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

II.

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:




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:




Invert the diagonal:




… Operator equation of the 2nd kind





… Neumann series





… 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)

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

Here

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

where


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

are defined recursively asHere

and for

where


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:

with



Periodic Phase Differences:


with




Periodic Ratios:


with




Periodic Ratios:

Periodic Ratios:


with


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 Approximate Currents:

Concerning Exact Currents:


Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:


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:




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:



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

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