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

Sébastien Tordeux and Patrick Joly

WAVES 2005, Brown University, July 2005

INRIA-Rocquencourt-Projet POEMS

ETH-SAM
<figure><img src="image_1.png" /></figure>

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

Sébastien Tordeux, Patrick Joly

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Sébastien Tordeux, Patrick Joly. Matching of asymptotic expansions for the wave propagation in media with thin slot. WAVES, Brown University, 2005, Providence, United States. inria-00527861

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Submitted on 20 Oct 2010

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A typical application

How can we study the scattering in media with **thin slot**?

<figure><img src="image_1.png" /></figure>

A physical problem with two **caractéristical** lengths

$$ \begin{cases} \text{The wavelength } \lambda \\ \text{The width of the slot } \varepsilon \end{cases} $$
P. Joly†, S. Tordeux‡,*

†UMR POems, INRIA-Rocquencourt, 78153 Le Chesnay Cedex, France,

‡SAM, ETHZ, 8092 Zurich, Switzerland,

*Email: tordeux@math.ethz.ch

Abstract

In this talk we will use the matching of asymptotic expansion to derive new slot models. This models will be mathematically validated via some error estimates.

Introduction

In practical applications concerning both electromagnetic or acoustic waves, many wave propagation problems involve the presence of structures whose at least one space dimension of characteristic length, denoted $\varepsilon$, is small with respect to the wave length $\lambda$: we think for instance to highly oscillating coefficients, thin layers, thin screens, wires or thin slots, which is the topic we address in this work. For the numerical simulation of such wave propagation problems, it is natural to look for approximate or “effective” models that should permit to avoid to mesh the computational domain, even locally, at the scale of $\varepsilon$. One meets a similar situation for the treatment of thin slots. Let us take the example of a thin slot in a 2D context. The geometry of the problem is represented by figure 1. A first application is the microwave shielding of

<figure><img src="image_2.png" /><figcaption>Figure 1: Geometry of the propagation domain.</figcaption></figure>

thin slots. From the physical point of view, the domain of validity is typically given by:

$$ \begin{cases} \lambda/1000 < \varepsilon < \lambda/10, \\ \varepsilon < L/10, \\ \lambda/10 < L < 10\lambda, \end{cases} \quad (1) $$

where $\lambda$ is the wave length, $\varepsilon$ is the width of the slot and $L$ the length of the slot. One can refer to [5] for exam-

ples. A second application in electromagnetism corresponds to the so-called flanged waveguide antenna which corresponds to a semi-infinite straight slot. This case, that has been studied in [1] in a non asymptotic context, will be analyzed in detail in this paper.

An intuitive idea is to try to derive an approximate “1D - 2D” model: a 1D model, posed on the curve that materializes the limit of the slot when $\varepsilon$ tends to 0, for the propagation inside the slot and a 2D model for the rest of the computational domain. Such models have been designed in the engineering literature (see [2], [3], [4], [5]) and are commonly used in various computational codes. One can find a mathematical analysis of a related model in [6].

An alternative way to obtain some approximate model comes from the analytical work by the “british and russian school” of applied mathematics on this type of problem [7], [8], [9]. The aim they pursue a priori is not to derive approximate models for numerical approximation. They wish to describe as accurately as possible the behaviour of the solution with the help of analytical formulas or at least solutions of simple canonical problems. The technique that is used is the method of matched asymptotics expansions which consists in separating the domain of propagation in several parts in which one makes different ansatz on the solution. For instance, for the slot problem, it would consist in considering three zones : the inner slot, the exterior region and a neighborhood of the end points of the slot. Such techniques permit for instance to take correctly into account the singularities of the solution. This type of technique is bere used to derive effective models for numerical computation.

One can refer to [6] to see this technique in a more complicated case where there is an obstacle on the half space.

The exact model

In this work, we shall consider the simplest possible propagation model, namely the 2D scalar wave equation, that can be used in acoustics of course, but also in electromagnetism if one separates the transverse electric or
# A typical application

How can we study the scattering in media with **thin slot**?

<figure><img src="image_2.png" /></figure>

An **asymptotic** case:

$$ \epsilon \ll \lambda $$
transverse magnetic polarizations:

$$ \frac{\partial^2 U^\varepsilon}{\partial t^2} - \Delta U^\varepsilon = F, \qquad (2) $$

where we have assumed, for simplicity, that the propagation velocity of waves is equal to 1. We shall look at the time harmonic regime, i.e. consider a source term of the form:

$$ F(x,t) = f(x) \cdot \exp(-i \omega t). \qquad (3) $$

where the pulsation $\omega > 0$ is a data of the problem, related to the wavelength $\lambda$ by:

$$ \omega \lambda = 2 \pi. \qquad (4) $$

In (3), $f$ denotes a compactly supported function whose support does not intersect the slot. We then look for solutions of the same form:

$$ U^{\varepsilon}(x,t) = u^{\varepsilon}(x) \cdot \exp(-i \omega t), \qquad (5) $$

which leads to the Helmholtz equation for the complex value function $u$:

$$ -\Delta u^{\varepsilon} - \omega^2 u^{\varepsilon} = f. \qquad (6) $$

A typical source in application is a regularized $\delta$-function an approximation to a point source.

For the boundary condition, we shall consider the homogeneous Neumann condition:

$$ \frac{\partial u^{\varepsilon}}{\partial n} = 0. \qquad (7) $$

In some sense, it is the “only” boundary condition that permits the propagation of waves inside the slot. This is due to the fact that the slot behaves as a waveguide whose first cut-off frequency is 0.

## The results

The matching of asymptotic expansions consists in approximating the exact solution with series in different zones. These series have to match in some transition zones (where two expansions are valid). Here one has to use three zones.

The first zone is a far field zone. This zone consists in the $(x < 0, y)$ far away (with respect to $\varepsilon$) from the end of the slot.

The second zone is a slot zone. It consists in the point in the slot which are far away (with respect to $\varepsilon$) from the end of the slot.

The third zone is a near field zone (or boundary layer zone). It consists in the point which are closed (with respect to 1 from the end of the slot)

Far field expansion. Firstly, one can expand the exact solution in the half-space with respect to the coordinates $x$ and $y$ using a formal series of functions independent of $\varepsilon$:

$$ u^{\varepsilon}(x,y) = u^0 + \sum_{i=1}^{+\infty} \sum_{k=0}^{i-1} \varepsilon^i \log^k \varepsilon \, u_i^k \qquad (8) $$

where $u^0$ is the outgoing solution of the slotless problem:

$$ \left\{ \begin{array}{ll} \Delta u^0 + \omega^2 u^0 = -f, & \text{for } x < 0 \\ \frac{\partial u^0}{\partial x} = 0, & \text{for } x = 0. \end{array} \right. \qquad (9) $$

The $u_i^k$'s are defined in the half-space. They satisfy Helmholtz condition with Neumann boundary condition but are singular at the origin ($u^0$ is regular):

$$ \left\{ \begin{array}{ll} \Delta u_i^k + \omega^2 u_i^k = 0, & \text{for } x < 0, \\ \frac{\partial u_i^k}{\partial n} = 0, & \text{for } x = 0 \text{ if } y \neq 0 \end{array} \right. \qquad (10) $$

For example, $u_1^0$ is given by:

$$ u_1^0 = -\frac{\omega}{2} u^0(0) H_0^{(1)}(\omega r) \qquad (11) $$

The following error estimates permit to understand the meaning of the formal expansion (8):

$$ \left\{ \begin{array}{l} \|u^{\varepsilon} - u^0 + \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^i \log \varepsilon^k u_i^k\|_{H^1(K)} \qquad (12) \\ \leq C \varepsilon^{n+1} \log^n \varepsilon \|f\|_{L^2(\Omega)} \end{array} \right. $$

where $K$ is any compact included in the closed half space $(x \leq 0)$ which do not contain the origin.

Slot field expansion. Secondly, one can expand the solution inside the slot with respect to the coordinates:

$$ x \text{ and } Y = y/\varepsilon \qquad (13) $$

The exact solution admits the following formal expansion:

$$ u^{\varepsilon}(x, \varepsilon Y) = \sum_{i=0}^{+\infty} \sum_{k=0}^{i} \varepsilon^i \log^k \varepsilon U_i^k(x, Y) \qquad (14) $$

where the $U_i^k$ are 1-D solutions of the Helmholtz equation:

$$ U_i^k(x, Y) = U_i^k(0) \exp i\omega x \qquad (15) $$
# The numerical difficulty

A **mesh step** smaller than $\varepsilon$

<figure><img src="image_3.png" /></figure>

This leads to **costly** computations
For example, one can write $U_0^0$ as:

$$U_0^0(x) = u^0(0) \exp i\omega x \quad (16)$$

The following error estimates allows us to understand the formal serie (14):

$$ \left\{ \begin{array}{l} \left\| u^{\epsilon}(x, \epsilon Y) - \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^{i} \log^{k} \epsilon U_{i}^{k} \right\|_{H^{1}(K)} \\ \le C \epsilon^{n+1} |\log \epsilon|^{n+1} \|f\|_{L^{2}(\Omega)} \end{array} \right. \quad (17) $$

where K is any compact such that:

$$K \subset ]0; +\infty[ \times [x \times [-\frac{1}{2}; \frac{1}{2}]_Y \quad (18)$$

Near field expansion. Finally, one can expand the solution in a neighborhood of the end of the slot with respect to the coordinates:

$$X = x/\epsilon \quad \text{and} \quad Y = y/\epsilon \quad (19)$$

This introduces the canonical domain

$$T = ] - \infty; 0]_X \times \mathbb{R}_Y \cup [0; +\infty[ \times [-\frac{1}{2}; \frac{1}{2}]_Y. \quad (20)$$

The exact solution admits the following formal expansion:

$$u^{\epsilon}(\epsilon X, \epsilon Y) = \sum_{i=0}^{+\infty} \sum_{k=0}^{i} \epsilon^{i} \log^{k} \epsilon U_{i}^{k}(X, Y) \quad (21)$$

where the $U_i^k$'s are defined on T. They are solutions of the Laplace equation with Neumann boundary condition and are growing at infinity:

$$ \left\{ \begin{array}{l} \Delta U_i^k = 0, \text{ in } T, \text{ for } i = k \text{ or } k + 1, \\ \Delta U_i^k + \omega^2 U_{i-2}^k = 0, \text{ in } T \text{ for } i > k + 1, \\ \frac{\partial U_i^k}{\partial n} = 0, \text{ on } \partial T \end{array} \right. \quad (22) $$

One can use the following error estimate to understand the meaning of (21):

$$ \left\{ \begin{array}{l} \left\| u^{\epsilon}(\epsilon X, \epsilon Y) - \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^{i} \log^{k} \epsilon U_{i}^{k}(X, Y) \right\|_{H^{1}(K)} \\ \le C \epsilon^{n+1} \log^{n+1} \epsilon \quad \|f\|_{L^{2}(\Omega)} \end{array} \right. \quad (23) $$

For numerical computations. All the $u_i^k$'s, $U_i^k$'s and $U_i^k$'s are solutions of PDE which can easily be solved numerically. As the terms of the asymptotic expansions do not

depend on $\epsilon$, one can use a mesh which does not also depend on $\epsilon$. For example, to have an approximation of order 2 of the exact solution in the far field zone, one has just to compute

$$u^0, u_1^0, u_2^0 \text{ and } u_2^1 \quad (24)$$

and to construct the truncated serie:

$$u^0 + \epsilon u_1^0 + \epsilon^2 u_2^0 + \epsilon^2 \log \epsilon u_2^1. \quad (25)$$

## References

[1] G.A. Kriegsmann, The flanged waveguide antenna: discrete reciprocity and conservation, Wave Motion, 1999, volume 29.

[2] C.M. Butler and D.R. Wilton, General analysis of narrow strips and slots, IEEE Trans Ant. and Propag., 1980, volume 28.

[3] J. Gilbert and R. Holland, Implementation of the thin-slot formalism in the finite-difference code THREEDII, IEEE Trans Nuc Science, 1981, volume 28.

[4] P. Harrington and D. Auckland, Electromagnetic transmission through narrow slots in thick conducting screens, IEEE Trans Antenna Propagation, 1980, volume 28.

[5] A. Taflove and K. Umashankar and B. Becker and F. Harfoush and K.S. Yee, Detailed FDTD analysis of electromagnetic fields penetrating narrow slots ans lapped joints in thick conducting screens, IEEE Trans Antenna and Propagation, 1988, volume 36,

[6] S. Tordeux, Méthodes Asymptotiques pour la Propagation des Ondes dans les Milieux Comportant des Fentes, Université de Versailles Saint-Quentin-en-Yvelines, 2004.

[7] P. McIver, and A. D. Rawlins, Two-dimensional wave-scattering problems involving parallel-walled ducts, Quart. J. Mech. Appl. Math., volume 46, 1993.

[8] D.G. Crighton and A.P. Dowling and J.E Ffowks Williams and M. Heckl and F.G. Leppington, Modern Methods in Analytical acoustics, Lecture Notes, Springer-Verla, 1992.

[9] A. M. Il'in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, volume 102, 1992.
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).
# A simple problem

<figure><img src="image_4.png" /></figure>

Scalar wave equation:

$$ \frac{\partial^2 p^\epsilon}{\partial t^2} - \Delta p^\epsilon = f $$

Harmonic solution:

$$ p^{\epsilon}(x, y, t) = \exp(-i\omega t) u^{\epsilon}(x, y) $$

Helmholtz Equation:

$$ \Delta u^{\epsilon} + \omega^2 u^{\epsilon} = -f \quad \text{in } \Omega^{\epsilon} $$
# A simple problem

<figure><img src="image_5.png" /></figure>

**Outgoing solution at infinity:**

$$ \frac{\partial u^{\epsilon}}{\partial n} - i\omega u^{\epsilon} \le \frac{C}{r^2}, \quad \text{for } r \text{ large,} $$

**Neumann limit condition**
(rigid wall)

$$ \frac{\partial u^{\epsilon}}{\partial n} = 0 \quad \text{on } \partial\Omega^{\epsilon} $$
# A simple problem

<figure><img src="image_6.png" /></figure>

**Outgoing solution at infinity:**

$$ \frac{\partial u^{\epsilon}}{\partial n} - i\omega u^{\epsilon} \le \frac{C}{r^2}, \quad \text{for } r \text{ large,} $$

**Neumann limit condition**
(rigid wall)

$$ \frac{\partial u^{\epsilon}}{\partial n} = 0 \quad \text{on } \partial\Omega^{\epsilon} $$

With the **Dirichlet** limit condition, the transmission inside the slot is **negligible** ($o(\epsilon^\infty)$).
# A numerical computation

<figure><img src="image_7.png" /></figure>

Numerical computation done with the **high order finite elements code** of (M. Duruflé, INRIA)
# A numerical computation

Neumann

<figure><img src="image_8.png" /></figure>

Dirichlet

<figure><img src="image_9.png" /></figure>

$$ \frac{\varepsilon}{\lambda} = 0.5 $$

<figure><img src="image_10.png" /><figcaption>Matching of asymptotic expansions for the wave propagation in media with thin slot - p.6/34</figcaption></figure>
# A numerical computation

Neumann

<figure><img src="image_11.png" /></figure>

Dirichlet

<figure><img src="image_12.png" /></figure>

$$ \frac{\varepsilon}{\lambda} = 0.2 $$

<figure><img src="image_13.png" /></figure>
# A numerical computation

Neumann

<figure><img src="image_14.png" /></figure>

Dirichlet

<figure><img src="image_15.png" /></figure>

$$ \frac{\varepsilon}{\lambda} = 0.1 $$

<figure><img src="image_16.png" /><figcaption>Matching of asymptotic expansions for the wave propagation in media with thin slot – p.8/34</figcaption></figure>
# Objectives

<figure><img src="image_17.png" /></figure>

* Introduce **accurate** numerical methods
Objectives

<figure><img src="image_18.png" /></figure>

Introduce **accurate** numerical methods

<figure><img src="image_19.png" /></figure>

We need an **intermediate zone**

<figure><img src="image_20.png" /></figure>
# Objectives

<figure><img src="image_21.png" /></figure>

* Introduce **accurate** numerical methods

<figure><img src="image_22.png" /></figure>

* We need an **intermediate zone**

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

* A technique **the matching of asymptotic expansions**

<figure><img src="image_25.png" /></figure>

* Define **new approximate models** to compute the solution.

<figure><img src="image_26.png" /></figure>

* Use effectively “universal” technique of numerical computation (mesh refinement).
Three zones

<figure><img src="image_27.png" /></figure>

● Far field (2D field)

- Near field (boundary layer)

● Slot field (1D field)
Three zones

<figure><img src="image_28.png" /></figure>

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_29.png" /></figure>

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_30.png" /></figure>

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_31.png" /></figure>

Near field

The asymptotic assumptions:

$$ \epsilon \ll \eta_H(\epsilon) \ll \lambda, $$

$$ \epsilon \ll \eta_S(\epsilon) \ll \lambda. $$

$$ \epsilon \to 0 $$

$$ \eta(\epsilon) \to 0 $$

$$ \eta(\epsilon)/\epsilon \to +\infty $$
Three zones

<figure><img src="image_32.png" /></figure>

Slot field

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_33.png" /></figure>

Far and near

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_34.png" /></figure>

Slot and near

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
# The different steps of the method

<figure><img src="image_35.png" /></figure>

* Derivate the asymptotic expansions:

<figure><img src="image_36.png" /></figure>

* Formal part

<figure><img src="image_37.png" /></figure>

* Several presentations are possible
# 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
# 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

* Error estimates
# Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_38.png" /></figure>
# Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_39.png" /></figure>

No normalization:

$$X = x,$$

$$Y = y.$$
# Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_40.png" /></figure>

No normalization:

$X = x,$

$Y = y.$
# Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_41.png" /></figure>

$$ u^{\varepsilon} = u^{0} + \sum_{i=1}^{+\infty} \sum_{k=0}^{i-1} \varepsilon^{i} (\log \varepsilon)^{k} u_{i}^{k} + o(\varepsilon^{\infty}), \quad \text{in } \Omega. $$
Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_42.png" /></figure>

where the $u_i^k$ satisfy the **homogeneous Helmholtz** equation

$$ \Delta u_i^k + \omega^2 u_i^k = 0. $$
Slot field

<figure><img src="image_43.png" /></figure>
Slot field

<figure><img src="image_44.png" /></figure>

$$u^{\epsilon}(x, y) = U^{\epsilon}(x, \frac{y}{\epsilon})$$
# Slot field

<figure><img src="image_45.png" /></figure>

The **asymptotic context**: $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = x, Y = \frac{y}{\varepsilon}$.
# Slot field

<figure><img src="image_46.png" /></figure>

The **asymptotic context**: $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = x, Y = \frac{y}{\varepsilon}$.
# Slot field

<figure><img src="image_47.png" /></figure>

The **asymptotic context**: $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = x, Y = \frac{y}{\varepsilon}$.
Slot field

<figure><img src="image_48.png" /></figure>

$$u^{\epsilon}(x, Y\epsilon) = U^{\epsilon}(x, Y) = \sum_{i=0}^{+\infty} \sum_{k=0}^{i} \epsilon^i (\log \epsilon)^k U_i^k(x, Y) + o(\epsilon^\infty),$$
Slot field

<figure><img src="image_49.png" /></figure>

where the $U_i^k$ satisfy the 1D Helmholtz equation:

$$ \frac{d^2 U_i^k}{dx^2} + \omega^2 U_i^k = 0 $$
Near field

<figure><img src="image_50.png" /></figure>
Near field

<figure><img src="image_51.png" /></figure>

$$u^{\epsilon}(x, y) = u_{p}^{\epsilon}\left(\frac{x}{\epsilon}, \frac{y}{\epsilon}\right)$$
# Near field

<figure><img src="image_52.png" /></figure>

The **Asymptotic context**: $\epsilon \ll \eta_H \ll \lambda$, $\epsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\epsilon}$, $Y = \frac{y}{\epsilon}$
Near field

<figure><img src="image_53.png" /></figure>

The **Asymptotic context**: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_54.png" /></figure>

The Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The normalization: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_55.png" /></figure>

The **Asymptotic context**: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_56.png" /></figure>

$$u^{\varepsilon}(\varepsilon X, \varepsilon Y) = u_{p}^{\varepsilon}(X, Y) = \sum_{i=0}^{+\infty} \sum_{k=0}^{i} \varepsilon^{i} (\log \varepsilon)^{k} (u_{p})_{i}^{k}(X, Y) + o(\varepsilon^{\infty})$$
Near field

<figure><img src="image_57.png" /></figure>

where the $(u_p)_i^k$ satisfy the (in)-homogeneous Laplace
equation.

$$
\begin{cases}
\Delta(\boldsymbol{u}_p)_i^k = 0, & \text{if } i = k \text{ or } k+1, \\
\Delta(\boldsymbol{u}_p)_i^k = -\omega^2 (\boldsymbol{u}_p)_{i-2}^k, & \text{else.}
\end{cases}
$$
**Order 0 :** $\underline{u}^0$, $(u_p)_0^0$, $U_0^0$

<figure><img src="image_58.png" /></figure>

**Far field:**

$$ egin{cases} 
\begin{array}{l} \text{Find } \boldsymbol{u}^0 \in H_{\text{loc}}^1(\Omega) \text{ such that :} \\ -\Delta \boldsymbol{u}^0 - \omega^2 \boldsymbol{u}^0 = f, & \text{in } \Omega, \\ \\ \frac{\partial \boldsymbol{u}^0}{\partial n} = 0, & \text{on } \partial\Omega, \\ \\ \boldsymbol{u}^0 \text{ is outgoing.} 
\end{array} 
\end{cases} $$
Order 0 : $u^0, (u_p)_0^0, U_0^0$

<figure><img src="image_59.png" /></figure>

Near field:

$$ (u_p)_0^0 (X, Y) = u^0(A), \quad \text{in } \mathcal{B}_1. $$
Order 0 : $u^0, (u_p)_0^0, U_0^0$

$O_1$

Slot field:

$U_0^0(x, Y) = u^0(A) \exp(i\omega x)$, in $O_1$.
Order 1 : $\underline{u_1^0}$, $(u_p)_1^0$, $(u_p)_1^1$, $U_1^0$, $U_1^1$

<figure><img src="image_60.png" /></figure>

Approximation of the exact Solution:

$$u^{\epsilon} \approx u^{0} + \epsilon u_{1}^{0}$$
Order 1 : $\underline{u_1^0}$, $(u_p)_1^0$, $(u_p)_1^1$, $U_1^0$, $U_1^1$

<figure><img src="image_61.png" /></figure>

explicit form of $\underline{u_1^0}$

$$ \underline{u_1^0}(r, \theta) = -\frac{\omega}{2} u^0(A) H_0^{(1)}(\omega r). $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_62.png" /></figure>

Approximation of the exact solution:

$$ \left\{ \begin{array}{l} u^{\epsilon}(\epsilon X, \epsilon Y) = u_{p}^{\epsilon}(X, Y), \\ u_{p}^{\epsilon} \approx (u_{p})_{0}^{0} + \epsilon (u_{p})_{1}^{0} + \epsilon \log \epsilon (u_{p})_{1}^{1}. \end{array} \right. $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_63.png" /></figure>

Near field:

Find $(u_p)_1^0 \in H_{loc}^1(\mathcal{B}_1)$ such that:

$$ egin{cases} 
\Delta(\color{red}{u_p})_1^0 = 0, & \text{in } \mathcal{B}_1 \\ 
\displaystyle\frac{\partial(\color{red}{u_p})_1^0}{\partial n} = 0, & \text{on } \partial\mathcal{B}_1. 
\end{cases} $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_64.png" /></figure>

Behavior at infinity in the half-space:

$$ (u_p)_1^0(\rho, \theta) - \frac{\partial u^0}{\partial y}(A) \rho \cos\theta + \frac{\omega}{2} u^0(A) \left[ 1 + \frac{2i}{\pi} (\log\rho + \gamma) \right] = O\left(\frac{1}{\rho}\right). $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_65.png" /></figure>

**Behavior at infinity in the half-space:**

$$ (u_p)_1^0(\rho, \theta) - \frac{\partial u^0}{\partial y}(A) \rho \cos\theta + \frac{\omega}{2} u^0(A) \left[ 1 + \frac{2i}{\pi} (\log\rho + \gamma) \right] = O\left(\frac{1}{\rho}\right). $$

**Behavior at infinity in the slot:**

$$ (u_p)_1^0 (X, Y) - i \omega u^0(A) X = O(1). $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_66.png" /></figure>

$$ (u_p)_1^1 = -\frac{\dot{i}\omega}{\pi} u^0(A) $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, \underline{U_1^0}, \underline{U_1^1}$

<figure><img src="image_67.png" /></figure>

Approximation of the exact solution:

$$ \begin{cases} u^{\epsilon}(x, \epsilon Y) = U^{\epsilon}(x, Y), \\ U^{\epsilon} \approx U_{0}^{0} + \epsilon U_{1}^{0} + \epsilon \log \epsilon U_{1}^{1}. \end{cases} $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, \underline{U_1^0}, \underline{U_1^1}$

<figure><img src="image_68.png" /></figure>

The slot field:

$$ \mathbf{U}_1^0(x) = \int_0^1 \mathbf{U}_1^0(0, Y) \, dY \exp(i\omega x), $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_69.png" /></figure>

The slot field:

$$U_1^1(x) = -\frac{\mathbf{i}\omega}{\pi} u^0(A) \exp(\mathbf{i}\omega x).$$
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_70.png" /></figure>
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_71.png" /></figure>

* The far fields $u_i^k$

* satisfy the **homogeneous Helmholtz** equation

* are **singular** at the neighborhood of the origin

* are **outgoing** at infinity
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_72.png" /></figure>

* $u_i^k = \sum_{p=0}^{+\infty} a_p H_p^{(1)}(\omega r) \cos p\theta$
# The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_73.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_74.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$

The $a_p$ are functions of **lower order** terms
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_75.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$

The $a_p$ are functions of **lower order** terms
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_76.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$

The $a_p$ are functions of **lower order** terms
The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_77.png" /></figure>
# The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_78.png" /></figure>

* by **Laplace** equation:

$$ \Delta(u_p)_i^k = 0, \quad (i = k \text{ ou } k + 1), $$

$$ \Delta(u_p)_i^k = -\omega^2 (u_p)_{i-2}^k, \quad (i \ge k + 2), $$
# The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_79.png" /></figure>

* 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**
# The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_80.png" /></figure>

* Proof of the **existence-unicity**:

* with truncature functions, we subtract the growing behavior at infinity of the $(u_p)_i^k$

* We use the “classical” **variational theory** (wheighted Sobolev spaces, Leroux, Hardy,...)
The slot field of order $i > 1$

* The $U_i^k$ are defined on the **canonical** domain:

<figure><img src="image_81.png" /></figure>
The slot field of order $i > 1$

* The $U_i^k$ are defined on the **canonical** domain:

<figure><img src="image_82.png" /></figure>

* The $U_i^k$ does not depend on $y$.
The slot field of order $i > 1$

* The $U_i^k$ are defined on the **canonical** domain:

<figure><img src="image_83.png" /></figure>

* The $U_i^k$ does not depend on $y$.

* $U_i^k(x) = \int_0^1 (u_p)_i^k(0, Y)dY \exp i\omega x$
# Some properties

We see that:

<figure><img src="image_84.png" /></figure>

More $i-k$ is **large** more $u_i^k$ is **singular** at the origin:

$r^{-p}$ terms, $p = 0, \dots, i-k-1$
# Some properties

We see that:

* More $i-k$ is **large** more $u_i^k$ is **singular** at the origin:

$r^{-p}$ terms, $p = 0, \dots, i-k-1$

* More $i-k$ is **large** more $(u_p)_i^k$ is **growing**:

$\rho^p$ terms, $p = 0, \dots, i-k$,
$X^p$ terms, $p = 0, \dots, i-k$,
# Some properties

We see that:

* More $i-k$ is **large** more $u_i^k$ is **singular** at the origin:

$r^{-p}$ terms, $p = 0, \dots, i-k-1$

* More $i-k$ is **large** more $(u_p)_i^k$ is **growing**:

$\rho^p$ terms, $p = 0, \dots, i-k,$
$X^p$ terms, $p = 0, \dots, i-k,$

* When the **order** $i$ grows, one has $O(\frac{i^2}{2}) (\times 3)$ terms to compute...
Devirvate the terms of the as. exp.

<figure><img src="image_85.png" /></figure>

We search for solutions of the form:

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k \qquad (\text{far field}) $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k \qquad (\text{near field}) $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k U_i^k \qquad (\text{slot field}) $$
Devirvate the terms of the as. exp.

<figure><img src="image_86.png" /></figure>

We search for solutions of the form:

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k \qquad (\text{far field}) $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k \qquad (\text{near field}) $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k U_i^k \qquad (\text{slot field}) $$

<figure><img src="image_87.png" /></figure>

We **inject** the equations (Helmholtz, Neumann)
# Devirvate the terms of the as. exp.

* We search for solutions of the form:

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k \quad \text{(far field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k \quad \text{(near field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k U_i^k \quad \text{(slot field)} $$

* We **inject** the equations (Helmholtz, Neumann)

* We obtain the **coupling** conditions: (**the difficulty**)
# Far-Near coupling

<figure><img src="image_88.png" /></figure>

In a **thick zone**:

$$ \epsilon \ll \eta_H \ll \lambda. $$

We write the coupling condition:

$$ u^{\epsilon}(\eta_H, \theta) = (u_p)^{\epsilon}(\frac{\eta_H}{\epsilon}, \theta). $$

$$ \begin{gathered} \sum_{i \in \mathbb{Z}}^{+\infty} \sum_{k \in \mathbb{Z}}^{\epsilon} (\log \epsilon)^k u_i^k(\eta_H, \theta) \approx \sum_{i \in \mathbb{Z}}^{+\infty} \sum_{k \in \mathbb{Z}}^{\epsilon} (\log \epsilon)^k (u_p)_i^k(\frac{\eta_H}{\epsilon}, \theta) \\ \eta_H \to 0 \qquad \frac{\eta_H}{\epsilon} \to +\infty \end{gathered} $$
# Far-Near coupling

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k(\eta_H, \theta) \approx \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k(\frac{\eta_H}{\varepsilon}, \theta) \\ \eta_H \to 0 \qquad \frac{\eta_H}{\varepsilon} \to +\infty $$

We **expand**

* the left serie according to $\eta_H$ near 0

* The right serie according to $\eta_H/\varepsilon$ tending ot **infinity**

We **identify** all the terms of the two series.
# The conclusion of the coupling

<figure><img src="image_89.png" /></figure>

The **far field-near field coupling**:

The **singular** behavior of the **far field** is coupled with the
**none growing** behavior of the **near field at infinity**.

<figure><img src="image_90.png" /></figure>
# The conclusion of the coupling

<figure><img src="image_91.png" /></figure>

The **far field-near field coupling:**

The **growing** behavior of the **near field at infinity** is coupled
with the **none singular** behavior of the **far field**.

<figure><img src="image_92.png" /></figure>
# The conclusion of the coupling

* The **far field**-**near field** coupling:

* The **near field**-**slot field** coupling:

The **growing** behavior of the **near field** is coupled with the **none growing** behavior of the **slot field** (derivative values)

<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>
Mathematical analysis

<figure><img src="image_95.png" /></figure>

$$ \left\| \mathbf{u}^{\varepsilon} - \mathbf{u}^{0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^{i} (\log \varepsilon)^{k} \mathbf{u}_{i}^{k} \right\|_{H^{1}(\Omega^{R,R'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^{n} \|f\|_{L^{2}(\Omega)}. $$
Mathematical analysis

<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

$$ \left\| \textcolor{red}{u^{\epsilon}} - \textcolor{red}{u^0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \textcolor{blue}{\epsilon^i (\log \epsilon)^k u_i^k} \right\|_{H^1(\Omega^{R,R'})} \leq C \textcolor{blue}{\epsilon^{n+1}} (\log \epsilon)^n \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{u_p^{\epsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \textcolor{blue}{\epsilon^i (\log \epsilon)^k (u_p)_i^k} \right\|_{H^1(\mathcal{B}_1^{L,L'})} \leq C \textcolor{blue}{\epsilon^{n+1}} (\log \epsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$
Mathematical analysis

<figure><img src="image_98.png" /></figure>

<figure><img src="image_99.png" /></figure>

<figure><img src="image_100.png" /></figure>

$$ \left\| \textcolor{red}{u^{\epsilon}} - \textcolor{red}{u^0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \epsilon^i (\log \epsilon)^k \textcolor{red}{u_i^k} \right\|_{H^1(\Omega^{R,R'})} \le C \epsilon^{n+1} (\log \epsilon)^n \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{u_p^{\epsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^i (\log \epsilon)^k (\textcolor{red}{u_p})_i^k \right\|_{H^1(\mathcal{B}_1^{L,L'})} \le C \epsilon^{n+1} (\log \epsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{U^{\epsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^i (\log \epsilon)^k \textcolor{red}{U_i^k} \right\|_{H^1(\mathcal{O}_1^{l,l'})} \le C \epsilon^{n+1} (\log \epsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$
1. Mathematical analysis of the finite slot (**resonance** phenomena)

<figure><img src="image_101.png" /></figure>

The difficulty: the **stability** result.
Perspectives

1. Mathematical analysis of the finite slot (**resonance** phenomena)

2. Comparison with the **multi-scale** technique
Perspectives

1. Mathematical analysis of the finite slot (**resonance**) phenomena

2. Comparison with the **multi-scale** technique

3. The 3D **Maxwell** equation

<figure><img src="image_102.png" /></figure>
# Perspectives

1. Mathematical analysis of the finite slot (resonance phenomena)

2. Comparison with the multi-scale technique

3. The 3D Maxwell equation

4. The time domain (evolution equation)

$$ \frac{\partial^2 u}{\partial t^2} - c^2 \Delta u = 0. $$