Fatih Ecevit Max Planck Institute for Mathematics in the Sciences V í ctor Dom í nguez Ivan Graham...

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

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

Combine…

Electromagnetic & Acoustic Scattering SimulationsI.

Integral Equation Formulations

Radiation Condition:

High-frequency Integral Equation MethodsII.

Boundary Condition:

Integral Equation Formulations

Radiation Condition:

Single layer potential:

High-frequency Integral Equation MethodsII.

Boundary Condition:

Double layer potential:

Integral Equation Formulations

Radiation Condition:

Single layer potential:

High-frequency Integral Equation MethodsII.

1st1stkindkind

2nd2ndkindkind

Boundary Condition:

Double layer potential:

2nd2ndkindkind

Single Convex Obstacle: AnsatzSingle layer density:

High-frequency Integral Equation MethodsII.

Double layer density:

Single Convex Obstacle: AnsatzSingle layer density:

High-frequency Integral Equation MethodsII.

Double layer density:

Single Convex Obstacle: AnsatzSingle layer density:

High-frequency Integral Equation MethodsII.

Double layer density:

Single Convex Obstacle: AnsatzSingle layer density:

High-frequency Integral Equation MethodsII.

Double layer density:

current

non-physicalis

Bruno, Geuzaine,Monro, Reitich (2004)

Single Convex Obstacle: AnsatzSingle layer density:

High-frequency Integral Equation MethodsII.

Single Convex Obstacle: AnsatzSingle layer density:

High-frequency Integral Equation MethodsII.

BGMR (2004)

Single Convex Obstacle

A Convergent High-frequency ApproachHighly oscillatory!

High-frequency Integral Equation MethodsII.

Single Convex Obstacle

A Convergent High-frequency Approach

Localized Integration:

Highly oscillatory!

High-frequency Integral Equation MethodsII.

for all n

BGMR (2004)

Single Convex Obstacle

A Convergent High-frequency Approach

High-frequency Integral Equation MethodsII.

(Melrose & Taylor, 1985)

Single Convex Obstacle

A Convergent High-frequency Approach

High-frequency Integral Equation MethodsII.

(Melrose & Taylor, 1985)

Change of Variables:

BGMR (2004)

Single Smooth Convex Obstacle

High-frequency Integral Equation MethodsII.

Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2006 …

Single Smooth Convex Obstacle

High-frequency Integral Equation MethodsII.

Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2006 …

Huybrechs, Vandewalle …….…… 2006 …

Single Smooth Convex Obstacle

High-frequency Integral Equation MethodsII.

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

Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2006 …

Huybrechs, Vandewalle …….…… 2006 …

Single Smooth Convex Obstacle

High-frequency Integral Equation MethodsII.

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

Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2006 …

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

Langdon, Melenk …………..……… 2006 ..

Single Convex Polygon

Huybrechs, Vandewalle …….…… 2006 …

Single Smooth Convex Obstacle

High-frequency Integral Equation MethodsII.

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

Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2006 …

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

Langdon, Melenk …………..……… 2006 ..

Single Convex Polygon

Huybrechs, Vandewalle …….…… 2006 …

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:

circler domains ……………

general smooth domains …

circler 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:

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:

DGS (2006)Melrose, Taylor (1985)

A High-frequency Galerkin Method DGS (2006)

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

Plane-wave Scattering ProblemII.

for some on the “deep” shadow

A High-frequency Galerkin Method DGS (2006)

Plane-wave Scattering ProblemII.

DGS (2006)

for some on the “deep” shadow

A High-frequency Galerkin Method DGS (2006)

Polynomial ApproximationII.

Illuminated Region Deep ShadowShadow Boundaries

A High-frequency Galerkin Method DGS (2006)

Polynomial ApproximationII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing together

A High-frequency Galerkin Method DGS (2006)

Polynomial ApproximationII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing together

A High-frequency Galerkin Method DGS (2006)

Polynomial ApproximationII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing together

… approximation by zero

A High-frequency Galerkin Method DGS (2006)

Polynomial ApproximationII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing together

is the optimal choice

A High-frequency Galerkin Method DGS (2006)

Galerkin MethodII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing together

Discrete space

A High-frequency Galerkin Method DGS (2006)

Galerkin MethodII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing togetherFinal Estimate

A High-frequency Galerkin Method DGS (2006)

Galerkin MethodII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing togetherFinal Estimate

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

A High-frequency Galerkin Method DGS (2006)

Galerkin MethodII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing togetherFinal Estimate

A High-frequency Galerkin Method DGS (2006)

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

Galerkin MethodII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing togetherFinal Estimate

A High-frequency Galerkin Method DGS (2006)

Galerkin MethodII.

Illuminated Region Deep ShadowShadow Boundaries

… gluing togetherFinal Estimate

A High-frequency Galerkin Method DGS (2006)

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

New Galerkin MethodsIII.

Illuminated Region Deep ShadowShadow Boundaries

… 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 for high-frequency scattering simulations

New Galerkin Methods

Illuminated Region Deep ShadowShadow Boundaries

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

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

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

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

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

New Galerkin Methods… optimal

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

Discrete space

DGS (2006)

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

Discrete space

DGS (2006)

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

… first algorithm

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

Discrete space

DGS (2006)

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

… first algorithm

III.New Galerkin methods for high-frequency scattering simulations

degrees of freedom

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

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

New Galerkin Methods

Idea: changeof variables

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

… change of variables

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

… change of variables

control: derivatives of

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

… change of variables

control: derivatives of

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

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

… change of variables

control: derivatives of

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

and

with affine st.

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

… change of variables

control: derivatives of

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

and

with affine st.

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

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

III.New Galerkin methods for high-frequency scattering simulations

New Galerkin Methods

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

III.New Galerkin methods for high-frequency scattering simulations

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.

New Galerkin methods for high-frequency scattering simulations

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