Finite Element Modelling of Radiation Conditions in Electromagnetic Scattering

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Indian Institute of Technology KanpurDepartment of Electrical Engineering

B.Tech. Project Report (EE 492)

Finite Element Modelling of Radiation Conditions

in Electromagnetic Scattering

Vivek Saxena (Y6549)[email protected]

Advisor: Dr. Naren NaikDepartment of Electrical Engineering

IIT Kanpur

Second Semester (2009-10)
http://localhost/var/www/apps/conversion/tmp/scratch_3/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/[email protected]


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Contents

Contents 2

Certificate 3

Abstract 4

Acknowledgments 5

List of Figures 6

1 Introduction 7

2 Motivation & Objectives 9

3 Literature Survey 10

4 Work Done in the Final Semester 11

5 Basic Theory 12

5.1 Scattering Boundary Value Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.1 Scalar Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.2 Vector Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3.1 Concentric Ring Meshing for circular objects . . . . . . . . . . . . . . . . . . 155.3.2 DistMesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3.3 COMSOL-based Meshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.4 Mesh Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Methodology Adopted 25

6.1 Statement of the Forward Problem (for Cylindrical PEC Object in Air) . . . . . . . 266.1.1 The Partial Differential Equation for the Total Electric Field . . . . . . . . . 26

6.2 Statement of the Forward Problem (for Buried Mines). . . . . . . . . . . . . . . . . 276.2.1 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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7 Results 29

7.1 Circular PEC Object centered at the origin. . . . . . . . . . . . . . . . . . . . . . . 297.2 Location of2nd order ABC vs. that of1st order ABC . . . . . . . . . . . . . . . . . 317.3 Circular PEC Object placed off center . . . . . . . . . . . . . . . . . . . . . . . . . 327.4 Concave PEC Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.5 Square PEC Object placed off center . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.6 Two PEC Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.7 Comparison between1st order ABC and PML PEC Cylinder in Air . . . . . . . . . 367.8 Comparison between ABC and PML for Dielectric Cylinder in Air . . . . . . . . . . 377.9 The Subsurface Problem (Half Space Scattering). . . . . . . . . . . . . . . . . . . . 387.10 PEC Cylinder buried under Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.11 Dielectric Cylinder buried under Soil . . . . . . . . . . . . . . . . . . . . . . . . . . 397.12 Scattering from a PEC Cylinder in Air using the First Order ABC (Effect of varying

Discretization). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8 Conclusions 41

References 42

Appendix A: Finite Element Method Formulation 43

Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Element Matrices and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Integration over an arbitrary triangular patch . . . . . . . . . . . . . . . . . . . . . . . . 47Enforcement of Dirichlet Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Assembly of Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Solution of the Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Appendix B: First Order Absorbing Boundary Condition 50

Appendix C: Second Order Absorbing Boundary Condition 52

Appendix D: COMSOL Results 59

Appendix E: MATLAB Codes for Meshing 2D Regions 63

8.1 Meshing Code for Cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.2 Meshing Code for Off-center Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 658.3 Meshing Code for Square Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658.4 Meshing Code for Concave Object. . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.5 Meshing Code for Two Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Appendix F: MATLAB FEM Codes 69

8.6 First Order ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.7 Second Order ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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Certificate

This is to certify that the work contained in this report entitled Finite Element Modelling of Ra-diation Conditions in Electromagnetic Scattering by Vivek Saxena has been carried out under mysupervision and that this work has not been submitted elsewhere for a degree or diploma.

Dr. Naren NaikDepartment of Electrical EngineeringIIT Kanpur

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Abstract

The present study develops Finite Element Method (FEM) based models for the scattering prob-lem involving electromagnetic waves, with radiation boundary conditions. The main focus of ourwork has been towards the development of Absorbing Boundary Conditions (ABC) and PerfectlyMatched Layer (PML) based solutions. This work is towards the larger objective of developing anFEM framework for the shallow subsurface inverse scattering problem.

In the present work, the following has been accomplished:

Mesh generation for two dimensional domains, in MATLAB (meshing generation codes forcircular, rectangular, and arbitrarily shaped objects in 2D.)

Development and validation of first orderand second order ABC codes for homogeneousbackgrounds, in MATLAB.

Using COMSOL Multiphysics, motivated

the exploration of PML as against ABCs for the half-space problem by simulations, and

the need to generate own codes to solve the scattering problem (for greater flexibility).

Second Order ABCs for homogeneous backgrounds have been derived and validated. Theseand the First Order ABC are needed for a basic validation of the PML in homogeneousbackgrounds before moving on to the half-space inverse problem.

The primary objective was to set up the scattering problem and develop FEM codes for it, andmost importantly, compare different mesh truncation schemes.

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Acknowledgments

I thank my advisor, Dr. Naren Naik, for his support and encouragement throughout the academicyear, and for introducing me to the finite element method, subsurface scattering, and other existingacademic issues and problems arising in computational electrodynamics.

Vivek SaxenaRoll Number: Y6549Fourth Year UndergraduateB.Tech. ProgrammeDepartment of Electrical EngineeringIIT [email protected]

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List of Figures

1.1 Basic geometry of a Landmine Detection Problem . . . . . . . . . . . . . . . . . . . 71.2 Block Diagram of the Forward Scattering Problem. . . . . . . . . . . . . . . . . . . 81.3 Block Diagram of the Inverse Scattering Problem . . . . . . . . . . . . . . . . . . . 8

5.1 Flowchart of the Finite Element Method-based solution process . . . . . . . . . . . 145.2 Triangular Mesh Pattern forh = 0.09 . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Set Theoretic Operations on Regions . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4 Circular Mesh generated using DistMesh . . . . . . . . . . . . . . . . . . . . . . . . 195.5 Mesh for a Concave PEC Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 The unbounded and PML-bounded problems. . . . . . . . . . . . . . . . . . . . . . 225.7 The three step-process for setting up a PML . . . . . . . . . . . . . . . . . . . . . . 23

7.1 First and Second Order Solutions for PEC Cylinder Scattering . . . . . . . . . . . . 297.2 Analytical and FEM Solutions for the PEC Cylindrical Scatterer at/25 . . . . . . 307.3 1st order ABC versus2nd order ABC . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Electric Fields near an off-center PEC Cylinder . . . . . . . . . . . . . . . . . . . . 327.5 Scattering from an off-center PEC Cylinder . . . . . . . . . . . . . . . . . . . . . . 327.6 Electric Fields near a Concave PEC Object. . . . . . . . . . . . . . . . . . . . . . . 33

7.7 Scattering from a Concave PEC Object . . . . . . . . . . . . . . . . . . . . . . . . . 337.8 Square PEC Object Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.9 Scattering from a Square PEC Object. . . . . . . . . . . . . . . . . . . . . . . . . . 347.10 Mesh for two PEC objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.11 Scattering from Two PEC Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.12 Scattered Fields from PEC Cylinder in Air: ABC vs PML . . . . . . . . . . . . . . 367.13 Scattering from Dielectric Cylinder in Air: ABC vs PML . . . . . . . . . . . . . . . 377.14 Scattering from PEC Cylinder buried under Soil . . . . . . . . . . . . . . . . . . . . 387.15 Scattering from Dielectric Cylinder buried under Soil . . . . . . . . . . . . . . . . . 39

8.1 FEM Local Node Numbering Convention . . . . . . . . . . . . . . . . . . . . . . . . 44

8.2 Evaluation of ABC matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.3 Scattered Fields from PEC Cylinder in Air: ABC vs PML . . . . . . . . . . . . . . 598.4 Scattering from Dielectric Cylinder in Air: ABC vs PML . . . . . . . . . . . . . . . 608.5 Scattering from PEC Cylinder buried under Soil . . . . . . . . . . . . . . . . . . . . 618.6 Scattering from Dielectric Cylinder buried under Soil . . . . . . . . . . . . . . . . . 62

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Chapter 1

Introduction

Subsurface imaging is the reconstruction of topological attributes of an object buried underneathor embedded within the volume of a dielectric material. It involves the use of data of the elec-tromagnetic fields scattered by the object(s) of interest, to construct a contour map of the totalelectromagnetic field distribution in the near region of the scatterer(s). From this contour map andthe field data, it is usually possible to recover important details of the structure, using reconstruc-tion algorithms.

Important examples of subsurface imaging are landmine detection, tumor detection, geophysicaand oil exploration. The basic problem is conveyed effectively by the following illustration:

Figure 1.1: Basic geometry of a Landmine Detection Problem

The figure shows a spherical/cylindrical mine (shown in red) buried underneath the ground (soil,shown in maroon), which is in general undulating. The mine is the object of interest here and thequantities of relevance in this problem can be:

the depth below surface, of the mine, and

the structure of the mine.

These quantities can quantify the extent of destruction an explosion of the mine could cause. Thedepth can be used to determine whether the mine can be safely defused without risking an explo-sion, and the structural (material) content can give hints for identifying it so as to enable the useof specifically practical methods to defuse it.

An analogous situation arises in biomedical imaging, where structural information about a tu-mor or an abnormal growth in the body can be used to advance prognosis, possibly allowing early

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detection and improving the chances of patient treatment and survival in the case of a malignanttumor. In this problem, the mine of the Figure 1.1 is replaced by a possibly malevolent growthinside the body, and the interface becomes the skin-air interface. It is extremely important that themethods of imaging be non-invasive; that is, the process of recovery of compositional informationmust not result in perturbation which causes damage or destruction. For instance, tumors cannotbe imaged using ionizing radiation which might cause them to grow in size or become malignant.

Of course, landmines should not be touched more if one can help it.

Subsurface imaging problems form actually a subset of a larger category of problems known asInverse Problems. In general, to predict the results of a measurement requires

1. a model of the system under investigation, and,

2. a physical theory linking the parameters of the model to the parameters being measured

This prediction of observations (in our case, the scattered fields), given the values of the parametersdefining the model (object topology, dielectric properties) constitutes the forward problem.

Figure 1.2: Block Diagram of the Forward Scattering Problem

Solving the inverse problem consists of using the results of actual observations (the scattered fields)to infer the values of the parameters characterizing the system under investigation (the shapedescriptorsof the mine and its depth below the surface).

Figure 1.3: Block Diagram of the Inverse Scattering Problem

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Chapter 2

Motivation & Objectives

Subsurface imaging is the non-invasive recovery of shape and topological attributes of an objectburied within a dielectric region, for example, a mine buried beneath the soil. The method of imag-ing referred to here involves propagating electromagnetic waves of known frequency and amplitudeon the ground, measuring the fields scattered by the dielectric surface (soil) and the object (mine)and estimating the internal electromagnetic parameters of the scatterer.

In both landmine detection as well as tumor detection, there are several practical issues to consider,namely

Need for a non-invasive detection.

Low False Alarm Rate.

Distinguishing between landmines and clutter, or malignant and benign tumors.

For such purposes, tomographic reconstruction of the subsurface object is essential. And as motivated earlier, subsurface reconstruction is an inverse scattering problem.

The objective of this project is towards the development of a computational model for study-ing and analyzing subsurface imaging of objects buried under the soil, using the Finite ElementMethod. Towards this objective, this study investigates the use of Absorbing Boundary Conditionsand Perfectly Matched Layers to model the radiation conditions at infinity.

Inverse scattering solutions typically need forward scattering models. Specifically, it is desirableto have complete control over the implementation aspects of the forward problem model, for in-stance the FEM routines and the boundary conditions, because these are used to generate Frechetderivatives required for shape reconstruction. Hence, writing our own code for solving the forwardproblem through FEM is preferred over usage of existing forward problem solvers such as COM-SOL Multiphysics. However, the power of these packages cannot be undermined, and ideally it isdesirable to use both approaches to solve the complete problem.

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Chapter 4

Work Done in the Final Semester

Development of completely general 2D nodal FEM codes to handle arbitrary two dimensionalobjects.

Development of a framework to mesh arbitrary two dimensional objects.

Derivation and validation of the Second Order Absorbing Boundary Condition (ABC) expres-sions in homogeneous backgrounds, in an FEM model.

Study of comparison between First Order and Second Order ABCs, for various 2D objects,including a double object scattering example.

Validation of the superiority of the Second Order ABC over the First Order ABC for theobjects considered, and a brief study of the limitations of the second order ABC.

Development of interface codes to use MATLAB in conjunction with DistMesh (an open sourcelibrary of geometric primitives and meshes) and COMSOL Multiphysics (a commercial FEMmodeling package, used this semester solely for meshing complex objects).

Work Done in the First Semester Familiarization with the 2D Nodal Finite Element Method and methods of mesh truncation

(ABC and Perfectly Matched Layers ).

Development of a framework to mesh circular and rectangular objects.

Development of codes to solve Laplace and Poisson equations in 2D for electrostatic problemsand extension of the code to solve the Helmoltz equation in 2D.

Derivation and validation of the First Absorbing Boundary Condition (ABC) expressions inhomogeneous backgrounds, in an FEM model.

Development of a program to solve for the total and scattered electric fields in the vicinity ofa PEC cylinder in air, using the FEM and a First Order ABC for mesh truncation.

Study of comparison between First Order ABC and PML via COMSOL Multiphysics (usedas an FEM Modeling package).

Validation of the improved performance of the PML over the First Order ABC for subsurfacescattering, through COMSOL Multiphysics.

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Chapter 5

Basic Theory

The work done on this project centered around the forward problem, specifically the implementa-tion of a Finite Element Method-based framework for solving two dimensional partial differentialequations (PDEs) that arise in time harmonic scattering problems in electromagnetics.

Time Harmonic analysis allows us to find the solution to the problem for one spot excitation fre-quency. The response due to a general excitation field can be obtained by Fourier decomposition ofthe excitation and determination of the response for each Fourier mode using the existing framework

In a nutshell, the objective of the forward problem is to determine the scattered fields and to-tal field distribution in the near zone of a scatterer whose geometry is known apriori.

5.1 Scattering Boundary Value Problem

The scattering boundary value problem (BVP) in two dimensions is of two kinds:

1. Scalar problem

2. Vector problem

5.1.1 Scalar Problem

The objective of the 2D scalar problem is to solve the partial differential equation,

x

x

x y y y + = f(x, y) (x, y) (5.1)

subject to the boundary conditions

= p on 1 (5.2)x

xx +y

yy

n+ = q on 2 (5.3)

where

denotes the principal unknown field (electric or magnetic).

x x(x, y),y y(x, y), (x, y) and f(x, y) are scalar functions ofx andy . = 1+ 2 denotes the contour or boundary enclosing the area .

ndenotes the outward normal unit vector of.

, p and qare known parameters associated with the physical properties of the boundary. Inparticular,p andqcan be considered to be the boundary source or boundary excitation.

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Boundary condition (4.1) is known as the Dirichlet condition, whereas boundary condition ( 4.2) isknown as the Robin condition or the generalized Neumann boundary condition.

For an open domain scattering problem, the appropriate boundary condition on the far field isthe Sommerfeld Radiation condition[1]

limr r

E

H

+jk0r EH = 0 (5.4)

which in two dimensions becomes

lim

EzHz

+jk0

EzHz

= 0 (5.5)

where =

x2 +y2, r =

2 +z2 and k0 is the free space propagation wavenumber.

But for a numerical solution to this BVP, we cannot store information for on a com-puter with finite memory size. The Sommerfeld Radiation condition as expressed in the form aboveis no longer valid, since the domain has to be truncated by some artificial boundary that is locateda finite distance away from the scatterer.

Hence, we need to enforce a suitable boundary condition on this artificial boundary such thata perfect absorbing surface is simulated. By the use of an artificial boundary, we want to ensurethat the EM modes see a transparent surface and do not suffer reflection. This gives the impres-sion that the domain is still unbounded whereas it is in fact finite, and hence representible on acomputer system with finite memory size. This is known as Mesh Truncation.

The boundary condition on the artificial boundary which truncates the mesh, is imposed through

the second boundary condition, i.e. the generalized Neumann Boundary condition. It is knownas an Absorbing Boundary condition (ABC) and is described below. An alternative approach totruncate the mesh involves the use of an artificial absorbing layer known as a Perfectly MatchedLayer (PML), which represents an imaginary lossy medium through which electromagnetic fieldstravel without significant reflection.

5.1.2 Vector Problem

The objective of the 2D vector problem is to solve the PDE

( F) +F =G(r) r (5.6)

subject to the boundary conditions

n F = F1(r) on 1 (5.7)n ( F) = F2(r) on 2 (5.8)

where

F denotes the principal unknown field (electric or magnetic).

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(x, y) and (x, y) are scalar functions ofx and y . G(r) is a known vector function.

F1(r) is a known vector function specifying the Dirichlet boundary condition on 1.

F2(r) is a known vector function (possibly involving F) specifying the generalized vectorNeumann boundary condition on2.

In this study, only the scalar problem was examined. The vector problem involves a more complexformulation of the ABC and PML, which needs to be formulated more precisely before it can besimulated using a vector generalization of the FEM known as the Edge Element Method.

5.2 The Finite Element MethodThe continuum BVP is described by the geometry, a PDE and a set of boundary conditions. Aswe are interested in an approximate solution to the PDE, we resort to the weak formulation of theBVP (in which the PDE is enforced locally at discrete points via a weaker integral equation). Thegeometry is represented by a triangular mesh and a set of basis functions used to approximatethe unknown field within each element of the mesh.

Since we employ triangular meshes, there are three such linear basis functions, which will beused to interpolate the value of the principal known unknown field within each triangular elementgiven its values at the three nodes (vertices) of the triangle. The weak form along with a continuitycondition is amenable to discretization via these chosen basis functions. This method of solvinga PDE approximately, is known as the Finite Element Method (FEM). A flowchart of the FEMsolution process is shown below.

Figure 5.1: Flowchart of the Finite Element Method-based solution process

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This discretization yields a set of linear equations, the solution to which gives the values of theprincipal unknown field at all the nodes of all triangular elements forming the mesh. Once thenodal values are known, the value of the unknown field at any point in the solution domain can bedetermined by interpolation using the basis functions. The FEM can thus be used to find the totalfield in the vicinity of a scatterer. Since the incident field in a scattering scenario is known apriori,the method can be used to determine the scattered field as well.

In summary, the FEM consists of the following four major steps:

Discretization: breaking up the solution region into a finite number of elements (meshes).

Derivation of Governing Equations: derivation of governing equations for a typical ele-ment.

Assembly: assembling of all elements in the solution region.

Solution: solving the system of equations obtained.

The mathematical details of the Finite Element Method and how it is to be set up are described inAppendix A.

5.3 MeshingThree approaches were used for meshing:

1. Concentric Ring Meshing: this approach was used for circular objects in 2D.

2. DistMesh based Meshing: this approach was used to mesh arbitrary objects in 2D for whicha signed distance functioncould be easily obtained analytically.

3. COMSOL based Meshing: for very complicated objects that cannot be described using signeddistance functions, the COMSOL Multiphysics interface with MATLAB was used as a com-putational front-end for meshing.

5.3.1 Concentric Ring Meshing for circular objects

In order to solve for the total electric field in the near region of a PEC Cylinder when an incidentplane wave strikes it, a meshing algorithm was evolved for circular solution domains. The details ofthis meshing algorithm are described below, with reference to the particular geometry under study

A PEC Cylinder of radius = a is centered at the origin. The solution domain extends from = a to = ABC, where ABC is the radius of the circular boundary over which the AbsorbingBoundary Condition is to be enforced. The region from = 0 to = ABC is divided into Nring

annular rings, with Nring given by

Nring=

ABC a

3h2

(5.9)

whereh is the discretization size. The differential step change in the radial direction is given by

=ABC a

Nring(5.10)

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At a fixed (where = a : r : ABC), the angular separation between two successive nodes isgiven by

= 2

Npts(5.11)

whereNpts= 2/h is the total number of nodes on the boundary of the segment at radius . Asincreases, the angular spacing between two successive nodes reduces, because the total number of

nodes on the boundary increases with its radius. By following this procedure, nodes are markedon the solution domain, and triangulation using the MATLAB function delaunay populates theannular regions by as many equilateral triangles of side length h as possible, with the remainingregion being filled up with skewed triangles.

Triangles located in the region interior to the PEC which are rendered by the triangulation sub-routine, must be removed as this region is excluded from the solution domain. The local nodenumbering generated by this subroutine is not always counterclockwise as required by the FEM for-mulation (c.f. Figure 7.1), so a reordering of the nodes is carried out to bring them to the canonicaorientation. A generalization to dielectric objects is described below.

The inputsto the meshing algorithm are

cyl (ora): the radius of the PEC

ABC: the radius of the circular boundary over which the ABC is to be imposed

h: the discretization size (a measure of the equilateral triangle side, in units of wavelength)

The outputs of the meshing algorithm are

nNodes: Number of Nodes

nElements: Number of elements

N(e, i): Connectivity array (described in Appendix A).

x(i), y(i): x- andy-coordinates of the nodes.A triangular mesh pattern for cyl = a = 0.5,ABC= 1.5and h = 0.09generated using the abovealgorithm is shown below.

Figure 5.2: Triangular Mesh Pattern for h = 0.09

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Points to note:

1. In order to generalize concentric ring meshing to non-PEC (i.e. dielectric) objects, all we needto do is set a= 0 in equations (4.9 4.11), with the understanding that the region interiorto the dielectric cylinder is also to be meshed. Different values ofh could be specified for thetwo regions, if desired.

2. For a mesh to be a valid input to the FEM code block, it is important that the local nodenumbering of each triangle be consistent with the counterclockwise numbering conventiondescribed in Appendix A, failing with the FEM matrices become singular, resulting in severelyerroneous solutions.

5.3.2 DistMesh

DistMesh is an open source MATLAB-based meshing library for two- and three-dimensional objectswritten by Persson and Strang [3]. DistMesh uses a signed distance function d(x, y) in twodimensions (ord(x,y,z)in three dimensions) as a shape descriptor. The property of this functionis that it is negative inside the region of interest (to be meshed) and non-negative outside. Hence

a subdomain R2 of interest can be described using the property

d(x, y) =

a negative quantity for(x, y) a nonnegative quantity for(x, y) / (5.12)

As a simple example, for an annular region 1< < 2, the distance function is

d(x, y) =

1+22

x2 +y2 +1 22 (5.13)

which is the same as

d(x, y) = max(d1(x, y), d2(x, y)) = d1(x, y) d2(x, y)2

+d1(x, y) +d2(x, y)2 (5.14)

where d1,2(x, y) =

x2 +y2 1,2 are the distance functions for the inner and outer circles re-spectively. This operation is equivalent to a set theoretic subtraction of two domains. That is, toconstruct an annular region, we can begin with a circular region of radius 2and remove the circularregion of radius 1 from it.

Generalizing this idea to arbitrary domains requires the introduction of two more analogous settheoretic operations on a pair of regions: Union and Intersection.

Denoting the signed distance functions of regions A and B by dA(x, y) and dB(x, y) respectivelythe three operations required for constructing a sub-region from a larger region in two dimensions,are listed below.

Union : dAB(x, y) = min(dA(x, y), dB(x, y)) (5.15)

Difference : dA\B(x, y) = max(dA(x, y), dB(x, y)) (5.16)Intersection : dAB(x, y) = max(dA(x, y), dB(x, y)) (5.17)

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The operations mentioned above are described geometrically in the figure below.

Figure 5.3: Set Theoretic Operations on Regions using Signed Distance Functions

DistMesh uses an algorithm that combines a physical principle of force equilibrium in a truss struc-ture with a mathematical representation of the geometry using signed distance functions [15]. Theiterative technique is based on the physical analogy between a simplex mesh and a truss structure

Meshpoints are nodes of the truss. Assuming an appropriate force-displacement function for the

bars in the truss at each iteration, the algorithm solves for an equilibrium condition. The forcesmove the nodes and iteratively, the Delaunay triangulation algorithm adjusts the topology.

The basic algorithm employed by DistMesh is described below, step-by-step.

1. Create a uniform distribution of nodes within the bounding box of the geometry, correspondingto equilateral triangles.

2. Remove all nodes outside the desired geometry (using the signed distance functions negativityas a criterion for interioricity).

3. Now, enter the main loop where the location of the nodes is iteratively improved until equi-

librium is reached.

(a) Perform a Delaunay triangulation on the existing node set to determine the topology ofthe truss at this iteration.

(b) Now, evaluate the force function. If equilibrium is attained, break. otherwise, determinethe extent of movement of the nodes in this iteration and reiterate until equilibrium.

4. Determine the list of triangles which is a ne 3 matrix (where ne is the total number oftriangular elements), and the node coordinatesx and y, after removing duplicate entries.

5. Determine the the boundary edges, for imposing Dirichlet, Neumann and Mesh Truncationconditions.

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An example of a mesh generated using DistMesh is shown below. A circular object with a center(0, 0.5) and radius0.5is placed inside the domain. The center of the original circular domain isplotted in red to emphasize that the object is off-center.

Figure 5.4: Circular Mesh generated using DistMesh, with a discretization size 0.2.

It should be noted that while using DistMesh, one has to ensure that the numbering convention ispreserved. This is easily ensured by using the signed area of each triangular element as an indicatorof local ordering. If the signed area is negative, swapping the global node numbering of any twonodes will result in a well ordered element for FEM purposes. Another issue that frequently cropsup while using DistMesh is the presence of multiple identical entries in the element connectivitymatrix. These have to be removed before the mesh is given as an input to the FEM code.

A major problem with DistMesh is that for certain geometries, the meshing algorithm may failto converge. This can happen if the domain has objects which have sharp corners. In such situa-tions, it is desirable to re-specify the object at the corners and make it more smooth. This couldbe achieved using Bezier curves or splines. Irrespective of the method used to specify the curveeventually every representation must yield an appropriate signed distance function d(x, y) whichresults in a convergent mesh. This is not always possible, as one might expect, and hence we resortto a more sophisticated meshing algorithm implemented in COMSOL Multiphysics.

5.3.3 COMSOL-based Meshing

COMSOL Multiphysics is a powerful FEM-solver, which comes in-built with a geometry drawingtool and a MATLAB interface with geometric primitives as MATLAB commands. The use of COM-SOL Multiphysics as an FEM solver was explored in the first semester.

In this section, we restrict ourselves to the use of COMSOL as a mesh generator.

The advantage of COMSOL over DistMesh is that it can be used to adaptively remesh a sub-domain more finely than the rest of the region. This is especially useful at corners and interfaceswhere DistMesh is likely to yield coarse meshes if at all it can avoid oscillations.

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An example of a mesh generated for a complicated concave PEC object in 2D is shown below:

Figure 5.5: Mesh for a concave PEC object generated using COMSOL, with a discretization size0.2.

The exact internal details of COMSOLs meshing algorithm are not known, and in this particularcase, COMSOL has been used more as a black box for meshing complicated objects, and transferringthe mesh to our own FEM codes, so that we have control over all the matrix elements. However, itis known that COMSOL uses Delaunay Triangulation for meshing.

5.4 Mesh TruncationIn order to apply the finite element method to open-region scattering or radiation problems, theinfinite solution domain must be truncated into a finite one. This is a fundamental requirementof any computational solution due to the restriction of finite memory on a computational engine

Geometric truncation implies the imposition of boundary conditions which force the solution toevolve within the reduced geometry. However, such forced terminations can introduce additionaartifacts into the solution due to reflection, scattering, etc. The problem then, is to determine sucha termination that does not introduce undesirable properties in the scattered fields of interest in thesolution domain, and at the same time, prevents them from differing considerably from the solutionin the physical unbounded space.

Now, assuming that all sources and objects are immersed in free space and located within a fi-nite distance from the origin of a coordinate system, the electric and magnetic fields are requiredto satisfy the Sommerfeld Radiation Condition[1],

limr

r

EH

+jk0r

EH

= 0 (5.18)

which in two dimensions becomes

lim

EzHz

+jk0

EzHz

= 0 (5.19)

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But this condition is no longer valid since the solution domain has been artificially truncated bysome boundary which is located at a finite distance from the scatterer. So, we need to enforcea suitable boundary condition on this artificial boundary such that a perfect absorbing surface issimulated. That is, the EM modes in the region see a transparent surface, and do not sufferreflection. This gives the impression that the solution domain is unbounded whereas it is in factfinite, and hence representible on a computer system with finite memory size.

The first order absorbing boundary condition in two dimensions is given by,

sc

+

jk0+

1

2

sc = 0 (5.20)

whereas the second order absorbing boundary condition is given by,

sc

n +

jk0+

2 j

2

8(j k0)

sc j2(j k0)

2sc

s2 = 0 (5.21)

where/ndenotes the derivative normal to the scattering surface (in 2D cylindrical coordinatesn = ) and 2/s2 denotes the second order tangential derivative (in 2D cylindrical coordinates,s = ). Also, = 1/ is the curvature of the 1-surface. This is the Bayliss-Gunzberger-TurkeABC [6]. In Appendix B, it is shown that the first order ABC can be directly included in theFEM formulation. With a suitable redefinition of the intrinsic boundary condition in the FEM, thesecond order ABC can also be similarly included, as described in Appendix C.

In two dimensions, the scattered field in the far field zone has the asymptotic form

sc(, ) =A()ejk

(5.22)

The important feature of this asymptotic solution is that it falls of as1/ and not exponentiallywith distance. The first order absorbing boundary condition (ABC) can be rewritten as

sc

=

jk 1

2

sc (5.23)

The convergent expansion for the solution of the Helmoltz equation satisfying the Sommerfeldradiation condition[1] is given by,

sc(, ) =H(2)0 (k)

n=0

fn()

n

+H(2)1 (k)

n=0

gn()

n

(5.24)

whereH(2)0 andH

(2)1 denote the zeroth- and first-order Hankel functions of the second kind, respec-

tively. Using the asymptotic form for the Hankel functions, we can write this as

sc(, ) =ejk

n=0

an()

n (5.25)

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From this we can obtain the second order ABC by neglecting terms of order O(9/2)in the followingexpression

sc

=

jk 1

2+

1

8jk2+

1

8k23

sc +

1

2jk2+

1

2k23

2sc

2 +O

1

9/2

(5.26)

There are limitations to the ABC however. Strictly,such an ABC can only be enforced on a

circular boundary. This is because of the ill-posed definition of a second order curvature termover nonsmooth boundaries.

Further, since an ABC aims to model a reflectionless surface, it cannot be very close to the scat-terer. In fact, the farther away from the scatterer the ABC is imposed, the more accurate theFEM solution will be. But doing so will increase the amount of data storageenormously. Althoughin principle it is possible to reduce the problem size by using a higher order ABC, the tradeoffbetween space complexity and time complexity will remain since higher order ABCs willrequire the computation of higher order field derivatives, both tangential as well as normal.

Also, it is known that the Absorbing Boundary Condition runs into difficulties with interfaces,because of the typical assumption of a homogeneous region around the scatterer. Finally, theABCis perfectly absorbing only for a plane wave, which is not physically realizable.

Hence, we resort to another form of mesh truncation, known as a Perfectly Matched Layer (PML)[8,10,11, 12]. Strictly, a PML is not a boundary condition but rather an additional domain thatabsorbs incident radiation without producing reflections.

Figure 5.6: (a) the unbounded problem, and (b) the geometry augmented with a PML.

The figure on the left indicates the geometry of the original BVP. On the right, the solution domainhas been truncated all round by the use of a perfectly absorbing layer that is reflectionless. Thuswe are looking for a layer of (artificial) material to enclose our region of interest in such a way thatthe fields under study in that region do not see this layer. This is a difficult problem in itself, asa reflectionless layer enclosing the entire solution domain is difficult to develop in the absence ofinformation about the very fields it must fail to reflect. It should be noted that this layer is artificial.

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A PML is developed using a three-step process, described by Johnson [8] and illustrated in thefigure below:

Figure 5.7: The three step-process for setting up a PML

1. In infinite space, we analytically continue the solutions and equations to a complexx contourThis changes oscillating waves into exponentially decaying waves outside the region of interestwithout reflections.

2. Still in infinite space, we perform a coordinate transformation to express the complex x as afunction of a real coordinate. In these new coordinates, we have real coordinates and complexmaterials.

3. We truncate the domain of this new real coordinate inside the complex-material region: sincethe solution is decaying there, as long as we truncate it after a long enough distance (wherethe exponential tails are small), the boundary condition used will not significantly impact thefinal solution. So, even hard-wall truncations are acceptable.

The hard-wall truncation at the outer surface of the layer can be effected using a simple Dirichletcondition for a PEC, for example, incurring minimal (of course, unavoidable) reflections there. It is

known [8] that the PML can be incorporated by the following differential operator transformations

x 1

1 +ix(x)

x (5.27)

y 1

1 +iy(y)

y (5.28)

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The artificial PML material is seen to have both an electrical as well as a magnetic conductivity dueto the existence of terms corresponding to currents of electric and magnetic charges. Despite thelimitations of the PML (multiple angle absorption, numerical reflections due to discretization, andthe existence of some inhomogeneous media where the PML fails to act as a reflectionless layer), itsfeatures and apparent simplicity in implementation over the standard homogeneous ABCs discussedabove are suggestive of its utility in the half-space subsurface scattering problem.

To summarize, imposing a PML for mesh truncation is equivalent to modifying the differentialoperator in the original BVP in a local region of the solution domain, so as to obtain an artificiamedium with finite electric and magnetic conductivity (and absorption). This medium convertsoscillating complex exponentials into decaying real exponentials due to the analytic continuationof space. To this extent, the PML is another way of enforcing an absorbing boundary conditionbut without forcing the scattered field itself to satisfy a mixed boundary condition explicitly inthe solution domain. The PML formulation can be deduced from Maxwells equations by introducing a complex-valued coordinate transformation with the additional requirement that the waveimpedance should remain unaffected.

In practice, the PML is typically used along with an absorbing boundary condition at its outer(unphysical) boundary. While a hard Dirichlet condition can also be used to terminate the fieldsit acts as a reflector for any fields that may leak into the PML region. As this is undesirable, anabsorbing boundary condition of the first order can be used to avoid reflections of any remnantfields in the PML region. As demonstrated by the simulated results, a combination of PML andABC reduces the reflections induced by mesh truncation more effectively than purely a first orderABC.

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Chapter 6

Methodology Adopted

In this project, we focused on solving theforward problemto compute the scattered electric fieldsand total field distributions for the following scenarios:

1. Perfect Electric Conducting (PEC) cylinder placed in air (vacuum).

2. PEC concave object placed in air.

3. PEC square object placed in air.

4. Two PEC objects (concave + square) placed in air.

5. Dielectric cylinder placed in air.

6. PEC cylinder buried under the soil.

7. Dielectric cylinder buried under the soil.

Mesh truncation schemes used were the first order ABC, the second order ABC and PerfectlyMatched Layers. Of these the first four problems were solved by writing our own computer pro-grams in MATLAB from scratch. For meshing circular and rectangular regions, we generated ourown codes. But for complicated geometries which can be described using signed distance functionsDistMesh can be used for meshing. For still more complicated geometries, such as concave objectsCOMSOLs meshing subroutines were used.

For the first four situations listed above, our own FEM codes were used with mesh truncationprovided by first order and second order ABCs. Both DistMesh and COMSOL were used solely asmesh generators in this class of problems. The mesh connectivity matrices generated by both these

libraries have to be suitably reordered for use in an FEM code. Therefore, interface code for bothDistMesh as well as COMSOL had to be written in MATLAB. The last three problems mentionedabove were formulated and simulated solely in COMSOL Multiphysics, an FEM-based modellingand visualization program.

Emphasis was placed on PEC objects to examine the performance of various mesh truncationschemes in the worst case scenario of a perfect reflecting surface in the path of an incidentfield. Specifically, PEC objects are near perfect reflectors of the incident field, and impose the sever-est possible boundary condition on the tangential electric field, that of forcing it to be identicallyzero on their surfaces. Naturally, if a mesh truncation scheme works well in case of PECs, it shouldwork well in case of dielectrics.

A contour plot of the electric field distribution for each of these cases is plotted in the next sectionIn this section, we formulate the scattering problem. Meshing has been described in section 4.3 ofthis report. The mathematical details of the Finite Element Method and the Absorbing BoundaryConditions, are described thoroughly in Appendices A through C. We now state the forward prob-lem formally for two inherently open region and closed region problems.

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6.1 Statement of the Forward Problem (for Cylindrical PEC

Object in Air)A perfect electric conductor, in the shape of a circular cylinder of radius a, is oriented in vacuumwith its axis along the zaxis. A plane z-polarized electromagnetic wave of wavelength = 2/k0with an electric field amplitude E0 impinges on the cylinder normally,

Einc =azE0ejk0x =azE0ejk0 cos (6.1)

and is scattered. It is required to find the totalelectric field (incident + scattered) as a function ofthe angle at a distance = from the center of the cylinder. It is given thata = /2and E0 = 1V/m.

6.1.1 The Partial Differential Equation for the Total Electric Field

The incident E field is polarized along the z-direction, so the scattered E field from the infinitelylong cylinder is also polarized along the z-direction. Therefore, the total electric field E will haveonly a z-directed component: E= Eaz, and there will be no z-variations in the field:

Ezz

= 0 = E=Ez Ez(x, y) (6.2)

Consider the following two time-Harmonic Maxwells equations,

E = j0rH (6.3) H = J +j0rE (6.4)

Solving forH from (6.3) and substituting into (6.4), we get

1

j0 1r E= J +j0rE (6.5)Due to (6.2) we also have

E=axEzy

ay Ezx

(6.6)

The curl term on the left hand side of (6.5) can be written as

1

r E

=

ax ay azx

y

z

1r

Ezy

1r

Ezx 0

(6.7)

= az

x

1r

Ezx

y

1r

Ezy

(6.8)

where, the condition r/z= 0has been used to arrive at the second equality. Substituting ( 6.8)into (6.4) and simplifying, we get the Helmoltz equation

x

1

r

Ezx

+

y

1

r

Ezy

+k20rEz =j0Jz (6.9)

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wherek20 =200. The source free version of this equation (for Jz = 0) has the same form as the

canonical scalar wave equation in the Galerkin formulation [1],

x

x

x

y

y

y

+ =f (6.10)

where,

x = y = 1r

(6.11)

= Ez (6.12)

= k20r (6.13)

f = 0 (6.14)

6.2 Statement of the Forward Problem (for Buried Mines)Landmines are modelled as small abnormalities embedded in an otherwise uniform media with an

air-ground interface. These abnormalities are characterized by the electrical permittivity and theconductivity , whose values differ from those of the host media.

The incident field is modeled here as a plane polarized wave with the electric field given by

Einc(r) =E0ei00y eit (6.15)

propagating in the negative y-direction in the half space{y >0}. Here is the angular frequencyof the stimulus signal (= 2f, where f 1 GHz). It is assumed that{y >0} is air and{y 0,2rr0(1 +jtan ) fory


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Chapter 7

Results

7.1 Circular PEC Object centered at the origin

The first object to be considered has a radius a = /2where is the free space propagation wave-length. The first order ABC is imposed at 3/2, a wavelength away from the object. The incidentfield has the form E0e

jk0x forE0 = 1 V /m, k0 = 2/. All distances are normalized with respectto. So without loss of generality, = 1.

For this particular problem, the concentric ring based meshing algorithm described above wasused. A plot of the total electric field as a function of the polar angle at a radius= , is shownbelow on the left, for various values of the mesh discretization parameter h (expressed in units of

).

Figure 7.1: First and Second Order solutions for PEC Cylinder Scattering for discretization sizes/2,/5,/10, and /25, all evaluated at = .

The same experiment was then repeated but with a second order ABC imposed on a mesh truncationboundary, rather than the first order condition. The corresponding plot of the total electric field isshown in the figure on the right above. This problem has an analytical solution[7], given by

Ez(, ) =E0

n=

jn

Jn(k0) Jn(k0a)H

(2)n (k0a)

H(2)n (k0)

ejn (7.1)

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where Ez denotes the z-component of the total electric field, a is the radius of the PEC cylinderand and are the cylindrical polar coordinates. Here,Jn denotes the Bessel Function of the FirstKind of order n and H

(2)n denotes the Hankel Function of the Second Kind of ordern.

The absolute values of the exact, first order and second order solutions are plotted in the figurebelow, for a discretization size of/25, on the boundary = .

Figure 7.2: Analytical and FEM Solutions for the PEC Cylindrical Scatterer, at discretization /25evaluated at =

From the plots, we can infer that

1. Both1st and 2nd order ABC solutions become smoother as the discretization size is lowered.2. For a discretization size of/25, the 2nd order ABC-based solution almost coincides with the

analytical solution, whereas the 1st order ABC-based solution deviates from the analyticalsolution more significantly at very small angles ( 0o), moderately large angles (100o and260o) and again at around 360o. The solution is naturally symmetric about the x-axis.

Clearly, the 2nd-order ABC-based solution more closely approximates the total electric field thanthe1st-order ABC based solution.

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7.2 Location of 2nd order ABC vs. that of 1st order ABC

The total electric field was computed using both the 1st and 2nd order ABC. The 1st order ABCwas imposed at a distance = 2 whereas the 2nd order ABC was imposed at = 1.5, with thediscretization for both situations taken to be 0.04.

The total electric field at = is plotted below, as a function of the polar angle. It is evi-dent that both the numerical solutions are almost comparable, with the 2nd order solution moreclosely approximating the exact (analytical) solution. This demonstrates that the 2nd orderABC gives a more accurate solution even when imposed at a boundary closer to the

scatterer.

Figure 7.3: Electric Field at =

The table below illustrates the advantage gained by using the 2nd order ABC over the1st order ABCAs is evident from the numerical values, the 2nd order ABC yields a solution comparable to (if notslightly better than) the1st order ABC, despite using about halfthe number of nodes and elements

Parameter First Order ABC at 2 Second Order ABC at1.5No. of nodes 1843 984No. of elements 3511 1828No. of Dirichlet Nodes 35 35No. of ABC Nodes 140 105

Table 2: Comparison of matrix sizes for 1st and 2nd order ABC.

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7.3 Circular PEC Object placed off center

The domain consists of a circular object of radius 0.5, centered at(0, 0.5). The mesh, polar andcontour plots of the field are shown below. The contour plot confirms that the 2nd order solutionis smoother, and better preserves the symmetry of the object. The field very close to the PEC isexpected to be very nearly zero, which is more evident when the 2nd order ABC is used for mesh

truncation.

Figure 7.4: Electric Fields near an off-center PEC Cylinder

Figure 7.5: Contour Plots of the Electric Field, using the 1st order ABC (left) and 2nd order ABC(right)

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7.4 Concave PEC Object

The object was modelled using two elliptical arcs centered at (0.2, 0), with semi-axes of lengtha = 0.2, b = 0.5. The total electric field was computed using the 1st and 2nd order ABCs, at= is plotted on the right, above.

Figure 7.6: Electric Fields near a Concave PEC Object

Contour plots of the total electric field in the vicinity of the concave PEC scatterer are plottedbelow. Although mostly comparable, the electric field in the near region of the PEC in the forwardscattering direction, is seen to spread backwards less in case of the 2nd order solution. This isconsistent with the fact that a PEC strongly suppresses forward scattering.

Figure 7.7: Electric Field Contours for Scattering from a Concave PEC Object, using the 1st orderABC (left) and the 2nd order ABC (right).

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7.6 Two PEC ObjectsThe framework developed also allows us to examine the field patterns due to the presence of multipleobjects - a more practical scenario. As an example, we consider two objects, as shown in the mesheddomain below. The domain consists of a concave object and a square of edge length 0.2with thetop left corner at the origin.

Figure 7.10: Mesh for double PEC object scattering

Contour plots of the total electric field are shown below. The 2nd order ABC-based solution is muchbetter than the one based on the 1st order ABC, because it is much more regular and smootherand evaluates a very small electric field in the immediate vicinity of the scatterer, an observation

consistent with the properties of a PEC.

Figure 7.11: Contour Plots for Scattering from two PEC objects, using the 1st order ABC (left)and 2nd order ABC (right)

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7.7 Comparison between 1st order ABC and PML PEC Cylin-

der in Air

Comsol Multiphysics was used for simulation, and the 1st order ABC and PML were implementedand compared at four frequencies: 0.5 GHz,1 GHz,1.5 GHzand3 GHz. In this case, the excitationplane wave is incident from the top. The results are shown below.

Figure 7.12: Scattered Fields from PEC Cylinder in Air: ABC vs PML

The solution is seen to be smoother with the PML in place, than just with the First Order ABC,due to better absorbing characteristics of the PML. Comsol also plots the scattered field within thePEC, which is not plotted conventionally. This should be ignored.

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7.8 Comparison between ABC and PML for Dielectric Cylin-

der in AirIn this case, a dielectric cylinder with relative permittivityr = 2.86 and conductivity = 2.864 104S/m, is placed in air. As before, the excitation plane wave propagates along the negativey-direction.

Figure 7.13: Scattering from Dielectric Cylinder in Air: ABC vs PML

As with the PEC, the solution is seen to be smoother with the PML in place, than just with the FirstOrder ABC, due to better absorbing characteristics of the PML. Comsol also plots the scatteredfield within the PEC, which is not plotted conventionally. This should be ignored.

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7.9 The Subsurface Problem (Half Space Scattering)In this case, a simplified version of the situation portrayed in Figure 1.1, was modeled and solved inComsol Multiphysics, for two possible objects buried under the soil (a PEC cylinder and a dielectriccylinder), and two kinds of soil (dry and wet soil). The results are given below. In the subsurfaceproblem, we use the combined ABC-PML method throughout. The frequency of the incident wave

is 1.0 GHzin each case.

7.10 PEC Cylinder buried under SoilA PEC Cylinder of radius r = 5 cmis placed at a depth of10 cmbelow the surface of the soil. Thesimulation was carried out for two kinds of soil, dry and wet. The mesh truncation scheme involvesusing PML on the inner rectangular boundaries, and a first order ABC on the outer rectangularboundaries.

Figure 7.14: Scattering from PEC Cylinder buried under Soil (z-component of total field for dry

soil (left) and wet soil (right).)

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7.11 Dielectric Cylinder buried under Soil

A Dielectric Cylinder of relative permittivity r = 2.86, conductivity = 2.864 104S/mhavinga radius r = 5 cm is placed at a depth of10 cm below the surface of the soil. The simulation wascarried out for two kinds of soil, dry and wet. The mesh truncation scheme involves using PML onthe inner rectangular boundaries, and a first order ABC on the outer rectangular boundaries.

Figure 7.15: Scattering from Dielectric Cylinder buried under Soil (z-component of total field fordry soil (left) and wet soil (right).

It is observed that the contrast of the mine increases as the soil becomes more wet, whether the mineis a PEC or a dielectric (TNT). This is consistent witht he increased conductivity and dielectricconstant of moist soil, tends to concentrate the scattered field.

Note: The contour plots in sections 7.7-7.11 are reproduced after magnification in

Appendix D, for greater ease of viewing and clarity.

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7.12 Scattering from a PEC Cylinder in Air using the First

Order ABC (Effect of varying Discretization)

For a PEC cylinder of radius /2, and an ABC boundary of radius 3/2, the analytical and FEMsolutions are compared at = , for different discretizations, to illustrate the effect of meshingThe mesh pattern, a contour plot of the total electric field in the region, and a graph of the FEM

and exact solution have been plotted for three discretization sizes, and the results are shown below.

Fig. a: Discretization Size= 0.2

Fig. b: Discretization Size= 0.09

Fig. c: Discretization Size= 0.04

It is seen that for cyl = a= 0.5,ABC= 1.5, the FEM solution is very nearly equal to the exactsolution for the discretization size 0.04. The entire source code for solving this problem is given inthe appendices of this report.

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Chapter 8

Conclusions

A framework for solving the Laplace, Poisson and Helmoltz equations in two dimensions has beenconstructed from scratch in MATLAB, comprising of meshing, preprocessing and FEM routines,with a feature to impose Mixed Boundary Conditions, in particular, the 1st and 2nd order ABCsAs a part of this project, a system for meshing of arbitrary shapes in two dimensions has been putin place by combining our own codes with DistMesh and COMSOL for meshing.

Completely general FEM codes have been written to compute the total and scattered field dis-tributions in the near region of one or more scatterers in two dimensional domains. Results fromseveral electromagnetic scattering situations have been mentioned and discussed in this report,comparing the 1st and 2nd order ABCs. Exact analytical expressions for 1st and 2nd order ABCs

have been derived in Appendices B and C. These are directly used in an FEM code, and allow theprogrammer to have complete flexibility while solving a problem computationally.

Using COMSOL Multiphysics as an FEM modelling package, contour plots of the scattered fieldsfor PEC and dielectric cylinders in air and soil were extracted by simulation, results of which havebeen stated here.

Further work needs to be done to include codes for a generalized Perfectly Matched Layers inthis framework, and validate results obtained using COMSOL-based PMLs, so as to have com-plete acesss to and control over all the element matrices and vectors in the formulation of theFinite Element Method (Appendix A). The following inferences can be made from analytical andcomputational studies:

1. The2nd order ABC is better than the 1st order ABC as for mesh truncation, as it introducesfewer artifacts in the computational solution, by capturing the curvature of the field.

2. The 2nd order ABC can be imposed at a smaller distance from the scatterer, thanthe 1st order ABC, without compromising fidelity of the solution.

3. The Second Order ABC used by us can be imposed only on circular boundariesWe observed that the second order curvature term contributes to large oscillations in the field,when evaluated over nonsmooth countours, leading to severe errors in the results.

4. For circular objects, the concentric ring meshing algorithm, described in section 4.3.1 yieldsbetter solutions than those provided by DistMesh and COMSOL. This is because of thesymmetry of the concentric ring mesh coinciding with the symmetry of the object geometry.

5. The PML-based mesh truncation scheme is better than the one using only a First Order ABC

6. The contrast of scattered field distribution in the near region of the scatterer is enhanced asthe moisture content in the soil increases.

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References

1. J. Jin, The Finite Element Method in Electromagnetics. New York: John Wiley & Sons, 1993

2. A.C. Polycarpou, Introduction to the Finite Element Method in Electromagnetics. San RafaelMorgan& Claypool Publishers, 2006.

3. P. Persson and G. Strang, DistMesh A Simple Mesh Generator in MATLAB, SIAM Reviewvol. 46, no. 2, pp. 329-345, June 2004.

4. J.L. Volakis, A. Chatterjee, and L.C. Kempel, Finite Element Method for ElectromagneticsAntennas, Microwave Circuits, and Scattering Applications. Hyderabad: Universities Press

(India) Pvt. Ltd., 2001.

5. S. Kesavan,Topics in Functional Analysis and Applications. New Delhi: New-Age Intl., 2008

6. A. Bayliss, M. Gunzburger and E. Turkel, Boundary conditions for the numerical solution ofelliptic equations in exterior regions, SIAM J. Appl. Math., vol. 42, no. 2, April 1982.

7. C.A. Balanis, Advanced Engineering Electromagnetics. New York: John Wiley& Sons, 1989

8. Steven G. Johnson, Notes on Perfectly Matching Layers (PMLs) [MIT Course 18.369/18.336],August 2007. [Online]. Available: http://www-math.mit.edu/~stevenj/18.369/pml.pdf.

9. J.L. Volakis and A. Chatterjee, A selective review of the finite element-ABC and the finiteelement-boundary integral methods for electromagnetic scattering, Ann. Telecommun., vol

50, no. 5-6, pg. 499-509, May 1995.10. J.P. Brenger, A Perfectly Matched Layer for the absorption of electromagnetic waves, J

Comput. Phys., vol. 114, no. 2, pp. 185-200, October 1994.

11. C.M. Rappaport and S. Winton, Using the PML ABC for air-soil wave interaction modelingin the time and frequency domains, Intl J. of Susburf. Sensing Tech. and Appl., vol. 1, no3, pp. 289-304, July 2000.

12. J. Jin and W.C. Chew, Combining PML and ABC for the Finite-Element analysis of scat-tering problems, Microwave Opt. Tech. Lett., vol. 12, no. 4, pp. 192-197, July 1996.

13. Y.A. Gryazin, M.A. Klibanov, and T.R. Lucas, Two numerical methods for an inverse problem

for the 2-D Helmoltz equation, J. Comput. Phys., vol. 184, no. 1, pp. 122-148, Jan. 2003.14. M.A. Gonzalez-Huici, U. Uschkerat, and A. Hoerdt, Numerical simulation of electromagnetic-

wave propagation for land mine detection using GPR, Geoscience and Remote Sensing Sym-posium, July 23-28, 2007, Barcelona, Spain. IGARSS 2007. IEEE International.

15. Allan P. Engsig-Karup, An introduction to Discontinuous-Galerkin FEM for partial dif-ferential equations, August 2009. [Online]. Available: http://www2.imm.dtu.dk/~apek/DGFEMCourse2009/Material.php .

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http://www-math.mit.edu/~stevenj/18.369/pml.pdfhttp://www2.imm.dtu.dk/~apek/DGFEMCourse2009/Material.phphttp://www2.imm.dtu.dk/~apek/DGFEMCourse2009/Material.phphttp://www2.imm.dtu.dk/~apek/DGFEMCourse2009/Material.phphttp://www2.imm.dtu.dk/~apek/DGFEMCourse2009/Material.phphttp://www-math.mit.edu/~stevenj/18.369/pml.pdf


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Appendix A: Finite Element MethodFormulation

We now examine the steps in a 2D FEM solution in some detail. We will assume for now thatthe meshing elements are triangular, with three nodes, which are the three vertices. The resultingmeshing scheme is known as Triangular Meshing. In the scalar situation, the objective is to solvethe boundary value problem,

x

x

x y y y + = f (x, y) (8.1)

where is the unknown function,x,y, andare known parameters associated with the physicalproperties of the domain, andfis the source or excitation function. The boundary conditions tobe considered are given by

= p on1 (8.2)

and x

xx+y

yy

n+ = qon 2 (8.3)

where = 1+2denotes the contour or boundary enclosing,nis its outward normal unit vector

and,p, andqare known parameters associated with the physical properties of the boundary. Thisis the Mixed Boundary Condition, and is a generalization of the Dirichlet and Neumann boundarycondition.

With a suitable meshing in place that breaks up the solution domain into Ntriangular meshes, weseek an approximation of = e within each element e(e= 1, 2, . . . , N ):

(x, y) Ne=1

e(x, y) (8.4)

A common approximation for

e

within elementeis the first order polynomial expansion,e(x, y) =ae +bex+cey (8.5)

whereae,be andce are constants to be determined. An additional property ofe is that it vanishesoutside elemente.

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where

Ne1 (x, y) = 1

2A[(x2y3 x3y2) + (y2 y3)x+ (x3 x2)y] (8.10)

Ne2 (x, y) = 1

2A[(x3y1 x1y3) + (y3 y1)x+ (x1 x3)y] (8.11)

N

e

3 (x, y) =

1

2A[(x1y2 x2y1) + (y1 y2)x+ (x2 x1)y] (8.12)and A is the area of the element e, given by

A=1

2

1 x1 y11 x2 y21 x3 y3

=1

2[(x2 x1)(y3 y1) (x3 x1)(y2 y3)] (8.13)

The value of A is positive if the nodes are numbered counterclockwise (starting from any node)as shown in the figure above. TheNeis are known as element shape functions and satisfy thefollowing properties:

Nei(xej , y

ej ) =ij = 1, i= j

0, i =j (8.14)

3i=1

Nei(x, y) = 1 (8.15)

Weak Form

The weak form of the differential equation

x

x

x

y

y

x

+ = f (8.16)

is

e

x

wx

x

+ ywy

y

dxdy +

e

wdxdye

w

x

x

nx+yy

ny

dl

=

e

w f dx dy(8.17)

wherenx and ny are the direction cosines of the normal to the surface. In the Galerkin approachthe weight functions w(x, y) are taken to be equal to the triangular interpolation functions used inthe FEM. 1

1The triangular shape functions are:

N1(, ) = 1 N2(, ) = N3(, ) =

The primary unknown quantity is interpolated within an element e as

e(x, y) =3

j=1

ejNj(x, y)

Here ej denotes the value of the unknown quantity at local node j of element e.

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After appropriate discretization, the weak form becomes e

x

Nix

3j=1

ejNjx

+y

Niy

3j=1

ejNjy

dxdy

+

eNi

3j=1

ejNj

dxdy=

eNigdxdy

eNi

xx n

x+y y ny

dl

fori= 1, 2, 3 (8.18)

wheree is the solution domain corresponding to element e and e denotes the boundary curve ofelement e (in this case, a triangle). This equation can be represented in a matrix form given by

Me11 Me12 Me13Me21 Me22 Me23Me31 M

e32 M

e33

e1e2e3

+

Te11 Te12 Te13Te21 Te22 Te23

Te31 Te32 T

e33

e1e2e3

=

ge1ge2ge3

+

pe1pe2pe3

(8.19)

where

Meij =

e

x

Nix

Njx

+y

Niy

Njy

dxdy (8.20)

Teij =

e

NiNjdxdy (8.21)

gei =

Nif dxdy (8.22)

pei =

e

Ni

x

xnx+y

yny

dl (8.23)

The matrix system (8.19) can be written more compactly as Ke11 Ke12 Ke13Ke21 Ke22 Ke23Ke31 K

e32 K

e33

e1e2e3

=

be1be2be3

(8.24)

where

Keij = Meij+ T

eij (8.25)

bei = gei +p

ei (8.26)

Element Matrices and Vectors

Let i j k be a cyclic permutation of1 2 3. Then, the elements of matrices M and T aregiven by the following expressions.

Meii =

x

(yjk)2

4Ae+y

(xkj)2

4Ae

(8.27)

Meij =

x

yjkyki4Ae

+yxkjxik

4Ae

(8.28)

Teij = Ae

12 (1 +ij) (8.29)

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HereAe denotes the area of the triangular element e, andxij =xei xej,yij =yei yej . The elements

of vectorge are given by

ge1 = f Ae

3 (8.30)

ge2 = f Ae

3

(8.31)

ge3 = f Ae

3 (8.32)

if f does not appreciably vary over the element. A more general set of matrix equations for ageometry withNe elements and Ms Dirichlet/Neumann segments is given by

Nee=1

([Ke]{e} {be}) +Mse=1

([Ks]{s} {bs}) = 0 (8.33)

where the matrices have been suitably augmented, and the individual elements are given by

Keij = 1

4Ae(exb

ei b

ej+

eyc

ei c

ej) +

Ae

12e(1 +ij) fori, j= 1, 2, 3 (8.34)

bei =

e

f(x, y)Nei(x, y)dxdy fori = 1, 2, 3 (8.35)

Ksij =

10

NsiNsj l

sd fori, j = 1, 2 (8.36)

bsi =

10

qNsilsd fori= 1, 2 (8.37)

This alternate system of equations is more useful if more general boundary conditionshave to be imposed. The values of the other matrix and vector elements are dependent on whethera first order ABC (Appendix B) is used or a second order ABC (Appendix C). Assuming that thematrix elements have been defined properly in accordance with the chosen absorbing boundarycondition, the remaining procedure is as follows.

Integration over an arbitrary triangular patch

The integral expression

bei =

e

f(x, y)Nei(x, y)dxdy fori= 1, 2, 3 (8.38)

is zero in the total field formulation because f = 0 there, but if one works in the scattered fieldformulation, this term arises frequently. Also, if orvary inside an element, we have to use theintegral expression forKeij itself, which involves a similar double integral. Since this integral is overan arbitrary triangular patch, the x and y integrals are coupled. We now give a nonlinear transfor-mation from an arbitrary triangular patch to a square which allows this integral to be carried outvery simply using numerical integration, for every element.

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Consider the integraleg(x, y)dxdy. In order to simplify the numerical integration and not have

to consider various cases for the slopes of the edges of the triangular element, and its orientationwith respect to the axes, we define the following nonlinear transformation

x(u, v) = (1 u)xe1+u [(1 v)xe2+vxe3] (8.39)y(u, v) = (1 u)y

e1+u [(1 v)y

e2+vy

e3] (8.40)

This transformation maps the arbitrarily shaped and oriented triangular region e in (x, y) spaceto a square in (u, v) space, that is{(u, v) : 0 u 1, 0 v 1}. The Jacobian matrix of thetransformation is given by

J=

xe1+ (1 v)xe2+vxe3 uxe2+uxe3ye1+ (1 v)ye2+vye3 uye2+uye3

(8.41)

Hence, the integral becomes

e

g(x, y)dxdy= 1

u=0

1

v=0

g(x(u, v), y(u, v))

|J

|dudv (8.42)

where

|J|(u, v) = (xe1+ (1 v)xe2+vxe3)(uye2+uye3) (ye1+ (1 v)ye2+vye3)(uxe2+uxe3) (8.43)

Dirichlet Boundary Conditions

Let the prescribed values of be stored in a vector C(i) and let there by N nodes on 1, whoseglobal node numbers are stored in a vector DBC(i). Then, the Dirichlet boundary condition isenforced by setting

bDBC(i) = C(i), KDBC(i),DBC(i) = 1, KDBC(i),j = 0 forj

=DB C(i) (8.44)

bj bj Kj,DBC(i)C(i), Kj,DBC(i)= 0 forj=DB C(i) (8.45)for each i = 1, 2, . . . , N . In this problem, C(i) = 0i = 1, 2, . . . , N .Assembly ofKe, Ks into K

The first step is the assembly ofKe intoK(fore = 1, 2,...nElements), the pseudocode for whichis given below.

AssembleKe into K

1 for e 1 to nElements2 do3 for i 1 to34 do5 forj 1 to 36 doK(N(e, i), N(e, j)) K(N(e, i), N(e, j)) +Ke(i, j)7 end for8 end for9 end for

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The connectivity array for ABC segments, ns(i, s), denoted byN S(s, i)in the code, is constructedonce the segmentss (fors = 1, 2, . . . ,nSegABC) lying on the ABC boundary have been identifiedEach such segment s consists of two nodes, and so NS(s, i) serves a similar purpose for the ABCas does N(e, i) for the Dirichlet condition and for general-purpose indexing of the nodes. Thepseudocode for assembling Ks into Kis given below.

AssembleKs

into K1 for s 1 to nSegABC2 do3 for i 1 to24 do5 forj 1 to 26 doK(NS(s, i), NS(s, j)) K(NS(s, i), NS(s, i)) +Ks(i, j)7 end for8 end for9 end for

Solution of the Linear System

After setting up K and b as described above, the nodal fields are determined by solving the linearsystem,

Ku= b (8.46)

Using element shape functions, the unknown field can be interpolated within any element given itsvalues at the three nodes of that element. A fine discretization of the solution domain yields a moreaccurate solution for the nodal fields than a relatively coarse discretization. The interpolation can beimproved by increasing the number of points within each element at which the linear interpolationscheme is employed. The code developed here allows the user to change both these parameters so

as to be able to refine the solution as required.

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Appendix B: First Order AbsorbingBoundary Condition

The first order Absorbing Boundary condition has the form

sc

+

jk0+

1

2

sc = 0 (8.47)

The mixed boundary condition on a curve 2 is given byx

xax+y

yay

an+ =q (8.48)

which, in this case, is used to enforce the first order absorbing boundary condition [3], given by

sc

+

jk0+

1

2

sc = 0 (8.49)

Here,sc is the scattered field. For this problem,sc =Escz =Ez Eincz . In free spacer = 1, andthe-direction is normal to the outer boundary 2 on which the ABC is to be imposed. Therefore,

Ezx

ax+ Ezy

ay an = Ez

(8.50)

= jk0+ 1

2 (8.51)

q = jk0Eincz (ax an) +

jk0+ 1

2

Eincz (8.52)

To summarize, a first order ABC on boundary 2 can be imposed through the use of themixed boundary condition with the assignment of and qaccording to the above equationsIn the program, these boundary conditions are imposed on the discretized solution domain, afterdetermining the connectivity arrays n(e, i) andns(s, i) [1]. The outer boundary 2 is chosen to bea circular boundary with radius ABC.

In enforcing the ABC, there are three cases to consider for each element e, accordingly as whichone of the three edges lies on the ABC. These cases are listed below, along with the correspondingvalues ofpei (fori = 1, 2, 3) that must be assigned. Here lij denotes the length of edge i j.

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Figure 8.2: Case I: Edge 1 2 lies on the ABC boundary

Case 1: Edge 1 2 lies on the ABC boundary

pe1 = E0q0l12ejk0xe1

1 jk0x21 ejk0x21

(k0x21)2

(8.53)


1 + (jk0x21+ 1)ejk0x21

(k0x21)2 (8.54)pe3 = 0 (8.55)

where

q0= j k0(ye2 ye1)l12

(8.56)

Case 2: Edge 3 1 lies on the ABC boundary


1 + (jk0x13+ 1)ejk0x13(k0x13)2

(8.57)

pe2 = 0 (8.58)

pe3 = E0q0l13ejk0xe3 1 jk0x13 e

jk0x13

(k0x13)2 (8.59)

where


(8.60)

Case 3: Edge 2 3 lies on the ABC boundarype1 = 0 (8.61)


1 jk0x32 ejk0x32

(k0x32)2

(8.62)


1 + (jk0x32+ 1)ejk0x32(k0x32)2 (8.63)

where


(8.64)

In the alternate formulation (8.33 8.37), the elements ofKs are given by

Ks(i, j) =ls

6(1 +ij) (8.65)

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Appendix C: Second Order AbsorbingBoundary Condition

In this appendix, the exact details of how the second order ABC is discretized and imposed on acircular boundary, are described.

Let 2 denote the surface over which the second order absorbing boundary condition (2-ABC)is to be imposed. The boundary condition of the third kind is given by

x

xnx+y

yny

+ = q on2 (8.66)

The absorbing boundary condition must be recast in this canonical form so as to determine theelements of the vector pe (c.f. Appendix A). The 2-ABC is expressible in this form if we split as= inc +sc, that is, as a sum of an incident field and a scattered field, and identify and q tobe given by2

=

jk0+

2 j

2

8(j k0) j

2(j k0)2

s2

(8.67)

q = inc

n + jk0+

2 j

2

8(j

k0)inc j

2(j

k0)

2inc

s2 (8.68)

In order to incorporate the general boundary condition of the third kind, contributions from thefunctional

Fb=

2

1

2 q

d (8.69)

must be accounted for in the weak form given. Here, has been sandwiched by the unknown fieldThis is becauseis self-adjoint, assuming2to be a smooth closed contour, and is now an operatorrather than a scalar function as it was in case of the first order ABC [ 1]. To further simplify theanalysis, we write

= 1+22

s2 (8.70)

with

1 =

jk0+

2 j

2

8(j k0)

(8.71)

2 = j2(j k0) (8.72)

2It is assumed hereafter that x = y =.

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We can write Fb as

Fb =

2

1

2

1

2 +22

s2

q

d (8.73)

= 2 1

21

2 12

2

s 2

q d (8.74)where integration by parts has been carried out on the second term of equation (8.73), assuming2 is indeed a smooth closed contour. The negative sign here is incorrectly stated as a positive signin Jin [1]. The next step is to discretize Fb over each element as was done for the 1-ABC:

Fb=Mss=1

Fsb(s) (8.75)

whereFsb denotes the integral over segment s. Differentiating with respect to si gives the matrix

system F

sb

s

= [Ks]{s} {bs} (8.76)

with the elements in[Ks]and{bs} given by

Ksij =

10

NsiNsj l

sd i, j = 1, 2 (8.77)

bsi =

10

qNsilsd i= 1, 2 (8.78)

withqgiven by (6.69). For the 2-ABC, the expression for Ksij is given by

Ksij =

10

1l

sNsiNsj

2ls

Nsi

Nsj

d (8.79)

HereNs1() = 1 and Ns2 () =, so that Ns1 () +Ns2 () = 1. This gives us

Ks11 = Ks22 =

s1

ls

3 +

s2ls

(8.80)

Ks12 = Ks21=

s1

ls

6

s2

ls (8.81)

Evaluation ofpeiThere are three cases to consider while imposing the 2nd order ABC on a circular boundary, de-pending on which edge of a triangular boundary element lies on the ABC contour. These threecases are similar to those considered in the 1nd order ABC in Appendix B, but the expressions for

pei are considerably more complex because of the structure of the function q in (8.68), and the factthatis now a differential operator.

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The unknown field is = Ez andinc =Eincz =E0e

jk0x. Now,qbecomes

q(x, y) = jk0E0

xejk0x+jk0+

2 j

2

8(j k0)

E0ejk0x j

2(j k0)

jk0x

2 k

20y

2

E0e

jk

(8.82)

Since p

e

i involves line integrals ofq, we can express the edge along which the line integral is to becomputed through the slope-intercept form of a straight line,

y= mx+c (8.83)

thereby convertingqto a function ofx alone along the edge under consideration. For simplificationwe define

a = jk0E0

(8.84)

b =

jk0+

2 j

2

8(j k0)

E0 (8.85)

c1 = k0E02(j k0)2 (8.86)

c2 = jk20E02(j k0) (8.87)

We assume that is cons

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