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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
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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
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Governing Equations
(TE, TM, Acoustic)
Maxwell Eqns. Helmholtz Eqn.
Electromagnetic & Acoustic Scattering SimulationsI.
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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
Electromagnetic & Acoustic Scattering SimulationsI.
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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…
Electromagnetic & Acoustic Scattering SimulationsI.
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Integral Equation Formulations
Radiation Condition:
High-frequency Integral Equation MethodsII.
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Integral Equation Formulations
Radiation Condition:
Single layer potential:
High-frequency Integral Equation MethodsII.
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Integral Equation Formulations
Radiation Condition:
Single layer potential:
High-frequency Integral Equation MethodsII.
current
Single layer density:
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Single Convex Obstacle: AnsatzSingle layer density:
High-frequency Integral Equation MethodsII.
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Single Convex Obstacle: AnsatzSingle layer density:
High-frequency Integral Equation MethodsII.
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Single Convex Obstacle: AnsatzSingle layer density:
High-frequency Integral Equation MethodsII.
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Single Convex Obstacle: AnsatzSingle layer density:
High-frequency Integral Equation MethodsII.
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Single Convex Obstacle: AnsatzSingle layer density:
High-frequency Integral Equation MethodsII.
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Single Convex Obstacle
A Convergent High-frequency Approach
LocalizedIntegration:
Highly oscillatory!
High-frequency Integral Equation MethodsII.
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for all n
Single Convex Obstacle
A Convergent High-frequency Approach
LocalizedIntegration:
Highly oscillatory!
High-frequency Integral Equation MethodsII.
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for all n
Single Convex Obstacle
A Convergent High-frequency Approach
LocalizedIntegration:
Highly oscillatory!
High-frequency Integral Equation MethodsII.
BoundaryLayers:
(Melrose & Taylor, 1985)
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for all n
Single Convex Obstacle
A Convergent High-frequency Approach
LocalizedIntegration:
Highly oscillatory!
(Bruno & Reitich, 2004)
High-frequency Integral Equation MethodsII.
BoundaryLayers:
(Melrose & Taylor, 1985)
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Single Smooth Convex Obstacle
High-frequency Integral Equation MethodsII.
Bruno, Geuzeine, Reitich ………….. 2004 …
Bruno, Geuzeine (3D) …………….. 2006 …
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Single Smooth Convex Obstacle
High-frequency Integral Equation MethodsII.
Bruno, Geuzeine, Reitich ………….. 2004 …
Bruno, Geuzeine (3D) …………….. 2006 … holy grail !!
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Single Smooth Convex Obstacle
High-frequency Integral Equation MethodsII.
Domínguez, Graham, Smyshlyaev … 2006 … (circle/sphere)
Bruno, Geuzeine, Reitich ………….. 2004 …
Bruno, Geuzeine (3D) …………….. 2006 … holy grail !!
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Single Smooth Convex Obstacle
High-frequency Integral Equation MethodsII.
Domínguez, Graham, Smyshlyaev … 2006 … (circle/sphere)
Bruno, Geuzeine, Reitich ………….. 2004 …
Bruno, Geuzeine (3D) …………….. 2006 … holy grail !!
Huybrechs, Vandewalle …….…… 2006 …
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Single Smooth Convex Obstacle
High-frequency Integral Equation MethodsII.
Domínguez, Graham, Smyshlyaev … 2006 … (circle/sphere)
Bruno, Geuzaine, Reitich ………….. 2004 …
Bruno, Geuzaine (3D) …………….. 2006 …
Chandler-Wilde, Langdon ………… 2006 …
Langdon, Melenk …………..……… 2006 …
Single Convex Polygon (2D)
holy grail !!
Huybrechs, Vandewalle …….…… 2006 …
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering Configurations
High-frequency Integral Equation MethodsII.
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
Multiply with theinverse of thediagonal operator
Invert the diagonal:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
Invert the diagonal:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
Invert the diagonal:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Component form:
Invert the diagonal:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:
… Operator equation of the 2nd kind
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:
… Operator equation of the 2nd kind
… Neumann series
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:
… Operator equation of the 2nd kind
… Neumann series
is the superposition over all infinite pathsof the solutions of the integral equations
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:
… Operator equation of the 2nd kind
… 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
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
Generalized Phase Extraction: (for a collection of convex obstacles)
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Generalized Phase Extraction: (for a collection of convex obstacles)
… given by GO
Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
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Rearrangement into Sums over Periodic Orbits:can be represented as the superposition of the solution of the above
integral equations over primitive periodic orbits
Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
High-frequency Integral Equation MethodsII.
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
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A Convergent High-frequency Approach
High-frequency Integral Equation MethodsII.
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A Convergent High-frequency Approach
High-frequency Integral Equation MethodsII.
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A Convergent High-frequency Approach
High-frequency Integral Equation MethodsII.
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A Convergent High-frequency Approach
High-frequency Integral Equation MethodsII.
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Iteration 1 Iteration 2 Iteration 3
Iteration 10
A Convergent High-frequency ApproachIterated Currents:
High-frequency Integral Equation MethodsII.
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A Convergent High-frequency Approach
1st reflections 2nd reflections 3rd reflections
Iterated Phase Functions:
High-frequency Integral Equation MethodsII.
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A Convergent High-frequency Approach
Noreflections
3rdreflections
1streflections
2ndreflections
Iterated Phases on Patches
High-frequency Integral Equation MethodsII.
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A Convergent High-frequency ApproachShadow Boundaries:
High-frequency Integral Equation MethodsII.
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Asymptotic Expansions in 2DOn Illuminated Regions: (E., Reitich, 2006)
and are defined recursively as
Here
and for
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 3DOn Illuminated Regions: (Anand, Boubendir, E., Reitich, 2006)
are defined recursively asHere
and for
where
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 3DOn Illuminated Regions: (Anand, Boubendir, E., Reitich, 2006)
are defined recursively asHere
and for
where
Asymptotic Expansions of Multiple Scattering IterationsIII.
Asymptotic expansions ofthe surface current for thevector electromagnetic case(E., Hackbusch, Kriemann, 2006)
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Intuition … Fermat’s principle
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
Thanks to Daan Huybrechs for teaching mehow to make movies in Matlab
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Asymptotic Expansions in 2D
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Periodic Phase Minimizer:
with
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Periodic Phase Differences:
Periodic Phase Minimizer:
with
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Periodic Phase Differences:
Periodic Ratios:
Periodic Phase Minimizer:
with
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Periodic Phase Differences:
Periodic Ratios:
Periodic Phase Minimizer:
with
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Periodic Phase Differences:
Periodic Ratios:
Periodic Ratios:
Periodic Phase Minimizer:
with
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Solutions of explicit quadratic equations
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Solutions of explicit quadratic equations
Asymptotic Expansions of Multiple Scattering IterationsIII.
2-Dimensions:
curvatures
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Solutions of explicit quadratic equations
Asymptotic Expansions of Multiple Scattering IterationsIII.
2-Dimensions:
curvatures 3-Dimensions:
principal curvatures matrix
rotation
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Concerning Approximate Currents:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Concerning Approximate Currents:
Concerning Exact Currents:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Rate of Convergence:
Concerning Approximate Currents:
Concerning Exact Currents:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:
Asymptotic Expansions of Multiple Scattering IterationsIII.
Extension of Rate of Convergence over the Entire Boundaries:
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Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:
Asymptotic Expansions of Multiple Scattering IterationsIII.
Extension of Rate of Convergence over the Entire Boundaries:
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Rate of Convergence on Periodic Orbits Behavior of Illuminated Regions:
Asymptotic Expansions of Multiple Scattering IterationsIII.
Extension of Rate of Convergence over the Entire Boundaries:
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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
Asymptotic Expansions of Multiple Scattering IterationsIII.
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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:
Asymptotic Expansions of Multiple Scattering IterationsIII.
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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:
Asymptotic Expansions of Multiple Scattering IterationsIII.
Displayed in Numerical Examples:
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IV. Numerical Examples & Acceleration of Convergence
2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
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2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2DIV. Numerical Examples & Acceleration of Convergence
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2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2DIV. Numerical Examples & Acceleration of Convergence
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2 Periodic Example:
Point SourceIllumination
Numerical Examples in 2DIV. Numerical Examples & Acceleration of Convergence
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3 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2DIV. Numerical Examples & Acceleration of Convergence
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3 Periodic Example:
Point SourceIllumination
Numerical Examples in 2DIV. Numerical Examples & Acceleration of Convergence
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Acceleration of ConvergenceIV. Numerical Examples & Acceleration of Convergence
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Acceleration of ConvergenceIV. Numerical Examples & Acceleration of Convergence
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Acceleration of ConvergenceIV. Numerical Examples & Acceleration of Convergence
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Acceleration of ConvergenceIV. Numerical Examples & Acceleration of Convergence
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Acceleration of ConvergenceIV. Numerical Examples & Acceleration of Convergence
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2 Periodic Example:
0.07240.07400.07850.0718
Iteration 1 Iteration 2 Iteration 3
Iteration 10
Numerical Examples in 3DIV. Numerical Examples & Acceleration of Convergence
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Numerical Examples in 3D
Two ellipsoids and with radii and centers and
IV. Numerical Examples & Acceleration of Convergence
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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)
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Thanks