Matching of asymptotic expansions for the wave propagation in media with thin slot

(1)

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Matching of asymptotic expansions for the wave

propagation in media with thin slot

Sébastien Tordeux, Patrick Joly

To cite this version:

Sébastien Tordeux, Patrick Joly. Matching of asymptotic expansions for the wave propagation in

media with thin slot. TiSCoPDE workshop (New Trends in Simulation and Control of PDEs), WIAS,

2005, Berlin, Germany. �inria-00528072�

(2)

Matching of asymptotic expansions

for the wave propagation in media

with thin slot

S ´ebastien Tordeux and Patrick Joly

TiSCoPDE workshop, Berlin, September 2005

INRIA-Rocquencourt-Projet POEMS

ETH-SAM

(3)

A typical application

How can we study the scattering in media with

thin slot

?

source

wave

reflected

transmitted

wave

A physical problem with two

caracteristical

lengthes

The

wavelength

The

width

of the slot

(4)

A typical application

How can we study the scattering in media with

thin slot

?

source

wave

reflected

transmitted

wave

An

asymptotic

case:

(5)

The numerical difficulty

A

mesh step

smaller than

(6)

Some references

- Thin slot:

Harrington

,

Auckland

(1980),

Tatout

(1996).

- Finite differences:

Taflove

(1995).

- Thin plates and junction theory,...

Ciarlet

,

Le Dret

,

Dauge

-

Costabel

.

- Matching of asymptotic expansions:

McIver

,

Rawlins

(1993),

Il’in

(1992).

- multiscale analysis

Maz’ya

,

Nazarov

,

Plamenevskii

(1991).

Oleinik

,

Shamaev

,

Yosifian

(1992).

(7)

A simple problem

Scalar

wave equation:

Harmonic

solution:

Helmholtz

Equation:

in

(8)

A simple problem

Outgoing

solution at infinity:

for

large

Neumann

limit condition

(rigid wall)

on

(9)

A simple problem

Outgoing

solution at infinity:

for

large

Neumann

limit condition

(rigid wall)

on

With the

Dirichlet

limit condition, the transmission inside the

slot is

negligible

(

(10)

A numerical computation

Dirichlet

Neumann

Numerical computation done with the

high order finite

elements code

of (

M. Duruflé

, INRIA)

(11)

A numerical computation

Dirichlet

Neumann

(12)

A numerical computation

Dirichlet

Neumann

(13)

A numerical computation

Dirichlet

Neumann

(14)

Objectives

Introduce

accurate

numerical methods

We need an

intermediate zone

A technique

the matching of asymptotic expansions

Define

new approximate models

to compute the

solution.

Use effectively “universal” technique of numerical

computation (mesh reffinement).

(15)

Objectives

Introduce

accurate

numerical methods

We need an

intermediate zone

A technique

the matching of asymptotic expansions

Define

new approximate models

to compute the

solution.

Use effectively “universal” technique of numerical

computation (mesh reffinement).

(16)

Objectives

Introduce

accurate

numerical methods

We need an

intermediate zone

A technique

the matching of asymptotic expansions

Define

new approximate models

to compute the

solution.

Use effectively “universal” technique of numerical

computation (mesh reffinement).

(17)

Three zones

Far field

(2D field)

Near field

(boundary layer)

(18)

Three zones

The

asymptotic assumptions:

.

(19)

Three zones

The

asymptotic assumptions:

.

(20)

Three zones

Far field

The

asymptotic assumptions:

.

(21)

Three zones

Near field

The

asymptotic assumptions:

.

(22)

Three zones

Slot field

The

asymptotic assumptions:

.

(23)

Three zones

Far and near

The

asymptotic assumptions:

.

(24)

Three zones

Slot and near

The

asymptotic assumptions:

.

(25)

The different steps of the method

Derivate

the asymptotic expansions:

Formal

part

(26)

The different steps of the method

Derivate

the asymptotic expansions:

Formal

part

Several presentations are possible

Describe

the asymptotic expansions

Rigorous

part

Definition

of the terms of the asymptotic expansions

(27)

The different steps of the method

Derivate

the asymptotic expansions:

Formal

part

Several presentations are possible

Describe

the asymptotic expansions

Rigorous

part

Definition

of the terms of the asymptotic expansions

Mathematical validation

of the asymptotic expansions

Rigorous

part

(28)

Far field

Asymptotic context:

.

(29)

Far field

Asymptotic context:

.

No

normalization

:

(30)

Far field

Asymptotic context:

.

No

normalization

:

(31)

Far field

Asymptotic context:

.

in

(32)

Far field

Asymptotic context:

.

where the

satisfy the

homogeneous Helmholtz

equation

(33)

(34)

Slot field

(35)

Slot field

The

asymptotic

context:

.

The

normalization:

(36)

Slot field

scaling

The

asymptotic

context:

.

The

normalization:

(37)

Slot field

scaling

The

asymptotic

context:

.

The

normalization:

(38)

Slot field

scaling

in

(39)

Slot field

scaling

where the

satisfy the

1D Helmholtz

equation:

(40)

Near field

(41)

Near field

(42)

Near field

The

Asymptotic

context:

The

normalization

:

(43)

Near field

The

Asymptotic

context:

The

normalization

:

(44)

Near field

scaling

The

Asymptotic

context:

The

normalization

:

(45)

Near field

scaling

The

Asymptotic

context:

The

normalization

:

(46)

Near field

scaling

(47)

Near field

scaling

where the

satisfy the

(in)-homogeneous Laplace

equation

.

if

or

else.

(48)

Order 0 :

,

Far field:

Find

such that

in

on

is outgoing.

(49)

Order 0 :

,

Near field:

in

(50)

Order 0 :

,

Slot field:

exp

in

(51)

Order 1 :

,

Approximation

of the exact Solution:

(52)

Order 1 :

,

explicit form of

(53)

Order 1 :

,

Approximation

of the exact solution:

(54)

Order 1 :

,

Near field:

Find

such that:

in

on

(55)

Order 1 :

,

Behavior at infinity

in the half-space:

! " # $ &%

(56)

Order 1 :

,

Behavior at infinity

in the half-space:

! " # $ &%

Behavior at infity

in the slot:

(57)

Order 1 :

,

(58)

Order 1 :

,

Approximation

of the exact solution:

(59)

Order 1 :

,

The slot field:

(60)

Order 1 :

,

The slot field:

(61)

The far field of order

The fields

are defined in the

half space

:

(62)

The far field of order

The fields

are defined in the

half space

:

The far fields

satisfy the

homogeneous Helmholtz

equation

are

singular

at the neighborhood of the origin

are outgoing at infinity

(63)

The far field of order

The fields

are defined in the

half space

:

(64)

The far field of order

The fields

are defined in the

half space

:

(65)

The far field of order

The fields

are defined in the

half space

:

The

(66)

The far field of order

The fields

are defined in the

half space

:

Im

The

are functions of

lower order

terms

(67)

The far field of order

The fields

are defined in the

half space

:

Im

The

(68)

The near fields of order

The

are defined in the

canonical

domain:

(69)

The near fields of order

The

are defined in the

canonical

domain:

by

Laplace

equation:

ou

(70)

The near fields of order

The

are defined in the

canonical

domain:

by

Laplace

equation:

by polynomial

growings

at infinity:

The

growings

in the half space are functions of

far

field of lower (or equal) order

The

growings

in the slot are functions of the slot

fields

of lower order

(71)

The near fields of order

The

are defined in the

canonical

domain:

Proof of the

existence-unicity

:

with truncature functions, we subtract the growing

behavior at infinity of the

We use the “classical”

variational theory

(wheighted

(72)

The slot field of order

The

are defined on the

canonical

domain:

(73)

The slot field of order

The

are defined on the

canonical

domain:

The

does not depend on

(74)

The slot field of order

The

are defined on the

canonical

domain:

The

does not depend on

.

(75)

Some properties

We see that:

More

is

large

more

is

singular

at the origin:

terms,

(76)

Some properties

We see that:

More

is

large

more

is

singular

at the origin:

terms,

More

is

large

more

is

growing

:

terms,

(77)

Some properties

We see that:

More

is

large

more

is

singular

at the origin:

terms,

More

is

large

more

is

growing

:

terms,

When the

order

grows, one has

(

) terms to

compute...

(78)

Mathematical analysis

! %

(79)

Mathematical analysis

! % ! %

(80)

Mathematical analysis

! % ! % ! %