Fatih Ecevit Max Planck Institute for Mathematics in the Sciences

Fatih EcevitMax Planck Institute for Mathematics in the Sciences

Víctor DomínguezIvan Graham

New Galerkin Methods forHigh-frequency Scattering Simulations

Universidad Pública de NavarraUniversity of Bath

Collaborations

Outline

High-frequency integral equation methods Main principles (BGMR 2004) A robust Galerkin scheme (DGS 2006) Required improvements

II.

New Galerkin methods for high-frequency scattering simulationsIII. Two new algorithms

Electromagnetic & acoustic scattering problemsI.

New Galerkin methods for high-frequency scattering simulations

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.

Boundary Condition:



Single layer potential:


Boundary Condition:

Double layer potential:



Single layer potential:


1st1stkindkind

2nd2ndkindkind

Boundary Condition:

Double layer potential:

2nd2ndkindkind

Single Convex Obstacle: AnsatzSingle layer density:


Double layer density:



Double layer density:

current

non-physicalis

Bruno, Geuzaine,Monro, Reitich (2004)



BGMR (2004)

Single Convex Obstacle

A Convergent High-frequency ApproachHighly oscillatory!



A Convergent High-frequency Approach

Localized Integration:

Highly oscillatory!


for all n

BGMR (2004)




(Melrose & Taylor, 1985)




(Melrose & Taylor, 1985)

Change of Variables:

BGMR (2004)

Single Smooth Convex Obstacle


Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2006 …





Huybrechs, Vandewalle …….…… 2006 …



Domínguez, Graham, Smyshlyaev … 2006 … (circler bd.)









Chandler-Wilde, Langdon ….…….. 2006 ..

Langdon, Melenk …………..……… 2006 ..

Single Convex Polygon







Chandler-Wilde, Langdon ….…….. 2006 ..

Langdon, Melenk …………..……… 2006 ..

Single Convex Polygon


Domínguez, E., Graham, ………… 2007 … (circler bd.)

The Combined Field Operator

A High-frequency Galerkin Method DGS (2006)II.

The Combined Field Operator

Continuity:

circler domains ……………

general smooth domains …

Giebermann (1997)

DGS (2006)

II. A High-frequency Galerkin Method DGS (2006)

The Combined Field OperatorII.

Continuity:

Coercivity:


general smooth domains …


general smooth domains … open problem

Giebermann (1997)

DGS (2006)

DGS (2006)

A High-frequency Galerkin Method DGS (2006)

Plane-wave Scattering ProblemII. A High-frequency Galerkin Method DGS (2006)

Plane-wave Scattering ProblemII.

is an explicitly defined entire function with known asymptotics

are smooth periodic functions

is not explicitly known but behaves like:



is an explicitly defined entire function with known asymptotics

are smooth periodic functions

is not explicitly known but behaves like:

DGS (2006)Melrose, Taylor (1985)


Plane-wave Scattering ProblemII. A High-frequency Galerkin Method DGS (2006)


for some on the “deep” shadow



DGS (2006)

for some on the “deep” shadow


Polynomial ApproximationII.

Illuminated Region Deep ShadowShadow Boundaries




… gluing together




… gluing together

… approximation by zero




… gluing together

is the optimal choice


Galerkin MethodII.


… gluing together

Discrete space


Galerkin MethodII.


… gluing togetherFinal Estimate


Galerkin MethodII.



Question Can one obtain a robust Galerkin method that works for higher frequencies as well as low frequencies?


Galerkin MethodII.




In other words higher frequencies: low frequencies: do an approximation on the deep shadow region??

Galerkin MethodII.




Galerkin MethodII.




In other words higher frequencies: low frequencies: do an approximation on the deep shadow region??

New Galerkin MethodsIII.


… gluing together … new Galerkin methods

Treat these four transition regions separatelyA straightforward extension of the Galerkinapproximation in DGS (2006) applies to deep shadow region


New Galerkin Methods


… gluing together … new Galerkin methods

Treat these four transition regions separatelyA straightforward extension of the Galerkinapproximation in DGS (2006) applies to deep shadow region

The highly oscillatory integrals arising in the Galerkin matrices can be efficiently evaluated as the stationary phase points are apriory known

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin MethodsIII.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods… optimal



Discrete space

DGS (2006)



Discrete space

DGS (2006)

DEG (2007)Discrete space defined in a similar wayincluding the deep shadow

… first algorithm



Discrete space

DGS (2006)

DEG (2007)Discrete space defined in a similar wayincluding the deep shadow

… first algorithm


degrees of freedom

New Galerkin MethodsIII.New Galerkin methods for high-frequency scattering simulations


Idea: changeof variables



… change of variables




control: derivatives of





… but how do we obtain an optimal change of variables?





… but how do we obtain an optimal change of variables? … mimic the algorithm

and

with affine st.



Discrete space

DGS (2006)

DEG (2007) … second algorithmDiscrete space defined in a similar way including the deep shadow while on the transition regions polynomials are replaced by



Discrete space

DGS (2006)

DEG (2007) … first algorithmDiscrete space defined in a similar way including the deep shadow

degrees of freedom

DEG (2007) … second algorithmDiscrete space defined in a similar way including the deep shadow while on the transition regions polynomials are replaced by


ReferencesO. P. Bruno, C. A. Geuzaine, J. A. Monro and F. Reitich:Prescribed error tolerances within fixed computational times forscattering problems of arbitrarily high frequency: the convex case,Phil. Trans. Roy. Soc. London 362 (2004), 629-645.


D. Huybrechs and S. Vandewalle:A sparse discretisation for integral equation formulations of highfrequency scattering problems, SIAM J. Sci. Comput., (to appear).

V. Domínguez, I. G. Graham and V. P. Smyshlyaev:A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering, Num. Math. 106 (2007) 471-510.

V. Domínguez, F. Ecevit and I. G. Graham:Improved Galerkin methods for integral equations arising in high-frequency acoustic scattering, (in preparation).

Thanks

Fatih Ecevit Max Planck Institute for Mathematics in the Sciences

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