Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent...

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