High frequency scattering and spectral methods

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Linnaeus University Dissertations

No 265/2016

Imad Akeab

High frequency scattering

and spectral methods

linnaeus university press

Lnu.se

isbn: 978-91-88357-40-3

Hi gh f req uen cy sc at ter in g a nd sp ect ra l me th ods

Ima

d Akea

b

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Linnaeus University Dissertations

No 265/2016

H

IGH FREQUENCY SCATTERING

AND SPECTRAL METHODS

I

MAD

A

KEAB

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Linnaeus University Dissertations

No 265/2016

H

IGH FREQUENCY SCATTERING

AND SPECTRAL METHODS

I

MAD

A

KEAB

(6)

Abstract

Akeab, Imad (2016). High frequency scattering and spectral methods, Linnaeus

University Dissertation No 265/2016, ISBN: 978-91-88357-40-3. Written in

English.

This thesis consists of five parts. The first part is an introduction with

references to some recent work on 2D electromagnetic scattering problems at

high frequencies. It also presents the basic integral equation types for

impenetrable objects and the standard elements of the method of moments.

An overview of frequency modulated radar at low frequencies is followed by

summaries of the papers.

Paper I presents an accurate implementation of the method of moments for a

perfectly conducting cylinder. A scaling for the rapid variation of the solution

improves accuracy. At high frequencies, the method of moments leads to a

large dense system of equations. Sparsity in this system is obtained by the

modification of the path in the integral equation. The modified path reduces

the accuracy in the deep shadow.

In paper II, a hybrid method is used to handle the standing waves that are

prominent in the shadow for the cylindrical TE case. The shadow region is

treated separately, in a hybrid scheme based on a priori knowledge about the

solution. An accurate method to combine solutions in this hybrid scheme is

presented.

In paper III, the surface current in the shadow zone of a convex or a concave

scatterer is approximated by extracting the dominant waves. An accurate

technique based on the symmetric discrete Fourier transform is used to extract

the complex wavenumbers and amplitudes for those waves. The dominant

waves constitute a concise form of scaling that is used to improve the

performance of the method of moments. The effect of surface curvature on the

dominant waves has been investigated in this work.

In paper IV, frequency modulated continuous wave radar (FMCW) at low

frequency is studied as a way to locate targets that are normally not detected by

conventional radar. Three separate platforms with isotropic antennas are used

for this purpose. The trilateration method is a way to locate the targets

accurately by means of spectral techniques.

The problem of ghost targets has been studied for monostatic and multistatic

radar. In the case of confluent echoes in the spectra, potentially missing echoes

are reinserted in order to locate all targets. The Capon method is used to

High frequency scattering and spectral methods

Doctoral dissertation, Department Physics and Electrical Engineering,

Linnaeus University, Växjö, Sweden, 2016

ISBN: 978-91-88357-40-3

Published by: Linnaeus University Press, 351 95 Växjö, Sweden

Printed by: Elanders Sverige AB, 2016

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Abstract

Akeab, Imad (2016). High frequency scattering and spectral methods, Linnaeus

University Dissertation No 265/2016, ISBN: 978-91-88357-40-3. Written in

English.

This thesis consists of five parts. The first part is an introduction with

references to some recent work on 2D electromagnetic scattering problems at

high frequencies. It also presents the basic integral equation types for

impenetrable objects and the standard elements of the method of moments.

An overview of frequency modulated radar at low frequencies is followed by

summaries of the papers.

Paper I presents an accurate implementation of the method of moments for a

perfectly conducting cylinder. A scaling for the rapid variation of the solution

improves accuracy. At high frequencies, the method of moments leads to a

large dense system of equations. Sparsity in this system is obtained by the

modification of the path in the integral equation. The modified path reduces

the accuracy in the deep shadow.

In paper II, a hybrid method is used to handle the standing waves that are

prominent in the shadow for the cylindrical TE case. The shadow region is

treated separately, in a hybrid scheme based on a priori knowledge about the

solution. An accurate method to combine solutions in this hybrid scheme is

presented.

In paper III, the surface current in the shadow zone of a convex or a concave

scatterer is approximated by extracting the dominant waves. An accurate

technique based on the symmetric discrete Fourier transform is used to extract

the complex wavenumbers and amplitudes for those waves. The dominant

waves constitute a concise form of scaling that is used to improve the

performance of the method of moments. The effect of surface curvature on the

dominant waves has been investigated in this work.

In paper IV, frequency modulated continuous wave radar (FMCW) at low

frequency is studied as a way to locate targets that are normally not detected by

conventional radar. Three separate platforms with isotropic antennas are used

for this purpose. The trilateration method is a way to locate the targets

accurately by means of spectral techniques.

The problem of ghost targets has been studied for monostatic and multistatic

radar. In the case of confluent echoes in the spectra, potentially missing echoes

are reinserted in order to locate all targets. The Capon method is used to

obtain high resolution spectra and thus reduce the confluence problem. The

need for bandwidth is also reduced.

High frequency scattering and spectral methods

Doctoral dissertation, Department Physics and Electrical Engineering,

Linnaeus University, Växjö, Sweden, 2016

ISBN: 978-91-88357-40-3

Published by: Linnaeus University Press, 351 95 Växjö, Sweden

Printed by: Elanders Sverige AB, 2016

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Thesis submitted for the degree of Doctor of Technology

High frequency scattering

and spectral methods

Imad Kassar Akeab

October 5, 2016

Department of Physics and Electrical Engineering

Faculty of Technology

Linnaeus University

Sweden

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Thesis submitted for the degree of Doctor of Technology

High frequency scattering

and spectral methods

Imad Kassar Akeab

October 5, 2016

Department of Physics and Electrical Engineering

Faculty of Technology

Linnaeus University

Sweden

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List of Figures

1.1 A cross section of an infinite PEC cylinder illuminated by a wave polarized

in the z-direction. . . 5

1.2 A window function χ. . . 12

1.3 Block diagram of a simple FMCW radar. . . 15

1.4 Geometry of a bistatic radar system. . . 17

1.5 An illustration of trilateration by means of intersecting circles. . . 19

1.6 Block diagram for an SDR radio. . . 20

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List of Figures

1.1 A cross section of an infinite PEC cylinder illuminated by a wave polarized

in the z-direction. . . 5

1.2 A window function χ. . . 12

1.3 Block diagram of a simple FMCW radar. . . 15

1.4 Geometry of a bistatic radar system. . . 17

1.5 An illustration of trilateration by means of intersecting circles. . . 19

1.6 Block diagram for an SDR radio. . . 20

1.7 Simple flow graph for a GNU simulation of FMCW. . . 22

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Acknowledgements

I would like to thank my supervisor, Professor Sven-Erik Sandström for his many helpful comments that led to a substantial improvement of this thesis. A special thanks also to my family for their great support during the period of study and preparation of this thesis.

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Acknowledgements

I would like to thank my supervisor, Professor Sven-Erik Sandström for his many helpful comments that led to a substantial improvement of this thesis. A special thanks also to my family for their great support during the period of study and preparation of this thesis.

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Abstract

This thesis consists of five parts. The first part is an introduction with references to some recent work on 2D electromagnetic scattering problems at high frequencies. It also presents the basic integral equation types for impenetrable objects and the standard elements of the method of moments. An overview of frequency modulated radar at low frequencies is followed by summaries of the papers.

Paper I presents an accurate implementation of the method of moments for a perfectly conducting cylinder. A scaling for the rapid variation of the solution improves accuracy. At high frequencies, the method of moments leads to a large dense system of equations. Sparsity in this system is obtained by the modification of the path in the integral equa-tion. The modified path reduces the accuracy in the deep shadow.

In paper II, a hybrid method is used to handle the standing waves that are promi-nent in the shadow for the cylindrical TE case. The shadow region is treated separately, in a hybrid scheme based on a priori knowledge about the solution. An accurate method to combine solutions in this hybrid scheme is presented.

In paper III, the surface current in the shadow zone of a convex or a concave scat-terer is approximated by extracting the dominant waves. An accurate technique based on the symmetric discrete Fourier transform is used to extract the complex wavenumbers and amplitudes for those waves. The dominant waves constitute a concise form of scaling that is used to improve the performance of the method of moments. The effect of surface curvature on the dominant waves has been investigated in this work.

In paper IV, frequency modulated continuous wave radar (FMCW) at low frequency is studied as a way to locate targets that are normally not detected by conventional radar. Three separate platforms with isotropic antennas are used for this purpose. The trilateration method is a way to locate the targets accurately by means of spectral tech-niques. The problem of ghost targets has been studied for monostatic and multistatic radar. In the case of confluent echoes in the spectra, potentially missing echoes are rein-serted in order to locate all targets. The Capon method is used to obtain high resolution spectra and thus reduce the confluence problem. The need for bandwidth is also reduced.

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Abstract

This thesis consists of five parts. The first part is an introduction with references to some recent work on 2D electromagnetic scattering problems at high frequencies. It also presents the basic integral equation types for impenetrable objects and the standard elements of the method of moments. An overview of frequency modulated radar at low frequencies is followed by summaries of the papers.

Paper I presents an accurate implementation of the method of moments for a perfectly conducting cylinder. A scaling for the rapid variation of the solution improves accuracy. At high frequencies, the method of moments leads to a large dense system of equations. Sparsity in this system is obtained by the modification of the path in the integral equa-tion. The modified path reduces the accuracy in the deep shadow.

In paper II, a hybrid method is used to handle the standing waves that are promi-nent in the shadow for the cylindrical TE case. The shadow region is treated separately, in a hybrid scheme based on a priori knowledge about the solution. An accurate method to combine solutions in this hybrid scheme is presented.

In paper III, the surface current in the shadow zone of a convex or a concave scat-terer is approximated by extracting the dominant waves. An accurate technique based on the symmetric discrete Fourier transform is used to extract the complex wavenumbers and amplitudes for those waves. The dominant waves constitute a concise form of scaling that is used to improve the performance of the method of moments. The effect of surface curvature on the dominant waves has been investigated in this work.

In paper IV, frequency modulated continuous wave radar (FMCW) at low frequency is studied as a way to locate targets that are normally not detected by conventional radar. Three separate platforms with isotropic antennas are used for this purpose. The trilateration method is a way to locate the targets accurately by means of spectral tech-niques. The problem of ghost targets has been studied for monostatic and multistatic radar. In the case of confluent echoes in the spectra, potentially missing echoes are rein-serted in order to locate all targets. The Capon method is used to obtain high resolution spectra and thus reduce the confluence problem. The need for bandwidth is also reduced.

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Introduction

High frequency scattering problems that are formulated with integral equations lead to linear systems with a large number of unknowns. The solution of these problems, by means of efficient and accurate numerical techniques, is the topic of many recent papers [1, 2, 3, 4, 5, 6].

The Fast multipole method (FMM) is a general technique that has been developed to handle a large number of unknowns (N ). The method requires O(NlogN), or O(N), operations. Obtaining high accuracy with the FMM requires a large N, also for 2D problems. With the O(NlogN) complexity [7], the computational time will be consid-erable and the balance between accuracy and execution time is a point of interest. The standard FMM methods are derived from approximations of the kernel, in the form of Taylor expansions, or as expansions in terms of eigenfunctions. In a recent study [7], one handles the kernel with Cauchy’s integral formula and the Laplace transform. This leads to diagonal multipole-to-local operators, which reduce the computational time without losing accuracy.

Methods based on wavelets have been developed to solve the 2D Helmholtz problem for large wavenumbers. The idea is to obtain a sparse matrix by means of a wavelet basis. The number of operations needed to solve this sparse matrix is O(Nlog2_{N )}_{. An} investigation showed that the sparsity in this method is lost for high frequencies [8].

In this thesis we seek a fast and accurate method to solve two dimensional Helmholtz scattering problems for both medium and high frequencies. A formulation in terms of an integral equation has the advantage that there is a reduction down to a 1D problem. Dirichlet and Neumann boundary conditions at the surface of the scatterer are considered. An accurate approximation for the solution in the shadow region of the scatterer is proposed.

At high frequencies, the solution is rapidly oscillating and requires a large number of basis functions to be represented correctly. Since the phase of the solution resembles that of the incident field [1], according to physical optics (PO), one could factor the total solution into a smooth part and a phase factor. Extracting the phase variation from the solution reduces the size of the linear system substantially. One way of handling this

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Introduction

High frequency scattering problems that are formulated with integral equations lead to linear systems with a large number of unknowns. The solution of these problems, by means of efficient and accurate numerical techniques, is the topic of many recent papers [1, 2, 3, 4, 5, 6].

The Fast multipole method (FMM) is a general technique that has been developed to handle a large number of unknowns (N ). The method requires O(NlogN), or O(N), operations. Obtaining high accuracy with the FMM requires a large N, also for 2D problems. With the O(NlogN) complexity [7], the computational time will be consid-erable and the balance between accuracy and execution time is a point of interest. The standard FMM methods are derived from approximations of the kernel, in the form of Taylor expansions, or as expansions in terms of eigenfunctions. In a recent study [7], one handles the kernel with Cauchy’s integral formula and the Laplace transform. This leads to diagonal multipole-to-local operators, which reduce the computational time without losing accuracy.

Methods based on wavelets have been developed to solve the 2D Helmholtz problem for large wavenumbers. The idea is to obtain a sparse matrix by means of a wavelet basis. The number of operations needed to solve this sparse matrix is O(Nlog2_{N )}_{. An} investigation showed that the sparsity in this method is lost for high frequencies [8].

In this thesis we seek a fast and accurate method to solve two dimensional Helmholtz scattering problems for both medium and high frequencies. A formulation in terms of an integral equation has the advantage that there is a reduction down to a 1D problem. Dirichlet and Neumann boundary conditions at the surface of the scatterer are considered. An accurate approximation for the solution in the shadow region of the scatterer is proposed.

At high frequencies, the solution is rapidly oscillating and requires a large number of basis functions to be represented correctly. Since the phase of the solution resembles that of the incident field [1], according to physical optics (PO), one could factor the total solution into a smooth part and a phase factor. Extracting the phase variation from the solution reduces the size of the linear system substantially. One way of handling this problem was investigated by Bruno, et al. in 2004 [1]. They used the principle of station-ary phase for local integration at high frequencies. In 2007 Huybrechs and Vandewalle [9] obtained a sparse linear system by modifying the integral equation by means of a

suit-1 ix

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Schlick. This radar operates at 3-30 MHz with a 16-element antenna array [14, 15]. An antenna array provides sufficient directivity in the azimuth direction if the array is big enough. The resolution of the antenna is not a problem at high frequencies, while at low frequencies the size of the antenna becomes impractical if a high resolution is required. A geometrical method called trilateration can be used to detect targets without relying on directivity. The trilateration method can locate most targets but there is a problem with false, or ghost targets. An FMCW radar for low frequencies that handles the problems with confluent echoes and ghost targets is studied in paper IV.

In 2005 Fölster, et al. [16] considered a 24 GHz automotive radar network with four short-range radars. The network is suitable only for short distances (about 30 m) and it covers an azimuth angle of 140 degrees. Since the observation area is limited, one deals with concentrated targets. The ghost target problem is reduced by processing the data derived from the trilateration method. Rabe, et al. [17] suggested a trilateration method based on only two antennas. The distance between the two antennas is adjusted in relation to the distance between two targets to avoid the occurrence of ghost targets. The limited observation angle of the radar reduces the number of ghost targets.

The accuracy of the trilateration is to a large extent given by the accuracy of the mea-sured ranges. Huang, et al. [18] studied the details of the intersection problem. With range errors there comes an ambiguity and the Lemoine point was proposed as the best estimate for the target postion. Samokhin, et al. [19] used trilateration for a 3D problem and applied sorting techniques to handle the problem with ghost targets.

1.1 Outline of the topics in the thesis

Some background and notation for the integral equations for the transverse electric and magnetic case is given. The combined field integral equation (CFIE) is used to elimi-nate internal resonances. Section (1.5) briefly introduces the uniform and nonuniform B-splines. Integration paths for numerical quadrature are discussed as a preparation for the implementation of sparsity. A few words on numerical quadrature and solvers for linear systems make up sections (1.7) and (1.8). An overview of an hybrid method to combine a known solution in the shadow zone, with the numerical solution in the lit zone, is presented in section (1.9). The discrete Fourier transform and its capacity to identify complex wave numbers is described briefly in section (1.10). An overview of how scaling can be used to obtain a suitable description for the current in the shadow zone is given in section (1.11).

Section (1.12) deals with FMCW estimation of the distance to a target and this is able complex integration contour. The decay of the kernel makes it possible to reduce

the number of basis functions along the complex integration path and obtain sparsity. Leaving the real axis and doing integration in the complex plane leads to a sparse matrix. This approach is accurate in the lit zone of the scatterer only [9]. The deep shadow is excluded with the argument that the field is approximately zero there at high frequencies. The problem with rapidly oscillating solutions can be handled by using a simple scaling in the lit zone and a more complicated scaling with amplitude variation in the shadow zone. By connecting the scalings properly, good results can be achieved. This technique and an investigation of sparsity and accuracy are presented in paper I.

One could also use a hybrid approach with entirely separate solutions in the lit zone and in the shadow zone. A solution that is known a priori can be used in the shadow zone. A crucial aspect in this context is the matching of the two solutions and a method to do this accurately is presented in paper II.

An example with hybrid techniques for a 3D perfect electric conductor (PEC) scatterer involves a combination of the method of moments and physical optics [10]. Another hy-brid technique is proposed by Engquist, et al. [11] for a two-dimensional high frequency scattering problem. They combine the solution obtained with the FMM, for the shadow zone, with a geometrical optics approach for the lit zone [1]. The two solutions overlap at the shadow boundaries in order to avoid discontinuities there. This method is fast, but it is restricted to simple convex geometries.

The surface currents at the shadow boundary were first studied systematically by Fock [12]. The corresponding Fock type wave numbers are accurate only for circular scatterers. In a numerical approach, Kwon, et al. extracted the phase of the surface current from a known current on a scatterer [13]. The known numerical solution was obtained with the method of moments in 3D. The efficiency of the method depends critically on the scatterer geometry. Accurate scalings for noncircular two-dimensional scatterers are studied in paper III.

If we now turn to the last part of this thesis, the focus shifts to a different applica-tion of spectral techniques. A radar concept known as frequency modulated continuous wave (FMCW) involves spectral algorithms and can be used for target location at low frequencies. The use of low frequencies is intended to extend the penetration capability of the radar and thereby allow detection of hidden targets. A 2D simulation for monostatic and multistatic radar systems is considered for the case with several targets. The type of radar that is considered here is intended for shorter ranges and for targets obscured by hills or targets hidden in a forest. An advantage with FMCW radar is that only low power is needed.

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Schlick. This radar operates at 3-30 MHz with a 16-element antenna array [14, 15]. An antenna array provides sufficient directivity in the azimuth direction if the array is big enough. The resolution of the antenna is not a problem at high frequencies, while at low frequencies the size of the antenna becomes impractical if a high resolution is required. A geometrical method called trilateration can be used to detect targets without relying on directivity. The trilateration method can locate most targets but there is a problem with false, or ghost targets. An FMCW radar for low frequencies that handles the problems with confluent echoes and ghost targets is studied in paper IV.

In 2005 Fölster, et al. [16] considered a 24 GHz automotive radar network with four short-range radars. The network is suitable only for short distances (about 30 m) and it covers an azimuth angle of 140 degrees. Since the observation area is limited, one deals with concentrated targets. The ghost target problem is reduced by processing the data derived from the trilateration method. Rabe, et al. [17] suggested a trilateration method based on only two antennas. The distance between the two antennas is adjusted in relation to the distance between two targets to avoid the occurrence of ghost targets. The limited observation angle of the radar reduces the number of ghost targets.

The accuracy of the trilateration is to a large extent given by the accuracy of the mea-sured ranges. Huang, et al. [18] studied the details of the intersection problem. With range errors there comes an ambiguity and the Lemoine point was proposed as the best estimate for the target postion. Samokhin, et al. [19] used trilateration for a 3D problem and applied sorting techniques to handle the problem with ghost targets.

1.1 Outline of the topics in the thesis

Some background and notation for the integral equations for the transverse electric and magnetic case is given. The combined field integral equation (CFIE) is used to elimi-nate internal resonances. Section (1.5) briefly introduces the uniform and nonuniform B-splines. Integration paths for numerical quadrature are discussed as a preparation for the implementation of sparsity. A few words on numerical quadrature and solvers for linear systems make up sections (1.7) and (1.8). An overview of an hybrid method to combine a known solution in the shadow zone, with the numerical solution in the lit zone, is presented in section (1.9). The discrete Fourier transform and its capacity to identify complex wave numbers is described briefly in section (1.10). An overview of how scaling can be used to obtain a suitable description for the current in the shadow zone is given in section (1.11).

Section (1.12) deals with FMCW estimation of the distance to a target and this is followed by a presentation of radar types in section (1.13). The Capon method is an alternative to FFT and section (1.14) describes how a spectrum can be obtained. The trilateration method is introduced briefly in section (1.15). An introduction to Software

3 able complex integration contour. The decay of the kernel makes it possible to reduce

the number of basis functions along the complex integration path and obtain sparsity. Leaving the real axis and doing integration in the complex plane leads to a sparse matrix. This approach is accurate in the lit zone of the scatterer only [9]. The deep shadow is excluded with the argument that the field is approximately zero there at high frequencies. The problem with rapidly oscillating solutions can be handled by using a simple scaling in the lit zone and a more complicated scaling with amplitude variation in the shadow zone. By connecting the scalings properly, good results can be achieved. This technique and an investigation of sparsity and accuracy are presented in paper I.

One could also use a hybrid approach with entirely separate solutions in the lit zone and in the shadow zone. A solution that is known a priori can be used in the shadow zone. A crucial aspect in this context is the matching of the two solutions and a method to do this accurately is presented in paper II.

An example with hybrid techniques for a 3D perfect electric conductor (PEC) scatterer involves a combination of the method of moments and physical optics [10]. Another hy-brid technique is proposed by Engquist, et al. [11] for a two-dimensional high frequency scattering problem. They combine the solution obtained with the FMM, for the shadow zone, with a geometrical optics approach for the lit zone [1]. The two solutions overlap at the shadow boundaries in order to avoid discontinuities there. This method is fast, but it is restricted to simple convex geometries.

The surface currents at the shadow boundary were first studied systematically by Fock [12]. The corresponding Fock type wave numbers are accurate only for circular scatterers. In a numerical approach, Kwon, et al. extracted the phase of the surface current from a known current on a scatterer [13]. The known numerical solution was obtained with the method of moments in 3D. The efficiency of the method depends critically on the scatterer geometry. Accurate scalings for noncircular two-dimensional scatterers are studied in paper III.

If we now turn to the last part of this thesis, the focus shifts to a different applica-tion of spectral techniques. A radar concept known as frequency modulated continuous wave (FMCW) involves spectral algorithms and can be used for target location at low frequencies. The use of low frequencies is intended to extend the penetration capability of the radar and thereby allow detection of hidden targets. A 2D simulation for monostatic and multistatic radar systems is considered for the case with several targets. The type of radar that is considered here is intended for shorter ranges and for targets obscured by hills or targets hidden in a forest. An advantage with FMCW radar is that only low power is needed.

An important long distance application of FMCW is referred to as over the horizon radar (OTH). An example of a long distance coastal radar is presented by Gurgel and

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�(θ)

ρ(θ) δ

�

��

�

′

��

�

��

θ

ψ

�

_{� �}

Figure 1.1: A cross section of an infinite PEC cylinder illuminated by a wave polarized in the z-direction.

Here, R =| ρ(θ) − ρ_(θ₎_{| is the distance between the observation and the source point}

and H(1)

0 is the Hankel function of the first kind and zero order.

From the extinction theorem, the total electric field inside the scatterer equals zero (E = 0)[21]. Thus the incident electric field is given by,

Ei(x, y) =− 2π 0 i 4H (1) 0 (kR) ∂E ∂n ρ2₊ dρ dθ 2 dθ. (1.7)

A surface current Jsis induced in the z direction. The normal derivative of the electric

field on the boundary equals,

∂E(x, y)

∂n = ikηJs. (1.8)

Here k and η are the wavenumber and the impedance of free space, respectively. The scatterer surface should be smooth, otherwise the normal derivative ∂E(x, y)/∂n_{is not}

Defined Radio (SDR) is given in section (1.16). Short summaries of papers I-IV and a bibliography concludes the introduction.

1.2 The Electric Field Integral Equation (EFIE)

In electromagnetic scattering problems, the total electric field E is the sum of the incident field Ei_{and the scattered field E}s_{. By treating 2D problems, the total electric field is,}

E(x, y) = Ei(x, y) + Es(x, y). (1.1) A wave with the electric field in the z direction is incident on an infinite PEC cylinder with an angle ψ as shown in Fig. 1.1. The magnetic field is then transverse (TM) to the z direction [20] and it is possible to derive the electric field integral equation (EFIE). Since the electric field is polarized in the z direction Ei_{= E}i_z_ˆ_{, one can deal with the problem}

as a scalar problem. For the interior case, where the observation point approaches the surface from inside and is separated by a short distance δ, the radii for the source and observation point are,

ρ(θ) = b(θ), (1.2)

ρ(θ) = b(θ)− δ. (1.3)

From Green’s theorem, it follows that the scattered electric field at a point (x, y) is Es(x, y) = 2π 0 G0(θ, θ)∂E ∂n ρ2₊ dρ dθ 2 dθ. (1.4)

Here ˆnis a unit normal to the scatterer surface. The scatterer boundary is defined by the function b(θ),

b(θ) = a1a2

(a2cosθ)2_{+ (a1sinθ)}2, (1.5) where a1and a2is the semi major and semi minor axis, respectively.

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�(θ)

ρ(θ) δ

�

��

�

′

��

�

��

θ

ψ

�

_{� �}

Figure 1.1: A cross section of an infinite PEC cylinder illuminated by a wave polarized in the z-direction.

Here, R =| ρ(θ) − ρ_(θ₎_{| is the distance between the observation and the source point}

and H(1)

0 is the Hankel function of the first kind and zero order.

From the extinction theorem, the total electric field inside the scatterer equals zero (E = 0)[21]. Thus the incident electric field is given by,

Ei(x, y) =− 2π 0 i 4H (1) 0 (kR) ∂E ∂n ρ2₊ dρ dθ 2 dθ. (1.7)

A surface current Jsis induced in the z direction. The normal derivative of the electric

field on the boundary equals,

∂E(x, y)

∂n = ikηJs. (1.8)

Here k and η are the wavenumber and the impedance of free space, respectively. The scatterer surface should be smooth, otherwise the normal derivative ∂E(x, y)/∂n_{is not}

well defined in particular at edges. The final result is an integral equation for a PEC surface (TM case),

5 Defined Radio (SDR) is given in section (1.16). Short summaries of papers I-IV and a

bibliography concludes the introduction.

1.2 The Electric Field Integral Equation (EFIE)

In electromagnetic scattering problems, the total electric field E is the sum of the incident field Ei_{and the scattered field E}s_{. By treating 2D problems, the total electric field is,}

E(x, y) = Ei(x, y) + Es(x, y). (1.1) A wave with the electric field in the z direction is incident on an infinite PEC cylinder with an angle ψ as shown in Fig. 1.1. The magnetic field is then transverse (TM) to the z direction [20] and it is possible to derive the electric field integral equation (EFIE). Since the electric field is polarized in the z direction Ei_{= E}i_z_ˆ_{, one can deal with the problem}

as a scalar problem. For the interior case, where the observation point approaches the surface from inside and is separated by a short distance δ, the radii for the source and observation point are,

ρ(θ) = b(θ), (1.2)

ρ(θ) = b(θ)− δ. (1.3)

From Green’s theorem, it follows that the scattered electric field at a point (x, y) is Es(x, y) = 2π 0 G0(θ, θ)∂E ∂n ρ2₊ dρ dθ 2 dθ. (1.4)

Here ˆn is a unit normal to the scatterer surface. The scatterer boundary is defined by the function b(θ),

b(θ) = a1a2

(a2cosθ)2_{+ (a1sinθ)}2, (1.5) where a1 and a2 is the semi major and semi minor axis, respectively.

The Green’s function G0(θ, θ₎_{in two dimensions simplifies in this case to,}

G0(θ, θ) = i 4H

(1)

0 (kR). (1.6)

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make it small. There are two types of errors. One error relates to the conditioning of Eq. 1.13 and increases with δ. So for this error, it would be best to have a δ that is as small as possible. Then the kernel in Eq. 1.9 is sharp enough to project out the solution accurately. However, this requires that the matrix elements in Z are calculated accurately. The other type of error relates to the numerical quadrature and this error increases when δ decreases. There is then an optimal value of δ that is found by ob-serving the condition number and the error in the computed current. In practice, good results are obtained for a range of δ values and it is not critical to determine an optimal δ. In spite of the ill-posedness, a high accuracy can be achieved provided that suitable δ values are used. No optimal value of N is observed in this context.

Typically one needs 10 basis functions per wavelength in order to get a good approxi-mation for the surface current on a boundary. This leads to a large number of unknowns and solving the system of linear equations with regular solvers will be very slow. New accurate numerical techniques to solve this problem for circular and non-circular bound-ary using a reduced number of basis functions are suggested in papers I-III.

The system of linear equations in Eq. 1.13 can be solved by means of Gaussian elimi-nation or iterative techniques like the Quasi-Minimal Residual (QMR).

1.3 The Magnetic Field Integral Equation (MFIE)

If a wave, with the magnetic field in the z direction Hi _{= H}i_z_ˆ_{, is incident on a PEC}

cylinder, the electric field is transverse (TE) to the z direction. The magnetic field only is needed to calculate the surface current. The magnetic field has a vanishing normal derivative on a PEC boundary [21],

∂H(x, y)

∂n = 0, (1.14)

and the magnetic field equals the surface current H = Js. If the observation point lies

inside the scatterer, the extinction theorem [23] and Green’s theorem yields,

Hi(x, y) = k 4i 2π 0 H(1)₁ (kR)∂R ∂nJs ρ2₊ dρ dθ 2 dθ. (1.15) since, Ei(x, y) =kη 4 2π 0 H(1)₀ (kR)Js ρ2₊ dρ dθ 2 dθ. (1.9)

Eq. 1.9 is an integral equation of the first kind. This type of integral equation is ill-posed when δ is larger than zero due to the smoothness of the integral kernel. A consequence is that the solution Jsof Eq. 1.9 is not continuous with respect to Ei[22].

Some kind of regularization process is required to obtain the solution numerically and such a process will be described below.

The ill-posedness related to the integral equation of the first kind is handled in the discretisation process. Galerkin methods can be used to obtain an approximate solution by solving a system of linear equations. To calculate an approximate solution for the sur-face current Js, one divides the boundary into cells. The surface current in each cell can

then be approximated as a superposition of sub-sectional basis functions. This method is called the method of moments (MoM). Spline functions can be used as basis functions. A B-spline of zero order (p = 0) is a simple pulse function.

Nj,0(θ) =

1 if θ∈ cell j

0 elsewhere . (1.10)

Higher order spline functions approximate the surface current more accurately, and will be discussed in section (1.5). The current in terms of N splines is written,

Js(θ) = N

j=1

cjNj,p(θ), (1.11)

and one obtains,

Ei(x, y) =kη 4 N j=1 cj 2π 0 H(1)0 (kR)Nj,p(θ) ρ2₊ dρ dθ 2 dθ. (1.12) By performing a testing (collocation) at the middle of each cell on the boundary, one extracts a system of linear equations. The matrix form, in terms of the usually dense matrix Zmj, is,

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make it small. There are two types of errors. One error relates to the conditioning of Eq. 1.13 and increases with δ. So for this error, it would be best to have a δ that is as small as possible. Then the kernel in Eq. 1.9 is sharp enough to project out the solution accurately. However, this requires that the matrix elements in Z are calculated accurately. The other type of error relates to the numerical quadrature and this error increases when δ decreases. There is then an optimal value of δ that is found by ob-serving the condition number and the error in the computed current. In practice, good results are obtained for a range of δ values and it is not critical to determine an optimal δ. In spite of the ill-posedness, a high accuracy can be achieved provided that suitable δ values are used. No optimal value of N is observed in this context.

Typically one needs 10 basis functions per wavelength in order to get a good approxi-mation for the surface current on a boundary. This leads to a large number of unknowns and solving the system of linear equations with regular solvers will be very slow. New accurate numerical techniques to solve this problem for circular and non-circular bound-ary using a reduced number of basis functions are suggested in papers I-III.

The system of linear equations in Eq. 1.13 can be solved by means of Gaussian elimi-nation or iterative techniques like the Quasi-Minimal Residual (QMR).

1.3 The Magnetic Field Integral Equation (MFIE)

If a wave, with the magnetic field in the z direction Hi_{= H}i_z_ˆ_{, is incident on a PEC}

cylinder, the electric field is transverse (TE) to the z direction. The magnetic field only is needed to calculate the surface current. The magnetic field has a vanishing normal derivative on a PEC boundary [21],

∂H(x, y)

∂n = 0, (1.14)

and the magnetic field equals the surface current H = Js. If the observation point lies

inside the scatterer, the extinction theorem [23] and Green’s theorem yields,

Hi(x, y) = k 4i 2π 0 H(1)₁ (kR)∂R ∂nJs ρ2₊ dρ dθ 2 dθ. (1.15) since, ∂R ∂n = ˆn· ∇R = ˆn· R R = ˆn _{· ˆ}_R. _(1.16) 7 Ei(x, y) =kη 4 2π 0 H(1)₀ (kR)Js ρ2₊ dρ dθ 2 dθ. (1.9)

Eq. 1.9 is an integral equation of the first kind. This type of integral equation is ill-posed when δ is larger than zero due to the smoothness of the integral kernel. A consequence is that the solution Jsof Eq. 1.9 is not continuous with respect to Ei[22].

Some kind of regularization process is required to obtain the solution numerically and such a process will be described below.

The ill-posedness related to the integral equation of the first kind is handled in the discretisation process. Galerkin methods can be used to obtain an approximate solution by solving a system of linear equations. To calculate an approximate solution for the sur-face current Js, one divides the boundary into cells. The surface current in each cell can

then be approximated as a superposition of sub-sectional basis functions. This method is called the method of moments (MoM). Spline functions can be used as basis functions. A B-spline of zero order (p = 0) is a simple pulse function.

Nj,0(θ) =

1 if θ∈ cell j

0 elsewhere . (1.10)

Higher order spline functions approximate the surface current more accurately, and will be discussed in section (1.5). The current in terms of N splines is written,

Js(θ) = N

j=1

cjNj,p(θ), (1.11)

and one obtains,

Ei(x, y) = kη 4 N j=1 cj 2π 0 H(1)0 (kR)Nj,p(θ) ρ2₊ dρ dθ 2 dθ. (1.12) By performing a testing (collocation) at the middle of each cell on the boundary, one extracts a system of linear equations. The matrix form, in terms of the usually dense matrix Zmj, is,

[E_mi] = [Zmj][cj]. (1.13)

A first requirement for Eq. 1.13 is that the determinant det(Z) of the N × N matrix Z is non-zero. Then this equation is well-posed since the dimension N is finite. Still, the error in the solution could be large and an optimization procedure is required to

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Nj,p(θ) =

θ− θj

θj+p− θjNj,p−1(θ) +

θj+p+1− θ

θj+p+1− θj+1Nj+1,p−1(θ). (1.21)

The initial value of the B-spline order is zero. If (p = 0), one has a pulse function with support [θj, θj+1]; θcould also be complex. Zero order splines are easy to apply in

the MoM for a closed surface, since there is no overlap between the basis functions at the end points. To obtain accurate results, we have used higher order B-splines for the surface current and the B-splines then connect near the end points of [0, 2π].

Dividing the boundary into equi-distant θ intervals leads to uniform B-splines. For certain problems, one could use nonuniform B-splines. Nonuniform B-splines are obtained by modifying the knot vector that defines the intervals for the splines. An increased resolution may be needed for problematic regions, such as the deep shadow (standing wave pattern). ’

1.6 Integration along a complex contour with a decaying

kernel

According to Cauchy’s integral theorem, the integration in the complex plane in Eqs. 1.12 and 1.17 does not depend on the path of integration, provided that the integrand is analytic. Doing the integration along a real path (as introduced in section 1.2 and 1.3) leads to a dense matrix. One could find a suitable path in the complex plane and get essentially the same results, but with a sparse matrix [4]. It is not so easy to find a suitable path when the observation point lies in the shadow of the scatterer (paper I), since the integrand does not decay rapidly. The consequence is that the contour has to pass via the support of a number of basis functions in order to reach small values. There are some iterative methods for the steepest descent path [27], but they converge slowly. An approximate path is defined by means of line segments, as discussed in paper I.

1.7 Numerical quadrature

Quadrature refers to any method that approximates an integral. A quadrature with n points includes nodes θj and weights wj,

b a f (θ)dθ≈b− a 2 n wjf ( b− a 2 θj+ b + a 2 ). (1.22)

The final result is an integral equation for a PEC surface (TE case), Hi_{(x, y) =} k 4i 2π 0 H(1)₁ (kR) ˆR· ˆnJs ρ2₊ dρ dθ 2 dθ. (1.17) H(1)₁ is the Hankel function of the first kind and first order. This formula is called the Magnetic field integral equation (MFIE). In this case the induced surface current lies in the plane shown in Fig. 1.1. A problem is that the singularity of the kernel is stronger than for the TM case, and this complicates the quadrature. By approximating the surface current with a spline basis one obtains a linear system as described in section 1.2.

1.4 The Combined Field Integral Equation (CFIE)

Surface integral equations like the EFIE and the MFIE introduced in sections 1.2 and 1.3, have internal resonance problems. At, or near, resonance frequencies, the kernel of the integral equation becomes singular and that leads to a bad condition number. A way to avoid this problem is to use a formulation known as the combined field integral equation CFIE. This can be achieved by adding the derivatives of the integrals [24] to obtain a well-conditioned system as given by,

P = ζ + ∂

∂n, (1.18)

where ζ is a complex constant. A suitable value for ζ is given by the wavenumber of the incident field |ζ| ≈ k. Since the resonance frequencies are removed, this type of integral equation has a unique solution and is stable for all ranges of frequencies by choosing the optimal value for δ. The CFIE for the TM and TE cases are given by:

PEi_{(x, y) =} kη 4 2π 0 PH (1) 0 (kR)Js ρ2₊ dρ dθ 2 dθ (1.19) and, PHi(x, y) = k 4i 2π 0 PH (1) 1 (kR) ˆR· ˆnJs ρ2₊ dρ dθ 2 dθ. (1.20)

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Nj,p(θ) =

θ− θj

θj+p− θjNj,p−1(θ) +

θj+p+1− θ

θj+p+1− θj+1Nj+1,p−1(θ). (1.21)

The initial value of the B-spline order is zero. If (p = 0), one has a pulse function with support [θj, θj+1]; θcould also be complex. Zero order splines are easy to apply in

the MoM for a closed surface, since there is no overlap between the basis functions at the end points. To obtain accurate results, we have used higher order B-splines for the surface current and the B-splines then connect near the end points of [0, 2π].

Dividing the boundary into equi-distant θ intervals leads to uniform B-splines. For certain problems, one could use nonuniform B-splines. Nonuniform B-splines are obtained by modifying the knot vector that defines the intervals for the splines. An increased resolution may be needed for problematic regions, such as the deep shadow (standing wave pattern). ’

1.6 Integration along a complex contour with a decaying

kernel

According to Cauchy’s integral theorem, the integration in the complex plane in Eqs. 1.12 and 1.17 does not depend on the path of integration, provided that the integrand is analytic. Doing the integration along a real path (as introduced in section 1.2 and 1.3) leads to a dense matrix. One could find a suitable path in the complex plane and get essentially the same results, but with a sparse matrix [4]. It is not so easy to find a suitable path when the observation point lies in the shadow of the scatterer (paper I), since the integrand does not decay rapidly. The consequence is that the contour has to pass via the support of a number of basis functions in order to reach small values. There are some iterative methods for the steepest descent path [27], but they converge slowly. An approximate path is defined by means of line segments, as discussed in paper I.

1.7 Numerical quadrature

Quadrature refers to any method that approximates an integral. A quadrature with n points includes nodes θj and weights wj,

b a f (θ)dθ≈b− a 2 n j=1 wjf ( b− a 2 θj+ b + a 2 ). (1.22)

There are many types of quadrature depending on the selection of nodes and weights:

9 The final result is an integral equation for a PEC surface (TE case),

Hi_{(x, y) =} k 4i 2π 0 H(1)₁ (kR) ˆR· ˆnJs ρ2₊ dρ dθ 2 dθ. (1.17) H(1)₁ is the Hankel function of the first kind and first order. This formula is called the Magnetic field integral equation (MFIE). In this case the induced surface current lies in the plane shown in Fig. 1.1. A problem is that the singularity of the kernel is stronger than for the TM case, and this complicates the quadrature. By approximating the surface current with a spline basis one obtains a linear system as described in section 1.2.

1.4 The Combined Field Integral Equation (CFIE)

Surface integral equations like the EFIE and the MFIE introduced in sections 1.2 and 1.3, have internal resonance problems. At, or near, resonance frequencies, the kernel of the integral equation becomes singular and that leads to a bad condition number. A way to avoid this problem is to use a formulation known as the combined field integral equation CFIE. This can be achieved by adding the derivatives of the integrals [24] to obtain a well-conditioned system as given by,

P = ζ + ∂

∂n, (1.18)

where ζ is a complex constant. A suitable value for ζ is given by the wavenumber of the incident field |ζ| ≈ k. Since the resonance frequencies are removed, this type of integral equation has a unique solution and is stable for all ranges of frequencies by choosing the optimal value for δ. The CFIE for the TM and TE cases are given by:

PEi_{(x, y) =}kη 4 2π 0 PH (1) 0 (kR)Js ρ2₊ dρ dθ 2 dθ (1.19) and, PHi(x, y) = k 4i 2π 0 PH (1) 1 (kR) ˆR· ˆnJs ρ2₊ dρ dθ 2 dθ. (1.20)

1.5 Uniform and nonuniform B-splines

There are many types of basis functions that can be used in the method of moments [25], [5]. Higher order B-splines are smooth polynomials and have good approximation properties. The recursive form for B-splines of order p can be written as [26]:

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match the solutions by means of overlapping smooth functions is devised.

Fock functions can also be used as a priori solutions. Even though the result is less accurate, there is the interesting aspect that very good accuracy can be obtained with imperfect a priori information, provided that the matching of the solutions is good.

1.10 The discrete Fourier transform (DFT)

A function f(t) can be sampled with N equidistant samples. The number of samples should be sufficient to capture the variation in the function. According to the Nyquist sampling theorem, there should be at least two samples per period for the highest fre-quency contained in the function/signal. For a highly oscillatory function, a large number of samples is needed.

The discrete Fourier transform (DFT) is a techniques based on samples. The DFT with samples from an interval [−T, T ] is of interest here. The Fourier coefficients Fj of

the function f(t) are,

Fj= 1 2N N n=1−N f (n∆t)e−iπnj/N, (1.24) for j = 1 − N to N, ∆t = T/N.

The endpoints of the interval [−T, T ] are matched so that the function f(t) is formally periodic. The performance of the DFT can be improved to obtain more accurate results, by means of a window function χq(t/T ). The windowed Fourier coefficients Fj(q)of the

function f(t) multiplied by the window function χq(t/T )of order q are,

F_j(q)= 1 2N N n=1−N f (n∆t)χq(n∆t/T )e−iπnj/N, (1.25)

where χqis given by,

χq t T =2 q_(q!)2 (2q)! 1 + cosπt T q . (1.26)

In the context that will be discussed here, a window function of order q = 4 is suitable Gauss-Legendre, Gauss-Chebyshev and Gauss-Hermite. Based on the type of the

inte-grand, the nodes can be distributed non-uniformly in order to calculate the integrand more accurately. Gauss-Legendre quadrature is used in this thesis for an integrand that is both oscillatory and near singular along the path of integration. The nodes θj are

given by the nulls of the Legendre polynomial Pnand the weights are given by,

wj=− 2 (1− θ2 j)[P n(θj)]2 . (1.23)

In order to evaluate highly oscillatory integrals accurately and efficiently, more advanced quadratures are of interest. Recent papers have presented methods that deal with this problem [28],[29]. The quadrature has not been the main focus in this thesis.

1.8 Direct and iterative solvers for linear systems

A system of linear equations can be solved with methods of two types: • Direct methods (Gaussian elimination).

• Iterative methods (GMRES, QMR).

Direct methods find the solution after a fixed number of operations. Direct methods are stable and accurate and could be used for banded systems.

If the matrix is well conditioned, an approximate solution can be obtained with fast iterative methods, but preconditioners are normally needed. Iterative methods are suit-able for sparse matrices. In papers I, II and III Gaussian elimination is used to solve the linear system of equations in order to obtain high accuracy.

1.9 Hybrid methods and the matching problem

A problem with sparsity is that it is not easily obtained for the shadow zone of the scatterer (see section 1.6). A way to avoid this problem is to use a hybrid scheme that involves a solution for the shadow zone that is known a priori. This scheme obviously eliminates the calculation of the current in the shadow zone and can be seen as a form of sparsity. There lies a problem in matching the solutions with high accuracy and this problem is treated in paper II.

The proposed hybrid method is tested for a simple smooth 2D scatterer. The sparse version of the method of moments, described in section 1.6, is suitable for the lit zone

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match the solutions by means of overlapping smooth functions is devised.

Fock functions can also be used as a priori solutions. Even though the result is less accurate, there is the interesting aspect that very good accuracy can be obtained with imperfect a priori information, provided that the matching of the solutions is good.

1.10 The discrete Fourier transform (DFT)

A function f(t) can be sampled with N equidistant samples. The number of samples should be sufficient to capture the variation in the function. According to the Nyquist sampling theorem, there should be at least two samples per period for the highest fre-quency contained in the function/signal. For a highly oscillatory function, a large number of samples is needed.

The discrete Fourier transform (DFT) is a techniques based on samples. The DFT with samples from an interval [−T, T ] is of interest here. The Fourier coefficients Fj of

the function f(t) are,

Fj = 1 2N N n=1−N f (n∆t)e−iπnj/N, (1.24) for j = 1 − N to N, ∆t = T/N.

The endpoints of the interval [−T, T ] are matched so that the function f(t) is formally periodic. The performance of the DFT can be improved to obtain more accurate results, by means of a window function χq(t/T ). The windowed Fourier coefficients Fj(q)of the

function f(t) multiplied by the window function χq(t/T )of order q are,

F_j(q)= 1 2N N n=1−N f (n∆t)χq(n∆t/T )e−iπnj/N, (1.25)

where χq is given by,

χq t T =2 q_(q!)2 (2q)! 1 + cosπt T q . (1.26)

In the context that will be discussed here, a window function of order q = 4 is suitable to deemphasise the endpoints of the interval, cf. Fig. 1.2. Increasing q does not lead to a substantial improvement in the results.

11 Gauss-Legendre, Gauss-Chebyshev and Gauss-Hermite. Based on the type of the

inte-grand, the nodes can be distributed non-uniformly in order to calculate the integrand more accurately. Gauss-Legendre quadrature is used in this thesis for an integrand that is both oscillatory and near singular along the path of integration. The nodes θj are

given by the nulls of the Legendre polynomial Pnand the weights are given by,

wj =− 2 (1− θ2 j)[P n(θj)]2 . (1.23)

In order to evaluate highly oscillatory integrals accurately and efficiently, more advanced quadratures are of interest. Recent papers have presented methods that deal with this problem [28],[29]. The quadrature has not been the main focus in this thesis.

1.8 Direct and iterative solvers for linear systems

A system of linear equations can be solved with methods of two types: • Direct methods (Gaussian elimination).

• Iterative methods (GMRES, QMR).

Direct methods find the solution after a fixed number of operations. Direct methods are stable and accurate and could be used for banded systems.

If the matrix is well conditioned, an approximate solution can be obtained with fast iterative methods, but preconditioners are normally needed. Iterative methods are suit-able for sparse matrices. In papers I, II and III Gaussian elimination is used to solve the linear system of equations in order to obtain high accuracy.

1.9 Hybrid methods and the matching problem

A problem with sparsity is that it is not easily obtained for the shadow zone of the scatterer (see section 1.6). A way to avoid this problem is to use a hybrid scheme that involves a solution for the shadow zone that is known a priori. This scheme obviously eliminates the calculation of the current in the shadow zone and can be seen as a form of sparsity. There lies a problem in matching the solutions with high accuracy and this problem is treated in paper II.

The proposed hybrid method is tested for a simple smooth 2D scatterer. The sparse version of the method of moments, described in section 1.6, is suitable for the lit zone since the integrand decays rapidly there. In the case of a circular object, the series so-lution can be used for the deep shadow, as information known a priori. A technique to

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where j = 1 − N to N and l = 1 − N to N. The explicit dependence on j, l and q is dropped from Aν and A−ν.

The accurate extracted wavenumbers kD,−kDand amplitudes AkD,−AkDwill be used in the next section to approximate the surface current in the shadow zone of a scatterer.

1.11 Scaling in the shadow zone of a scatterer

At high frequencies, the numerical solution of the integral equation leads to a linear system with a large number of unknowns. One approach to tackle this is to reduce the oscillation (variation) that needs to be resolved in the numerical solution. This can be done by introducing a scaling. The scaling is applied over the whole boundary b(θ) but special attention is given to the shadow region of the scatterer. The surface current can be factored so that there is an oscillatory part foscand a smooth part J0,

Js(θ) = J0(θ)fosc(θ). (1.29)

From physical optics one has the simple standard scaling given by the incident field with an angle ψ (cf. Fig. 1.1),

fosc(θ) = eikb(θ)cos(θ+ψ). (1.30)

This scaling is suitable for the lit zone of the scatterer. A scaling where the complex Fock wave numbers are used in the shadow zone is introduced in paper I. This scaling is accurate for the current on a circular scatterer. Noncircular scatterers could be treated by using the arc length along the surface.

For the TE case, the series solution Jse for the current on a circular scatterer with radius a is given by,

Jse(θ) = 2i πka ∞ m=−∞ eim(θ+π 2) H m(ka) . (1.31)

For scaling purposes, and in order to handle the shadow zone accurately, this solution can be analysed by means of an extension of the DFT [30]. The parameters for the scaling

-1.0 -0.5 0.0 0.5 1.0 0 1 2 3 t/ T

χ

4[t/ T]

Figure 1.2: A window function χ.

For scaling purposes it is of interest to approximate the function f(t) with complex exponentials f(t) = Aνeiνt+ A−νe−iνt. These exponentials can be used to approximate

the surface current in the deep shadow of a scatterer. The DFT [30] can be used to extract the wavenumbers ν and amplitudes Aν that are needed in the complex

exponen-tials. By comparing the ratio of two adjacent windowed Fourier coefficients F(q)

j for f(t)

and G(q)

j (ν)for an elementary function gν(t) = eiνt, one can extract these wavenumbers

and amplitudes by using

G(q)_j_m₋₁(ν) G(q)_j_m(ν) =

F_j(q)_m₋₁

F_j(q)_m . (1.27)

The best result is obtained for a value j = jmthat corresponds to the largest coefficient

F_j(q)

m. With an acceptable first estimate for ν, root finding applied to Eq. 1.27 provides an accurate value ν = KD, where KDcan be complex.

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where j = 1 − N to N and l = 1 − N to N. The explicit dependence on j, l and q is dropped from Aν and A−ν.

The accurate extracted wavenumbers kD,−kDand amplitudes AkD,−AkDwill be used in the next section to approximate the surface current in the shadow zone of a scatterer.

1.11 Scaling in the shadow zone of a scatterer

At high frequencies, the numerical solution of the integral equation leads to a linear system with a large number of unknowns. One approach to tackle this is to reduce the oscillation (variation) that needs to be resolved in the numerical solution. This can be done by introducing a scaling. The scaling is applied over the whole boundary b(θ) but special attention is given to the shadow region of the scatterer. The surface current can be factored so that there is an oscillatory part fosc and a smooth part J0,

Js(θ) = J0(θ)fosc(θ). (1.29)

From physical optics one has the simple standard scaling given by the incident field with an angle ψ (cf. Fig. 1.1),

fosc(θ) = eikb(θ)cos(θ+ψ). (1.30)

This scaling is suitable for the lit zone of the scatterer. A scaling where the complex Fock wave numbers are used in the shadow zone is introduced in paper I. This scaling is accurate for the current on a circular scatterer. Noncircular scatterers could be treated by using the arc length along the surface.

For the TE case, the series solution Jse for the current on a circular scatterer with radius a is given by,

Jse(θ) = 2i πka ∞ m=−∞ eim(θ+π 2) H m(ka) . (1.31)

For scaling purposes, and in order to handle the shadow zone accurately, this solution can be analysed by means of an extension of the DFT [30]. The parameters for the scaling are obtained from Eqs. 1.25 and 1.26, based on an interval 2T = θ2− θ1 corresponds to arc length on the deep shadow.

13 -1.0 -0.5 0.0 0.5 1.0 0 1 2 3 t/ T

χ

4[t/ T]

Figure 1.2: A window function χ.

For scaling purposes it is of interest to approximate the function f(t) with complex exponentials f(t) = Aνeiνt+ A−νe−iνt. These exponentials can be used to approximate

the surface current in the deep shadow of a scatterer. The DFT [30] can be used to extract the wavenumbers ν and amplitudes Aν that are needed in the complex

exponen-tials. By comparing the ratio of two adjacent windowed Fourier coefficients F(q)

j for f(t)

and G(q)

j (ν)for an elementary function gν(t) = eiνt, one can extract these wavenumbers

and amplitudes by using

G(q)_j_m₋₁(ν) G(q)_j_m(ν) =

F_j(q)_m₋₁

F_j(q)_m . (1.27)

The best result is obtained for a value j = jmthat corresponds to the largest coefficient

F_j(q)

m. With an acceptable first estimate for ν, root finding applied to Eq. 1.27 provides an accurate value ν = KD, where KDcan be complex.

The amplitudes that correspond to these wavenumbers are,

Aν= F_j(q) G(q)_j (ν), A−ν= F_l(q) G(q)_l (−ν), (1.28) 12