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Fatih EcevitBogazici University, Istanbul
Convergent Scattering Algorithms
Joint work with: Fernando Reitich, University of Minnesota
Integral Equation Formulations
Radiation Condition:
Single layer potential:
current
Single layer density:
Integral Equation Formulations: AnsatzSingle layer density:
Integral Equation Formulations: AnsatzSingle layer density:
Integral Equation Formulations: AnsatzSingle layer density:
: 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
: open subset of: open conic subset of i.e.
where as
A little bit of microlocal analysis:Asymptotic Expansions
Asymptotic Expansion of :
for
andwhere
: 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
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{
Theorem (Domínguez, Graham, Smyshlyaev ‘07): … derivative estimates
… arclength parametrization
… shadow boundaries
… resembles the behavior of
Several Numerical Algorithms
Domínguez, Graham, Smyshlyaev … 2007 …
Bruno, Geuzaine, Monro, Reitich … 2004 …
Bruno, Geuzaine (3D) ……………. 2007 …
Huybrechs, Vandewalle …….…… 2007 …
Domínguez, E., Graham, ………… 2007 …
Multiple Scattering Configurations
Multiple Scattering Configurations
Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
Disjoint Scatterers:Component form:
Multiply with theinverse of thediagonal operator
Invert the diagonal:
Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
Disjoint Scatterers:Component form:
Invert the diagonal:
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
Multiple Scattering FormulationIntegral Equation of the 2nd Kind:
Disjoint Scatterers:Reduction to the Interaction of Two-substructures:
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:
Visibility:
No-occlusion:
Broken rays: well-defined, existence, uniqueness
Broken rays: well-defined, existence, uniqueness
Broken rays: illuminated regions (IL)shadow regions (SR)shadow boundaries (SB)
… convexwave fronts
Asymptotic Expansions of Scattered Fields
Theorem (E., Reitich 2009):
Theorem (E., Reitich 2009):planewave incidence…ansatz………………….
On the illuminated region:
… Hoermander classes
……… asymptotic expansions
Theorem (E., Reitich 2009):planewave incidence…ansatz………………….
Over the entire boundary:
……………………. Hoermander classes
… asymptotic expansions
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
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
GeneralizedGeometrical OpticsApproximations
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
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
Acoustic Asymptotic Expansions in 3DTheorem: (Anand, Boubendir, E., Reitich)
where
and
Here, defining
we have set
and
Finally
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.
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
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:
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:
2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
2 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
2 Periodic Example:
Point SourceIllumination
Numerical Examples in 2D
3 Periodic Example:
PlanewaveIllumination
Numerical Examples in 2D
3 Periodic Example:
Point SourceIllumination
Numerical Examples in 2D
2 Periodic Example:
0.07240.07400.07850.0718
Iteration 1 Iteration 2 Iteration 3
Iteration 10
Numerical Examples in 3D
Numerical Examples in 3D
Numerical Examples in 3D
Thanks