fatih ecevit bogazici university, istanbul · fatih ecevit bogazici university, istanbul convergent...
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Fatih EcevitBogazici University, Istanbul
Convergent Scattering Algorithms
Joint work with: Fernando Reitich, University of Minnesota
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Integral Equation Formulations
Radiation Condition:
Single layer potential:
current
Single layer density:
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Integral Equation Formulations: AnsatzSingle layer density:
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Integral Equation Formulations: AnsatzSingle layer density:
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Integral Equation Formulations: AnsatzSingle layer density:
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: open subset of: open conic subset of i.e.
: Hoermander Class of order and
(multi-indices), s.t. compact,
A little bit of microlocal analysis:Hoermander Classes
invariant under diffeomorphisms in the x variable
generalizes to the case where is a smooth manifold
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: open subset of: open conic subset of i.e.
where as
A little bit of microlocal analysis:Asymptotic Expansions
Asymptotic Expansion of :
for
andwhere
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: compact, smooth, strictly convex
compact
Asymptotic Expansions of
Theorem (R.Melrose & M.Taylor - ‘85) :
i.e.
On the illuminated region
i.e.
On the shadow region
decays rapidly in the sense of Schwarz as
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compact
i.e.
i.e.
On a vicinity of the shadow boundary
Positive on the illuminated regionNegative on the shadow regionVanishes precisely to first order at the shadow boundary{
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Theorem (Domínguez, Graham, Smyshlyaev ‘07): … derivative estimates
… arclength parametrization
… shadow boundaries
… resembles the behavior of
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Several Numerical Algorithms
Domínguez, Graham, Smyshlyaev … 2007 …
Bruno, Geuzaine, Monro, Reitich … 2004 …
Bruno, Geuzaine (3D) ……………. 2007 …
Huybrechs, Vandewalle …….…… 2007 …
Domínguez, E., Graham, ………… 2007 …
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Multiple Scattering Configurations
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Multiple Scattering Configurations
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
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:
Disjoint Scatterers:Component form:
Invert the diagonal:
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Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
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:
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
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Generalized Phase Extraction: (for a collection of convex obstacles)
… given by GO
Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
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Visibility:
No-occlusion:
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Broken rays: well-defined, existence, uniqueness
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Broken rays: well-defined, existence, uniqueness
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Broken rays: illuminated regions (IL)shadow regions (SR)shadow boundaries (SB)
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… convexwave fronts
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Asymptotic Expansions of Scattered Fields
Theorem (E., Reitich 2009):
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Theorem (E., Reitich 2009):planewave incidence…ansatz………………….
On the illuminated region:
… Hoermander classes
……… asymptotic expansions
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Theorem (E., Reitich 2009):planewave incidence…ansatz………………….
Over the entire boundary:
……………………. Hoermander classes
… asymptotic expansions
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Theorem (E., Reitich 2009):planewave incidence…ansatz………………….
Over the entire boundary:
……………………. Hoermander classes
… asymptotic expansions
On the illuminated region:
… Hoermander classes
……… asymptotic expansions
Extends in the same way to 3D
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Theorem (E., Reitich 2009): … derivative estimates
… arclength parametrization
… shadow boundaries
… resembles the behavior of
…extension of single scattering results in DGS (2006) to multiple scattering
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GeneralizedGeometrical OpticsApproximations
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Asymptotic Expansions in 2DTheorem: (E., Reitich) For any , the iterated density satisfies
on any compact subset of the m-th illuminated region as Here, isdefined over the entire boundary by
withand
and defined recursively as
where
and
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Acoustic Asymptotic Expansions in 3DTheorem: (Anand, Boubendir, E., Reitich)density satisfies
on any compact subset of the m-th illuminated region as Here, isdefined over the entire boundary by
withand
and defined recursively as
where
and
For any , the iterated
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Acoustic Asymptotic Expansions in 3DTheorem: (Anand, Boubendir, E., Reitich)
where
and
Here, defining
we have set
and
Finally
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Electromagnetic Asymptotic Expansions in 3D
Radiation Condition: Silver-Muller radiation condition
in
Perfect Conductor: on
onthe scattered electromagnetic field can be recovered through theclassical Stratton-Chu formulae.
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Electromagnetic Asymptotic Expansions in 3DTheorem: (E., Hackbusch)
on any compact subset of the m-th illuminated region asHere, is defined over the entire boundary by
with
For any , the iterated surface current satisfies
and
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Rate of Convergence on Periodic Orbits Periodic Phase on:
Periodic Phase Minimizer:
with Rate of Convergence:
Solutions of explicitquadratic equations
curvatures
principalcurvatures matrix
rotation
3D:
2D:
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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:
Displayed in Numerical Examples:
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2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
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2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
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2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
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2 Periodic Example:
Point SourceIllumination
Numerical Examples in 2D
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3 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
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3 Periodic Example:
Point SourceIllumination
Numerical Examples in 2D
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2 Periodic Example:
0.07240.07400.07850.0718
Iteration 1 Iteration 2 Iteration 3
Iteration 10
Numerical Examples in 3D
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Numerical Examples in 3D
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Numerical Examples in 3D
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Thanks