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