University of Cape Town

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1 The copright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private stud or noncommercial research purposes onl. Published b the (UCT) in terms of the non-eclusive license granted to UCT b the author.

2 Finite Element Method using Vector Finite Elements Applied to Edd Current Problems Leila Adams A dissertation submitted to the Department of Electrical Engineering,, in fulfilment of the requirements for the degree of Master of Science in Engineering. Cape Town, March 2

3 Declaration I know the meaning of plagiarism and declare that all the work in the document, save for that which is properl acknowledged, is m own. This dissertation is being submitted for the degree of Master of Science in Engineering in the. It has not been submitted before for an degree or eamination in an other universit. Signature of Author Cape Town 8 March 2 i

4 Abstract Vector fields found in electromagnetics are fundamentall different to vector fields found in other research areas such as structural mechanics. Electromagnetic vector fields possess different phsical behaviour patterns and different properties in comparison to the other vector fields and therein lies the necessit of the development of a finite element which would be able to cater for these differences. The vector finite element was then developed and used within the finite element method specificall for the approimation of electromagnetic problems. This dissertation investigates the partial differential equation that governs edd current behaviour. A finite element algorithm is coded and used to solve this partial differential equation and produce vector field simulations for fundamental edd current problems. Edd current phenomena belong to a particular branch of electromagnetic theor, thus the vector field solutions of edd current problems would possess electromagnetic properties and therefore the vector element proved to be highl desirable choice to use within the implementation of this finite element algorithm. The dissertation covers research theor concerning the partial differential equation that governs edd current phenomena and vector finite elements. All the knowledge and concepts gained from this research are then used for the implementation process of a finite element algorithm. An analtical solution to a simple theoretical edd current problem was compared to the simulation of the finite element solution of the same edd current problem. Good phsical and behavioural similarities between the simulation and the analtical solution provided proof of the successful implementation of the finite element algorithm and therefore established confidence in the capabilit of the finite element algorithm to approimate solutions to other edd current problems. The finite element algorithm was then used on various arbitrar edd current problems and the simulated vector field solutions were compared to the research theor that was covered in previous chapters and electromagnetic field theor. The analsis shows that the research theor and the field theor is in good agreement with the simulated solutions and thus served as more proof that the implementation of the finite element algorithm was successful. ii

5 The ink of a scholar is mightier that the blood of a martr. Prophet Mohammad (P.B.U.H.) Seek knowledge from the cradle to the grave. Prophet Mohammad (P.B.U.H.) iii

6 Acknowledgements I would like to epress m thanks to those who assisted me during the course of this research project. In particular: to m supervisors Prof. B.D.Redd and Associate Prof. A.J.Wilkinson for their assistance and help. to m famil for their support, encouragement and patience during this time. to Mr. S. Adams, for some advice regarding the graphical representation of certain vector field results. to the and CERECAM for awarding a Postgraduate Scholarship Bursar. iv

7 Contents Declaration Abstract i ii Acknowledgements List of Smbols Nomenclature Nomenclature Introduction. Aim and Objective of the Dissertation Scope and Limitations of the Dissertation Contents of the Dissertation Mathematical Model of Edd Current Behaviour 4 2. Differential Vector Form of Mawell s Equations Phasor Form of Mawell s Equations Phsical Description of the Edd Current Model Model Restrictions Derivation of the Partial Differential Equation that Governs Edd Current Behaviour - Differential Vector Form Derivation of the Partial Differential Equation that Governs Edd Current Behaviour - Phasor form Interpretation of the Mathematical Equations Governing Edd Current Behaviour iv iii iv v 3 Analtical Solution of the PDE that Governs Edd Current Behaviour 2 3. PDE Modelled without a Source Region Included in the Domain Problem Statement Analtical Solution of Problem Skin Depth and Skin Effect Phenomena Analsis of Problem PDE Modelled with a Source Region Included in the Domain Problem Statement v

8 3.2.2 Analtical Solution of Problem Analsis of Problem Electric Field Induced within a Conducting Bod through a Uniform Time- Varing Harmonic Magnetic Field Problem Statement Analtical Solution of Problem Analsis of Problem Nodal Finite Elements Scalar Field Simple Elements Two Dimensional Triangular Scalar Finite Element Triangular Coordinate Transformation Shape Functions Vector Finite Elements 3 5. Preliminar Theor Vectors Vector Fields Boundar Conditions for Electromagnetic Fields Two Dimensional Triangular Node-Based Vector Element Shape Functions Construction of a Vector Field within a Triangular Node-Based Vector Element Continuit of the Node-based Vector Element Two Dimensional Triangular Vector Element Vector Shape Functions Construction of a Vector Field within a Triangular Element Tangential Continuit of the Vector Element Calculation of a Vector Field Across a Triangular Finite Element Construction of a Rotational Field Construction of a Constant Field Using Triangular Vector Elements to Create a Two-Dimensional Vector Field across a FE Mesh Creating a Two-Dimensional Vector Field Properties of the Vector Field Produced b the Vector Element and the Node-based Vector Element Investigation of the Tpes of Vector Fields that The Vector Elements are Capable and Incapable of Constructing Limitations of the Vector Element Higher-Order Vector Elements vi

9 6 Finite Element Method Domain Discretization Two Dimensional Domain Discretization Data-Structures Formulation of the Sstem of Linear Equations Formulation of the Weighted-Integral Equation for a Three-Dimensional Finite Element Formulation of Linear Equations for a Two-Dimensional Finite Element Discretization of FE Matrices Assembl of the Global Matri Sstem of Equations Implementation of the Assembl Process Solving the Linear Sstem of Equations Building the Solution Vector Field Implementation Issues Concerning the FEM 9 7. Element Connectivit within a FE Mesh Data Sorting and Handling Boundar Element Data Information Edge Value Calculation Implementation of the Assembl Process of the Global Matri Sstem Appling Linear Algebra to Solve the Sstem of Equations Application of Boundar Conditions Solving the Linear Sstem of Equations Iterative Techniques for Solving Linear Sstems Construction of the Solution Vector Field Brief Overview of FE Algorithm FEM Solutions of the PDE that Governs Edd Current Behaviour 8. Finite Element Meshes FEM Solutions of an Analticall Solved Problem PDE Modelled Without a Source Region Included in the Domain FEM Simulated Solution Analsis of FEM Generated Solutions Summar FEM Solutions of Arbitrar Edd Current Configurations FEM Solution of a Boundar-Driven Edd Current Configuration FEM Solution of a Boundar-Driven Edd Current Configuration FEM Solution of a Force Driven Edd Current Configuration Conclusion Chapter Summar Future Work vii

10 A Analtical Solutions 7 A. Derivation of the Analtical Solution Found in Section A.2 Derivation of the Analtical Solution Found in Section A.3 Derivation of the Analtical Solution Found in Section A.3. Analtical Calculation of the Current Densit or Edd Currents. 74 Bibliograph 76 viii

11 List of Figures 2. Principle Edd Current Formation within a Conductor Schematic illustration of metallic bonding [2] Illustration of the Problem Statement of section Graphical Solution of Equation (3.) Illustration of the Problem Statement of section Graphical Solution of Equation (3.) Illustration of the Problem Statement of section Graphical Solutions of Equation (3.38) Coordinate Transformation of a Triangle in the Cartesian Coordinate Sstem Coordinate Transformation of Multiple Coordinate Points within a Triangle Triangle Finite Element Two Dimensional Scalar Finite Element Components of a vector Electric Field Deflection Across a Material Interface Two Dimensional Nodal-Based Vector Element Continuit of the node-based Vector Element Across the Element Interface Continuit of the node-based Vector Element Across the Material Interface Two Dimensional Vector Element Vector Shape Functions of a Unit Right Angle Triangle Element Inter-Connectivit Information Tangential Continuit and Normal Discontinuit of the Vector Field Across an Element Interface Tangential Continuit and Normal Discontinuit of the Vector Field Across a Material Interface Located Between Two Elements Construction of a Rotational Vector Field Across a Element Rotational Vector Field Across Element Construction of a Constant Vector Field Across a Element Constant Vector Field Across Element Flow Chart Diagram of a Vector Field Simulation Finite Element Mesh A Rotational Vector Field Built using Four Vector Elements An Arbitrar Vector Field Built using Four Vector Elements Finite Element Mesh i

12 5.2 Divergent Vector Field Vector Field having both Curling and Divergent Properties Vector Field Representation of Blocks B and B3 of Table Vector Field Representation of Block A2 of Table Vector Field Representation of Block A of Table Vector Field Representation of Block A3 of Table Vector Field Representation of Block A3 of Table Vector Field Representation of Block A of Table Vector Field Representation of Block A3 of Table Vector Field Representation of Block A3 of Table Linear Vector Field Representation of Block C2 of Table Vector Field Representation of Block C of Table Vector fields of Representation of Block C3 of Table Vector fields of Representation of Block C3 of Table Vector fields of Representation of Block C3 of Table Vector fields of Representation of Block C3 of Table Discretization Error using Rectangular Elements [] Discretization Error using Triangular and Quadrilateral Elements [] A Generic Three Element Mesh Flow Diagram of the Triangle Database Generation Two Element FE Mesh Flow Diagram of the Generation of the Boundar Database Flow Diagram of Boundar Edge Value Computation Assembl of the Global Matri Sstem of a Force-driven Configuration. 7.6 Assembl of the Global Matri Sstem of a Boundar-driven Configuration2 7.7 Solving the Linear Sstem of Equations for a Boundar-Driven Configuration Solving the Linear Sstem of Equations for a Force-Driven Configuration Flow Diagram of Global Field Component and Coordinate Data Computation General Overview of the Information Flow Process of the FE Algorithm Finite Element Mesh of the Domain Solution Vector Field E E Simulated at a frequenc of f = Hz Solution Vector Field E E Simulated at a frequenc of f = 5Hz Solution Vector Field E E Simulated at a frequenc of f = 2Hz Solution Vector Field E E within a Copper Conductor Simulated at a frequenc of f = Hz Geometr of the Problem Configuration Finite Element Mesh of the Domain Solution Vector Field E E and J E Simulated at a frequenc of f =.5Hz Solution Vector Field E E and J E Simulated at a frequenc of f = Hz.. 24

13 8. Solution Vector Field E E and J E Simulated at a frequenc of f = 5Hz Solution Vector Field E E and J E Simulated at a frequenc of f = 2Hz Solution Vector Field E E and J E Simulated at a frequenc of f = 5Hz Solution Vector Field E E within a Copper Conductor Simulated at a frequenc of f = Hz Geometr of the Problem Configuration Finite Element Mesh of the Domain Solution Vector Field E T Simulated at a frequenc of f =.5Hz Solution Vector Field J T Simulated at a frequenc of f =.5Hz Solution Vector Field E T Simulated at a frequenc of f = Hz Solution Vector Field J T Simulated at a frequenc of f = Hz Solution Vector Field E T Simulated at a frequenc of f = 5Hz Solution Vector Field J T Simulated at a frequenc of f = 5Hz Solution Vector Field E T Simulated at a frequenc of f = 2Hz Solution Vector Field J T Simulated at a frequenc of f = 2Hz Geometr of the Problem Configuration Finite Element Mesh of the Domain Solution Vector Field E T Simulated at a frequenc of f =.5Hz Solution Vector Field J T Simulated at a frequenc of f =.5Hz Solution Vector Field E T Simulated at a frequenc of f = Hz Solution Vector Field J T Simulated at a frequenc of f = Hz Solution Vector Field E T Simulated at a frequenc of f = 5Hz Solution Vector Field J T Simulated at a frequenc of f = 5Hz Solution Vector Field E T Simulated at a frequenc of f = 2Hz Solution Vector Field J T Simulated at a frequenc of f = 2Hz Solution Vector Field E T Simulated at a frequenc of f = 5Hz Solution Vector Field J T Simulated at a frequenc of f = 5Hz Solution Vector Field E T Simulated at a frequenc of f = Hz Finite Element Mesh i

14 List of Tables 5. Calculation of the Vector Field E (,) at each Node of the Element Calculation of the Vector Field E (,) at each Node of the Element Table of Vector Field Representations Connectivit between Elements and Edges Table of Numbering Combination Sequences [22] Local Edge and Node Numbering Sstem of a Vector Element Element Connectivit Information used to Construct Global Matrices Connectivit between Elements and Nodes Coordinate Location of the Nodes Connectivit between Edges and Nodes Connectivit between Elements and Edges Connectivit between Elements and Cartesian Points Boundar and Non-boundar Edges Boundar and Non-boundar Elements Connectivit between Boundar Elements and Edges Connectivit between Non-Boundar Elements and Edges Connectivit between Boundar Elements and Cartesian Points Connectivit between Non-Boundar Elements and Cartesian Points Connectivit between Boundar Edges and Nodes Connectivit between Non-Boundar Edges and Nodes Boundar and Non-Boundar Nodes ii

15 List of Smbols B Magnetic Flu Densit D Displacement Flu Densit E Electric Field Intensit E E Induced Electric Field Intensit E S Ecitation Electric Field Intensit Ẽ E Phasor Form of Induced Electric Field Intensit Ẽ S Phasor Form of Ecitation Electric Field Intensit H Magnetic Field Intensit H E Induced Magnetic Field Intensit H S Ecitation Magnetic Field Intensit H E Phasor Form of Induced Magnetic Field Intensit H E Phasor Form of Ecitation Magnetic Field Intensit J Current Densit J E Edd Currents β Attenuation Constant δ Skin Depth ε Permittivit λ Wavelength µ Permeabilit µ m Mobilit Constant ρ v Electron Charge Densit σ Conductivit ω Angular Frequenc f Frequenc Differential Operator t Time iii

16 Nomenclature Finite Element Method A numerical technique used to solve partial differential equations subject to certain boundar conditions. Matlab A scientific programing language used mostl for design and simulation. Skin Depth The distance after which the induced electric field or edd currents has attenuated to approimatel 36.8% of it s original value as the field penetrates into a conducting bod. Skin Effect Relationship shared between the skin depth and the frequenc. iv

17 Acronms BVP Boundar Value Problem BC Boundar Condition CEM Computational Electromagnetics EM Electromagnetic FEA- Finite Element Analsis FEM Finite Element Method FE Finite Element PDE Partial Differential Equation UCT v

18 Chapter Introduction The finite element method (FEM) is one of the best-known methods for the solution of partial differential equations (PDE s) in applied mathematics and computational mechanics [2]. It is a numerical technique for obtaining an approimate solution to PDE s subject to certain boundar conditions [2, 3]. FEM has been widel used in structural mechanics and thermodnamics since the 95 s approimatel [2, 3]. The earliest finite elements called nodal finite elements were used to approimate scalar field problems and the structural design of this element had its degrees of freedom (unknown parameters) represent the values of the scalar field at its nodes [7]. The nodal element was used to approimate vector field problems as well [8] but there were slight structural modifications made to the element in order to perform this task and this modified element was known as the node-based vector element. The structural design of this element has its degrees of freedom representing a cartesian component [2] at each of its nodes. The first application for electromagnetics (EM) problems was undertaken in the late 96s [2]. In solving EM problems b the FEM, these node-based vector elements were recognised to not work ver well [2, 3]. The structural design of the node-based vector element makes it impractical and difficult for the element to handle electromagnetic vector field properties and the structural design of the element also allowed for wasteful computational calculations concerning the degrees of freedom when the element approimated vector fields [3, 3, 2]. In the late 98 s the vector (edge) element was introduced in computational electromagnetics (CEM) [2] and this element avoided and solved man of the problems encountered when solving EM problems [3] in contrast to when nodebased vector elements were used. In 98, the French mathematician J.C.Nedelec published a paper which investigated the structure of polnomial spaces that the basis functions of a finite element should span in order to reduce the computation of wasted degrees of freedom [2]. The structural design of the vector element has its degrees of freedom represented as the tangential field component along its edges [8, 2] and this construction allowed for fewer degrees of freedom in contrast to the node-based vector element. J.P.Webb in his paper [3] and J.Jin in his book [3] both state that in vector electromagnetics problems, the curl of the field is often as important as the field itself. The vector element has b construction reduced degrees of freedom that allows for the element to model the curl-space more efficientl as shown b Nedelec [2] thereb approimating curling vector fields more effectivel. Thus, the vector element possesses unique properties which favoured EM behaviour. The dissertation project will focus on the use of this vector element within the FEM to solve a particular branch of EM problems known as edd current problems [6, 7]. The vector PDE that governs edd current behaviour is derived using Mawell s equations and this

19 PDE is used to solve certain fundamental edd current configurations.. Aim and Objective of the Dissertation The objective of this dissertation is to successfull implement a Finite Element (FE) algorithm that can be used to model a vector PDE that governs edd current behaviour for certain fundamental theoretical problems. The FE algorithm makes use of the FEM to approimate a solution to the PDE and then uses the vector elements to graphicall construct the solution. Investigation into the properties of the vector element was also done to gain a better understanding of how the structural design contributes to the function capabilit of the element to favour EM behaviour in contrast to node-based vector elements..2 Scope and Limitations of the Dissertation The dissertation covers edd current problems under investigation that are linear and twodimensional. This dissertation is a first attempt at research that covers vector finite elements at UCT and therefore a lot of emphasis was placed on the investigation into understanding the properties of the vector elements and on the successful implementation of a FE algorithm which makes use of these vector finite elements. The dissertation therefore approaches the algorithm from a theoretical perspective and thus focuses on implementing the FE algorithm on theoretical edd current problems and not on real world edd current problems. The dissertation in this wa tries to provide a solid foundation for further research and development into vector finite elements at UCT..2. Contents of the Dissertation In Chapter 2 the PDE that models edd current behaviour is derived from Mawell s equations in differential vector form as well as in phasor form. A brief interpretation of the PDE is given in the last section of this chapter. Chapter 3 deals with the calculation of analtical solutions of three fundamental edd current problems. The solution to one of these problems are simulated through the FE algorithm in Chapter 8, where this analtical solution serves the purpose of being a reliable means to test whether the FE algorithm was implemented correctl. Also, the behaviour pattern of the analtical solutions also aids in the interpretation of simulated solutions of the FE algorithm for arbitrar edd current problems in Chapter 8. Chapter 4 deals with the theor used to construction a two-dimensional nodal finite element. Derivation of coordinate transformation between two triangular sstems and the shape functions are illustrated. The construction of the nodal finite element mainl consists of the use of coordinate transformation and the shape functions. The knowledge of coordinate transformation and shape functions of the nodal finite element are latter incorporated into the construction of the vector finite element. Chapter 5 begins with some preliminar theor (given in sub-sections 5.., 5..2 and 5..3) that is needed to eplain certain concepts concerning the vector elements. The two-dimensional node-based vector element and two-dimensional vector (edge) element are derived along with their shape functions. The design structure of both elements are different and this results in contrasting differences in the functionalit and properties of each element when approimating vector fields and EM behaviour. Illustrations and calculations are done as to how the vector element achieves approimating vector fields over 2

20 a single element. The vector elements are then used to construct a two-dimensional vector field across a finite element mesh b using an algorithm especiall coded for this purpose. Lastl, a short investigation is done to analse which tpe of vector fields the vector element is capable and incapable of constructing and a brief overview on the limitations of the vector element that is used for this dissertation is mentioned. Chapter 6 gives a brief overview of the FEM. A description of each procedure within the FEM is given as well as the derivation of the FE equations and matrices. Chapter 7 discusses the theor behind the implementation process of the FE algorithm. Concepts, knowledge and findings covered throughout the research found in Chapters 4, 5, 2, 3, 6 and 7 is used in this implementation process. In Chapter 8 solutions to edd current problems are simulated where the results of these simulations are analsed. The chapter begins b comparing an analtical solution of a simple fundamental edd current problem in Chapter 3 to the simulation of the FE solution of the same problem. Simulations of other arbitrar edd current problems are also done. Chapter 9 draws a few conclusions and a few recommendations. Appendi A provides the full derivation of analtical solutions found in Chapter 3. 3

21 Chapter 2 Mathematical Model of Edd Current Behaviour The model gives a mathematical description b means of a partial differential equation (PDE) of the formation and phsical behaviour of edd currents. Mawell s equations are used to construct this PDE that governs the edd current behaviour. 2. Differential Vector Form of Mawell s Equations The phsical behaviour of electromagnetic fields can be adequatel summarised mathematicall b four differential equations known as Mawell s equations [5, ]. Phsicall these Mawell s equations describe time-varing electromagnetic fields [5]. Mawell s equations in differential form are defined as follows [5]: E = µ H t H = σ E + ε E t (2.) (2.2) ε E = ρ v (2.3) µ H = (2.4) where E [ v ] [ ] [ ] m is the electric field intensit, H Am is the magnetic field intensit, µ Hm [ ] [ ] [ ] is the permeabilit, σ Sm is the conductivit, ε Fm is the permittivit, ρ C v m 3 is the electron charge densit and t smbolises time. Constitutive relations equations are defined as follows [5]: D = ε E (2.5) B = µ H (2.6) J = σ E (2.7) 4

22 [ ] where D C m 2 is the displacement flu densit, B[T] is the magnetic flu densit and [ ] J A m 2 is the current densit. The constitutive relations for this dissertation will be represented b linear equations. Material constants are defined as follows [5]: µ = µ µ r (2.8) ε = ε ε r (2.9) where µ = 4π 7[ H m ] and ε = 36π 9[ F m ]. The material tpes specified in this dissertation will posses linear behaviour. The values of the constant terms permeabilit µ r, permittivit ε r and conductivit σ depends on the tpe of material being modelled [5]. The following vector identit written in terms of the electric field intensit, E [5]: E = ) ( E 2 E (2.) will be used in the derivation of the PDE (epressed in differential form) that governs edd current behaviour. In this dissertation, the focus will be on a on linear behaved sstem; that is, the equations will be considered in their linearised form as well as describe linear behaviour in phsical applications. 2.2 Phasor Form of Mawell s Equations Time-varing electromagnetic fields are ver often either sinusoidal or else periodic in nature [8]. The mathematical handling of sinusoidal functions is made easier b carring the amplitude and phase information about the sinusoid in the form of a comple number known as a phasor [8]. The primar advantage of using phasors in analsis of timeharmonic sstems is the simplification that results in the differentiation and integration with respect to time in mathematical equations [5]. It is important to remember that the use of a phasor can onl be applied to a sstem of mathematical equations when the sstem under consideration is linear [3, 4]. Mawell s equations in phasor form are defined as follows [8]: Ẽ = jωµ H (2.) H = σẽ + jωεẽ (2.2) εẽ = ρ v (2.3) µ H = (2.4) where in phasor form, Ẽ [ v ] [ ] m is the electric field intensit, H Am is the magnetic field ] intensit, ω[ rads s is the angular frequenc and j represents the imaginar number. Constitutive relations equations in phasor form are defined as follows [8]: D = εẽ (2.5) 5

23 B = µ H (2.6) [ ] where in phasor form, D C [ ] m 2 densit and J A m 2 J = σẽ (2.7) is the displacement flu densit, B[T] is the magnetic flu is the current densit. The constitutive relations for this dissertation will be represented b linear equations. The following vector identit written in terms of the electric field intensit in phasor form, Ẽ [8]: Ẽ = ) ( Ẽ 2 Ẽ (2.8) will be used in the derivation of the PDE (epressed in phasor form) that governs edd current behaviour. 2.3 Phsical Description of the Edd Current Model The phsical description of the edd current model is shown pictoriall in Figure 2.. Figure 2.: Principle Edd Current Formation within a Conductor Primar magnetic field induces edd currents, J E within a conductor. Edd Currents, J E [ A m 2 ], are induced electrical currents [, 9, 6, 7]. A time-varing harmonic magnetic field (primar magnetic field, H S [ Am ]) develops when a coil is eited b a low frequenc sinusoidal current (source current, J S [ A m 2 ]) [, 9, 6, 7] as illustrated in Figure 2.. The source current J S is modelled b the following equation [, 9, 6, 7]: J S = σ E S (2.9) where σ, represents the conductivit of the coil and E S [ vm ] represents the electrical field present within the coil. Another electrical conductor ma come into contact with this time-varing magnetic field, H S. The magnetic field, H S, then induces edd currents J E, within this conductor [, 9, 6, 7] as illustrated in Figure 2.. The edd currents are modelled b the following equation [, 9, 6, 7]: J E = σ E E (2.2) where σ, represents the conductivit of the electrical conductor that is in contact with the primar magnetic field H S, and E E [ vm ] represents the electrical field induced b the 6

24 primar magnetic field H S, within the second conductor [, 9, 6, 7]. Equation (2.2) can also be written alternativel as [5, 2]: J E = ρ v v dri ft (2.2) where v dri ft = µ m E E and ρ v, represents the electron charge densit. The term v dri ft represents the electron drift velocit [5, 2] within the (secondar) conductor and µ m, represents the mobilit constant of a particular material [5]. Thus: J E = ρ v ( µ m E E ) The conductivit term in Equation (2.2) is given b: = (ρ v µ m ) E E (2.22) σ = ρ v µ m (2.23) In conductors, the mobilit constant is relativel high due to the chemical atomic bonding structure of a conducting material, known as metallic bonds [2]. The metallic bonds model stipulates the following atomic bonding scheme:. conductors (metallic material) have at most one, two, or three valence electrons [2]. 2. these valence electrons are not bound to an particular atom within the solid and are more or less free to drift throughout the entire metal [2]. 3. these electrons ma be thought of belonging to the metal as a whole, or forming a sea of electrons or an electron cloud [2]. 4. the remaining non valence electrons and atomic nuclei form what are called ion cores, which posses a net positive charge equal in magnitude to the total valence electron charge per atom [2]. Figure 2.2 is a schematic illustration of metallic bonding. Figure 2.2: Schematic illustration of metallic bonding [2]. The presence of the induced electrical field E E, causes a force to be applied to the unbound electrons within the conductor [9]. The movement of these electrons (mobilit of the electrons) in response to this applied force, E E gives rise to electrical currents (edd currents, J E ) within the conducting material [5, 9] as illustrated in Figure 2.. These edd currents create a secondar magnetic field, H E [ Am ] that opposes the effect of the applied magnetic field, H S [9, 3]. The induced edd currents and the creation of secondar magnetic field is known as Lenz s Law [9]. 7

25 2.3. Model Restrictions The following phsical restrictions will be placed on the model for this dissertation:. the entire partial differential equation (PDE) that describes edd current behaviour is formulated in terms of the electric vector field component, E E [6, 7]. 2. the model will be assumed to be entirel a linear sstem. Due to the assumption of a linear sstem, the vector field E E can thus be represented in phasor form Ẽ [4, 3]. 3. a time-varing harmonic magnetic field, which is the eternal forced ecitation that induces edd currents within a conductor/material [9]. 4. no net free charge densit, ρ v will eists so ρ v = because the total number of carriers (electrons) in a given volume V (of the conductor) equals the number of positivel-charged nucleons [8,, 5], thus giving the divergence constraint of.ε E = for Equation (2.3) [6, 7, 3]. Therefore, term ) (. E Equation (2.) is neglected, and Equation (2.) becomes [8,, 5]: = ( ρ v ε ) in E 2 E (2.24) Similarl, the term ) (.Ẽ = ( ρ v ) ε in Equation (2.8) is neglected, therefore Equation (2.8) becomes [8]: Ẽ 2 Ẽ (2.25) 5. the materials will have a linear behaviour in response to a forced eternal stimulus [5]. The material constants are written mathematicall as in Equations (2.8), (2.9) [5]. The constitutive relations equations are written mathematicall as in Equations (2.5), (2.6) and (2.7) in vector form and the Equations (2.5), (2.6) and (2.7) in phasor form respectivel [5]. 6. rates of time variation is sufficientl slow, the displacement current term, ε E t in Equation (2.2) is neglected [8, 3]. Thus Equation (2.2) can be approimated as: H = σ E (2.26) 7. for edd currents modelled within a conductor the relation σ ωε will alwas hold [8,, 5], since conductors posses ver high conductivit σ, values and ver low dielectric ε, values [5]. This relation causes the term jωεẽ in Equation (2.2) to be neglected [6, 7, 5]. Thus the Equation (2.2) can be approimated as [8, 6, 7, 3]: H = σẽ (2.27) 2.4 Derivation of the Partial Differential Equation that Governs Edd Current Behaviour - Differential Vector Form Derivation of the PDE begins with Equation (2.) which is stated here again for the readers convenience [5, 6, 7, 3] 8

26 E E = µ H t Performing a curl operation on the RHS and LHS of Equation (2.), gives [5, 6, 7, 3]: E E = ( ) µ H t (2.28) According to [6, 7, 3] the edd current model mathematicall possesses two magnetic intensit field variables: H = H E + H S (2.29) as was also discussed in Section 2.3. Substituting Equation (2.29) into Equation (2.28) and using Equation (2.26) as well, produces: E E = (( ) ( )) µ H E + µ H S t = ( ) µ H E ( ) µ H S t t = µ ( ) H E µ ( ) H S t t = µ ) (σ E E µ ) (σ E S t t = µσ t E E µσ t E S (2.3) Substituting Equation (2.24) into RHS of Equation (2.3) ields: 2 E E = µσ t E E µσ t E S 2 E E µσ t E E = µσ t E S (2.3) Equation (2.3) is the general non-homogeneous vector wave equation that represents edd current behaviour mathematicall [6]. It is epected that the solution of Equation (2.3) will ield a vector function of the following general algebraic structure in threedimensions [6, 5, 4]: E E (,,z,t) = U (,,z,t)î+v (,,z,t) ĵ+w (,,z,t) ˆk (2.32) and in two-dimensions [6, 5, 4]: E E (,,t) = U (,,t)î+v (,,t) ĵ (2.33) where î, ĵ and ˆk are orthogonal unit vectors [6, 5, 4]. The forcing function which is known, is also a vector function and has the following general algebraic structure in threedimensions: and in two-dimensions: E S (,,z,t) = U S (,,z,t)î+v S (,,z,t) ĵ+w S (,,z,t) ˆk (2.34) E S (,,t) = U S (,,t)î+v S (,,t) ĵ (2.35) Equations (2.32), (2.34) are three-dimensional vector functions and Equations (2.33), (2.35) are two-dimensional vector functions [4, 6]. According to Equation (2.3), the term E S acts as the forcing function that induces the electric field, E E in a domain [5, 6, ]. The presence of the electric field, E E in a conductor causes edd currents J E to form (refer to section 2.3 for the eplanation of how edd currents are induced in a conductor) [6, 7]. 9

27 2.5 Derivation of the Partial Differential Equation that Governs Edd Current Behaviour - Phasor form Phasor form signifies that the time variable within an equation will be absent, therefore onl spatial variables will appear in the equation. Derivation of the PDE in phasor form begins with Equation (2.). Equation (2.) is stated here for the readers convenience [8]. E E = jωµ H Similar to Equation (2.29), the magnetic intensit field phasor variable is written as [6, 7, 4]: H = H E + H S (2.36) Performing a curl operation the RHS and LHS of Equation (2.), and at the same time substituting Equation (2.36) into Equation (2.) produces [6, 7]: ( ) ( ) ẼE = jωµ H E jωµ H S (2.37) Substituting Equation (2.2) into Equation (2.37) produces [6, 7]: ẼE = jωµσe E jωµσe S (2.38) Substituting Equation (2.25) into RHS of Equation (2.38) ields [6, 7]: 2 Ẽ E 2 Ẽ E jωµσ E E = jωµσe E jωµσ E S = jωµσ E S (2.39) Equation (2.39) is the general non-homogeneous vector wave equation in phasor form that represents edd current behaviour mathematicall [6, 7]. It is epected that the solution of Equation (2.39) will ield a vector function of the following general algebraic structure in three-dimensions [6, 5, 4, 5]: and in two-dimensions [6, 5, 4, 5]: Ẽ E (,,z) = U (,,z)î+v (,,z) ĵ+w (,,z) ˆk (2.4) Ẽ E (,) = U (,)î+v (,) ĵ (2.4) The forcing function which is known, is a vector function and has the following general algebraic structure in three-dimensions: and in two-dimensions: Ẽ S (,,z) = U S (,,z)î+v S (,,z) ĵ+w S (,,z) ˆk (2.42) Ẽ S (,) = U S (,)î+v S (,) ĵ (2.43) [6, 5, 4]. Similar to section 2.4, Equations (2.4) and (2.42) are known as vector functions in phasor form [6, 5, 4, 5]. According to Equation (2.39) (the electric field formulation of the edd current problem in phasor form) the term Ẽ S acts as the forcing function that induces the electric field, Ẽ E in a domain [5, 6, ]. The presence of the electric field, Ẽ E in a conductor causes edd currents J E to form (refer to section 2.3 for the eplanation of how edd currents are induced in a conductor) [6, 7].

28 2.6 Interpretation of the Mathematical Equations Governing Edd Current Behaviour The derivation of the edd current PDE thus begins with Equations (2.) and (2.) as seen in Sections 2.4 and 2.5 respectivel. The phsical behaviour of edd currents induced in a conductor in response to an eternal forced ecitation (a time-varing harmonic magnetic field ) is described mathematicall b the PDE given b Equation (2.3) in the time domain and Equation (2.39) in the frequenc domain respectivel. Both Equations (2.3) and (2.39) are able to support problems: with source-free region, in this case the PDE is homogeneous and boundar-driven [5]. that contain the source region, in this case the PDE is inhomogeneous and forcedriven [5]. The edd current PDE is homogeneous and boundar-driven when the geometr of the problem involves the penetration of a time-varing field directl onto the boundar of the domain of a specific problem. When considering boundar-driven problems, the BC can be imposed on the different regions of the domain of the problem [5]. The boundar of the domain can be: the material/conductor itself [5]. small air space which surrounds the material/conductor [5]. The edd current PDE is inhomogeneous and force-driven when the domain of the problem contains the the source region [5, 6]. The source region is the spatial position within the domain that the magnetic field is prescribed on.

29 Chapter 3 Analtical Solution of the PDE that Governs Edd Current Behaviour Analtical solutions to Equation (2.39) of three fundamental problems were done for the following reasons: a FE algorithm was coded to model Equation (2.39) which governs edd current behaviour. The analtical solution of Section 3. will be compared to the FE simulated solution of the same problem, therefore the analtical solutions serves as a reliable means to test whether the algorithm was implemented correctl. the behaviour pattern of the analtical solutions ma aid in the interpretation of simulated solutions of the FE algorithm for arbitrar edd current problems which is simulated b the FE program in Chapter 8. to gain a better understanding of how the phsical behaviour of edd currents relates to the mathematical equations that govern them. Producing an analtical solution is achieved b using Equation (2.39). To produce a boundar value problem the RHS of Equation (2.39) is set to zero, the term jωµσe S =. Equation (2.39) becomes [8]: 2 E E jωµσe E = (3.) Equation (3.) is homogeneous. This homogeneous Equation (3.) will be subject to certain boundar conditions that will model certain edd current situations [8]. Equation (3.) can be written as: 2 E E k 2 E E = (3.2) 2

30 where k 2 = jωµσ. Simplification of the term k [8, ]: k = jωµσ = ( jωµσ) 2 = (ωµσe j π 2 ) 2 = ωµσe j 4 π ( ) ωµσ + j ωµσ = 2 = (+ j) ωµσ (3.3) 2 = ωµσ (+ j) 2 = Let π f µσ = β, therefore [8, ]: The wave number k is comple [8, ]. ωµσ (+ j) 2 = π f µσ(+ j) k = β(+ j) (3.4) 3. PDE Modelled without a Source Region Included in the Domain In this section, the analtical solution of a simple boundar-driven problem is derived and analsed. The the electric field E E, which produces the edd currents as discussed in Section 2.3, is induced in the conducting bod through a uniform time-varing harmonic electric field located at the boundar of the conducting bod. 3.. Problem Statement The aim is to find an analtical solution to Equation (2.39) with a source-free region. The problem is thus boundar-driven as discussed in Section 2.6. The time-varing field: E S (t) = E S cos(ωt)î (3.5) where E S, is the constant amplitude and ω, is the angular frequenc. Equation (3.5) is specified on the boundar of the domain space at position z = on the z ais and this field is polarised in the positive î direction ( ais). The domain of this problem is the z plane and is shown in Figure 3.. 3

31 X Boundar of the Conductor where the BC is Imposed on +z Z Figure 3.: Illustration of the Problem Statement of section 3.. Convert Equation (3.2) to phasor form [5, 8]: E S (t) = Re ( E S e jωt) î (3.6) E S = E S (3.7) where E S = E S e jθ, but θ =. Equation (3.7) will form the boundar condition equation of this problem Analtical Solution of Problem Assume that the, electric vector field component E E is polarised in the direction and travels in the z direction [8, 3]. Equation (3.2) will take on the following form [8, 3]: E 2 z 2 k 2 E = (3.8) Equation (3.8) is subjected to the following boundar conditions [8]: E () = E S (3.9) E ( ) = (3.) The analtical solution to Equation (3.8) subject to the boundar conditions given b Equations (3.9) and (3.) is derived to be: E (z,t) = E S e βz cos(ωt βz)î (3.) where β = π f µσ. Please refer to Appendi A for the full derivation of how the solution given b Equation (3.) was obtained. Equation (3.), gives the solution of a spatial propagating vector wave E, with an eponential deca in the wave amplitude [5]. The constant β, within the eponential decaing term e βz is also known as the attenuation constant [2, 5]. The phase constant has the same value as the attenuation constant [2, 5] Skin Depth and Skin Effect Phenomena The electric field vector wave E, travelling in a conducting media is attenuated b a factor e βz as the wave travels along the direction of propagation (the z ais), of Equation (3.) [2]. The attenuation constant β = π f µσ is be written as: δ = β = π f µσ (3.2) 4

32 where δ is a constant with dimension in meters [m], this length is called the skin depth of the material [5]. The skin depth δ is defined to be the distance after which E, the magnitude of the electric field vector wave E, has decreased to eactl e (approimatel 36.8%) of it s initial value as the wave penetrates into the domain. For a given medium (eg. µ, σ = constant), the skin depth will decrease with increasing frequenc, this is known as the skin effect [5]. In the particular case of a perfect conductor (or superconductor when σ ), the skin depth becomes zero, and it is independent of frequenc [5]. The electric and magnetic fields do not penetrate into the medium at all, this is known as the Meissner s effect [9, 9]. The skin depth of a particular material depends on the frequenc of the electromagnetic wave and on the conductivit of the material itself [5]. Edd currents tend to develop on the outer surfaces of conductors, a phenomenon known as skin effect [7]. The penetration depth is a ver important parameter in edd current phenomenon, and epresses the abilit of penetration of the electromagnetic field in conducting bodies [7]. The skin effect has a direct relationship to frequenc [7]. The higher the frequenc, the smaller the penetration depth [7] Analsis of Problem. Equation (3.) mathematicall describes a spatial propagating sinusoidal wave travelling in the positive z direction [5].The electric field is also perpendicular to the direction of the wave propagation and is thus the wave is referred to as a transverse wave [9]. 2. The vector wave as it travels along the z ais, decas eponentiall (attenuates) in amplitude due to the damping term e βz in the positive z direction [5, 2]. The decaing eponential term e βz, where β = π f µσ, controls the attenuation depth of the wave as it travels along the z ais as described b Equation (3.), thus the attenuation depth of the wave mathematicall depends directl on the variables such as, frequenc ( f), permittivit (µ) and conductivit (σ) [5, 2]. The attenuation term e βz, is directl responsible for the skin effect behaviour described in subsection 3..3 when the frequenc term f is varied [5, 2]. 3. The sinusoidal wave originates from the boundar located at coordinate, z =, due to the ecitation boundar condition given b Equation (3.9) [8, ]. 4. The electric field E, eerts a force on the free electrons within the conducting bod and thus causes the motion of these electrons [9,, 5]. The movement of the electrons produces a current [9,, 5] within the conducting bod and thus edd currents J E are formed [9], refer to section 2.3. The analtical solution, Equation (3.) is illustrated in Figure 3.2a at a frequenc of f = 5Hz, time t = s and Figure 3.2b at a frequenc of f =.5Hz, time t = s. 5

33 z 3 z (a).5.5 Figure 3.2: Graphical Solution of Equation (3.) Sub-figure 3.2a, simulated at a frequenc of f = 5Hz. At this frequenc, the electric vector wave E does not propagate far along the z ais. This is an eample of skin depth phenomena (refer to subsection 3..3). Sub-figure 3.2b, simulated at a frequenc of f =.5Hz. At a lower frequenc, the electric vector wave E propagates further along the z ais in comparison to the wave in sub-figure 3.2a. 3.2 PDE Modelled with a Source Region Included in the Domain In this section, the analtical solution of a simple force-driven problem is derived and analsed. The electric field E E, which produces the edd currents, as discussed in Section 2.3 is induced in the conducting bod through a uniform time-varing harmonic electric field. The electric field is located at a source/ecitation region within the conducting bod Problem Statement The aim is to find an analtical solution to Equation (2.39) that contains a source region. An ecitation time-varing field: (b) E S (t) = E s cos(ωt)î (3.3) is specified on the source region, this field is polarised in the positive î direction ( ais). The domain of this problem is the z plane and the location of the source region is at the centre of the domain, at position z = as shown in Figure

34 -z Source/Ecitation Region X +z Z Figure 3.3: Illustration of the Problem Statement of section 3.2. Convert Equation (3.2) to phasor form: E S (t) = Re ( E s e jωt) î (3.4) Ẽ = E s (3.5) where Ẽ = E s e jθ, but θ =. Equation (3.5) will form the boundar condition equation of this problem Analtical Solution of Problem Assume that the, E E -field is polarised in the î direction and travels in the ˆk direction (z ais). Equation (3.2) will take on the following form [8, 3]: E 2 z 2 Equation (3.6) is subjected to the following forced condition: k 2 E = (3.6) E () = E s (3.7) The analtical solution to Equation (3.6) subject to the boundar condition given b Equation (3.7) is derived to be: E (z,t) = { Es 2 eβz cos(ωt + βz) f or z < 2 e βz cos(ωt βz) f or z E s = E s 2 e β z cos(ωt β z ) (3.8) where β = π f µσ. Please refer to Appendi A for the full derivation of how the solution given b Equation (3.8) was obtained. Equation (3.8), gives the solution of two propagating waves travelling in opposite directions along the z ais. The waves emerge from a source point and decas in amplitude in the directions of wave propagation [5]. The constant β, within the eponential decaing term e βz is also known as the attenuation constant [2, 5]. The phase constant has the same value as the attenuation constant [2, 5] Analsis of Problem Equation (3.8) describes a pair of sinusoidal waves travelling in the positive z direction and negative z direction respectivel. 7

35 The sinusoidal waves, as the travel are decaing eponentiall due to the damping terms e βz in the positive z direction and e βz in the negative z direction respectivel. At the boundaries of the domain the following boundar conditions are implicitl implied: E ( ) = (3.9) E ( ) = (3.2) Both sinusoidal waves originate from a spatial source region located at co-ordinate, z =. The attenuating wave behaviour throughout the domain described b Equation (3.8) was caused b a time-varing electric field E S (t) given b Equation (3.3), located at the source region. The analtical solution, Equation (3.8) is illustrated in Figure 3.4a at a frequenc of f = 5Hz, time t = s and Figure 3.4b at a frequenc of f =.5Hz, time t = s. z (a).5.5 Figure 3.4: Graphical Solution of Equation (3.) Sub-figure 3.4a, simulated at a frequenc of f = 5Hz. At this frequenc, the electric vector wave E does not propagate far along the positive z ais and the negative z ais. Sub-figure 3.4b, simulated at a frequenc of f =.5Hz. At a lower frequenc, the electric vector wave E propagates further along the positive z ais and the negative z ais in comparison to the wave in sub-figure 3.4a. z (b) 3.3 Electric Field Induced within a Conducting Bod through a Uniform Time-Varing Harmonic Magnetic Field In this section, the analtical solution of a boundar-driven problem is derived and analsed. The the electric field E E, which produces the edd currents, as discussed in Section 2.3 is induced in the conducting bod through a uniform time-varing harmonic magnetic field located at the boundar of the conducting bod. 8

36 3.3. Problem Statement The aim is to find an analtical solution to Equation (2.39) with a source-free region, the problem is thus boundar-driven. The time-varing magnetic field: B S (t) = B cos(ωt) ˆk (3.2) is specified on the boundar plane ( plane) at the position z = on the z ais of the domain space, this field is polarised in the positive ˆk direction (z ais) as shown in Figure 3.5. Figure 3.5: Illustration of the Problem Statement of section 3.3. The magnetic field B(t), is uniform over a circular area centred on the origin (,) = (,), on the surface of the boundar plane ( plane). Convert the time derivative of Equation (3.2) to phasor form: B S (t) = B cos(ωt) ˆk = Re ( B e jωt) (3.22) B S (t) = Re ( B e jωt) (3.23) B S (t) t = Re ( jωb e jωt) B S (t) t = Re ( B S (ω)e jωt) (3.24) B S (ω) = jωb (3.25) where B S (ω) = jωb e jθ, but θ =. Appling Equation (2.), Farada s law to Equation (3.25) ields: ( 2 jωb ) E E E E ( 2 jωb ) 9 = jωb (3.26) = jωb = jωb

37 E = 2 jωb (3.27) E = 2 jωb (3.28) Thus, the magnetic field, Equation (3.2) produces a circulating time-varing electric field, given in phasor form as: Ẽ S = E î+ E ĵ = ( 2 jωbî+ ) 2 jωb ĵ = 2 jωb ( î ĵ ) (3.29) This circular/rotational E S -field given b Equation (3.29), is centred about the origin of the boundar plane ( plane) and will form the boundar condition equation of this problem Analtical Solution of Problem Assume that the, E E -field is polarised in the î direction and ĵ direction and travels in the ˆk direction (z ais). Equation (3.2): 2 E E k 2 E E = will take on the following forms when decomposed into its vector components: in the î direction and in the ĵ direction. E 2 z 2 E 2 z 2 k 2 E = (3.3) k 2 E = (3.3) Vector Component in î direction. Beginning with Equation (3.3): 2 E z 2 k 2 E = in the î direction and. Equation (3.3) is subjected to the following boundar conditions: E () = 2 jωb (3.32) E ( ) = (3.33) The analtical solution to Equation (3.3) subject to the boundar conditions given b Equations (3.32) and (3.33) is derived to be: where β = ωµσ. E (z,t) = 2 jωb e βz [cos(ωt βz) jsin(ωt βz)] (3.34) = 2 ωb e βz sin(ωt βz) 2

38 Vector Component in ĵ direction. Beginning with Equation (3.3): 2 E z 2 k 2 E = in the ĵ direction. Equation (3.3) is subjected to the following boundar conditions: E () = E s (3.35) E ( ) = (3.36) The analtical solution to Equation (3.3) subject to the boundar conditions given b Equations (3.35) and (3.36) is derived to be: E (z,t) = 2 jωb e βz [cos(ωt βz) jsin(ωt βz)] (3.37) = 2 ωb e βz sin(ωt βz) where β = ωµσ. The full solution of the problem is obtained b adding together the solutions of Equations (3.34) and (3.37), this gives: E E (,,z,t) = E (,z,t)î+e (,z,t) ĵ = 2 ωb e βz sin(ωt βz)î+ 2 jωb e βz sin(ωt βz) ĵ ( = 2 ωbî+ ) 2 ωb ĵ e βz sin(ωt βz) = 2 ωb ( î+ĵ ) e βz sin(ωt βz) (3.38) Please refer to Appendi A for the full derivation of how the solution given b Equation (3.38) was obtained Analsis of Problem. Equation (3.38) describes a sinusoidal vector wave (having î directed and ĵ directed vector components) travelling in the positive z direction. The vector wave, as it travels along the z ais, decas eponentiall (attenuates) due to the damping terms e βz in the positive z direction. The wave originates from the boundar located at co-ordinate, z =, due to the ecitation boundar condition (time-varing magnetic field) given b Equation (3.2). The analtical solution, Equation (3.38) is illustrated in Figure 3.6a. 2. Equation (3.38) in phasor form: Ẽ E (,,z) = 2 ωb ( î+ĵ ) e βz e jβz (3.39) The Ẽ E -field wave Equation (3.39), has an anti-clockwise polarised rotational direction ( î+ĵ ), that is opposite to the time-varing Ẽ S -field, given b Equation (3.29), which posses a clockwise polarised rotational direction ( î ĵ ). The clockwise time-varing Ẽ S -field was produced b a time-varing magnetic field that was polarised in the positive ˆk direction given b Equation (3.2) (refer to section 3.3.), here the magnetic field is rewritten for the reader s convenience: B S (t) = B cos(ωt) ˆk 2

39 Equation (3.2) can also be written as: H S (t) = H µcos(ωt) ˆk (3.4) where B = H µ. The Ẽ E -field wave given b Equation (3.39), will produce its own magnetic field H E (refer to section 2.3). The anti-clockwise rotation direction of the Ẽ E -field wave given b Equation (3.39), suggests according to Farada s law (refer Equation (3.26)) that this time-varing H E -field would be polarised in the negative ˆk direction. Equation (3.29) and Equation (3.39), illustrates Lenz s law analticall. Two simulations of the analtical solution given b Equation (3.38) were simulated at frequencies of f = 5Hz and f = Hz shown in Figure 3.6a and 3.6b respectivel. z (a) Figure 3.6: Graphical Solutions of Equation (3.38) Sub-figure 3.6a, simulated at a frequenc of f = 5Hz. Single frame picture taken from a movie simulation that runs for approimatel 43 seconds. The high frequenc value causes the following effects to the vector wave: The wave onl travels a short distance along the ais of wave propagation (the z ais) due to an increase in attenuation strength, refer to sub-section Wave propagation could be seen because the wavelength λ, is small in comparison to the dimension of the domain along the z ais. Sub-figure 3.6b, simulated at a frequenc of f = Hz. Single frame picture taken from a movie simulation that runs for approimatel 43 seconds. The low frequenc value causes the following effects to the vector wave: The wave travels a longer distance along the ais of wave propagation (the z ais) in comparison to the wave in Figure 3.6a due to an decrease in attenuation strength, refer to sub-section z (b) 22

40 Chapter 4 Nodal Finite Elements The earliest finite elements were designed to approimate scalar fields [7] and these finite elements are also referred to in this dissertation as scalar finite elements. This chapter eplains the theor used in the construction of scalar finite elements. 4. Scalar Field A two-dimensional scalar field can be written algebraicall as: F (,) = f + f + f 2 (4.) where Equation (4.) can be classified as a polnomial function of two variables (F (,)) [4, 6] and also as a linear (first-order) polnomial function [4, 6]. The variables f, f and f 2 are the coefficients (which are constants) of the polnomial function [4, 6]. 4.2 Simple Elements Geometries such as lines in one dimension, triangles in two dimension and tetrahedrons in three dimension are often called simple elements because the are the simplest possible shapes, with the minimum number of vertices, in two and three dimensions respectivel, that is found in space [2, 3]. An polgon no matter how irregular, can be represented eactl as a union of triangles, and an polhedron can be represented as a union of tetrahedrons [2]. It is thus reasonable to emplo the triangle as the fundamental element shape when subdividing a domain in two dimensions, and to etend a similar treatment to three dimensional domain problems b using tetrahedrons [2]. 4.3 Two Dimensional Triangular Scalar Finite Element In the scope of this dissertation, the scalar finite element is capable of constructing scalar fields of the algebraic form as in Equation (4.) [7] Triangular Coordinate Transformation The coordinates of an point within a unit right angled triangle can be transformed to an point within an arbitrar triangle within the cartesian coordinate sstem b the use of a 23

41 transformation matri [5]. The transformation matri is derived as follows: P = 2 3 (4.2) 2 3 and U = u u 2 u 3 v v 2 v 3 (4.3) where Equation (4.2) is called the coordinate matri which contains the coordinate points (, ), ( 2, 2 )and ( 3, 3 ) located at the vertices (nodes) of an arbitrar triangular element [5]. Another matri given b Equation (4.3) called the unit coordinate matri contains the coordinate points (u, v ), (u 2, v 2 )and (u 3, v 3 ) located at the vertices (nodes) of a unit right angled triangle [5]. Equation (4.4) computes the transformation matri: T = PU (4.4) = Equation (4.5) illustrates how the transformation matri given b Equation (4.4) is used to transform coordinates located in the unit right angled triangle to coordinates located in an arbitrar triangular element: = u (4.5) v = L u (4.6) v where RHS of Equation (4.5) is a arbitrar set of coordinates located within an arbitrar triangle of the cartesian coordinate sstem. The LHS of Equation (4.5) contains the coordinate transformation matri as well as the known coordinates of the unit right angled triangle. In Equation (4.6) the transformation matri is smbolised b the linear operator L. The transformation matri in Equation (4.5) is calculated b using the known cartesian points (, ) = (3, 4), ( 2, 2 ) = (, 2)and ( 3, 3 ) = (5, ) located at the vertices (nodes) of the arbitrar triangle in sub-figure 4.b. The transformation of the cartesian points from the unit right angle in sub-figure 4.a to the arbitrar triangle in sub-figure 4.b is as follows: L (u, v ) = L (,) = (, ) = (3, 4) (4.7) and L (u 2, v 2 ) = L (,) = ( 2, 2 ) = (, 2) (4.8) 24

42 and L (u 3, v 3 ) = L (,) = ( 3, 3 ) = (5, ) (4.9) The operator L, transforms coordinates from one coordinate sstem, the unit right angled triangle coordinate sstem, to another coordinate sstem, an arbitrar triangle. 5.8 n 4 n v n n (a) u Figure 4.: Coordinate Transformation of a Triangle in the Cartesian Coordinate Sstem Sub-figure 4.a illustrates the cartesian points (u, v ) = (,) is located at node n, (u 2, v 2 ) = (,) is located at node n 2 and (u 3, v 3 ) = (,) is located at node n 3 on the unit right angle triangle. Sub-figure 4.b illustrates that using Equation (4.5) the cartesian points located at the nodes of the unit right angle triangle were transformed to the cartesian points located at the nodes of the arbitrar triangle, where (, ) = (3, 4) is located at node n, ( 2, 2 ) = (, 2) is located at node n 2 and ( 3, 3 ) = (5, ) is located at node n 3 on the arbitrar triangle.. The Figure 4.2 illustrates the transformation of a greater number of points from the unit right angle triangle to the arbitrar triangle b using the transformation matri in Equation (4.5) n 2 (b) 3 n n n 5 v n 2 n n n (a) u (b) Figure 4.2: Coordinate Transformation of Multiple Coordinate Points within a Triangle Sub-figure 4.2a illustrates multiple points that are numbered across the right angle triangle. Sub-figure 4.2b illustrates the transformation of the points in sub-figure 4.2a to the points in the arbitrar triangle. The numbers indicate the transformation position. 25

43 4.3.2 Shape Functions An linear function, f e (,), within a triangle element can be computed b [2, 5]: f e (,) = a+b+c (4.) = [ ] a b (4.) c where coefficients a, b and c have to be known [5]. Equation (4.) can also be written in the following mathematical form [2, 5]: f e (,) = λ f + λ 2 f 2 + λ 3 f 3 (4.2) = [ ] f λ λ 2 λ 3 f 2 (4.3) f 3 where the function values at the nodes f (, ) = f, f 2 ( 2, 2 ) = f 2 and f 3 ( 3, 3 ) = f 3 are known [2, 5] and λ = λ (,), λ 2 = λ 2 (,) and λ 3 = λ 3 (,) are linear interpolation functions that span the entire element []. A linear interpolation function spanning a triangle must be linear in two orthogonal directions []. Since the function values at the nodes are known, Equation (4.) can be written as a sstem of linear equations for the three known function values [2, 5]: f (, ) = a+b + c (4.4) f 2 ( 2, 2 ) = a+b 2 + c 2 (4.5) f 3 ( 3, 3 ) = a+b 3 + c 3 (4.6) 2 n 2 n n Figure 4.3: Triangle Finite Element Nodes, n where coordinate (, ) is located, n 2 where coordinate ( 2, 2 ) is located and n 3 where coordinate ( 3, 3 ) is located. The nodes are labelled at the vertices of the triangle. Equations (4.4), (4.5) and (4.6) are represented in matri notation [2, 5] as: f a f 2 = 2 2 b (4.7) f c 26

44 where the matri , contains the coordinates located at the nodes of the triangle element as seen in Figure 4.3, these coordinates are known. Calculation of the a unknown coefficients b, is achieved b [2, 5]: c a b c = f f 2 f 3 (4.8) but substituting Equation (4.8) into Equation (4.) gives [2, 5, 3]: f e (,) = [ ] f 2 2 f f 3 where (4.9) [ ] λ λ 2 λ 3 = [ ] 2 2 (4.2) 3 3 The interpolation functions are also called shape functions or basis functions []. Using Å ÌÄ ËÝÑ ÓÐ : where the area of the triangle element is: = A e a a 2 a 3 (4.2) = b b 2 b 3 (4.22) c c 2 c 3 A e = 2 ( ) (4.23) Equation (4.22) contains the coefficients of the shape functions [3]. Substituting Equation (4.22) into Equation (4.2) gives: λ (,) = a + b +c (4.24) λ 2 (,) = a 2 + b 2 +c 2 (4.25) λ 3 (,) = a 3 + b 3 +c 3 (4.26) Substituting Equation (4.2) into Equations (4.24), (4.25) and (4.26) gives [5]: λ (,) = ( )+( 2 3 )+( 3 2 ) 2A e (4.27) λ 2 (,) = ( 3 3 )+( 3 )+( 3 ) 2A e (4.28) λ 3 (,) = ( 2 2 )+( 2 )+( 2 ) 2A e (4.29) 27

45 The shape functions have been derived in complete smbolic notation. Equation (4.27), (4.28) and (4.29) shows that the coefficients of the shape functions are constructed from the known coordinates at the nodes of the triangle element in Figure 4.3. Substituting Equation (4.22) into Equation (4.8) produces: a b c = a a 2 a 3 b b 2 b 3 c c 2 c 3 f f 2 f 3 (4.3) Equation (4.3) represents the coefficients of Equations (4.24), (4.25) and (4.26) as a f linear combination of the known function values at the nodes f 2, and the coefficients f 3 a a 2 a 3 of the shape functions b b 2 b 3. c c 2 c 3 Using the above equations a mathematical proof was derived showing that an function value f e (,), within a triangle element can be calculated from a linear combination of the three shape functions given b Equations (4.27), (4.28) and (4.29) and the three known function values f (, ) = f, f 2 ( 2, 2 ) = f 2 and f 3 ( 3, 3 ) = f 3 at the nodes of the element: f e (,) = a+b+c = [ ] a b c = [ ] a a 2 a 3 b b 2 b 3 c c 2 c 3 = [ ] a a 2 a 3 b b 2 b 3 c c 2 c 3 = [ ] f λ λ 2 λ 3 f 2 f 3 f f 2 f 3 (4.3) f f 2 f 3 = λ f + λ 2 f 2 + λ 3 f 3 The end product of Equation (4.3) is written in full notation as: f e (,) = λ (,) f (, )+λ 2 (,) f 2 ( 2, 2 )+λ 3 (,) f 3 ( 3, 3 ) (4.32) Equation (4.2) states that the function f e (,), can be calculated at an arbitrar point (,), within the triangle element, b the direct summing of the shape functions λ (,), λ 2 (,) and λ 3 (,) of the element along with the known function values f (, ), f 2 ( 2, 2 ) and f 3 ( 3, 3 ) at the nodes of the element. Visualisation of the Scalar Finite Element and its Corresponding Shape Functions A small algorithm was coded to visuall displa a single finite element along with its associated shape functions using the information of sub-sections 4.3. and The 28

46 coordinate transformation is calculated according to Equations (4.2), (4.3), (4.4) and (4.5). The shape functions are calculated according to the Equations (4.27), (4.28) and (4.29). (a) Shape Function: λ (,) (b) Shape Function: λ 2 (,) (c) Shape Function: λ 3 (,) (d) Scalar Field: f e (,) Figure 4.4: Two Dimensional Scalar Finite Element Figure 4.4d, is a graphical illustration of Equation (4.32). The transparent triangular plate indicates the computed scalar field across the nodal finite element. The blue points located on this transparent triangular plate are the scalar field values f e (,), that are computed through Equation (4.32), using the cartesian points (, ) within the triangle element of figure

47 Chapter 5 Vector Finite Elements Vector finite elements were specificall designed to approimate electromagnetic vector fields, also known mathematicall as vector functions [2, 3, 7], that obe the Mawell curl equations [2]. The PDE that models edd current behaviour is derived from Mawell s equations in Chapter 2 and the solution to this PDE equation is a vector field (refer to Section 2.5) thus making the vector finite element a highl desirable choice to use within the FE algorithm. 5. Preliminar Theor Some important preliminar theor is provided concerning vectors, vector fields, and EM vector field properties. 5.. Vectors An vector can be decomposed into a normal component and tangential component with reference to a certain interface, for eample : E = E î+e ĵ (5.) where with reference to the -ais, E is the component perpendicular to the -ais called the normal component and E is the component parallel to the -ais called the tangential component as illustrated in Figure 5.. 3

48 Figure 5.: Components of a vector Vector E, is decomposed into its normal component E, and its tangential component E. The components E and E are both constants Vector Fields A two-dimensional vector function can be written in terms of component functions as follows: E (,) = F (,)î+g(,) ĵ (5.2) where F (,) and G(,) are the component functions of the vector function E (,) in the i direction and j direction respectivel. A tpical linear vector field in two dimensions has component functions comprising of linear (first-order) complete polnomial epansion [4, 6, 2]: F (,) = f + f + f 2 (5.3) where in the î direction: G(,) = g + g +g 2 (5.4) and in the ĵ direction. A first-order complete approimation for a two-dimensional function in and has three terms [2]. One term is a constant and the other two terms are linear in and respectivel [2], as seen in Equations (5.3) and (5.4). A complete polnomial epansion can also be referred to as a full-order epansion [3]. These component functions are also known as scalar fields [7] Differential Relations for a Scalar Field A differential operation can be performed on a scalar field (scalar function) [, 9, 6]. This differential operation is: Gradient of a Scalar Field : T = F (5.5) where F, is a scalar field and T is a vector field that is produced when the differential operator, acts upon F []. 3

49 Gradient of a Scalar Field This differential relation phsicall measures the direction of the fastest change of a scalar field in magnitude and direction [, 9, 6] Differential Relations for Vector Fields Two important differential operations that can be performed on vector fields [, 9, 6]. These differential operations are: Divergence of a Vector Field : A = E (5.6) where E, is a vector field and A, is a scalar field that is produced when the differential operator, acts upon E via the dot product []. Curl of a Vector Field : B = E (5.7) where E, is a vector field and B, is a vector field that is produced when the differential operator, acts upon E via the cross product []. Divergence of a Vector Field This differential relation phsicall measures the divergent (spreading out) capabilit of the vector field E (in Equation 5.6), from a point in question or towards a point in question [, 9, 6]. A divergent vector field can be identified as having a source point from where the vector field seems to emerge from or a sink point towards where the vector field seems to be heading [, 9, 6]. A purel divergent vector field has zero curling capabilit [, 9, 6] that is: E = (5.8) Curl of a Vector Field This differential relation phsicall measures the rotational (curling) capabilit of the vector field E (in Equation 5.7) around a point in question [, 9, 6]. A curling vector field can be identified as having no source point or sink point [, 9, 6]. A purel rotational vector field has zero divergence capabilit [, 9, 6] that is: E = (5.9) 5..3 Boundar Conditions for Electromagnetic Fields The boundar conditions are valid for both time-independent and time-dependent electromagnetic fields [5]. In two dimensions and in the cartesian coordinate sstem, electromagnetic fields can be separated into a component that is parallel to an interface (tangential component) and a component that is perpendicular to an interface (normal component) [5]. 32

50 Figure 5.2: Electric Field Deflection Across a Material Interface The diagram illustrates that the electric field will bend (deflect) as it passes from one medium (material) to another medium, because the tangential components of electric field are continuous and the normal components are discontinuous across the material interface [5, ]. This bending of electromagnetic waves is called refraction [5, 8]. The tangential components of an electric vector field across a material interface are continuous, that is [5, 8]: E t = E t2 (5.) There is a discontinuit between the normal components of an electric vector field across a material interface (different mediums), that is the normal components of an electric vector field across a material interface are not continuous [5, 8]: E n E n2 (5.) For eample, the normal components of an electric vector field E, changes abruptl across a dielectric interface because of the dielectric discontinuit [8]: E n = ε 2 ε E n2 (5.2) Similarl, the normal components of an electric vector field E, changes abruptl across a conductor interface because of the conductor discontinuit [8]: E n = σ 2 σ E n2 (5.3) In electromagnetics, there is a lack of normal continuit of the electric field across different material mediums [7] that is, there is normal discontinuit of the electromagnetic field across different material mediums [8]. Due to the discontinuit of the normal component of Electromagnetic vector fields, the field vectors will thus change in magnitude and direction across a material interface [5]. 33

51 5.2 Two Dimensional Triangular Node-Based Vector Element To represent a vector field E (, ), with node-based vector elements, the natural approach was to treat the vector field as two coupled scalar fields in two dimensions, E in the î direction and E in the ĵ direction [7]. The unknown parameters are also called the degrees of freedom and are the two coupled scalar fields located at each node (verte) of the triangular finite element [7] as illustrated in Figure 5.3. Y E ( (,), ) n E E 2 n 2 ( 2, 2) (,) E 2 E 3 n 3 ( 3, 3)E (,) 3 Figure 5.3: Two Dimensional Nodal-Based Vector Element The scalar fields: E and E is located at node n, E 2 and E 2 is located at node n 2, E 3 and E 3 is located at node n Shape Functions The shape (basis) functions given b Equations (4.27), (4.28) and (4.29), which are used for the scalar elements are also used for the node-based vector elements. These shape functions are repeated for the readers convenience: X λ = ( )+( 2 3 )+( 3 2 ) 2A e λ 2 = ( 3 3 )+( 3 )+( 3 ) 2A e λ 3 = ( 2 2 )+( 2 )+( 2 ) 2A e The algebraic epression of these shape functions can be referred to as complete linear (first-order) polnomials according to sub-section 5..2, where this terminolog was discussed. 34

52 5.2.2 Construction of a Vector Field within a Triangular Node-Based Vector Element The vector field E N (,), across the element can be computed b the following equation [7]: E N (,) ( E î+e ĵ ) λ (,)+ ( E 2 î+e 2 ĵ ) λ 2 (,)+ ( E 3 î+e 3 ĵ ) λ 3 (,) (5.4) where λ i (,) for i = : 3 are the nodal shape functions (refer to Equations (4.24), (4.25) and (4.26)) and E i and E i for i = : 3 are the two coupled scalar fields [7] and E N (,) signifies a vector field approimated b a node-based vector element. Equation (4.24), (4.25) and (4.26) is substituted into Equation (5.4). Using Å ÌÄ ËÝÑ ÓÐ, the complete general smbolic notation of the vector field across the element is: E N (,) ( (E ( 3 2 )+E 2 ( 3 )+E 3 ( 2 )) 2A e + (E ( 2 3 )+E 2 ( 3 )+E 3 ( 2 )) 2A e + (E ) ( )+E 2 ( 3 3 )+E 3 ( 2 2 )) î+ 2A e (( E ( 3 2 )+E 2 ( 3 )+E 3 ( 2 ) ) 2A e ( E ( 2 3 )+E 2 ( 3 )+E 3 ( 2 ) ) + 2A e ( E ( )+E 2 ( 3 3 )+E 3 ( 2 2 ) ) ) + 2A e ĵ(5.5) where the variables (, ), ( 2, 2 ), ( 3, 3 ) are the coordinates at the nodes of the triangle element, A e is the area of the triangle element given b Equation (4.23), and E, E 2, E 3, E, E 2 and E 3 are the coupled scalar fields of the triangle element. Using Å ÌÄ ËÝÑ ÓÐ to collect all the terms in, the terms in and the constant terms in Equation (5.46), it was found that: The first component function of the field, E N (,) takes on the following algebraic structure: F (,) = f + f + f 2 (5.6) in the î direction. Comparing Equation (5.3) to Equation (5.5), it is found that: and and f = (E ( )+E 2 ( 3 3 )+E 3 ( 2 2 )) 2A e (5.7) f = (E ( 2 3 )+E 2 ( 3 )+E 3 ( 2 )) 2A e (5.8) f 2 = (E ( 3 2 )+E 2 ( 3 )+E 3 ( 2 )) 2A e (5.9) The second component function of the field, E N (,) takes on the following algebraic structure: G(,) = g + g +g 2 (5.2) 35

53 in the ĵ direction. Comparing Equation (5.4) to Equation (5.5), it is found that: ( E ( )+E 2 ( 3 3 )+E 3 ( 2 2 ) ) g = 2A e (5.2) and ( E ( 2 3 )+E 2 ( 3 )+E 3 ( 2 ) ) g = 2A e (5.22) and ( E ( 3 2 )+E 2 ( 3 )+E 3 ( 2 ) ) g 2 = 2A e (5.23) Thus, the vector field has the following general algebraic structure: E N (,) = ( f + f + f 2 )î+(g + g +g 2 ) ĵ (5.24) where Equation (5.24) is a linear vector field that is, the component functions are complete linear (first-order) polnomials [4, 6, 2] which are similar to Equations (5.3) and (5.4). Thus, the vector field E N (,), takes on the general algebraic structure of Equation (5.2). The element as a whole is considered to be full-order []. The node-based vector element supports vector field epansions of the form of Equation (5.24) across a triangular element as derived through Equation (5.5) Continuit of the Node-based Vector Element The unknown parameters (degrees of freedom) are the two coupled scalar fields located at each node of the element E i, for i = : 4, and E j for j = : 4 [7] as illustrated in Figure 5.4a. E 3 n 3E 3 Et=Et 2 E n F E 2 n 2 Element Interface E n =E n2 E E 2 F 2 E 4 n 4E 4 Element Interface (a) (b) Figure 5.4: Continuit of the node-based Vector Element Across the Element Interface Sub-figure 5.4a illustrates that the components E i, for i = : 4 and E j for j = : 4 can be viewed as normal and tangential vector components respectivel in reference to the element interface. Continuit conditions are placed on these components. Labelling convention: Node numbering n, n 2, n 3 and n 4 - black, element number one, F -red, element number two, F 2 -blue. Sub-figure 5.4b illustrates that when continuit conditions are imposed on each cartesian component (normal and tangential) at nodes n and n 2, the result is a vector field that is continuous [7] across both finite elements, thus there is no change in magnitude and direction of the vectors [5] across the element interface. 36

54 Figure 5.4a illustrates that the two elements are connected b and share nodes n and n 2. At node n, the components E and E of the field E N (,), in element F can be made equal to the components E 2 and E2 of the field E 2 N (,), in element F 2 b imposing the following continuit condition between the elements: for component in the î direction and E = E2 (5.25) E = E2 (5.26) for component in the ĵ direction. The superscript indicates that the components E and E and the field E N (,) all belong to element F and the superscript 2 indicates that the components E 2 and E2 and the field E 2 N (,) all belong to element F 2. Similarl at node n 2, the same continuit conditions is applied, that is: for component in the î direction and E 2 = E2 2 (5.27) E 2 = E2 2 (5.28) for component in the ĵ direction. The continuit conditions of Equation (5.25), (5.26), (5.27) and (5.28) forces each cartesian component, normal and tangential, at nodes n and n 2 to be continuous therefore, the entire vector field is continuous [7] across both finite elements and thus there is no normal discontinuit of this constructed vector field which is a natural propert for electromagnetic vector fields [2]. Node-based vector elements impose full-continuit of the vector field across elements as illustrated in sub-figure 5.4b [7, 3]. Therefore, the use of such elements to solve electromagnetic problems involving changes in material interfaces as discussed in sub-section 5..3, where there is a definite discontinuit of the normal part of the vector field as illustrated in Figure 5.2, is not a straightforward task. Certain techniques, which are not in the scope of this dissertation, have been developed to impose tangential continuit and normal discontinuit for node-based vector elements to be used in EM problems [3], but these techniques can be awkward to implement [7]. Material Material Interface Material 2 E n =E n2 E E n (a) E2 Et E n2 Et2 Et =Et 2 E 3 n 3 E Material 3 Et=Et 2 E n F E 2 n 2 Material Interface E n =E n2 E E 2 (b) F2 Material 2 E 4 n 4 E 4 Material Material 2 Material Interface (c) Figure 5.5: Continuit of the node-based Vector Element Across the Material Interface Sub-figure 5.5a illustrates a vector field will be continuous across an material interface when continuit conditions are placed on both normal and tangential components of this vector field that is, En = E2 n and Et = Et 2. Sub-figures 5.5b and 5.5c illustrates how node-based vector elements impose full-continuit of the vector field across material interfaces, thus there is no change in magnitude and direction of the vectors [5] across the material interface. 37

55 As a summar to sub-sections and 5.2.3, these node-based vector elements are both full-order and full-continuous [] respectivel, where the latter is undesirable for electromagnetic field computations [3]. The full-order refers to the component functions possessing complete polnomials epansions as was shown in sub-section 5..2 [, 2]. 5.3 Two Dimensional Triangular Vector Element A two-dimensional vector field E (, ) can be interpolated across a vector element through the use of three shape functions and the edges of the vector element, where the edges are recognised to be the degrees of freedom of the vector finite element, in comparison to the node-based vector element where the coupled scalar fields located at the nodes were the degrees of freedom (refer to Section 5.2)[7]. The edges can be visualised as the borders of the vector element that connects the vertices of the element together. Figure 5.6 illustrates a vector finite element consisting of three edges labelled E 2, E 3 and E 23 and three nodes labelled n, n 2 and n 3 [2, 2]. B inspection, the indeing of the edges (shown b their subscripts) can follow the connectivit information of the element, which describes how the edges are connected to the nodes of the element [2, 2]. As illustrated in Figure 5.6: Edge E 2, is connected to node n and node n 2 Edge E 3, is connected to node n and node n 3 Edge E 23, is connected to node n 2 and node n 3 In the above, E i j is the edge directed from node n i to node n j [2, 2]. The edges can also be labelled with the following notation of E i for i = : 3 [2, 2]. As illustrated in Figure 5.6: Edge E 2 = E Edge E 3 = E 2 Edge E 23 = E 3 Note that the inde i in the notation E i is different to the inde i in the notation E i j. Y (,) n (, ) E =E 2 E =E2 3 n 2 (,) ( 2, 2 ) E =E3 23 n 3 ( 3, 3) (,) X Figure 5.6: Two Dimensional Vector Element 38

56 5.3. Vector Shape Functions The vector shape functions of a triangular finite element are defined as follows [2, 3, 7]: w i j = λ i λ j λ j λi (5.29) There are three such vector functions per triangle [2], which can also be given the following notation of N i for i = : 3. Each shape function w i j, is associated with an edge E i j directed from n i to node n j as follows [2, 2]: the edge E 2, is associated with the shape function w 2 = N the edge E 3, is associated with the shape function w 3 = N 2 the edge E 23, is associated with the shape function w 23 = N 3 Note that the inde i in the notation N i is different to the inde i in the notation w i j. The shape functions used for the nodal finite elements shown in sub-section are also used to construct the vector shape functions [2, 2]. These shape functions are given b Equations (4.24), (4.25) and (4.26). The gradients of these shape functions are [2, 3]: λ = ( 2 3 ) î+ ( 3 2 ) ĵ (5.3) 2A e 2A e λ2 = ( 3 ) î+ ( 3 ) ĵ (5.3) 2A e 2A e λ3 = ( 2 ) î+ ( 2 ) ĵ (5.32) 2A e 2A e for i, j = : 3. The area of the triangular element A e is given b Equation (4.23) [5]. Using Å ÌÄ ËÝÑ ÓÐ, the complete smbolic notation of the three vector shape functions for an arbitrar triangle within the cartesian plane was found, using the definition of Equation (5.29): N = w 2 = λ λ 2 λ 2 λ ( = ( ) 4 A 2 ( ) 4 A 2 ( ( ) 4 A 2 + ( ) 4 A 2 = 4A 2 ( ( ) ( ) ) î+ 4A 2 ( ( )+( ) ) ĵ (5.33) ) ĵ ) î+ N 2 = w 3 = λ λ 3 λ 3 λ ( = (2 4A )+( )î+ 2 ( (2 4A ) ( ) ) ĵ (5.34) N 3 = w 23 = λ 2 λ 3 λ 3 λ 2 ( = (2 4A ) ( ) ) î+ ( (2 4A )+( ) ĵ (5.35) 39

57 from Equations (5.33), (5.34) and (5.35) it can easil be seen that these shape functions take on the basic algebraic form of: N i (,) = N i î+n i ĵ = (a+b)î+(c+d) ĵ (5.36) for i = : 3. Substituting the coordinates located at the nodes of the unit right angled triangle of Figure 5.6 into Equations (5.33), (5.34) and (5.35), the vector shape functions for this finite element calculated b Å ÌÄ ËÝÑ ÓÐ are [2]: N = w 2 (,) = î+(+) ĵ (5.37) N 2 = w 3 (,) = î ĵ (5.38) N 3 = w 23 (,) = ( )î+ĵ (5.39) Figure 5.7 displas the vector shape functions given b Equations (5.37), (5.38) and (5.39) on a unit right angle triangle element ShapeFunc n E E n 2 E 23 n 3 (a) Vector Shape Function: N = w 2 (,) ShapeFunc 3 n ShapeFunc 2 n E E n 2 E n 23 3 (b) Vector Shape Function: N 2 = w 3 (,) E2 2 5 E n E 2 23 n (c) Vector Shape Function: N 3 = w 23 (,) Figure 5.7: Vector Shape Functions of a Unit Right Angle Triangle 4

58 Properties of the Vector Shape Functions There are some important features that can be noticed concerning the vector shape functions [2]. The shape function, N 3 = w 23 (,), given b Equation (5.39) is chosen to be analsed. Visuall, along edge E 2 as seen in Figure 5.7c, the vector function (vector field) given b Equation (5.39) is purel normal that is, onl normal components eist and the tangential components are zero in reference to edge E 2 [2]. These normal components of this vector function also increases linearl from node n to node n 2 along edge E 2 as illustrated in Figure 5.7c [2]. Similarl along edge E 3, the vector function is also purel normal and increases linearl from node n to node n 3 along this edge as illustrated in Figure 5.7c [2]. On edge E 23 however, the function has both normal and tangential components as illustrated in Figure 5.7c b the vectors located on this edge, therefore these vectors can be separated into both normal components and tangential components, that is: N 3 = (5.4) is the tangential component function of Equation (5.39) in the î direction with reference to edge E 23 and N 3 = (5.4) is the normal component function of Equation (5.39) in the ĵ direction with reference to edge E 23. Looking carefull at Equation (5.4) it can be seen that along edge E 23, all -coordinates are zero. The tangential component function in the î direction given b Equation (5.4) thus becomes: ( ( ) = )î = î (5.42) for = along edge E 23, edge E 23 is located on the = line, see Figure 5.7c. Similarl the normal component function in the ĵ direction given b Equation (5.4) remains: ĵ (5.43) for =, but increases linearl in along edge E 23, see Figure 5.7c. Thus along edge E 23, the tangential components of shape function N 3 = w 23 (,), is constant as given b Equation (5.42) and the normal components are linear as given b Equation (5.43). Vector function N 3 = w 23 (,), can be described as a function possessing a constant tangential component along edge E 23 and linear normal component along all other edges (E 2 and E 3 ) as is clearl seen in Figure 5.7c [2]. The function has a continuous tangential/linear normal behaviour abbreviated as CT/LN [2] and due to these properties the value calculated for edge E 23 is the tangential field along edge E 23 [2]. Analsing the other shape functions N = w 2 (,) and N 2 = w 3 (,), the same behaviour pattern will be found along edges E 2 and E 3 respectivel [2]. Also the values calculated for edges E 2 and E 3 will represent the tangential field along edges E 2 and E 3 respectivel [2]. This behaviour pattern directl coincides with the association of a vector shape function with a particular edge as mentioned in sub-section Due to this mied-order behaviour of the shape functions as given b Equations (5.42) and (5.43) [2], the vector element is also known as a CT/LN element or a mied-order element [2].The curl of the shape functions are calculated as follows: ( Ni N i (,) = N ) i ˆk = (c a) ˆk (5.44) for i = : 3. The curls of the shape functions are nonzero constants and hence the shape functions and their curls are referred to as being complete to zeroth-order [2]. 4

59 5.3.2 Construction of a Vector Field within a Triangular Element The vector field E V (,), can be computed at an point of the triangle finite element b directl summing the shape functions within each element along with the edges [2], that is: E V (,) E 2 w 2 + E 3 w 3 + E 23 w 23 (5.45) where E 2, E 3 and E 23 are the edges of the triangular element, w 2, w 3 and w 23 are the vector shape functions [2] and E V (,) signifies a vector field approimated b the vector element. Equation (5.33), (5.34) and (5.35) is substituted into Equation (5.45). Using Å ÌÄ ËÝÑ ÓÐ, the complete general smbolic notation of the vector field E V (,) is: ( E V (,) = ( E 2 + E 3 E 23 ) 4A e ( )E 2 4A 2 e + ( )E 3 4A 2 e ) 2 2 )E 23 4A 2 î+ e ( (E 2 E 3 + E 23 ).. 4A e + ( )E 2 4A 2 e ( )E 3 4A 2 e + ( )E 23 4A 2 e ) ĵ (5.46) where the variables (, ), ( 2, 2 ), ( 3, 3 ) are the coordinates at the nodes of the triangle element, A e is the area of the triangle element, and E 2, E 3, E 23 are the edges of the triangle element. Using Å ÌÄ ËÝÑ ÓÐ to collect all the terms in, the terms in and the constant terms in Equation (5.46), it is found that: The first component function of the field E V (,) has the algebraic structure: F (,) = f + f 2 (5.47) in the î direction, where f = in relation to Equation (5.3) and comparing Equation (5.3) to Equation (5.46), it is found that: and f 2 = 4A e ( E 2 + E 3 E 23 ) (5.48) f =.. ( )E 2 4A 2.. e..+ ( )E 3 4A 2.. e.. ( )E 23 4A 2 e (5.49) 42

60 The second component function of the field has the algebraic structure: G(,) = g + g (5.5) in the ĵ direction, where g 2 = in relation to Equation (5.4) and comparing Equation (5.4) to Equation (5.46), it is found that: and g = 4A e (E 2 E 3 + E 23 ) (5.5) g =..+ ( )E 2 4A 2.. e.. ( )E 3 4A 2.. (5.52) e..+ ( )E 23 4A 2 e Thus, the vector field that the vector element is capable of constructing has the following general algebraic structure: E V (,) = ( f + f 2 )î+(g + g ) ĵ (5.53) The vector element supports vector field epansions of the form of Equation (5.53) across a triangular element, there are however a few eceptions and further eplanation will be given in sub-section The element can be referred to as being complete to zerothorder [, 2] due to the shape functions themselves being complete to zeroth-order as shown b Equation (5.44) Tangential Continuit of the Vector Element The unknown parameters (degrees of freedom) are the edges E i j, of the triangular vector finite element [7]. Assembling the elements together across a mesh requires the element connectivit information [, 2, 3]. The element connectivit information is a table that shows the information of how each element within the mesh is connected to one another though the edges E i j of the elements, that is, an edge can be shared b at most two elements [, 2, 3]. 43

61 .5 n E n F. E E E F n.5 4 E 5 n (a) No. Faces Local Edge Local Edge 2 Local Edge 3 F E E 2 E 3 2 F 2 E 4 E 3 E 5 (b) Figure 5.8: Element Inter-Connectivit Information Sub-figure 5.8a demonstrates that edge E 3, connects elements F and F 2. Sub-figure 5.8b displas the connectivit data of the vector finite elements generated b Å ÌÄ. This connectivit data is presented in the form of a table called Ø. Sub-figure 5.8a illustrates that the two elements are connected b and share edge number E 3. The table referred to in this dissertation as Ø in sub-figure 5.8b also clearl tabulates this element inter-connectivit information which states the following: the column labelled as Faces, contains the global element numbering [2, 2]. The mesh in sub-figure 5.8a contains two elements, element number one, labelled as F, and element number two, labelled as F 2. the net three columns labelled as Local Edge, Local Edge 2 and Local Edge 3, contain the global edge numbering sstem [2, 2]. Each triangular element is associated with three edges [2, 2]. Element F is associated with edges E, E 2, E3 and element F 2 is associated with edges E4 2, E2 3, E2 5 as illustrated b the table of Ø in sub-figure 5.8b. The superscript of the edge terms above indicates which element a particular edge belongs to. 44

62 E 3 Et =Et 2 n E n E 2 E n 2 E 4 n 3 F E 23 n 2 F2 E 24 n 4 Element Interface Element Interface (a) (b) Figure 5.9: Tangential Continuit and Normal Discontinuit of the Vector Field Across an Element Interface Sub-figure 5.9a illustrates that the tangential components of the vector field on edge E 2 is continuous that is Et = Et 2, and the normal components are discontinuous, that is En En. 2 Sub-figure 5.9b illustrates that there is a change in direction and magnitude of the vector field across the element interface (refer to sub-section 5..3). Now, sub-figure 5.9a illustrates that two elements share and are connected b edge numbered E 2. The component of the field E (,), in element numbered one, tangent to edge numbered E 2 can be made equal to the component of the field E 2 (,), in element numbered two, tangent to the same edge numbered E 2 b imposing the following continuit condition between the elements: E 2 = E2 2 (5.54) where the superscripts indicates the global finite element numbering sstem. Equation (5.54) states that edge E2 belonging to finite element F, is equal to edge E2 2 belonging to finite element F 2. Appling Equation (5.54) to the two element mesh as illustrated in Figure 5.9b, makes the field tangentiall continuous across edge E 2. The field so constructed is not normall continuous that is, the part of the field perpendicular with reference to the element interface (normal component of vector field) is discontinuous across the element interface because no continuit conditions were placed on the normal components. The vector finite element thus has the abilit to impose tangential continuit but not normal continuit on the vector field [7, ]. Depending on the edge values E i j, chosen for each finite element F and F 2 and using Equation (5.45) applied to both elements: E V (,) E2 w 2 + E 3 w 3 + E 23 w 23 (5.55) and E 2 V (,) E2 2 w2 2 + E2 4 w2 4 + E2 24 w2 24 (5.56) The vector field E V (,), approimated across both elements: E V (,) E V (,)+ E 2 V (,) = ( E 2 w 2 + E 3 w 3 + E 23 w 23) + ( E 2 2 w E2 4 w2 4 + E2 24 w2 24) (5.57) but, according to Equation (5.54), E2 = E2 2 therefore Equation (5.57) becomes: E V (,) = ( w 2 + w 2 ) 2 E2 + ( E 3 w 3 + E 23 w ) ( 23 + E4 w E 24 w 2 24) (5.58) 45

63 where both elements F and F 2 share edge E 2, the vector field E V (,) will have the abilit to change direction and magnitude across the element interface as illustrated b sub-figure 5.9b because the vector element has the abilit to interpolate the vector field in such a wa that the tangential continuit between the adjacent elements are enforced as shown b the term ( w 2 + w2 2) E2 in Equation (5.58), while the normal components of the vector field are allowed to be discontinuous []. This propert makes the vector element ver useful for when boundar conditions need to be imposed between different material interfaces [3, 7] as illustrated b Figure 5.b because the vector field E (, ), phsicall alwas changes (direction and intensit/magnitude) across an material interface as shown in Section n 3 Material E 3 Et =Et n 2 E 23 E 2 n 2 E n E n 2 Material Interface (a) Material 2 E E 24 4 n 4 Material Material Interface (b) Material 2 Figure 5.: Tangential Continuit and Normal Discontinuit of the Vector Field Across a Material Interface Located Between Two Elements Sub-figure 5.b illustrates tangential continuit and normal discontinuit for the vector located on edge E 2 (which is the material interface between the two elements). Sub-figure 5.b illustrates how there is a change (in direction and magnitude) of the vector field E V (,), across the material interface (each element represents a different material) due to the normal part of the vector field changing abruptl across this interface (normal discontinuit) [7]. The name CT/LN element used in sub-section 5.3. describes the vector element the best, because essentiall the construction of the vector field E V (,) within the element using the vector shape functions in Equation (5.45) allows the separation of the normal and tangential components along the edges of an element, the element is thus able to construct vector fields that are normall discontinuous and tangentiall continuous across element boundaries [, 7] as shown b Figure 5.2. As a summar to sub-sections and 5.3.3, these vector elements dropped the fullorder propert and the full-continuous propert respectivel [3] where the latter is desirable for electromagnetic field computations [7]. 5.4 Calculation of a Vector Field Across a Triangular Finite Element Vector elements onl provide a phsicall meaningful vector field when summed together [2] as given b Equation (5.45). An algorithm was coded to displa vector fields across a single finite element. The purpose of this eercise was so to gain a better understanding of how the vector element is able to approimate and displa vector fields. 46

64 To displa certain vector fields would depend on the value of the edges (average tangential field along a particular edge [2]) chosen on a triangular element as given b Equation (5.45). The finite element used is a unit right angled triangle, thus the shape functions are given b Equations (5.37), (5.38) and (5.39) Construction of a Rotational Field On a unit right angled triangle the author wished to displa the rotational field: E (,) = î+ĵ (5.59) Figure 5.: Construction of a Rotational Vector Field Across a Element On each edge of the triangle a local vector v i j can be calculated. This vector v i j, is formed b the subtraction of coordinate values as follows []: v i j = ( j, j ) (i, i ) (5.6) these coordinate values are located along an edge E i j on the element (refer to Figure 5.). Calculation of these vectors v i j, on the unit triangle using Equation (5.6) are as follows: v 2 = (,) (,) = (, ) = î ĵ (5.6) v 3 = (,) (,) = (, ) = î ĵ (5.62) v 23 = (,) (,) = (,) = î+ ĵ (5.63) The lengths of each vector vi j, on each edge of the triangle can also be calculated where vi j = Li j [], which is the actual length of the edges L i j, of the triangle. Therefore: v 2 = L 2 = (5.64) v 3 = L 3 = 2 (5.65) v 23 = L 23 = (5.66) 47

65 The unit vectors on each edge of the triangle can be calculated b the well known equation []: ê i j = v i j vi j (5.67) Substituting Equations (5.6), (5.62), (5.63), (5.64), (5.65), and (5.66) into Equation (5.67) gives: ê 2 = v 2 v 2 = î ĵ (5.68) ê 3 = v 3 v 3 = î ĵ 2 (5.69) ê 23 = v 23 v 23 = î+ ĵ (5.7) Equation (5.45) approimates a vector field E (, ), across a triangular element [2], therefore substituting Equations (5.37), (5.38) and (5.39) into Equation (5.45) produces: E (,) E V (,) ( = E ) ( 2 î+(+) ĵ + E ) ( 3 î ĵ + E ) 23 ( )î+ĵ = E 2 î+e 2 (+) ĵ+ E 3 î E 3 ĵ+ E 23 ( )î+e 23 ĵ = ( E 2 +E 3 +E 23 ( ))î+(e 2 (+) E 3 +E 23 ) ĵ (5.7) To find the tangential component of the field E (,), along edge E 3 (refer to Figure 5.) Equation (5.72) is used [, 9, 4, 2] : E tang/e3 = ê 3 E (,) (5.72) Substituting Equations (5.69) and (5.7) into Equation (5.72) gives: ( E tang/e3 = î ) ĵ [( E 2 +E 3 E 23 ( ))î+(e 2 (+) E 3 +E 23 ) ĵ ] 2 2 = ( E 2 +E 3 E 23 ( )) (E 2 (+) E 3 +E 23 ) (5.73) 2 2 Then substituting cartesian points = and = at node n = into Equation (5.73): E tang/e3 = ê 3 E (,) = ) 2 ( E 2 + E 3 + ) ( E = 2 ( E 2 + E 3 + E 2 ) = E 3 2 (5.74) Following the same procedure as the above, the tangential component of the field E (,), on edge E 2 and edge E 23 are : E tang/e2 = E 2 (5.75) and E tang/e23 = E (5.76)

66 respectivel. It can be deduced from Equations (5.74), (5.75) and (5.76) that the edge values E i j, of an arbitrar triangle can be calculated b the following well known equation [2, 7]: E tang/ei j = ê i j E (,) = E i j L i j (5.77) where E i j is the edge value, E tang/ei j is the tangent vector of the vector field E (,), located on edge E i j of the triangle and L i j is the length of a particular edge on the triangle. Equation (5.77) states that E i j, controls the tangential field E tang/ei j, on the local edge of the element numbered b i j [7] and this also makes it simple to constrain the tangential field to a prescribed boundar value [7]. Table 5. tabulates the calculation of the vector field E (,) at each node of the element (refer to Figure 5.). Node Co-ordinate Point E (,) = î+ĵ (, ) = (,) E (,) = î+ ĵ 2 ( 2, 2 ) = (,) E (,) = î+ ĵ 3 (, ) = (,) E (,) = î+ ĵ Table 5.: Calculation of the Vector Field E (,) at each Node of the Element To calculate the tangent vector E tang/e3 on edge E 3, of right-angled triangle Equation (5.77) is used: E tang/e3 = ê 3 E (,) ( ) î ĵ = ( î+ ĵ ) 2 = 2 (5.78) where L 3 = 2 and E 3 =. Using Equation (5.77), it is calculated that: where L 2 = and E 2 = and E tang/e2 = E tang/e23 = (5.79) (5.8) where L 23 = and E 23 =. Substituting Equations (5.78), (5.79) and (5.8) into Equation (5.7), it is verified that: E (,) E V (,) ( = E ) ( 2 î+(+) ĵ + E ) ( 3 î ĵ + E ) 23 ( )î+ĵ = ( î+(+) ĵ ) + ( î ĵ ) + ( ( )î+ĵ ) = î+ĵ Figure 5.2 displas the vector field E (,) = î + ĵ given b Equation (5.59). The shape functions given b Equations (5.37), (5.38) and (5.39) together with the edge value 49

67 calculations given b Equations (5.78), (5.79) and (5.8) created the vector field given b Equation (5.59). n.8.6 E2 2 5 E n 2 E 23 n Figure 5.2: Rotational Vector Field Across Element A rotational vector field E (,) = î+ ĵ, created b the vector element (Note: the ÉÙ Ú Ö function used b Å ÌÄ to displa the vector field, has automaticall scaled the vectors) Construction of a Constant Field On a unit right angle triangle the author wished to displa a constant field: E (,) = î (5.8) Figure 5.3: Construction of a Constant Vector Field Across a Element Alternativel Equation (5.77) can be written in a more convenient wa to calculate the 5

68 edges of a triangular finite element. Rearranging Equation (5.77) it is found that: E i j = L i j ( E tang/ei j ) ) = L i j (ê i j E (,) = ( L i j ê i j ) E (,) = v i j E (,) (5.82) where E tang/ei j = ê i j E (,) from Equation (5.77) and v i j = L i j ê i j from Equation (5.67). The vector v 2, v 3 and v 23 has been calculated b Equations (5.6), (5.62) and (5.63) respectivel, and the vector field E (, ), across the unit triangular element is approimated b Equation (5.7) again. Table 5. tabulates the calculation of the vector field E (,) at each node of the element (refer to Figure 5.3) Node Co-ordinate Point E (, ) = î (, ) = (,) E (,) = î+ ĵ 2 ( 2, 2 ) = (,) E (,) = î+ ĵ 3 (, ) = (,) E (,) = î+ ĵ Table 5.2: Calculation of the Vector Field E (,) at each Node of the Element To calculate the tangent vector E tang/e3 on edge E 3, of right-angled triangle Equation (5.82) is now used: Using Equation (5.82), it is calculated that: and E 3 = v 3 E (,) ( = ) î ĵ (î+ ĵ ) = (5.83) E 2 = (5.84) E 23 = (5.85) Substituting Equations (5.83), (5.84) and (5.85) into Equation (5.7), it is verified that: E (,) E V (,) ( = E ) ( 2 î+(+) ĵ + E ) ( 3 î ĵ + E ) 23 ( )î+ĵ = ( î+(+) ĵ ) + ( î ĵ ) + ( ( )î+ĵ ) = î ĵ + î î+ĵ = î Figure 5.4 displas the vector field E (,) = î given b Equation (5.8). The the shape functions given b Equations (5.37), (5.38) and (5.39) together with the edge value calculations given b Equations (5.83), (5.84) and (5.85) created the vector field given b Equation (5.8). 5

69 n E2 E n E n Figure 5.4: Constant Vector Field Across Element A constant vector field E (,) = î, created b the vector element (Note: the ÉÙ Ú Ö function used b Å ÌÄ to displa the vector field, has automaticall scaled the vectors). It is important to notice that the edges (degrees of freedom) calculated b Equations (5.79), (5.78), (5.8) in Section 5.4. and Equations (5.83), (5.84), (5.85) in sub-section respectivel are scalar, but the edges can also be signed [2] as demonstrated b Equation (5.78). 5.5 Using Triangular Vector Elements to Create a Two- Dimensional Vector Field across a FE Mesh Similar to section 5.4, an algorithm was coded to displa vector fields across a finite element mesh. The purpose of this eercise was so to use this algorithm as a tool to investigate and at the same time gain a better understanding of the tpes and properties of vector fields that these vector elements are capable of constructing or building Creating a Two-Dimensional Vector Field The author attempted to simulate a rotational vector field of the form as given b Equation (5.59) (repeated here again for the readers convenience): E (,) = î+ĵ across a four element mesh shown b sub-figure 5.6. The algorithm can displa an two-dimensional vector field E (, ) of the form given b Equation (5.53) across a finite element mesh. The algorithm consists of several data processing steps and the knowledge of section 5.4 was incorporated into these steps in order to construct a vector field. 52

70 Node Points & Face Nodes () Gen Triangle Database (2) Triangle Data Base (3) Domain Edge Value Calc (4) Global Edge Value E_,f_,f _,f_,g _,g_,g _ (5.2) Vec Field Component Calc (7) (8.) Disp Vec Plot - Gen Vec Field Over Entire Mesh (9) (5.) Global Tri- Coordina tes Global Vec Compon ent Data (6) (8.2) Figure 5.5: Flow Chart Diagram of a Vector Field Simulation A description with a brief eplanation of the process of information flow illustrated in Figure 5.5 is given below: Step : Data is processed and sorted into useful data structures. Step 2: The generated data is collected and stored. The stored data information is used in step 3. and step 5.2. The constants f, g, f, f 2, g and g 2 are used to set the tpe of vector field that one wishes the vector elements to displa according to the Equation (5.2). The 53

71 constants are set as f =, g =, f =, f 2 =, g = and g 2 = to give Equation (5.59). Step 3. takes the stored data information together with step 3.2, which takes the constants information to be processed. The edge values of each element within the entire mesh is computed and these edge values are calculated in a similar wa to the process shown in sub-sections and Step 4: The computed edge value information is then stored in a data structure. Step 5.2 takes the stored data information together with step 5., which takes the edges value information to be processed. The components E in the î direction and E in the ĵ direction of the vector field E (,) located at each coordinate (,) with in each element of the entire mesh are computed. Step 6 and step 7: The vector field component information and associated coordinate information are then stored in data structures. Step 8. takes the component data information together with step 8.2, which takes the coordinate information to be processed. The component information along with the coordinate information will be used to displa the vector field over the entire mesh. Step 9: Once the intended vector field is simulated, all computation stops..5 n E E 2 F n 5 F 2 E 3 E 7 F 3 F 4 E 5 E 6 n 2 E 4 n.5 4 E 8 n Figure 5.6: Finite Element Mesh A four element mesh is illustrated. The data is labelled as BLACK-Nodes, RED-Edges and GREEN- Elements. 54

72 (a) (b) Figure 5.7: A Rotational Vector Field Built using Four Vector Elements Sub-figure 5.7a illustrates that Equation (5.45) is applied to each element within the mesh to build a vector field, E (,) = î+ ĵ across the four element mesh. The finite elements are clearl capable of constructing this vector field. In sub-figure 5.7b, the same vector field, E (,) = î + ĵ is produced manuall b using the Å ÌÄ function Å Ö and ÉÍÁÎ Ê. In sub-figure 5.7a, the vector elements are clearl capable of constructing a vector field of the form given b Equation (5.59), across the FE mesh. Edge values are computed to construct the vector field E (,) = î+ĵ across each element within the FE mesh b using Equation (5.45) (refer to Section 5.4). Recall from sub-section 5.3.3, that the vector elements are structurall designed to be capable of producing vectors that have tangential component continuit but normal component discontinuit across an element interface and tangential component continuit between vectors of different elements (located on an element interface) are enforced through the edge values. However, the edge values computed (to construct the vector field E (,) = î+ĵ) allowed for a continuous vector field to be produced across the FE mesh that is, located on all the element interfaces throughout the FE mesh, there eists tangential component continuit as well as normal component continuit between vectors belonging to different finite elements. Therefore, the vectors situated on the element interfaces (which belong to different finite elements) possess the same magnitudes and directions. Thus, all the vectors situated on the element interface (gre lines) of each element (within the mesh) coincide with each other, therefore forming a continuous vector field simulation on all element interfaces across the FE mesh. In sub-figure 5.7b, the same vector field is plotted manuall using the Å ÌÄ function Å Ö and ÉÍÁÎ Ê. The vector field simulations of sub-figure 5.7a and sub-figure 5.7b are identical. The author then attempted to simulate the following vector field: E (,) = î+ĵ (5.86) across a four element mesh shown b Figure 5.6. The constants are set as f =, g =, f =, f 2 =, g = and g 2 = to give Equation (5.86). 55

73 (a) (b) Figure 5.8: An Arbitrar Vector Field Built using Four Vector Elements In sub-figure 5.8a, equation (5.45) is applied to each element within the mesh to build the vector field, E (,) = î+ĵ across the four element mesh. The finite elements are clearl incapable of constructing this vector field. In sub-figure 5.8b, the same vector field, E (,) = î+ ĵ is produced manuall b using the Å ÌÄ function Å Ö and ÉÍÁÎ Ê placed over the same four element mesh of Figure 5.6. In sub-figure 5.8a, the vector elements fail to construct the vector field given b Equation (5.86), across the FE mesh. The edge values computed to produce the vector field E (,) = î+ ĵ (refer to Section 5.4) across the FE mesh, created normal component discontinuit between vectors located on all element interfaces belonging to different finite elements of the entire FE mesh and therefore, some of the vectors situated on the element interfaces possess different magnitudes and directions. Thus, a discontinuous vector field is produced across the mesh and sub-figure 5.8a clearl shows how the vectors situated on the element interface (gre lines) of each element (within the mesh) diverge from each other instead of coinciding with each other as shown in sub-figure 5.7a and so, a continuous vector field simulation across the FE mesh could not be constructed. In subfigure 5.8b, the same vector field (given b Equation (5.86)) is plotted manuall using the Å ÌÄ function Å Ö and ÉÍÁÎ Ê. In summar, there are certain vector fields that the vector elements are capable of approimating (as shown b sub-figure 5.7a) and certain vector fields that the vector elements are incapable of approimating (as shown b sub-figure 5.8a). This subject will be further investigated in Section 5.6 and sub-section

74 Figure 5.9 shows the FE mesh that will be used for the simulated vector fields of Section 5.6 and sub-section Figure 5.9: Finite Element Mesh The FE mesh consists of 52 finite elements. 5.6 Properties of the Vector Field Produced b the Vector Element and the Node-based Vector Element The following properties of the vector field approimated b the vector element and the node-based vector element can be drawn from Sections 5.2 and 5.3:. An approimation of a linear vector field E (, ) as given b Equation (5.2), across the node-based vector element is given b Equation (5.4) (repeated here again for the readers convenience): E N (,) ( E î+e ĵ ) λ (,) + ( E 2 î+e 2 ĵ ) λ 2 (,) + ( E 3 î+e 3 ĵ ) λ 3 (,) = (E λ (,) + E 2 λ 2 (,)+E 3 λ 3 (,))î+ ( E λ (,) + E 2 λ 2 (,) + E 3 λ 3 (,) ) ĵ Here the unknown parameters (degrees of freedom) are associated with the two coupled scalar fields located at each of the three nodes of the element [7], thus giving a total of si unknown parameters for an element E, E 2, E 3, E, E 2 and E 3. An approimation of a linear vector field E (,) as given b Equation (5.2), across the vector element is given b Equation (5.45) (repeated here again for the readers convenience): E V (,) E 2 w 2 + E 3 w 3 + E 23 w 23 Here there are three unknown parameters (degrees of freedom) associated with the edges of the element E 2, E 3 and E 23 [7, 2]. The vector element has less degrees of freedom in comparison to the node-based vector element [7], therefore alread a huge computational advantage can be concluded when using vector elements because, there will be less degrees of freedom for the FE algorithm to compute for a particular problem [7]. 57

75 2. J.P.Webb in his paper [] and J.Jin in his book [2] directl stated that, in vector electromagnetics, the curl of the field E (, ), is as important as the field itself E (,) [, 2]. Therefore, if the vector field is represented b component functions of polnomial of order p, the curl of this vector field will then be represented b a polnomial of degree p [, 2]. The overall accurac of the solution will then be dominated b this lower degree of p [, 2], therefore the accurac of the solution will not be affected if the terms (called the gradient terms of order p) that do not contribute to the curl representation are removed while keeping the vector field representation complete to order of p [, 2]. This results in an element with fewer degrees of freedom, but with a better balance in accurac of representation of the field and its curl [, 2]. The vector element is the result of appling this idea to an element complete to first-order []. Appling the knowledge of the above paragraph to Equation (5.2) which is a polnomial order (degree) p =, the curl of the vector field E (,) is: ( G E (,) = F ) ˆk ( = (g + g +g 2 ) ) ( f + f + f 2 ) ˆk (5.87) = (g f 2 ) ˆk with polnomial order of p =, the curl is simpl a constant with direction. Clearl, the terms f and g 2 present in the polnomial epansion (of a linear vector field in two dimensions given b Equation (5.2)) do not contribute to Equation (5.87) and therefore, the terms f and g 2 within the polnomial epansion (of Equation (5.2)) does not affect the curl of the vector field E (,) as given b Equation (5.87) [, 2]. The terms f and g 2, are called the gradient terms and are of order p = [, 2]. The vector element approimates a vector field of the form as given b Equation (5.53) (repeated here again for the readers convenience): E V (,) = ( f + f 2 )î+(g + g ) ĵ The vector element thus preserves the terms f 2 and g that contribute directl to the curl of the vector field E (,), given b Equation (5.87) while at the same time removes the terms f and g 2 that do not have an effect on the curl of the vector field E (, ) given b Equation (5.87) [, 2]. The divergence of the vector field E (,) is: E (,) = F + G ( = ( f + f + f 2 ) ) (g + g +g 2 ) = f + g 2 (5.88) The terms f and g 2 contributes directl to the divergence of the vector field E (,), given b Equation (5.88), so the vector element has no divergence since these terms are removed [2]. Thus, it is eas to see b mathematical deduction that the vector field represented b Equation (5.53) has no divergence, that is: E V (,) = ( f + f 2 )+ (g + g ) = (5.89) 58

76 and the vector elements are therefore incapable of constructing such a field. The removal of the divergence terms f and g 2, makes the vector element incapable of constructing an vector field that possesses a divergence propert. This fact is verified graphicall in sub-sections 2. and 2.2 (refer to sub-section 5.5.). 2. Divergent Vector Fields When the constants are set as f =, g =, f =, f 2 = g = and g 2 =, the vector elements failed to represent a purel divergent vector field, of the algebraic form: E (,) = î+ĵ (5.9) The vector elements are not capable of supporting divergent fields as illustrated in Figure 5.2, where the vectors (belonging to different finite elements) situated on the element interfaces (of the FE mesh) diverge from each other as eplained in sub-section (a) Figure 5.2: Divergent Vector Field Sub-figure 5.2a illustrates a purel divergent vector field produced b the Å ÌÄ function Å Ö and using the equation E (,) = î + ĵ. This vector field is then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.2b illustrates the same vector field ( E (,) = î + ĵ) which was approimated b using the vector elements. Clearl the vector elements are incapable of constructing the vector field ( E (,) = î+ĵ) due to there being onl divergent terms f = and g 2 = present within the vector field equation (refer to point number (2) of Section 5.6 and sub-section 5.5.) (b) 2.2 Curling and Divergent Vector Fields When the constants are set as f =, g =, f = 2, f 2 = 3 g = 2 and g 2 = 3, the vector elements failed to represent a vector field, of the algebraic form: E (,) = (2 3)î+(2+3) ĵ (5.9) The vector elements are not capable of supporting vector fields possessing curling and divergent differential properties as illustrated in Figure 5.2, where the vectors (belonging to different finite elements) situated on the element interfaces (of the FE mesh) diverge from each other as eplained in sub-section

77 (a) Figure 5.2: Vector Field having both Curling and Divergent Properties Sub-figure 5.2a illustrates a vector field possessing curling and divergent properties and is produced b the Å ÌÄ function Å Ö and using the equation E (,) = (2 3)î+(2+3) ĵ. This vector field is then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.2b illustrates the same vector field ( E (,) = (2 3)î + (2+3) ĵ) which was approimated b using the vector elements. Clearl the vector elements are incapable of constructing the vector field ( E (,) = (2 3)î+(2+3) ĵ) due to the divergent terms f = 2 and g 2 = 3 present within the vector field equation (refer to point number (2) of Section 5.6 sub-section 5.5.). 3. The node-based vector element approimates a vector field of the form given b Equation (5.24) (repeated here again for the readers convenience):.5 (b) E N (,) = ( f + f + f 2 )î+(g + g +g 2 ) ĵ through its three shape functions of complete polnomial epansion and its si associated degrees of freedom given b Equation (5.4). The shape functions and the associated degrees of freedom preserves the gradient terms f and g 2 within the approimation of the vector field E N (,) within the element (as seen b Equation (5.24). Refer to Section 5.2 for the derivation of this equation) and therefore b design, this element contains wasted degrees of freedom because a degree of freedom tpicall represents a field component at a particular node [2] and not all field components for electromagnetic purposes need to be continuous across an inter-element or inter-material boundaries as shown in sub-sections 5..3 and [2,, 7]. Therefore, the normal components are the wasted degrees of freedom because the together with the nodal shape functions help to preserve the terms f and g 2 as shown in sub-section which are not required since the do not contribute to the curl of the vector field E (,) [, 2] as given b Equation (5.87). In comparison, the construction of the vector element is onl capable of approimating curling vector fields (as stated above through Equation (5.53)) through its three vector shape functions of mied-order [2] and the three associated degrees of freedom called edges given b Equation (5.45). The design of the vector element has resulted in the use of fewer degrees of freedom (edges) [2] that onl preserves the terms f 2 and g which contribute towards the curl of the vector 6

78 field (as stated above through Equation (5.87)). Thus, it is eas to see b mathematical deduction that the vector field represented b Equation (5.53) is capable of rotation: ( E V (,) = (g + g ) ) ( f + f 2 ) ˆk = (g f 2 ) ˆk (5.92) The vector fields approimated b the vector elements given b Equation (5.53) is subject to a few eceptions as alread mentioned in sub-section Sub-section 5.6. will investigate these eceptions. Equation (5.89) demonstrates that the vector element cannot represent a divergent vector field and Equation (5.92) demonstrates that the vector element is capable of representing rotational/circulating vector fields. The latter propert of the vector element make them ver useful and attractive to use for electromagnetic field vectors because the electromagnetic field vectors not onl obe the Mawell curl equations, but the are also constrained b the divergence equations [2] (refer to Chapter 2, specificall Section 2.3. point number 2). As a summar, in vector electromagnetics the electromagnetic behaviour is governed b the Mawell curl equations as shown in Chapters 2 and 3, therefore the curl of the vector field is of importance [, 2] and not the divergence of the vector field. From a phsical view point, the curl of the electric field is the time-rate of change of the magnetic field [3]. It seems that the purpose of the mied-order/zeroth-order vector element is to remove the terms f and g 2 from the polnomial epansion that make up the electric vector field E (,) given b Equation (5.2), which do not contribute to the magnetic field [3] (which is the curl of the electric field given b Equation (5.87)) Investigation of the Tpes of Vector Fields that The Vector Elements are Capable and Incapable of Constructing The vector element provides a vector field representation given b Equation (5.53) that is complete to order p =, which is zeroth-order [, 2]. However, there are certain vector field representations that the vector elements cannot approimate (refer to sub-section 5.5.) that fall under the general vector field representation given b Equation (5.53). The algorithm presented in Section 5.5 is used to investigate and verif the above statement. Equation (5.53) allows for siteen combinations of vector field representations due to the equation containing the four constant variables of f, f 2, g and g. Of the siteen combinations of vector fields in Table 5.3, the vector element is unable to approimate the eight combinations of vector fields found in column C, the are able to approimate the four combinations of vector fields found in column B and the are capable of approimating the four combinations of vector fields found in column A when a constrain is applied to the constants f 2 and g. 6

79 A B C f 2 AND g f 2 = AND g = f 2 OR g E V (,) = ( f + f 2 )î+(g + g ) ĵ E (,) = f î+g ĵ E V (,) = ( f + f 2 )î+g ĵ E V (,) = f î+(g + g ) ĵ f AND g A B C E V (,) = f 2 î+g ĵ E V (,) = E V (,) = f 2 î E V (,) = g ĵ f = AND g = 2 A2 B2 C2 E V (,) = f + f 2 î E V (,) = ( f + f 2 )î+g ĵ E (,) = f î E V (,) = f 2 î+g ĵ E V (,) = f 2 î+(g + g ) ĵ E (,) = g ĵ E V (,) = f î+g ĵ f OR g 3 E V (,) = g + g ĵ A3 B3 C3 Table 5.3: Table of Vector Field Representations Vector Field Approimations of Column B The vector elements are capable of approimating constant vector fields, of the algebraic form as seen in block B and block B3 of Table 5.3. These fields were simulated and displaed in Figure (a) (b) Figure 5.22: Vector Field Representation of Blocks B and B3 of Table 5.3 Sub-figure 5.22a illustrates the vector field E (,) = f î+g ĵ. The constant are set as f =, f 2 =, g =, f =, g = and g 2 =. Sub-figure 5.22b illustrates the vector field E (,) = f î. The constant are set as f =, f 2 =, g =, f =, g = and g 2 =. Sub-figure 5.22c illustrates the vector field E (,) = g ĵ. The constant are set as f =, f 2 =, g =, f =, g = and g 2 =. (c) When all the constants are set as zero that is, f =, f 2 =, g =, f =, g = and g 2 =, then the vector elements displas no vector field as given b the equation in block B2 of Table 5.3. Vector Field Approimations of Column A The vector elements are well suited to approimating rotational vector fields and are capable of approimating vector fields of the form as given b the equation in block A2 of Table 5.3 if and onl if the following constraint is applied: g = f 2 (5.93) 62

80 where Equation (5.93) when applied to the equation in block A2 of Table 5.3, produces a uniform rotational vector field. This can be seen in Sub-figure 5.7a which displas the rotational vector field given b Equation (5.59). Consequentl if the following constraint is applied to the equation in block A2 of Table 5.3: g f 2 (5.94) while choosing g > f 2 or either g < f 2, then a non-uniform rotational field is produced which the vector elements are not capable of approimating as illustrated in Figure (a) (b) (c) (d) Figure 5.23: Vector Field Representation of Block A2 of Table 5.3 Sub-figure 5.23a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = 2î+ 4 ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.23b illustrates the vector field E (,) = f 2 î + g ĵ where g > f 2. The constants are set as f =, f 2 = 2, g = 4, f =, g = and g 2 =. Sub-figure 5.23c illustrates a vector field made b the Š̹ Ä function Å Ö and using the equation E (,) = 6î+3 ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.23d illustrates the vector field E (,) = f 2 î+g ĵ where f 2 > g. The constants are set as f =, f 2 = 6, g = 3, f =, g = and g 2 =. 63

81 Onl when Equation (5.93) is applied to the equations located in block A and A3, are the vector elements capable of approimating these fields as seen in Figures 5.24, 5.25 and (a) Figure 5.24: Vector Field Representation of Block A of Table 5.3 Sub-figure 5.24a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = (4+3)î (2+3) ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.24b illustrates the vector field E (,) = (4+3)î (2+3) ĵ where g = f 2. The constants are set as f =, f 2 = 3, g = 3, f = 4, g = 2 and g 2 = (b) (a) (b) Figure 5.25: Vector Field Representation of Block A3 of Table 5.3 Sub-figure 5.25a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = ( ) î 3 ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Subfigure 5.25b illustrates the vector field E (,) = ( ) î 3 ĵ where g = f 2. The constants are set as f =, f 2 = 3, g = 3, f = 3 4, g = and g 2 =. 64

82 (a) Figure 5.26: Vector Field Representation of Block A3 of Table 5.3 Sub-figure 5.26a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = 4î+(5+4) ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Subfigure 5.26b illustrates the vector field E (,) = 4î+(5+4) ĵ where g = f 2. The constants are set as f =, f 2 = 4, g = 4, f =, g = 5 and g 2 =. However if, the other constraint of Equation (5.94) is applied to the same equations located in block A and A3, then the vector elements are not capable of approimating these fields as seen in Figures 5.27, 5.28 and (b) (a) (b) Figure 5.27: Vector Field Representation of Block A of Table 5.3 Sub-figure 5.27a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = (3 2)î + (+4) ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. The con- Sub-figure 5.27b illustrates the vector field E (,) = (3 2)î + (+4) ĵ where g f 2. stants are set as f =, f 2 = 2, g = 4, f = 3, g = and g 2 =. 65

83 (a) Figure 5.28: Vector Field Representation of Block A3 of Table 5.3 Sub-figure 5.28a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = ( 3)î + 4 ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Subfigure 5.28b illustrates the vector field E (,) = ( 3)î+4 ĵ where g f 2. The constants are set as f =, f 2 = 3, g = 4, f =, g = and g 2 = (b) (a) (b) Figure 5.29: Vector Field Representation of Block A3 of Table 5.3 Sub-figure 5.29a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = 3î (2+4) ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Subfigure 5.29b illustrates the vector field E (,) = 3î (2+4) ĵ where g f 2. The constants are set as f =, f 2 = 3, g = 4, f =, g = 2 and g 2 =. Vector Field Approimations of Column C Equations located in block C, C2 and C3 all possess a curling differential propert, but the vector elements are incapable of approimating these vector fields as seen in Figures 66

84 5.3, 5.32, 5.33, 5.34, 5.35 and 5.3. J.P. Webb stated in his paper [], that the field interpolation that the vector element provides is not even first-order [], and therefore linear fields as given b the equations located in block C2 of Table 5.3 cannot be represented b the vector element eactl [] (a) (c) (b) (d) Figure 5.3: Linear Vector Field Representation of Block C2 of Table 5.3 Sub-figure 5.3c illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.3d illustrates the vector field E (,) = ĵ. The constants are set as f =, f 2 =, g =, f =, g = and g 2 =. Sub-figure 5.3a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = î. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.3b illustrates the vector field E (,) = î. The constants are set as f =, f 2 =, g =, f =, g = and g 2 =. The above statement of J.P.Webb can be etended to include the variations of linear fields (as seen in blocks C and C3) supported b Equation (5.53). 67

85 (a).5.5 (c) (b) Figure 5.3: Vector Field Representation of Block C of Table 5.3 Sub-figure 5.3a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = (+2)î + 3 ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Subfigure 5.3b illustrates the vector field E (,) = (+2)î+3ĵ. The constants are set as f =, f 2 = 2, g =, f =, g = 3 and g 2 =. Sub-figure 5.3c illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = î + (3+4) ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.3d illustrates the vector field E (,) = î + (3+4) ĵ. The constants are set as f =, f 2 =, g = 4, f =, g = 3 and g 2 =. (d) 68

86 (a) Figure 5.32: Vector fields of Representation of Block C3 of Table 5.3 Sub-figure 5.32a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = (+)î. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.32b illustrates the vector field E (,) = (+)î. The constants are set as f =, f 2 =, g =, f =, g = and g 2 = (b) (a) (b) Figure 5.33: Vector fields of Representation of Block C3 of Table 5.3 Sub-figure 5.33a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = î+ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.33b illustrates the vector field E (,) = î+ĵ. The constants are set as f =, f 2 =, g =, f =, g = and g 2 =. 69

87 (a) Figure 5.34: Vector fields of Representation of Block C3 of Table 5.3 Sub-figure 5.33a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = 2î 4 ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.33b illustrates the vector field E (,) = 2î 4 ĵ. The constants are set as f =, f 2 =, g = 4, f = 2, g = and g 2 = (b) (a) (b) Figure 5.35: Vector fields of Representation of Block C3 of Table 5.3 Sub-figure 5.35a illustrates a vector field made b the Å ÌÄ function Å Ö and using the equation E (,) = ( 4) ĵ. The vectors are then plotted with the Å ÌÄ function ÉÍÁÎ Ê. Sub-figure 5.35b illustrates the vector field E (,) = ( 4) ĵ. The constants are set as f =, f 2 =, g = 4, f =, g = and g 2 =. 7

88 5.7 Limitations of the Vector Element The triangular vector elements for this dissertation posses two main limitations which is directl related to how these shape functions were designed mathematicall. The limitations are:. the element rate of convergence is rather poor [2]; the solution will approach the eact one slowl as the mesh is refined [2]. As in the case of nodal (scalar) elements, a better convergence rate can be achieved using higher-order elements. 2. the element is onl capable of approimating simple vector fields as given b Equation (5.53) and is therefore unable to approimate more comple mathematicall structured vector fields. Although the element can approimate vector fields such as those given b Equation (5.53), there were however some eceptions that was shown in sub-section Higher-order elements falls outside the scope of this dissertation and therefore onl a brief overview will be given in the net section Higher-Order Vector Elements There are two competing approaches to higher-order vector elements which are [2, 3]: Interpolator Vector Elements Hierarchical Vector Elements The difference between these elements lies in their construction [2, 3]. Interpolator Vector Elements The interpolator basis functions of these elements are defined on a set of points on the element, such that the basis function vanishes at all points ecept one [2]. There are advantages associated with these basis functions: the functions have good linear independence, thus resulting in a better conditioned matri sstem [2]. the interpolate the tangential component of the vector field, making enforcement of boundar conditions eas [2]. the have a unified epression which simplifies the implementation of computer codes for the generation of arbitrar-order basis functions [2]. These elements can be used for h adaption (mesh refinement). Disadvantages of the basis functions: the interpolator basis functions of a given order are all different from those of the lower orders [2]. Hence different order basis functions cannot be used together (in the same mesh), which makes it impossible to implement p adaption (iterative increase of the element orders in different regions (of the mesh) until the solution is converged to a specified accurac) [2]. 7

89 Hierarchical Vector Elements The basis functions are not defined on a set of points [2]. Higher-order hierarchical basis functions are formed b adding new basis functions to lower-order basis functions [2, 3, 2]. For eample, a first-order basis function contain those of the zeroth-order basis functions, and the second-order basis function will in turn contain those of the zerothorder and first-order basis functions [2], thus a specific higher-order basis function set contains all the lower-order basis functions [2]. This distinct advantage of these basis functions is that the permit the use of different order of basis functions (hence elements) within the same problem, and hence can be used for p adaption [2]. A good deal of time has been invested in studing the vector element for the following reasons: it is important because the vector element is the building block (foundation) of a vector field simulation. good knowledge is gained of the properties and limitations of the element. the knowledge is needed in order to code an entire finite element (FE) algorithm. 72

90 Chapter 6 Finite Element Method The finite element method (FEM) is a numerical technique used to solve problems that are described mathematicall b ordinar differential equations (ODE) and partial differential equations (PDE). []. The main idea behind the method is the representation of the domain into smaller sub-domains called finite elements []. These finite elements can be represented b [2]: lines in one dimension triangles in two-dimensions tetrahedrons in three-dimensions The numerical solution (of the ODE or PDE) within a finite element is calculated b using the values computed b FEM at the nodes (when using nodal finite elements) or edges (when using vector finite elements) of the discretized domain, along with the corresponding basis functions associated with the particular finite element [, 2]. The values of the nodes or edges are obtained after solving a sstem of linear equations which the FEM sets up [, 2]. The accurac of the solution can depend on, among other factors: the order of the basis functions, which ma be linear, quadratic or higher order []. domain discretization. Increasing the number of elements inside the domain (creating a finner mesh) increases the accurac of the solution in comparison to a domain discretized b less elements (producing a course mesh) []. Finite element analsis (FEA) can handle two different tpes of problems [2]: eigenanalsis (source-free) problems deterministic (force driven) problems Deterministic problems analsed using FEA involve a source [2]. Equation (2.3) of Section 2.4 and Equation (2.39) of Section 2.5 respectivel describe a deterministic problem, where the source can take the form of a boundar condition or the form of an actual source region present within the domain [6, 5]. In the case where the source takes the form of a boundar condition, the problem can be classified as boundar-driven, also known as a boundar value problem (BVP) [6, 5]. In the case where the source takes the form of actual source region, the problem can be classified as force-driven (refer 73

91 to Section 2.6 where these concepts were discussed with reference to Equations (2.3) and (2.39) [6, 5]). The response of the structure (conductor), to a forced ecitation (the source) is described b the solution to Equation (2.3) and Equation (2.39), as discussed in Section 2.6. The FEA is able to compute this solution [2]. The FEM is applied to Equation (2.39) [3, 6, 7]. Summar Outline of FEM The basic steps involved in the application of FEM are as follows:. Create a FE mesh b discretizing the entire domain into sub-domains and the subdomains are referred to as finite elements []. 2. Processing of the mesh data information into useful data-structures [2, 2, 3]. 3. Obtain a sstem of linear equations for a single element b appling the Galerkin method to the PDE under investigation []. 4. Formation of the global matri sstem of equations through the assembl of all elements within the domain and incorporation of the boundar conditions (BC s) into this global matri sstem []. 5. Solve the linear sstem of equations to obtain the unknown edge values of the domain []. 6. Build the solution vector field (solution to the PDE under investigation) over the entire domain []. The dissertation is focused mainl on solving problems over a two-dimensional domain. 6. Domain Discretization The entire domain is divided up into smaller sub-domains, the sub-domains are also called finite elements []. The union of all the sub-domains approimatel equals the full domain [3]. 6.. Two Dimensional Domain Discretization The domain of a two-dimensional BVP or force-driven problem, usuall has an irregular shape, as shown in Figure 6.(a) []. Using the FEM, the first step is to accuratel represent the phsical domain of the problem b a set of basic shapes called finite elements []. The use of a rectangle, for eample, as a basic finite element to discretize an irregular domain is certainl the simplest but not the most suitable choice because an assembl of rectangles cannot accuratel represent the arbitrar geometrical shape of a domain []. In such a case the discretization error is significant, as shown in Figure 6.(b), although it tends to decrease as the size of the rectangles in the domain becomes smaller []. However if a triangle element is used instead as the basic element for the meshing of the two-dimensional domain, the discretization error would be effectivel much smaller []. This is illustrated graphicall in Figure 6.2(a) []. A course mesh of an irregular domain 74

92 using quadrilateral elements is shown in Figure 6.2(b). Similar to the triangular element, the quadrilateral element results in a smaller discretization error []. Figure 6.: Discretization Error using Rectangular Elements [] (a)irregular two dimensional domain. (b)finite element mesh using rectangular elements. Figure 6.2: Discretization Error using Triangular and Quadrilateral Elements [] (a)finite element mesh using triangular elements. (b)finite element mesh using quadrilateral elements. 6.2 Data-Structures The domain is sub-divided into finite elements, thus forming a FE mesh and all finite elements within the entire mesh are connected to one another [, 5]. The data instructing how the elements are connected to each other through their edges, nodes and coordinates are referred to as the element connectivit information [, 2]. The element connectivit information is stored in lists which are referred to as data-structures [2, 23]. 75