwell Equations: (with Hackbusch & Kriemann)

Ongoing Work and Future Directions

• Magnetic field integral equation (MFIE) for the “surface current”• Numerical implementation utilizing Hierarchical Matrices (ongoing)

Overall Acceleration of Convergence: (ongoing)

• Comparison of different periodic orbit contributions• Acceleration of convergence via a “generalized” Pade approximation

Alternative Acceleration Strategies: (with Boubendir & Reitich)

• A Krylov subspace approach • Preconditioning based on asymptotic analysis

New Artificial Boundary Conditions: (with Boubendir)• For use with FEM utilizing asymptotic analysis (ongoing)

Thanks

fatih ecevit max planck institute for mathematics in the sciences

methods fmm

numerical methods

asymptotic methods

methods fdtd

ansatzsingle layer density

mismax planck institute

helmholtz eqn

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