Highfrequency Scattering

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New Galerkin Methods forHigh-frequency
Scattering Simulations
Fatih Ecevit Max Planck Institute for Mathematics
in the Sciences
Collaborations
Víctor Domínguez Ivan Graham
Universidad Pública de Navarra University of Bath
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New Galerkin methods for high-frequency
scattering simulations
Outline
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I.
Electromagnetic Acoustic Scattering Simulations
Governing Equations
Maxwell Eqns. Helmholtz Eqn.
(TE, TM, Acoustic)
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I.
Electromagnetic Acoustic Scattering Simulations
Scattering Simulations
Basic Challenges
Fields oscillate on the order of wavelength

• Computational cost
• Memory requirement

Numerical Methods
Convergent (error-controllable)

• Variational methods (MoM, FEM, FVM,)
• Differential Eqn. methods (FDTD,)
• Integral Eqn. methods (FMM, H-matrices,)
• Asymptotic methods (GO, GTD,)

Demand resolution of wavelength
Discretization independent of frequency
Non-convergent (error )
5
I.
Electromagnetic Acoustic Scattering Simulations
Scattering Simulations
Basic Challenges
Fields oscillate on the order of wavelength

• Computational cost
• Memory requirement

Numerical Methods
Convergent (error-controllable)

• Variational methods (MoM, FEM, FVM,)
• Differential Eqn. methods (FDTD,)
• Integral Eqn. methods (FMM, H-matrices,)
• Asymptotic methods (GO, GTD,)

Demand resolution of wavelength
Discretization independent of frequency
Non-convergent (error )
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II.
High-frequency Integral Equation Methods
Integral Equation Formulations
Boundary Condition
Radiation Condition
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II.
High-frequency Integral Equation Methods
Integral Equation Formulations
Boundary Condition
Radiation Condition
Single layer potential
Double layer potential
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II.
High-frequency Integral Equation Methods
Integral Equation Formulations
Boundary Condition
Radiation Condition
Single layer potential
1st kind
2nd kind
Double layer potential
2nd kind
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle Ansatz
Single layer density
Double layer density
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle Ansatz
Single layer density
Double layer density
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle Ansatz
Single layer density
Double layer density
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle Ansatz
Single layer density
Bruno, Geuzaine, Monro, Reitich (2004)
Double layer density
is
non-physical
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle Ansatz
Single layer density
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle Ansatz
Single layer density
BGMR (2004)
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle
A Convergent High-frequency Approach
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle
A Convergent High-frequency Approach
Localized Integration
for all n
BGMR (2004)
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle
A Convergent High-frequency Approach
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II.
High-frequency Integral Equation Methods
Single Convex Obstacle
A Convergent High-frequency Approach
Change of Variables
BGMR (2004)
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II.
High-frequency Integral Equation Methods
Single Smooth Convex Obstacle

• Bruno, Geuzaine, Monro, Reitich 2004

• Bruno, Geuzaine (3D) . 2006

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II.
High-frequency Integral Equation Methods
Single Smooth Convex Obstacle

• Bruno, Geuzaine, Monro, Reitich 2004

• Bruno, Geuzaine (3D) . 2006

• Huybrechs, Vandewalle . 2006

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II.
High-frequency Integral Equation Methods
Single Smooth Convex Obstacle

• Bruno, Geuzaine, Monro, Reitich 2004

• Bruno, Geuzaine (3D) . 2006

• Huybrechs, Vandewalle . 2006

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

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II.
High-frequency Integral Equation Methods
Single Smooth Convex Obstacle

• Bruno, Geuzaine, Monro, Reitich 2004

• Bruno, Geuzaine (3D) . 2006

• Huybrechs, Vandewalle . 2006

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

Single Convex Polygon

• Chandler-Wilde, Langdon ... 2006 ..

• Langdon, Melenk .. 2006 ..

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II.
High-frequency Integral Equation Methods
Single Smooth Convex Obstacle

• Bruno, Geuzaine, Monro, Reitich 2004

• Bruno, Geuzaine (3D) . 2006

• Huybrechs, Vandewalle . 2006

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

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

Single Convex Polygon

• Chandler-Wilde, Langdon ... 2006 ..

• Langdon, Melenk .. 2006 ..

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II.
A High-frequency Galerkin Method DGS (2006)
The Combined Field Operator
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II.
A High-frequency Galerkin Method DGS (2006)
The Combined Field Operator
Continuity
Giebermann (1997)
circler domains
DGS (2006)
general smooth domains
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II.
A High-frequency Galerkin Method DGS (2006)
The Combined Field Operator
Continuity
Giebermann (1997)
circler domains
DGS (2006)
general smooth domains
Coercivity
DGS (2006)
circler domains
general smooth domains open problem
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II.
A High-frequency Galerkin Method DGS (2006)
Plane-wave Scattering Problem
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II.
A High-frequency Galerkin Method DGS (2006)
Plane-wave Scattering Problem
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II.
A High-frequency Galerkin Method DGS (2006)
Plane-wave Scattering Problem
Melrose, Taylor (1985)
DGS (2006)
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II.
A High-frequency Galerkin Method DGS (2006)
Plane-wave Scattering Problem
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II.
A High-frequency Galerkin Method DGS (2006)
Plane-wave Scattering Problem
for some on the deep shadow
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II.
A High-frequency Galerkin Method DGS (2006)
Plane-wave Scattering Problem
for some on the deep shadow
DGS (2006)
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II.
A High-frequency Galerkin Method DGS (2006)
Polynomial Approximation
Deep Shadow
Illuminated Region
Shadow Boundaries
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II.
A High-frequency Galerkin Method DGS (2006)
Polynomial Approximation
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
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II.
A High-frequency Galerkin Method DGS (2006)
Polynomial Approximation
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
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II.
A High-frequency Galerkin Method DGS (2006)
Polynomial Approximation
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
approximation by zero
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II.
A High-frequency Galerkin Method DGS (2006)
Polynomial Approximation
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
is the optimal choice
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II.
A High-frequency Galerkin Method DGS (2006)
Galerkin Method
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
Discrete space
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II.
A High-frequency Galerkin Method DGS (2006)
Galerkin Method
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
Final Estimate
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II.
A High-frequency Galerkin Method DGS (2006)
Galerkin Method
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
Final Estimate
Question Can one obtain a robust Galerkin method
that works for higher frequencies as well as low
frequencies?
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II.
A High-frequency Galerkin Method DGS (2006)
Galerkin Method
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
Final Estimate
In other words higher frequencies
low frequencies do an approximation on the
deep shadow region??
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II.
A High-frequency Galerkin Method DGS (2006)
Galerkin Method
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
Final Estimate
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II.
A High-frequency Galerkin Method DGS (2006)
Galerkin Method
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
Final Estimate
In other words higher frequencies
low frequencies do an approximation on the
deep shadow region??
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
new Galerkin methods
Treat these four transition regions separately
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
Deep Shadow
Illuminated Region
Shadow Boundaries
gluing together
new Galerkin methods
Treat these four transition regions separately
The highly oscillatory integrals arising in the
Galerkin matrices can be efficiently evaluated as
the stationary phase points are apriory known
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
49
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
52
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
optimal
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
DGS (2006)
Discrete space
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
DGS (2006)
Discrete space
DEG (2007)
first algorithm
Discrete space defined in a similar way including
the deep shadow
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
DGS (2006)
Discrete space
DEG (2007)
first algorithm
Discrete space defined in a similar way including
the deep shadow
degrees of freedom
56
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
57
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
58
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
Idea change of variables
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
change of variables
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III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
change of variables
control derivatives of
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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?
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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
with affine st.
and
63
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
with affine st.
and
64
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
DGS (2006)
Discrete space
DEG (2007)
second algorithm
Discrete space defined in a similar way including
the deep shadow while on the transition regions
polynomials are replaced by
65
III.
New Galerkin methods for high-frequency
scattering simulations
New Galerkin Methods
DGS (2006)
Discrete space
DEG (2007)
first algorithm
Discrete space defined in a similar way including
the deep shadow
degrees of freedom
DEG (2007)
second algorithm
Discrete space defined in a similar way including
the deep shadow while on the transition regions
polynomials are replaced by
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New Galerkin methods for high-frequency
scattering simulations
References
O. P. Bruno, C. A. Geuzaine, J. A. Monro and F.
Reitich Prescribed error tolerances within fixed
computational times for scattering 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 high frequency 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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Thanks