Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique

187 Pages·2016·3.8 MB·English

by  Andrew Seagar (auth.)

Andrew Seagar
Application of
Geometric Algebra
to Electromagnetic
Scattering
The Clifford-Cauchy-Dirac Technique
Application of Geometric Algebra
to Electromagnetic Scattering
Andrew Seagar
Application of Geometric
Algebra to Electromagnetic
Scattering
The Clifford-Cauchy-Dirac Technique
123
AndrewSeagar
GoldCoast, QLD
Australia
ISBN978-981-10-0088-1 ISBN978-981-10-0089-8 (eBook)
DOI 10.1007/978-981-10-0089-8
LibraryofCongressControlNumber:2015954612
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To Cathy
Preface
Electromagnetism immediatelycapturedmyinterestwhen Iwas first introduced to
it as a student at university. Complete mastery of one of the fundamental phe-
nomena of our physical world seemed within reach, as promised in the form of
Maxwell’s equations [1]1. However, the realisation of that promise was not as
straightforward as I might have first imagined.
Exposure to electromagnetic theory in the period during and immediately fol-
lowing university, with its application to impedance imaging, left me with some
skills—largely restricted to two dimensions and quasi-static fields. I found the
conventionaltreatments atthat time for fullythree-dimensionaltime-varying fields
clumsy and unpalatable. I took that to reflect my own inadequacy in grasping the
methods rather than reflecting any inadequacy in the methods themselves. As it
turns out, I was mistaken.
The first inkling that there might really be a simpler way came for me in 2002
with the paper of Axelsson, Grognard, Hogan and McIntosh [2]. At that time
I started to seriously follow the literature relating to the theory of monogenic
functions expressed in Clifford’s algebra [3, 4] based on his interpretation of
Hamilton’s quaternions [5] in terms of Grassmann’s extension theory [6, 7], and
applied through the Cauchy integral [8, 9] to provide a solution to Maxwell’s
equations, and thereby to electromagnetic problems.
Althoughmuchiswritten,thereareonlyafewworkswhichIconsideressential.
FirstaretheworksofMcIntosh[10,11]providingasolidmathematicaljustification
on the validity of the Cauchy integral in multiple dimensions for monogenic
functionsinLipschitzdomains.SecondisthepaperofAxelsson,Grognard,Hogan
and McIntosh [2] giving the formulation of electromagnetic problems within the
Clifford–Cauchyframework.ThirdisthethesisofAxelssoncontainingmanymore
of the details [12].
Giventhatmybackgroundwasinelectricalengineeringandtheaforementioned
documents were written by mathematicians, they probably would have remained
1Section617.
vii
viii Preface
foreverbeyondmyunderstandingwereitnotthatIhadworkedformanyyearswith
one of the authors. With his support and guidance I was slowly able to master the
conceptsandcastthem,finallyasthisbook,inaformbywhichtheycanbeshared
with others having a background at a level similar to my own.
TheworksinthebookfollowultimatelybacktoCauchy,Hamilton,Grassmann,
Maxwell and Clifford rather than conventionally to Green, Hamilton, Maxwell,
Gibbs [13] and Heaviside [14]2 (also [15, 16]). The differences from conventional
techniques are that (i) Hamilton’s quaternions are used in whole to construct
Clifford’salgebraratherthanbeingstrippedaparttodeliverGibbs’svectorcalculus,
(ii) Clifford’s algebra is used instead of Gibbs’s vector calculus and (iii) integrals
are used involving Cauchy kernels instead of Green’s functions.
The ultimate outcome is an integral formulation which lends itself to a more
directandefficientsolutionthanisconventionallythecase.Thatleadsbynecessity
on any particular machine to either a faster solution for a given problem or the
abilitytosolveproblemsofgreatercomplexity.Onthebasisofefficiencyalone,the
new technique is destined in time to supplant at least some existing techniques.
Thebookiswrittenwiththeintentionthatanywhoareinterestedcanalsocome
to master, over a much shorter time-span than in my case, techniques which offer
simplerandmoreefficientsolutionstoproblemsofelectromagnetic scatteringthan
are currently in use.
During the development of the material and preparation of this work many
peoplehavebeenofassistanceinvariousways,eitherconsciouslyorunknowingly,
directly or indirectly. The various ways include the penetrating questions of stu-
dents,thesoundadviceandencouragementofcolleaguesandpeers,theperceptive
counsel of elders and betters, the professional guidance of the editorial team, and
the stoic forbearance and unfaltering faith of my immediate family.
Gold Coast Andrew Seagar
September 2015
2Sections51and66.
Contents
Part I Preparation
1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Grassmann’s Outer Product. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Outer Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Three Kinds of Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Constraint Equations. . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Interdependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Complex Numbers and Quaternions . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Quaternions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Clifford’s Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Vector Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Synchronicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Casting Physical Problems into Clifford Algebra . . . . . . . . . . . 13
1.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Numbers and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Structure of Clifford Numbers . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Inversion of Signs . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Addition and Subtraction. . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
ix
x Contents
2.4 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Dimensional Recursion . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Recursive Product. . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.3 Machine Computation. . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Reflection, Projection and Rotation Operators . . . . . . . . . . . . . 37
3.2 Reflection and Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 Application to Bivector Fields . . . . . . . . . . . . . . . . . 43
3.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.3 Application to Bivector Fields . . . . . . . . . . . . . . . . . 48
3.4 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.1 Affine Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.2 Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.3 Transformations Between Coordinate Systems . . . . . . 51
3.4.4 Multi-Dimensional Axes . . . . . . . . . . . . . . . . . . . . . 53
3.5 Intersection by Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Space and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Points in Space–Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Transformation Between Frames . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Related Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Related Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Derivatives in Space–Time . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Physical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Classical Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.1 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 Clifford Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Splitting the Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Frequency Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4.1 Leibniz Product Rule . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Contents xi
Part II Formulation
6 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Single Regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Multiple Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.1 Extreme Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.2 Components of Solutions. . . . . . . . . . . . . . . . . . . . . 89
6.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.1 Perfect Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4.2 Perfect Transmission. . . . . . . . . . . . . . . . . . . . . . . . 92
6.4.3 Partial Transmission and Reflection. . . . . . . . . . . . . . 93
6.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7 Cauchy Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1 Boundary Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2.1 Three Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Off the Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.1 Reproducing Formula . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.2 Cauchy Extension. . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 On the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.4.1 Cauchy Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.5 Example Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.5.1 Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.5.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.5.3 Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . 106
7.5.4 Practical Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8 Hardy Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.1 Perfect Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2 Perfect Transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3 Partial Transmission and Reflection . . . . . . . . . . . . . . . . . . . . 111
8.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9 Construction of Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.1 Perfect Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.2 Perfect Transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.3 Partial Transmission and Reflection . . . . . . . . . . . . . . . . . . . . 116
9.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.3.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.3.3 Computational Effort. . . . . . . . . . . . . . . . . . . . . . . . 120
9.3.4 Joint Interface Function. . . . . . . . . . . . . . . . . . . . . . 122
9.4 Convergence and Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123