fredrik edelvik hybrid solvers for the maxwell equations...

FREDRIK EDELVIK

Hybrid Solvers for the Maxwell Equations in

Time-Domain

Dissertation for the Degree of Doctor of Technology in Scientific Computingpresented at Uppsala University in 2002

Abstract

Edelvik, F. 2002. Hybrid Solvers for the Maxwell Equations in Time-Domain. ActaUniv. Ups. Uppsala Dissertations from the Faculty of Science and Technology 40. 164pp. Uppsala. ISBN 91-554-5354-6.

The most commonly used method for the time-domain Maxwell equations is the Finite-Difference Time-Domain method (FDTD). This is an explicit, second-order accuratemethod, which is used on a staggered Cartesian grid. The main drawback with theFDTD method is its inability to accurately model curved objects and small geometricalfeatures. This is due to the Cartesian grid, which leads to a staircase approximationof the geometry and small details are not resolved at all.

This thesis presents different ways to circumvent this drawback, but still take ad-vantage of the benefits of the FDTD method. An approach to avoid staircasing errorsbut still retain the efficiency of the FDTD method is to use a hybrid grid. A fewlayers of unstructured cells are used close to curved objects and a Cartesian grid isused for the rest of the domain. For the choice of solver on the unstructured grid twodifferent alternatives are compared: an explicit Finite-Volume Time-Domain (FVTD)solver and an implicit Finite-Element Time-Domain (FETD) solver.

The hybrid solvers calculate the scattering from complex objects much more ef-ficiently compared to using FDTD on highly resolved Cartesian grids. For the sameaccuracy in the solution roughly a factor of 10 in memory requirements and a factorof 20 in execution time are gained.

The ability to model features that are small relative to the cell size is often im-portant in electromagnetic simulations. In this thesis a technique to generalize a well-known subcell model for thin wires, in order to take arbitrarily oriented wires in FETDand FDTD into account, is proposed. The method gives considerable modeling flex-ibility compared to earlier methods and is proven stable. The results show excellentconsistency and very good accuracy on different antenna configurations.

The recursive convolution method is often used to model frequency dispersive ma-terials in FDTD. This method is used to enable modeling of such materials in theunstructured FVTD and FETD solvers. The stability of both solvers is analyzed andtheir accuracy is demonstrated by computing the radar cross section for homogeneousas well as layered spheres with frequency dependent permittivity.

Keywords: Maxwell’s equations, time-domain, finite volume methods, finite elementmethods, hybrid solver, dispersive materials, thin wires.

Fredrik Edelvik, Department of Information Technology, Scientific Computing, UppsalaUniversity, Box 337, SE-751 05 Uppsala, Sweden

c© Fredrik Edelvik 2002

ISSN 1104-2516ISBN 91-554-5354-6

Printed in Sweden by Elanders Gotab, Stockholm 2002

Acknowledgments

First of all I would like to thank Dr. Gunnar Ledfelt for all our endless discussionson the material in this thesis. I have really enjoyed working together with you!I thank my advisor Professor Per Lotstedt for sharing his deep knowledge innumerical analysis. I am grateful to Dr. Ulf Andersson for having the mainresponsibility of the FDTD code which is the basis for the time-domain solversin GEMS and for answering my questions regarding the computer system atPDC.

I thank Dr. Bo Strand for leading the GEMS project and for fruitful discussionson frequency dispersive materials. I am grateful to Anders Alund for his hardwork on the GEMS solvers and for always finding a compiler where my codedoes not work.

Furthermore, I would like to express my sincere gratitude to:

Dr. Douglas Riley for promptly answering my questions on computational elec-tromagnetics and for our joint work on thin wires. Thanks also for pushing meinto the world of finite elements.

All the people involved in the GEMS and GEMS2 projects. It has been inspiringto work together with you and to understand that there is an industrial need forthe results in this thesis. In particular I would like to thank Jonas Gustafssonfor making most of the hybrid grids used in the results sections.

Lars Eriksson for implementing parts of the FETD solver.

Martin Nilsson for his work on the fast multipole method. To be able to validatemy results against yours has been very valuable.

All the people involved in tdboll activities.

My wife Anna for love and support. I love you!

This work was financed by the Parallel and Scientific Computing Institute (PSCI),the Swedish Agency for Innovation Systems (VINNOVA) and the National Aero-nautical Research Program (NFFP). I have been able to participate in a numberof conferences thanks to: PSCI, Bernt Jarmarks stiftelse for vetenskaplig forskn-ing, the European Space Agency, Ericsson’s Research Foundation and C F Lil-jewalchs resestipendiefond.

Contents

1 Introduction 11.1 Computational Electromagnetics . . . . . . . . . . . . . . . . . . 11.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Frequency-domain methods, integral formulation . . . . . 21.2.2 Frequency-domain methods, PDE formulation . . . . . . . 21.2.3 Time-domain methods, integral formulation . . . . . . . . 31.2.4 Time-domain methods, PDE formulation . . . . . . . . . 3

1.3 The GEMS and GEMS2 projects . . . . . . . . . . . . . . . . . . 41.4 Outline and main results . . . . . . . . . . . . . . . . . . . . . . . 51.5 List of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Governing equations 92.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Reduction to two dimensions . . . . . . . . . . . . . . . . . . . . 112.4 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Finite Volume method in 2D 133.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Preservation of divergence . . . . . . . . . . . . . . . . . . . . . . 183.4 Dispersion analysis on triangular grids . . . . . . . . . . . . . . . 193.5 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Workload and memory requirements . . . . . . . . . . . . . . . . 26

4 Finite Volume method in 3D 294.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Creating the dual grid . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Preservation of divergence . . . . . . . . . . . . . . . . . . . . . . 334.5 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

v

4.6 Workload and memory requirements . . . . . . . . . . . . . . . . 38

5 Finite Element method in 3D 415.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Hybridization 476.1 Coupling of FDTD and FVTD in 2D . . . . . . . . . . . . . . . . 476.2 Coupling of FDTD and FVTD in 3D . . . . . . . . . . . . . . . . 496.3 Coupling of FDTD and FETD in 3D . . . . . . . . . . . . . . . 50

7 Numerical experiments in 2D 537.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.3 PMC wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8 A comparison of the hybrid solvers 618.1 Grid reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.3 Convergence in vacuum . . . . . . . . . . . . . . . . . . . . . . . 638.4 Scattering results . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.4.1 A PEC sphere . . . . . . . . . . . . . . . . . . . . . . . . 658.4.2 A dielectric sphere . . . . . . . . . . . . . . . . . . . . . . 678.4.3 The generic aircraft RUND . . . . . . . . . . . . . . . . . 70

8.5 Implicit versus explicit methods . . . . . . . . . . . . . . . . . . 728.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9 Frequency dispersive materials 779.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779.2 Recursive convolution . . . . . . . . . . . . . . . . . . . . . . . . 78

9.2.1 Finite Volume solver . . . . . . . . . . . . . . . . . . . . . 799.2.2 Finite Element solver . . . . . . . . . . . . . . . . . . . . 81

9.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.3.1 Finite Element solver . . . . . . . . . . . . . . . . . . . . 839.3.2 Finite Volume solver . . . . . . . . . . . . . . . . . . . . . 85

9.4 Scattering results . . . . . . . . . . . . . . . . . . . . . . . . . . 859.4.1 A Debye sphere . . . . . . . . . . . . . . . . . . . . . . . . 869.4.2 A Lorentz sphere . . . . . . . . . . . . . . . . . . . . . . 879.4.3 A General sphere . . . . . . . . . . . . . . . . . . . . . . . 889.4.4 A Layered sphere . . . . . . . . . . . . . . . . . . . . . . . 91

9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

10 Modeling thin wires in FETD 9310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

10.2 Thin wires in FETD . . . . . . . . . . . . . . . . . . . . . . . . . 9410.3 Field-wire coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 97

10.3.1 The interpolation operator . . . . . . . . . . . . . . . . . 9710.3.2 Bent wires . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

10.4 Iterative method . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.5 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

10.6.1 Receiving dipole antenna . . . . . . . . . . . . . . . . . . 10410.6.2 Transmitting dipole antenna . . . . . . . . . . . . . . . . 10710.6.3 Receiving loop antenna . . . . . . . . . . . . . . . . . . . 11010.6.4 Shielded enclosure . . . . . . . . . . . . . . . . . . . . . . 110

10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

11 Modeling thin wires in FDTD 11511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11511.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 11611.3 Field-wire coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 11711.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12011.5 Thin wires in the hybrid solvers . . . . . . . . . . . . . . . . . . 12111.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

11.6.1 Receiving dipole antenna . . . . . . . . . . . . . . . . . . 12211.6.2 Transmitting dipole antenna . . . . . . . . . . . . . . . . 12411.6.3 Receiving loop antenna . . . . . . . . . . . . . . . . . . . 12711.6.4 Transmitting loop antenna . . . . . . . . . . . . . . . . . 12811.6.5 Shielded enclosure . . . . . . . . . . . . . . . . . . . . . . 131

11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

12 Complex applications in 3D 13312.1 Scattering from a metallic business card . . . . . . . . . . . . . . 13312.2 Scattering from a metallic cone sphere . . . . . . . . . . . . . . . 13412.3 Scattering from the generic aircraft RUND . . . . . . . . . . . . 13612.4 Scattering from a military aircraft . . . . . . . . . . . . . . . . . 13912.5 Scattering from the Saab Trainer aircraft . . . . . . . . . . . . . 14012.6 Shielded enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . 14412.7 Lightning strike to a Saab 2000 aircraft . . . . . . . . . . . . . . 144

Bibliography 149

Chapter 1

Introduction

1.1 Computational Electromagnetics

The rapid increases in computing capabilities and the development of new nu-merical methods have facilitated the solutions of larger, more complex electro-magnetic scattering and radiation problems. Numerical methods are today ata level where they can be used to accurately simulate the electromagnetic re-sponse from very complicated objects such as aircraft, satellites, cars and variouselectronic devices. A numerical simulation can often replace expensive and verytime consuming measurements especially in the design and construction phase.

Computational Electromagnetics (CEM) can for instance be used to deter-mine the Radar Cross Section (RCS) for objects. The RCS is a far-field param-eter, which characterizes the scattering properties of radar targets. For a target,there are monostatic and bistatic RCS. For monostatic RCS the transmitter andthe receiver are at the same location, which is not the case for bistatic RCS.When designing a low-observable target, the RCS is the parameter you attemptto minimize.

Electronic systems are increasing rapidly in numbers and complexity as wellas their sensitivity to electromagnetic disturbances. Such a disturbance could forexample be direct or indirect effects of lightning. The high surface currents thatare induced can damage safety critical electronic systems in the aircraft. Test-ing of Electromagnetic Compatibility (EMC) properties for large and complexobjects is an area where CEM is used successfully.

Antenna design is another important field of application for CEM. For wire-less communication systems the antenna is one of the most critical componentsand a good design of the antenna is crucial for the overall system performance.Software that can be used to simulate an entire antenna including transmissionline accurately is an important tool for engineers.

1

2 CHAPTER 1. INTRODUCTION

1.2 Numerical methods

Electromagnetic fields are governed by the Maxwell equations, see Chapter 2.They can be solved either in the time-domain or the frequency-domain. Fur-thermore, the numerical method can be applied either to the Partial DifferentialEquation (PDE) formulation of the Maxwell equations or to an integral formu-lation.

Since, our focus is on time-domain methods we will concentrate on themand only briefly comment upon frequency-domain methods. The advantage oftime-domain methods is that we can solve for a broad range of frequencies atone single simulation.

1.2.1 Frequency-domain methods, integral formulation

The most well-known frequency-domain method is the Method of Moments (MoM)[68]. It is based on integral equations describing interactions between surface cur-rents. The integral equations transform the Maxwell equations into the problemof finding currents on the surface of the object. This implies that the dimen-sion of the problem is reduced by one. Another advantage is that several anglesof incidence are easily treated through the use of multiple right-hand sides inthe resulting system of equations. A drawback is that problems for which thevolume to surface ratio is low, which is often the case for EMC problems, andproblems involving varying material properties, are not solved efficiently.

MoM results in a dense linear system of equations. Solving this systemwith Gaussian elimination requires O(N3) arithmetic operations, where N isthe number of unknowns. If we keep the number of elements per wavelengthconstant, N is proportional to f2, where f is the frequency. The work to solvethe MoM system directly is therefore O(f6). One way to diminish this work-load is to solve the system using iterative methods. Iterative methods are usuallybased on matrix-vector multiplication, which has a complexity of O(N2). Hence,the work to solve the MoM system iteratively is O(f4) if the iterative methodconverges rapidly. An even better complexity can be achieved if the communica-tion between the unknown current sources is based on multipole expansions. Inthis case the linear system of equations can be solved with O(N log(N)) arith-metic operations if a multilevel method is used, which results in a workload ofO(f2 log(f2)) [12, 13, 49, 61].

1.2.2 Frequency-domain methods, PDE formulation

The most common PDE formulation of the Maxwell equations in the frequency-domain is the vector Helmholtz equation. For the electric field it is given by

∇× (1µ∇×E) = ω2εE , (1.1)

1.2. NUMERICAL METHODS 3

where ω is the angular frequency, and ε and µ are space-dependent materialproperties. Helmholtz equation is usually solved with Finite Element Meth-ods (FEM) [66] due to their geometrical flexibility and ability to handle varyingmaterial properties. FEM in frequency-domain is efficiently used for calculatingthe RCS of deep open cavities [43] and it is also commonly used within the mi-crowave technology industry. The widespread commercial code HFSS [32] usesFEM.

1.2.3 Time-domain methods, integral formulation

Time-domain methods for the integral formulation of the Maxwell equationshave so far not been widely used. However, in the last few years there hasbeen increased efforts in this area. Most methods are so-called Marching-On-in-Time (MOT) methods. The complexity of traditional MOT methods isO(NtN

2s ), where Nt is the number of time steps and Ns is the number of sur-

face unknowns. This complexity can be improved by using so-called Plane-WaveTime-Domain (PWTD) methods [12, 24]. PWTD methods are an extension ofthe frequency-domain fast multipole method to the time-domain. The complex-ity of the multilevel PWTD is O(NtNs log(Ns)).

Integral equation methods in time-domain have some advantages comparedto PDE methods in the time-domain: they do not suffer from dispersion errors,they only discretize a surface and no absorbing boundary condition is needed. Infact, recently time-domain integral equation methods have been used to createabsorbing boundary conditions for FDTD and FETD [25, 36].

The drawback with MOT methods is that they have been shown prone tolate time instabilities. This issue has been studied in detail by Walker’s groupand they demonstrate in [9, 17] that MOT schemes for solving magnetic fieldintegral equations can be stabilized for “all practical purposes” by relying onaccurate spatial integration rules and implicit timestepping.

1.2.4 Time-domain methods, PDE formulation

The most commonly used method for the time-domain Maxwell equations is theFinite-Difference Time-Domain method (FDTD) [72]. It has several advantagescompared to other time-domain methods. The simple data structures makeit straightforward to implement and facilitate vectorization and parallelization.The main drawback with the FDTD method is its inability to accurately modelcurved objects and small geometrical features. This is due to the Cartesiangrid, which leads to a staircase approximation of the geometry and small detailsare not resolved at all. In many applications, e.g. RCS predictions for complexobjects, the staircased approximations of the objects are not accurate enough.

There are several suggested remedies to circumvent the effects of the er-rors introduced by the staircase approximations. These include Cartesian sub-gridding [50, 64], conformal modeling [15], staircase free Cartesian grid meth-


ods [16, 18], overlapping grids [73, 74, 75, 76], multiblock body fitted structuredgrids [2, 60] and unstructured grids [31, 41].

Cartesian subgridding suffers from stability problems and the interpolationsrequired at grid interfaces violate the divergence relations. The idea of the con-formal modeling approach is to change the updating stencils in all cells that areintersected by the boundary. The main drawback with this method is its extremecomplexity in 3D. Furthermore, the question of stability is not sufficiently inves-tigated. The staircase free Cartesian grid methods are also based on a change ofthe FDTD updating stencil close to metallic or dielectric boundaries. However,stability in 3D remains to be solved.

The overlapping grids approach has serious stability problems due to theinterpolations between the grids [74, 76]. Another disadvantage is the grid gen-eration which often requires a high user intervention. The methods based onmultiblock body fitted structured grids originate from CFD. They are designedto be able to capture shocks and are in general very dissipative, which is nota good property for a solver of Maxwell’s equations. Moreover, the generationof the grid is a major hassle for complex objects and could take several manmonths even for a skilled engineer. Using an unstructured grid for the wholedomain makes the grid generation considerably easier for complex objects. Butan unstructured solver generally implies a more complex data structure andmore arithmetic operations per cell compared to a structured solver. However,the higher-order discontinuous Galerkin method presented in [31] has shownpromising results.

Another approach to avoid staircasing that has gained popularity lately isto use unstructured grids near curved objects and around small geometricaldetails, but revert to Cartesian grids as quickly as possible for the rest of thecomputational domain [20, 21, 55, 57]. This combines the efficiency of Cartesiangrids with the flexibility of unstructured grids. Another advantage of using ahybrid grid is that it gives access to accurate absorbing boundary conditionsdeveloped for Cartesian grids [6, 26]. However, recently a Perfectly MatchedLayer (PML) absorbing boundary condition has been developed also for thefinite element method in time-domain [35].

1.3 The GEMS and GEMS2 projects

All the research in this thesis has been performed in close cooperation withSwedish industry in the Parallel and Scientific Computing Institute (PSCI)projects GEMS and GEMS2. GEMS is an abbreviation for General Electro-Magnetic Solvers and was a Swedish three-year code development project thatwas supported by an extensive research program [62]. A substantial part of thefunding was supplied by the National Aeronautical Research Program (NFFP).The GEMS project has now been superceded by the GEMS2 project.

The main objective of the GEMS project was to develop a software suite

1.4. OUTLINE AND MAIN RESULTS 5

for solving the Maxwell equations. This software suite aims to be state-of-the-art and to form a platform for future development by Swedish industry andacademia. The codes are currently used in an industrial environment.

The core of the software suite is two hybrid codes, one for the time-domainand one for the frequency-domain. The time-domain code is a hybrid betweenFDTD, an explicit Finite-Volume Time-Domain (FVTD) solver and an implicitFinite-Element Time-Domain (FETD) solver. The frequency-domain code is ahybrid between MoM, Physical Optics (PO) and Geometrical Theory of Diffrac-tion (GTD) [7, 22, 23].

The industrial partners in GEMS were Ericsson Microwave Systems, SaabEricsson Space and Saab Avionics. Code developers were PSCI, the SwedishInstitute of Applied Mathematics (ITM) and the Swedish Defence ResearchAgency (FOI). The GEMS2 project involves PSCI, Ericsson Microwave Systems,Saab Avionics and ITM.

1.4 Outline and main results

The main part of this thesis is devoted to the unstructured FVTD and FETDsolvers. The description and analysis of the FVTD solvers were the basis of myLicentiate thesis presented in October 2000 [19]. In addition to the material inthe Licentiate thesis this thesis includes the FETD solver and subcell modelingof thin wires. The coupling of the solvers has been improved and the chapter onfrequency dispersive materials now includes FETD and a stability analysis.

Chapters 2, Governing equations

Electromagnetic fields are governed by the Maxwell equations which are statedin Chapter 2.

Chapters 3 and 4, The FVTD solvers

In Chapters 3 and 4 the FVTD solvers in 2D and 3D are presented and an-alyzed. The 2D FVTD solver is in space very similar to the solver used byRiley and Turner [55], whereas the solver in 3D differs in several ways. Themain improvements of the 3D solver compared to [55] include the generationof the dual grid, the update of the node values and the treatment of mate-rial interfaces. The time integration method we use is a third-order staggeredAdams–Bashforth scheme (ABS3) proposed by Ghrist et al. [27] for scalar waveequations. Stability analysis of this scheme shows that it works very well in ourcase. Furthermore, using ABS3 makes the time-coupling between FDTD andthe FVTD solver straightforward. A spatial filter of Laplace type is introducedin 3D to improve the stability characteristics of the solver.


Chapter 5, The FETD solver

Chapter 5 is devoted to the FETD solver. The solver is based on a discretiza-tion of the second-order wave equation for the electric field using edge-elementsas basis functions [37, 47]. A stability proof is given which shows that theFETD solver is unconditionally stable when the Newmark-Beta scheme is usedfor timestepping.

Chapter 6, Hybridization

In Chapter 6 the coupling between the unstructured solvers and FDTD is pre-sented. Coupling of two stable methods does not necessarily result in a stablehybrid method and in fact, the hybridization technique described in Section 6.2is not stable. The source of the instabilities is the diagonal component that isnot present in FDTD and therefore is calculated by interpolation. This breaksthe symmetry of the hybrid method and makes it unstable.

To develop a symmetric interface for the coupling of FDTD and FETD wefollow [57] and use a two-cell thick layer of hexahedral elements, such that cellsneighboring the tetrahedral region are subdivided into pyramidal and tetrahedralelements. The stability of the coupling follows from the fact that the resultinghybrid solver can be viewed as an Explicit-Implicit algorithm. Such algorithmsare proven stable in [34].

Chapter 7, Numerical experiments in 2D

In Chapter 7 we show that the staircasing errors in FDTD destroy the second-order accuracy on metallic, dielectric and diamagnetic cylinders in 2D. Further-more, we show that second-order accuracy is retained by our hybrid solvers witha correct averaging of the permeability and permittivity parameters. Finally, weperform an experiment on a PMC wall, which is a classical case demonstratingthe staircasing errors in FDTD [10].

Chapter 8, A comparison of the hybrid solvers

In this chapter we compare the two hybrid solvers in terms of accuracy, efficiencyand stability. Results from several scattering cases that show the superiority ofthe hybrid solvers compared to stand-alone FDTD are presented. The conclusionfrom the comparison is that the hybrid solver combining FETD and FDTD isthe preferred choice. The major reason is that it is stable as long as the time stepsatisfies the CFL condition in the FDTD region, whereas the solver combiningFVTD and FDTD may exhibit late time instabilities.

1.4. OUTLINE AND MAIN RESULTS 7

Chapter 9, Frequency dispersive materials

The ability to model frequency dispersive materials is an important feature fora general purpose electromagnetic code. They are for instance used as radarabsorbing materials to minimize the RCS for an aircraft. One approach to modelthese materials in FDTD is to use the Recursive Convolution (RC) method [44].In Chapter 9 the piecewise constant centered RC method presented in [59] isgeneralized to our FVTD and FETD solvers. A RC method for FETD has beenproposed in [77]. However, our method includes fundamental improvementsbecause it preserves the definiteness properties of the matrices as well as reducingto FDTD on hexahedral elements. Both these issues are important for thestability of the FDTD-FETD solver.

Chapters 10 and 11, Modeling thin wires in FETD andFDTD

The ability to model features that are small relative to the cell size is oftenimportant in electromagnetic simulations. Thus, the development of accuratemodels that characterize the physics of the features without the need for a highlyresolved grid is essential.

Thin wires are often important parts of electromagnetic compatibility andantenna problems. A subcell model for thin wires in the FDTD method usingmodified telegraphers equations has been developed [33]. A generalization ofthis model to arbitrarily oriented wires has been proposed by Ledfelt [40]. Rileyhas developed a method that incorporates thin wires into the FETD method bydiscretizing a second-order wave equation for the current [54].

In the FETD chapter we present a new method, which is also based on asecond-order equation for the current derived from modified telegraphers equa-tions. A discretization of this equation with linear nodal basis functions andthe use of a radial weighting function result in a symmetric spatial coupling be-tween field and wire. We prove using the energy method that the fully discretefield-wire system is unconditionally stable. Furthermore, our method allowsfor arbitrarily located and oriented wires with respect to the tetrahedral grid,which give considerable modeling flexibility. Results on different configurationsof dipole and loop antennas show good accuracy and excellent consistency. Notethat by consistency in this context we mean that the results are independent ofthe location and orientation of the antenna within the grid.

In the FDTD chapter we present a similar method for modeling arbitrarilyoriented wires in FDTD. The method is shown to be stable under a CFL limit.The consistency of the results are also in this case excellent, and the accuracy isvery similar to the FETD results.


Chapter 12, Complex applications in 3D

In this chapter results from a few very challenging scattering cases are pre-sented. The results obtained by the FDTD-FETD hybrid solver are comparedto measurements, frequency-domain solutions and highly resolved FDTD solu-tions. Two EMC examples are also included namely a visualization of a shieldingenclosure and a simulation of a lightning strike to a Saab 2000 aircraft.

1.5 List of papers

This thesis in partly based on material from the following papers:

I F. Edelvik, Analysis of a Finite Volume Solver for Maxwell’s Equations, InR. Vilsemeier (editor), Finite Volumes for Complex Applications II, IVG,University of Duisburg, Germany, pp. 141-148, July 1999.∗

II E. Abenius, U. Andersson, F. Edelvik, L. Eriksson, G. Ledfelt, HybridTime Domain Solvers for the Maxwell Equations in 2D, International Jour-nal for Numerical Methods in Engineering, 53(9):2185-2199, March 2002.∗∗

III F. Edelvik, G. Ledfelt, Explicit Hybrid Time Domain Solver for the MaxwellEquations in 3D, Journal of Scientific Computing, 15(1):61-78, March2000.∗ ∗ ∗

IV F. Edelvik, B. Strand, Frequency Dispersive Materials for 3D Hybrid Solversin Time Domain, to appear in IEEE Transaction on Antennas and Prop-agation, 2002.∗ ∗ ∗∗

V F. Edelvik, G. Ledfelt, A comparison of time-domain hybrid solvers forcomplex scattering problems, International Journal of Numerical Modeling,15(5), Sept/Oct 2002.∗ ∗ ∗

VI F. Edelvik, G. Ledfelt, P. Lotstedt, D. J. Riley, An Unconditionally StableSubcell Model for Arbitrarily Oriented Thin Wires in the FETD Method,Technical Report 2002-007, Department of Information Technology, Up-psala University, Sweden. Submitted to IEEE Transaction on Antennasand Propagation.∗ ∗ ∗∗

VII F. Edelvik, A New Technique for Accurate and Stable Modeling of Arbitrar-ily Oriented Thin Wires in the FDTD Method, Technical Report 2002-016,Department of Information Technology, Uppsala University, Sweden. Sub-mitted to IEEE Transaction on Electromagnetic Compatibility.∗ ∗ ∗∗

∗Copyright c© 1999 Hermes, Paris, France∗∗Copyright c© 2000 Plenum Publishing Corporation

∗ ∗ ∗Copyright c© 2002 John Wiley & Sons, Ltd∗ ∗ ∗∗Copyright c© 2002 IEEE

Chapter 2

Governing equations

2.1 The Maxwell equations

Electromagnetic fields are governed by the Maxwell equations. In general formthey are given by

∇ ·D = ρf , (2.1)∂B

∂t+∇×E = 0 , (2.2)

∇ ·B = 0 , (2.3)∂D

∂t−∇×H + Jf = 0 , (2.4)

where D is the electric flux density vector, E is the electric field, B is themagnetic flux density vector, H is the magnetic field, ρf is the free charge, andJf is the total free current density. Equation (2.1) is Gauss’ law for the electricfield and it summarizes Coulomb’s law of force between point charges plus theelectrical effects of matter, while (2.2) represents Faraday’s law of induction. Thethird equation reflects that free magnetic charges have not yet been observed.Finally, (2.4) includes Ampere’s law of force between currents plus the magneticeffects of matter and conservation of free charge.

For linear, isotropic and non-dispersive media we have

D = εE , B = µH , Jf = σE + J , (2.5)

where ε is the electric permittivity, µ is the magnetic permeability, σ is theelectric conductivity and J is the free current density arising from sources other

9

10 CHAPTER 2. GOVERNING EQUATIONS

than conductivity. Substituting these relations into (2.1–2.4) implies

∇ · ε E = ρf , (2.6)

µ∂H

∂t+∇×E = 0 , (2.7)

∇ · µ H = 0 , (2.8)

ε∂E

∂t−∇×H + σ E = −J . (2.9)

Hence, the Maxwell equations in E and H are a system of eight scalar equa-tions in six variables. At a first glance it may seem that the system is overdeter-mined, but it is readily shown that if the two divergence relations are satisfiedinitially they are always satisfied. Together with boundary conditions and ini-tial values the Maxwell equations constitute a well posed system of hyperbolicpartial differential equations.

If we take the time derivative of (2.9) and use (2.7) we obtain the vector waveequation

ε∂2E

∂t2+∇×

(1µ∇×E

)+ σ

∂E

∂t= −∂J

∂t. (2.10)

For lossless homogeneous materials without sources this reduces to

∂2E

∂t2− c2∆E +

σ

ε

∂E

∂t= 0 , (2.11)

where c = 1/√

µε is the speed of propagation for the electromagnetic wave. In asimilar manner, we can show that

∂2H

∂t2− c2∆H +

σ

ε

∂H

∂t= 0 , (2.12)

for lossless homogeneous materials without sources.

2.2 Integral formulation

Using Gauss’ divergence theorem and Stokes’ law the integral formulation ofMaxwell’s equations is obtained

∮

A

ε E · n dA = ρf , (2.13)

∂

∂t

∫

A

µ H · n dA +∮

Γ

E · dl = 0 , (2.14)∮

A

µ H · n dA = 0 , (2.15)

∂

∂t

∫

A

εE · n dA−∮

Γ

H · dl +∫

A

σ E · n dA = −∫

A

J · n dA , (2.16)

2.3. REDUCTION TO TWO DIMENSIONS 11

where A is an arbitrary area with unit normal n and Γ is the contour thatencloses A.

2.3 Reduction to two dimensions

If we have no variation in the z-direction all derivatives with respect to z equalzero. In that case the Maxwell curl equations in Cartesian coordinates reduceto

µ∂Hx

∂t= −∂Ez

∂y, (2.17)

µ∂Hy

∂t=

∂Ez

∂x, (2.18)

ε∂Ez

∂t=

∂Hy

∂x− ∂Hx

∂y− σ Ez − Jz , (2.19)

ε∂Ex

∂t=

∂Hz

∂y− σEx − Jx , (2.20)

ε∂Ey

∂t= −∂Hz

∂x− σEy − Jy , (2.21)

µ∂Hz

∂t=

∂Ex

∂y− ∂Ey

∂x, (2.22)

where the first three equations constitute the 2D Transverse Magnetic (TM)Maxwell’s equations and the last three constitute the 2D Transverse Electric (TE)Maxwell’s equations. The TM and TE modes are decoupled, i.e. they containno common field component. The two modes are completely independent forisotropic materials and they can exist simultaneously with no mutual interac-tion.

2.4 Material properties

In the Maxwell equations we have three parameters. For vacuum they are µ0 =4π ·10−7, ε0 ≈ 8.8541878 ·10−12 and σ = 0. For other materials the permeabilityand permittivity are defined relative to those of vacuum, i.e. we have ε = εrε0and µ = µrµ0. For most materials, εr and µr are frequency dependent, seeChapter 9. Materials for which we assume that εr and µr are independent offrequency are referred to as simple materials. Materials for which µr 6= 1 aremetals with high conductivity and we treat them as perfect electric conductors.

12 CHAPTER 2. GOVERNING EQUATIONS

2.5 Boundary conditions

At the interface between two lossless media we have [11]

n · (D1 −D2) = 0 ,

n× (E1 −E2) = 0 ,

n · (B1 −B2) = 0 ,

n× (H1 −H2) = 0 ,

(2.23)

where the subscripts indicate which region the field belongs to, and n is theinterface normal. Using the relations in (2.5), we get

n · (ε1E1 − ε2E2) = 0 ,

n× (E1 −E2) = 0 ,

n · (µ1H1 − µ2H2) = 0 ,

n× (H1 −H2) = 0 .

(2.24)

For Perfect Electric Conductors (PEC) we have [11]

n · εE = ρs ,

n×E = 0 ,

n ·H = 0 ,

n×H = Js ,

(2.25)

where ρs is the surface charge density and Js is the surface current density. Thenormal n is pointing out of the PEC region.

The tangential electric field is zero at the surface of a PEC. This is a conse-quence of the term perfect conductor. A nonzero tangential electric field woulddrive an infinite surface current which is clearly unphysical. However, this doesnot imply that the surface current must be zero. In fact, an external field alwaysimplies nonzero surface current since the magnetic field only has tangential com-ponents at the PEC surface and the surface current is related to the tangentialmagnetic field through the fourth condition in (2.25).

Chapter 3

Finite Volume method in2D

The Finite-Volume Time-Domain (FVTD) method is based on the followingintegral formulation of Faraday’s and Ampere’s laws:

∂

∂t

∫

A

µ H dA = −∮

Γ

n×E dl , (3.1)

∂

∂t

∫

A

εE dA =∮

Γ

n×H dl −∫

A

σE dA , (3.2)

where A is an arbitrary area with unit normal n and Γ is the path that enclosesA. We have chosen to focus on the TM Maxwell’s equations (2.17–2.19). Due tothe duality of the Maxwell equations a solver for the TE equations (2.20–2.22)could be obtained through a change of variables.

3.1 Spatial discretization

The integral formulations (3.1) and (3.2) are discretized on a staggered grid byintroducing a dual grid to the primary triangular grid. The magnetic componentsare situated at the nodes of the primary grid and the electric components, in the2D TM case only Ez, are situated at the nodes of the dual grid. The dual gridis created at the preprocessing stage by defining dual nodes at the barycentersof the primary cells; see [19] for a detailed description.

In the 2D TM case integrating the magnetic and electric fields over each dual

13

14 CHAPTER 3. FINITE VOLUME METHOD IN 2D

and primary cell gives the following integral form:

∂

∂t

∫

Adj

µdj H dA = −

∑

k

∫

Γdj,k

ndj,k × Ez dl , (3.3)

∂

∂t

∫

Api

εpi Ez dA =

∑m

∫

Γpi,m

npi,m ×H dl −

∫

Api

σpi Ez dA , (3.4)

where Adj is the area of dual cell j, Γd

j is the path that encloses Adj and nd

j,k arethe unit edge normals for the dual edges k in dual cell j. The variables belongingto the primary cell i are defined similarly.

All materials are defined relative to primary grid cells. For dielectric ma-terials that is a natural definition. However, the magnetic permeability, µ, isassociated with the dual grid cells. Therefore, µ requires averaging. The aver-age permeability of dual cell j is computed as

µdj =

∑q

µpqA

dj,q

Adj

, (3.5)

where Adj,q is the part of the area Ad

j that is inside primary cell q. Performingthe material averaging in this manner preserves the second-order accuracy of thesolver for inhomogeneous materials, see Section 7.2.

The area integrals in (3.3) and (3.4) are evaluated by taking the averagevalues of the fields multiplied by the areas of the respective cells. Simplifyingthe two integrands in the line integrals implies

µdj

∂

∂tHj =

1Ad

j

∑

k

∫

Γdj,k

Ez|j,k tdj,k dl , (3.6)

εpi

∂

∂tEz|i + σp

i Ez|i =1

Api

∑m

∫

Γpi,m

H · tpi,m dl , (3.7)

where tdj,k are unit vectors for the dual edges k in dual cell j and tp

i,m are unitvectors for the edges m in primary cell i. The line integral in (3.6) is evaluated byassuming that the electric field is piecewise linear along the dual edges. Hence,Ez|j,k is computed by taking the arithmetic mean value of the electric field atthe two nodes defining the dual edge, td

j,k. However, we cannot use the sameapproach when calculating the integral in (3.7) since that does not guaranteethat the divergence is preserved on a local cell level. This has been found to bevery critical if spurious modes in the numerical solution are to be suppressed.The divergence is preserved if we incorporate an “FDTD”-correction along theedges in the primary grid (see Section 3.3 for a proof). Following Riley et al. [55],

3.1. SPATIAL DISCRETIZATION 15

the magnetic field projected along the primary edge tpi,m is evaluated as

H · tpi,m = H · nd

j,k

(nd

j,k · tpi,m

)

+12

[(Hj + Hr)−

((Hj + Hr) · nd

j,k

)nd

j,k

] · tpi,m ,

(3.8)

where Hj and Hr are the magnetic field at the two nodes defining the primaryedge and H · nd

j,k is the “FDTD”-correction in the direction ndj,k orthogonal

to the dual edge tdj,k, which crosses the primary edge tp

i,m, see Figure 3.1. The“FDTD”-correction is updated according to

∂ H · ndj,k

∂t=

Ez|q − Ez|i∆d

j,k µdj,k

, (3.9)

where ∆dj,k is the length and µd

j,k is the average permeability of the dual edgetdj,k. The average permeability, µd

j,k, is approximated as

µdj,k =

µpq ∆q + µp

i

(∆d

j,k −∆q

)

∆dj,k

, (3.10)

where i and q are the two primary cells sharing dual edge tdj,k, ∆q is the part of

the length of the dual edge that is inside primary cell q, see Figure 3.1.

ndj,k

tp

i,mHj Hr

Ez i

Ez q

∆ q

tj,kd

Figure 3.1: The magnetic field along tpi,m is approximated by Hj , Hr and an

“FDTD”-correction in direction ndj,k.

Taking a closer look at (3.8) we see that if the primary and dual edges areorthogonal, the vectors nd

j,k and tpi,m are parallel and the second part of (3.8)


vanishes. Hence, the name “FDTD”-correction is somewhat misleading sincethat term is actually the important one. The magnetic node values are onlyused to give a better approximation of the edge-projected field in the case whennd

j,k and tpi,m do not align.

The boundary condition for a Perfect Magnetic Conductor (PMC) gives usthat the tangential component of the magnetic field at the object is zero. Acomplication occurs whenever the computation of H · tp

i,m does not reduce toH · nd

j,k, where tpi,m denotes a primary edge with one node on the boundary.

When that is not the case the following alternative is used (see Figure 3.2):

H · tpi,m = H · nd

j,k

(nd

j,k · tpi,m

)

+12

[Hj + (Hj · nr) nr − (Hj + (Hj · nr) nr) · nd

j,k

] · tpi,m ,

(3.11)

where nr denotes an average normal at the boundary node. This normal isdefined by taking the average direction of the two boundary edges including theboundary node and then taking the cross product with that direction and z.

nj,kd

~nr

p

H

t

r

i,m

Hj

Ez i

PMC boundary

Figure 3.2: Primary and dual cells at a PMC boundary.

For a Perfect Electric Conductor (PEC) the tangential electric components,in the TM case only Ez, should equal zero at the boundary. However, theEz components are not situated on the boundary. The boundary condition isenforced by setting the electric field below the conductor equal to the value of theelectric field directly above the conductor with a change of sign, see Figure 3.3.

3.2. TIME DISCRETIZATION 17

Ez1

PEC boundary

Ez2

-E z1 -E z2

Figure 3.3: The PEC boundary condition is enforced using the method of images.

These uniquely determined virtual image nodes are identified when the dual gridis constructed in the preprocessing phase, see [19].

3.2 Time discretization

We use a third-order staggered Adams–Bashforth scheme (ABS3) [27],

Hn+ 1

2j = H

n− 12

j (3.12)

+∆t

µdj Ad

j

∑

k

(2524

Ez|nj,k −112

Ez|n−1j,k +

124

Ez|n−2j,k

)tdj,k∆d

j,k ,

Ez|n+1i = Ez|ni +

∆t

εpi Ap

i

∑m

(2524

H · tpi,m|n+ 1

2− (3.13)

112

H · tpi,m|n−

12 +

124

H · tpi,m|n−

32

)∆p

i,m

+σp

i ∆t

2 εpi

(Ez|n+1

i + Ez|ni)

,

H · ndj,k|n+ 1

2 = H · ndj,k|n−

12 + ∆tF

(2524

Enz −

112

En−1z +

124

En−2z

), (3.14)

where F is an operator taking care of the update of H ·ndj,k according to (3.9).

Since ABS3 is a staggered time integrator the time-coupling with FDTD isstraightforward, see Chapter 6. Furthermore, its stability properties are superiorcompared to the traditionally used leap-frog scheme, see Section 3.5.


3.3 Preservation of divergence

The FVTD solver is based on an integral formulation of Ampere’s and Faraday’slaws. However, the Maxwell equations also include the Gauss’ laws that have tobe satisfied in order to ensure a physical solution. In vacuum the divergence ofthe magnetic displacement vector, B, as well as that of the electric displacementvector, D, should equal zero.

Proposition 3.3.1 For the 2D TMz Maxwell equations, ∇ ·D = 0 is automat-ically satisfied.

Proof∇ ·D = ∇ · εE =

∂ εEx

∂x+

∂ εEy

∂y+

∂ εEz

∂z= 0 .

The first two terms equal zero since E only has a z-component in the TMz caseand the last term is zero because we do not have any variation in the z-directionin 2D. Hence, the divergence of the D field will always equal zero. ¥

We now shift focus to the B field. As previously stated it is important thatthe divergence of the magnetic field is preserved on a local cell level in order tosuppress spurious modes.

Proposition 3.3.2 The FVTD solver preserves ∇ · B on a local cell level tomachine precision.

Proof For dual cell j we have

∂

∂t

∫

Adj

∇ ·B dA =∂

∂t

∫

Adj

∇ · µ H dA =∂

∂t

∮

Γdj

µ H · ndj dl ,

where Gauss’ divergence theorem is used to get the last equality. Splitting theintegral path into k segments gives us

∂

∂t

∑

k

∫

Γdj,k

µ H · ndj,k dl .

So far we are still using continuous variables. If we assume that H · ndj,k and µ

are constant along each dual edge we get

∂

∂t

∑

k

∫

Γdj,k

µ H · ndj,k dl =

∂

∂t

∑

k

µdj,k H · nd

j,k ∆dj,k ,

where ∆dj,k is the edge length. The permeability is time independent and we can

swap the sum and time derivative operators to obtain

∑

k

µdj,k ∆d

j,k

∂H · ndj,k

∂t=

∑

k

(Ez|q − Ez|i) = 0 ,

3.4. DISPERSION ANALYSIS ON TRIANGULAR GRIDS 19

where we have used (3.9). The last expression equals zero because in the sumover k, each electric node value occurs twice with opposite signs. Thus, thetime derivative of the divergence is equal to zero and hence the divergence ispreserved to machine precision in each cell. ¥

Note that if H · ndj,k instead had been computed as a projection of the

arithmetic mean value of the two magnetic node values the divergence would ingeneral have differed from zero.

3.4 Dispersion analysis on triangular grids

A frequently used technique to characterize the errors of a numerical scheme isFourier analysis. Assuming that the fields are periodic in space and neglectingboundary conditions, we make the following ansatz for the three unknown fieldcomponents:

Ψ|nI,J = Ψ0 ej(kxI∆+kyJ∆−ωn∆t) , (3.15)

where kx and ky are the x- and y-components of the numerical wave-vector,respectively. The numerical wave-vector k will in general differ from the physicalwave-vector k satisfying |k| = ω/c, where ω is the angular frequency and c is thespeed of light. That difference depends on how well the wave is resolved in thegrid, the type of grid and the angle of propagation in the grid. The differencebetween k and k gives rise to numerical phase and group velocities that departfrom the analytical values, which cause numerical errors that accumulate withdistance. Hence, the dispersion characteristics of a numerical scheme used forwave propagation problems are crucial for the accuracy of the solution.

In Figure 3.4 the different uniform grids used in the analysis are shown.Note that the normals and edge vectors align in the equilateral and diamondgrid. Thus, only H · nd

j,k survives expression (3.8), so for these two grids wedo not have to take the magnetic field in the nodes into account. For the one-directional grid, however, we have to consider both the magnetic field at thenodes and the “FDTD”-correction. Since the approximation of the dispersionrelation is in the midpoint of the respective building blocks shown in Figure 3.4the Ez components are interpolated from the barycenters to the midpoint of thebuilding blocks. Note that this interpolation is only performed in the analysis.

Let us first look at the errors introduced by the spatial discretization for theequilateral grid. Substituting (3.15) into (3.13) and assuming that no error isintroduced by the time discretization result in

jωEz =4√3 ε∆

[sin

(− ky∆

2√

3

)H1 + sin

(kx∆

4+

ky∆4√

3

)H2

+ sin

(− kx∆

4+

ky∆4√

3

)H3

].

(3.16)


Ez Ez

Ez

∆ ∆ ∆

3H H2

1H1H

H2

H3

Figure 3.4: The different uniform triangular grids used in the analysis, the equi-lateral grid (left), the one-directional grid (middle) and the diamond grid (right).The respective dual grids are indicated by dashed lines.

Similarly, from (3.14) we have for the three edges (see Figure 3.4)

jωH1 =2√

3jEz

µ∆sin

(− ky∆

2√

3

), (3.17)

jωH2 =2√

3jEz

µ∆sin

(kx∆

4+

ky∆4√

3

), (3.18)

jωH3 =2√

3jEz

µ∆sin

(− kx∆

4+

ky∆4√

3

). (3.19)

Inserting (3.17–3.19) into (3.16) leads to the numerical dispersion relation

(ω

c

)2

=8

∆2

[sin2

(ky∆2√

3

)+ sin2

(kx∆

4+

ky∆4√

3

)

+ sin2

(− kx∆

4+

ky∆4√

3

)].

(3.20)

Letting ∆ → 0 we obtain

(ω

c

)2

=(k2

x + k2y

)− 1

48

(k2

x + k2y

)2

∆2 + O(∆4) . (3.21)

Thus, the dispersion relation is approximated to second-order accuracy. Thesame analysis on a square grid with edge length ∆, where the solver is identicalto the FDTD scheme, gives an error constant equal to 1

12 [63]. The dispersionrelations are also approximated to second-order accuracy for the grids with right

3.4. DISPERSION ANALYSIS ON TRIANGULAR GRIDS 21

triangles. For the one-directional grid the relation reads

(ω

c

)2

=(k2

x + k2y

)−

(1

108k4

x −790

k2xk2

y −1

108k4

y

+2

135k3

xky +2

135kxk3

y

)∆2 + O(∆4) ,

(3.22)

and for the diamond grid it is given by

(ω

c

)2

=(k2

x + k2y

)− 1

36

(k4

x + k4y +

12

k3xky

+12

kxk3y +

43

k2xk2

y

)∆2 + O(∆4) .

(3.23)

The dispersion characteristics can be presented in terms of the error in thephase speed of waves of different grid resolution and propagation angle. Letkx = k cos(α) and ky = k sin(α), where α is the angle of propagation relative tothe x-axis and k = |k|. Substituting these two expressions into the dispersionrelation (3.20) and solving the nonlinear equation for k, result in a relationbetween the numerical wave speed, vp, and c, since vp/c = k/k. Figure 3.5illustrates the numerical phase velocity as a function of propagation angle andgrid resolution, where λ is the wavelength. The results obtained by the FVTDsolver are in excellent agreement with the true wave speed for all three gridseven for a moderate grid resolution of ten points per wavelength. The dispersioncharacteristics are clearly better for triangular grids than for square grids. Itis interesting to note that the numerical phase velocity is independent of thepropagation direction for the uniform equilateral grid.

0 30 60 90 120 150 1800.975

0.98

0.985

0.99

0.995

1

Wave angle α

Nor

mal

ized

pha

se v

eloc

ity

Equilateral tri grid − λ/10Square grid − λ/10Equilateral tri grid − λ/20Square grid − λ/20

0 30 60 90 120 150 1800.99

0.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1

Wave angle α

Nor

mal

ized

pha

se v

eloc

ity

Equilateral tri grid − λ/10Diamond tri grid − λ/10One directional tri grid − λ/10

Figure 3.5: Variation of numerical phase velocity for different grids as a functionof grid resolution and propagation angle.

We now turn our attention to the fully discrete case. Substituting (3.15)


into (3.14) and using (3.16)

− 2j

∆tsin

(ω∆t

2

)Ez =

4j√3ε∆

(2524− 1

12e−jω∆t +

124

e−2jω∆t

)

[H1 sin

(− ky∆

2√

3

)+ H2 sin

(kx∆

4+

ky∆4√

3

)+ H3 sin

(− kx∆

4+

ky∆4√

3

)] (3.24)

Similarly for the update of the magnetic fields we substitute (3.15) into (3.13),using (3.17–3.19)

− 2j

∆tsin

(ω∆t

2

)

H1

H2

H3

=

2j√

3Ez

µ∆

(2524− 1

12e−jω∆t +

124

e−2jω∆t

)

sin(− ky∆

2√

3

)

sin(

kx∆4 + ky∆

4√

3

)

sin(− kx∆

4 + ky∆

4√

3

)

(3.25)

Finally, substituting (3.25) into (3.24) we obtain the following fully discretedispersion relation for the grid with equilateral triangles:

4∆t2

sin2

(ω∆t

2

)=

8c2

∆2

(2524− 1

12e−jω∆t +

124

e−2jω∆t

)2

[sin2

(ky∆2√

3

)+ sin2

(kx∆

4+

ky∆4√

3

)+ sin2

(− kx∆

4+

ky∆4√

3

)].

(3.26)

The time step is chosen according to the stability limit for ABS3 (see nextsection) as c∆t = 6

7∆√6. As in the semi-discrete case we solve (3.26) for k.

A plot of vp/c shows almost identical results to those in Figure 3.5 due to thefact that ABS3 is a third-order accurate scheme and the errors from the spatialdiscretization dominate. If we use the leap-frog scheme for timestepping we geta small improvement in the dispersion characteristics since that scheme is onlysecond-order accurate and the errors from the spatial and time discretizationcancel to some extent. However, the errors are small in either case and, as willbe discussed in the next section, the stability properties of the leap-frog schememake it less suitable to use for general problems.

3.5 Stability analysis

An important issue for FVTD solvers of the Maxwell equations is how to achievelong-term stability without adding too much artificial dissipation. The stabilityregion for ABS3 for the scalar test equation u′ = λu is given in Figure 3.6.

3.5. STABILITY ANALYSIS 23

−0.2 −0.1 0 0.1 0.2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 3.6: Stability region for ABS3.

The scheme is stable for λ between ±12/7 along the imaginary axis comparedto the leap-frog scheme, which is stable between ±2. That implies that we haveto use a shorter time step for ABS3. However, the main disadvantage with leap-frog is that it is only stable on the imaginary axis and becomes unstable as soonas we have eigenvalues with a nonzero real part, which we are likely to have onunstructured grids and when boundaries are taken into account.

By introducing operators A and B that take care of the spatial discretizationwe are able to write the semi-discrete problem on matrix form as

∂

∂t

(H

Ez

)=

(0 AB 0

)(H

Ez

), (3.27)

where H =(H H · nd

).

We begin by looking at the uniform triangular grids used in the analysis ofthe dispersion error. For the grid consisting of equilateral triangles the operatorA is obtained from (3.25),

A =2j√

3µ∆

sin(− ky∆

2√

3

)

sin(

kx∆4 + ky∆

4√

3

)

sin(− kx∆

4 + ky∆

4√

3

)

, (3.28)


whereas the operator B is obtained from (3.24),

B =4j√3ε∆

sin(− ky∆

2√

3

)

sin(

kx∆4 + ky∆

4√

3

)

sin(− kx∆

4 + ky∆

4√

3

)

T

. (3.29)

The eigenvalues of the spatial operator are purely imaginary and given by

√BA = ±j

(8c2

∆2

[sin2

(ky∆2√

3

)+ sin2

(kx∆

4+

ky∆4√

3

)

+ sin2

(− kx∆

4+

ky∆4√

3

)]) 12

,

(3.30)

The sin2 terms are between zero and one, therefore the eigenvalues are boundedby

−j2√

6 c

∆≤√BA ≤ j

2√

6 c

∆. (3.31)

To obtain stability a necessary condition is that all eigenvalues of the spatial op-erator are within the stability region for the time discretization operator. Hence,the stability condition for the leap-frog scheme on the equilateral triangular gridequals

c∆tLF ≤ ∆√2· ∆min

∆, (3.32)

where the first part in the right-hand side is the stability condition on Cartesiangrids and ∆min equals the shortest edge length in the primary and dual grids.On the equilateral grid ∆min = ∆/

√3.

We now analyze how ABS3 behaves in our case. Using the operators A andB we are after some straightforward algebra able to write (3.12–3.14) on matrixform as

H

n+ 12

Ez|n+1

=

( I 2524∆tA

2324∆tB I + 625

576∆t2BA

) H

n− 12

Ez|n

+

(0 − 1

12∆tA124∆tB − 25

288∆t2BA

)H

n− 32

Ez|n−1

+

(0 1

24∆tA0 − 25

576∆t2BA

) H

n− 52

Ez|n−2

.

(3.33)


The plane wave ansatzH

n+ 12

Ez|n+1

=

H

12

Ez|1

ρn , (3.34)

leads to the following generalized eigenvalue problem

ρn−3(Iρ3 − Aρ2 − Bρ− C

) (H

12

Ez|1

)= 0 , (3.35)

where A, B and C are the matrices in (3.33), respectively. When we solve thegeneralized eigenvalue problem (3.35) we observe that stability is obtained if thetime step suggested by (3.32) is decreased by a factor 6/7, which is exactly therelative difference in stability along the imaginary axis (see Figure 3.6) betweenthe leap-frog and ABS3 schemes. Hence, the stability condition for ABS3 equals

c∆tABS3 ≤ 67

∆√2· ∆min

∆. (3.36)

This stability condition holds true also for the other two triangular grids, whicha similar analysis verifies.

To analyze the eigenvalues for a general unstructured grid including itsboundaries we can no longer use Fourier analysis. Instead, let

zn =(

Ez|n Hn− 1

2 Ez|n−1 Hn− 3

2 Ez|n−2

)T

, (3.37)

and after some straightforward rearrangements we are able to write (3.12–3.14)on matrix form as zn+1 = P(A,B)zn, where

P =

0BBBBBBBB@I + 625

576∆t2BA 23

24∆tB − 25

288∆t2BA 1

24∆tB 25

576∆t2BA

2524

∆tA I − 112

∆tA 0 124

∆tAI 0 0 0 0

0 I 0 0 0

0 0 I 0 0

1CCCCCCCCA . (3.38)

Note that the operators A and B are now taken from the code, since no analyticalexpressions exist for a general grid. Analyzing the eigenvalues of the companionmatrix P, for the grid shown in Figure 3.7, reveals that if we choose the timestep for ABS3 in the same manner as above we get the eigenvalue spectrumshown in Figure 3.7, where all eigenvalues are within the unit circle. If we usethe leap-frog scheme with the same time step, the largest eigenvalue is of theorder 1.0003. Hence the leap-frog scheme is unstable even for a time step well


within the stability limit along the imaginary axis. The eigenvalues close to theorigin in Figure 3.7 have to do with the fact that ABS3 is a multi-step scheme,but only one of the roots is relevant and we get a number of parasitic roots.However, since these roots are all close or equal to zero they are quickly dampedaway.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

0 2 4 6 80

2

4

6

8

Figure 3.7: Eigenvalues of the companion matrix P using ABS3 for a PMCcylinder scattering case. Primary grid around the cylinder is shown to the right.

3.6 Workload and memory requirements

To implement the solver efficiently we have chosen to compute as much as pos-sible initially. Hence, the update of the fields is accomplished using matricesacting on the respective vectors. Due to the fact that the matrices will be sparsebut without structure we store them in compress sparse row format. Hence,for each matrix we store three vectors. The first vector holds the position ofthe first nonzero on each row, the second vector keeps track of the columns ofthe nonzeros and the last vector contains their values. After the matrices havebeen created, all grid variables can be written to disk. Hence, this approach ismuch more efficient than recomputing the expressions needed to update the fieldvariables at every time step or using indirect addressing in several levels.

To be able to obtain a theoretical estimate of the memory requirements andnumber of arithmetic operations of the solver we have to make a few assumptions.First of all we neglect boundary conditions. We put σ equal to zero, but ε andµ may vary in space. The following is assumed about the grid:

• There are n triangles.

• There are three edges in each triangle and each edge is shared by twotriangles ⇒ 1.5 n edges.

3.6. WORKLOAD AND MEMORY REQUIREMENTS 27

• There are three nodes in each triangle and each node belongs to six trian-gles ⇒ 0.5n nodes and six dual edges per dual cell.

The field variables that we need to store are Ez, H · nd, H and H · tp. UsingABS3, Ez and H · tp are stored on three time levels, whereas H ·nd and H onlyare stored on the latest time level.

Variable Floating point numbersEz 3 · nH · nd 1.5 nH 2 · 0.5 n

H · tp 3 · 1.5 n∑10 n

Table 3.1: Memory requirements to store the field variables.

The memory requirements for the matrix operators used to update the fieldvariables can be calculated from (3.6–3.9).

Variable Floating point numbers IntegersEz n · 3 = 3 n 3 n + n = 4 nH · nd 1.5 n · 2 = 3n 3 n + 1.5 n = 4.5 nH n · 6 = 6 n 6 n + n = 7 n

H · tp 1.5 n · 5 = 7.5 n 7.5 n + 1.5 n = 9 n∑19.5 n 24.5 n

Table 3.2: Memory requirements to store the matrices.

Hence, using the results in Tables 3.1 and 3.2 the total memory requirementsfor the FVTD solver is 29.5 floating point numbers and 24.5 integers per cell.Using 64-bit precision for the floating point numbers and 32-bit precision for theintegers this means that approximately 334 bytes per cell are needed.

The number of arithmetic operations used by the solver are calculated from (3.6–3.9) and (3.12–3.14).

Variable Arithmetic operationsEz n · 11 = 11 nH · nd 1.5 n · 4 = 6 nH n · 12 = 12 n

H · tp 1.5 n · 14 = 21 n

Ez n · 5 = 5 n∑55 n

Table 3.3: Number of arithmetic operations per cell.


Thus, from Table 3.3 we conclude that the FVTD solver performs approx-imately 55 arithmetic operations per cell for each time step. This should becompared with FDTD, which needs eleven arithmetic operations per cell and36 bytes of memory. Hence, the FVTD solver is a factor five more expensive interms of arithmetic operation per cell and a factor of nine in terms of memoryper cell compared to FDTD.

Note that in the above calculations we have not used the fact that when theprimary and dual edges are orthogonal the second part of (3.8) vanishes andthe H values are not needed. A case where this will happen is the equilateralgrid. An implementation of the solver should of course take advantage of thisand the memory requirements and workload will go down considerably. Onan equilateral grid, or any other grid where orthogonality occurs, the memoryrequirements are 16.5 floating point numbers and 10 integers per cell. The totalmemory requirement is in this case 172 bytes using the same precision. Thenumber of arithmetic operations goes down to 31 per cell for such a grid. Thus,it is possible to reduce the memory requirements and the number of arithmeticoperations by approximately a factor of two for the special case when the gridsare mutually orthogonal.

Chapter 4

Finite Volume method in3D

The FVTD method is based on the following integral formulations of Faraday’sand Ampere’s laws:

∂

∂t

∫

A

µ H · n dA = −∮

Γ

E · dl , (4.1)

∂

∂t

∫

A

εE · n dA =∮

Γ

H · dl−∫

A

σ E · n dA , (4.2)

where A is an arbitrary area with unit normal n and Γ is the path that enclosesA.


The integral formulations are discretized on a staggered grid. The primarygrid consists of tetrahedral elements, which are generated by a Delaunay gridalgorithm. Associated with the unstructured tetrahedral grid is a mutually or-thogonal dual grid, the Dirichlet tessellation. The nodes in the dual grid are thecenters of the circumscribed spheres of each tetrahedron. However, for a generaltetrahedral grid there are circumcenters that lie outside the corresponding tetra-hedra. Such nodes deteriorate the accuracy of the method and they are hencereplaced by the barycenters of these tetrahedra. Replacing a circumcenter witha barycenter implies that the mutual orthogonality is lost locally. Note thatin the 2D solver all dual nodes are situated at the barycenters of the triangles.However, since the approach taken here is better circumcenters should be usedalso in 2D.

The normal components of the magnetic field are stored at each triangularface, whereas the normal components of the electric field are stored at each dual

29


face, see Figure 4.1.

E1

E2

E3

E4

ndj

H

Hr

q

t d

i,mt p

nip

j,k

Figure 4.1: A cell in the primary grid and a dual face.

Discretizing (4.1) and (4.2) over a primary and a dual face, respectively, resultsin

µpi

∂

∂t

∫

Api

H · npi dA = −

∑m

∫

Γpi,m

E · tpi,m dl , (4.3)

εdj

∂

∂t

∫

Adj

E · ndj dA =

∑

k

∫

Γdj,k

H · tdj,k dl − σd

j

∫

Adj

E · ndj dA , (4.4)

where Api is the area of the primary face i, Γp

i is the path that encloses Api , np

i

is the unit normal for face i and tpi,m are the unit vectors for the edges in face i.

The variables belonging to the dual face j are defined in the same way.In the case when the primary and dual grids are mutually orthogonal the

normals and edge vectors of the respective grids are parallel and the method isbasically a generalization of FDTD to unstructured grids. The normal compo-nent of the electric field, E · nd

j , is updated by circulating the magnetic fieldalong the edges of the dual face. The normal component of the magnetic field,H · np

i , at each triangular face is updated in the dual way by circulating theelectric field along the edges of the triangular face.

However, for non-orthogonal grids the normals and edge vectors do not alignand we need an approximation of the electric and magnetic fields projectedalong the respective edges. A straightforward projection of the fields normal tothe respective faces is not accurate enough. Therefore, this is accomplished byfirst calculating approximate values of the fields at the primary and dual nodes,respectively. The magnetic fields at the dual nodes, e.g. Hr in Figure 4.1, arecalculated as a least square fit of the magnetic fields normal to the four faces of


each tetrahedron. Electric node values are defined by doing a similar fit for theelectric fields normal to the dual faces of each dual cell.

Note that calculating the node values in this way differs from how it is donein 2D. In 2D the node values are updated using the two area integral formula-tions (3.1) and (3.2). However, using the corresponding volume integral formula-tions in 3D leads to significant node depolarization. For a plane wave travellingin a grid along one of the Cartesian directions only one of the components ineach vector field is supposed to be different from zero for perfect propagation.If any of the other components is different from zero depolarization of the wavehas occured. Looking at the vector fields at the nodes alone, we have observeddepolarization as high as on the order of ten percent. The amount of depolar-ization depends on the grid and is also present using the least squares approachbut to much less extent. However, this is not the whole story because it is theedge projected fields that are important and they tend to be accurate, but thenode values are not [53].

Following Riley et al. [55], the magnetic field projected along the dual edgetdj,k, see Figure 4.1, is evaluated as

H · tdj,k = (H · np

i )(np

i · tdj,k

)

+12

[(Hr + Hq)− ((Hr + Hq) · npi )np

i ] · tdj,k .

(4.5)

The electric field projected along the primary edge tpi,m is evaluated accordingly

as

E · tpi,m =

(E · nd

j

) (nd

j · tpi,m

)

+12

[(E1 + E2)−

((E1 + E2) · nd

j

)nd

j

] · tpi,m .

(4.6)

Taking a closer look at (4.5) and (4.6) we note that if the normal vectors, npi

and ndj , are parallel to the edge vectors, td

j,k and tpi,m, the second parts of (4.5)

and (4.6) vanish. Evaluating the fields projected along the edges in this wayguarantees that the divergence of the electric and magnetic fields are preservedon a local cell level up to machine precision; see Section 4.4 for a proof. Thishas been found crucial in order to suppress spurious modes in the solution.

All materials are defined relative to primary grid cells. Hence, the materialparameters require averaging to get approximate values at the faces. The averagepermeability on primary face i, µp

i , is defined by

µpi =

21

µpr

+ 1µp

q

, (4.7)

where r and q are the tetrahedral elements sharing face i. At a boundary face,µp

i equals the permeability of the only tetrahedron associated with the face. Theaverage electric material parameters on dual face j, εd

j and σdj , are constructed

as


εdj =

∑q

εpqA

dj,q

Adj

, σdj =

∑q

σpqAd

j,q

Adj

, (4.8)

where Adj,q is the part of the area of the dual face which is inside tetrahedron q

and the sums are taken over the tetrahedral elements associated with the dualface. Performing the material averaging in this manner preserves the second-order accuracy of the method for inhomogeneous materials. The reason for thedifferent averaging of magnetic and electric materials is that the normal magneticfield is discontinuous at a material interface, whereas the tangential electric fieldis continuous. Therefore, harmonic averaging is used for µ and area weightedarithmetic averaging is used for ε and σ [3].

The boundary condition for a PEC is implemented in the same way as thePMC boundary condition in 2D according to (3.11). For a PMC the tangen-tial magnetic components, should equal zero at the boundary. However, theH components are not situated on the boundary. The boundary condition isenforced by setting the magnetic field below the conductor equal to the valueof the magnetic field directly above the conductor with a change of sign in thetangential component. The PMC boundary condition is non-physical, but usefulas a symmetry boundary condition.


The equations (4.3) and (4.4) are discretized in time using the third-order stag-gered Adams–Bashforth scheme (ABS3) [27]

H · npi |n+ 1

2 = H · npi |n−

12 (4.9)

+∆t

µpi Ap

i

Xm

25

24E · tpi,m|n−

1

12E · tpi,m|n−1+

1

24E · tpi,m|n−2

∆p

i,m ,

E · ndj |n+1 = E · nd

j |n (4.10)

+∆t

εdj Ad

j

Xk

25

24H · tdj,k|n+ 1

2 − 1

12H · tdj,k|n−

12 +

1

24H · tdj,k|n−

32

∆d

j,k

+∆tσd

j

2 εdj

E · nd

j |n+1+E · ndj |n

,

where ∆pi,m are the lengths of the primary edges tp

i,m and ∆dj,k are the lengths of

the dual edges tdj,k. Note that the notation E ·nd

j |n means E ·ndj at time step n

and accordingly for the other components. The stability properties for the ABS3scheme are analyzed in Section 4.5. The FVTD solver has been parallelized usingMPI and OpenMP [51].

4.3. CREATING THE DUAL GRID 33

4.3 Creating the dual grid

The creation of the dual grid from the primary grid is performed in the pre-processing phase. To simplify the description the dual grid at a PMC, usedas symmetry boundary condition, is not treated here. Each tetrahedron has acorresponding dual node. The dual node is situated at the center of the circum-scribed sphere if the center is inside the tetrahedron. As discussed previouslythe circumcenters are preferred due to orthogonality properties, however sincedual nodes outside the tetrahedra will deteriorate the accuracy of the solver suchnodes are replaced by the barycenters.

A dual edge passes through every inner triangular face, e.g. tdj,n in Figure 4.1.

Similarly, each inner primary edge, e.g. tpi,m in Figure 4.1, has a correspond-

ing dual face. The dual edge tdj,n belongs to at most three dual faces, namely

the faces that are associated with the three edges of the triangular face npi in

Figure 4.1. The dual edges in each dual face are sorted to specify a certaincirculation around the perimeter of the face, which in turn defines a normaldirection.

Finally, each inner node has a surrounding dual element. The dual facesbelong to two dual elements, e.g. the dual face with normal nd

j in Figure 4.1belongs to the dual elements which surround the nodes 1 and 2. The face ispositive for one of these elements, i.e. the normal of the face is defined outwards,and negative for the other, i.e. the normal is defined inwards. Hence, there isa one-to-one correspondence between the nodes, edges, faces and cells of theprimary grid and the nodes, edges, faces and cells of the dual grid.

The definition of dual grid parameters such as length and direction of dualedges is trivial. The areas and normals of the dual faces are slightly morecomplicated due to the fact that the dual faces are not necessarily planar. Theface area normal, Nd

j , is defined as Ndj =

∫Ad

jnd

j dA. It is calculated by splittingthe face into triangles using the barycenter of the face. The area of the face isthen calculated as a dot product between the unit normal, nd

j , and the face areanormal, Nd

j . Hence, the areas and normals are calculated by a piecewise planarapproximation of the faces.

4.4 Preservation of divergence

Here we prove that the FVTD solver preserves the divergence of the magneticflux density vector, B. The proof for the divergence of the electric flux densityvector, D, is analogous.

Proposition 4.4.1 The FVTD solver preserves ∇ · B on a local cell level tomachine precision.


Proof Consider,

∂

∂t

∫

V pq

∇ ·B dV =∂

∂t

∫

V pq

∇ · (µH) dV =∂

∂t

∮

Ap

µ H · np dA ,

where Gauss’ divergence theorem is used to get the last equality. Splitting thearea integral path into i parts gives

∑

i

∂

∂t

∫

Api

µH · npi dA .

So far we are using continuous variables. If we assume that H · npi and µ are

constant on each primary face we obtain

∑

i

µpi

∂

∂t

∫

Api

H · npi dA = −

∑

i

∑m

∫

Γpi,m

E · tpi,m dl = 0 ,

where (4.3) has been used. The last expression is zero because in the doublesum the electric field projected along the edges belonging to the faces i appeartwice with opposite signs. Hence, the divergence of the magnetic flux densityvector is preserved up to machine precision. ¥

Note that the proof does not assume that the faces have to be planar, whichis not necessarily the case for the dual faces.


In Section 3.5 we analyzed the ABS3 scheme on uniform triangular grids and on ageneral triangular grid. We have also shown in experiments, see Section 7.1, thatif the time step is chosen according to (3.36) ABS3 is stable for general triangulargrids. In this section we will analyze ABS3 in 3D. The great advantage of usingABS3 compared to the commonly used leap-frog scheme is that we obtain astable and accurate solution even if the highest frequency eigenvalues of thespace operator have a small real part. Using ABS3 will suppress the amplitudeof the highest frequency components, but they are not accurately resolved bythe grid and are therefore considered parasitic.


−0.2 −0.1 0 0.1 0.2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.01 −0.005 0 0.005 0.01−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 4.2: The stability region for ABS3 is shown to the left. To the right theeigenvalues for the perturbed 4 × 4 × 4 grid. Note the different x-scales in thefigures.

To analyze the eigenvalues in vacuum for the semi-discrete case, we write thesolver on matrix form as

∂hn

∂t= −Mµ Ce Pe en , (4.11)

∂en

∂t= Mε Ch Ph hn , (4.12)

where Mµ and Mε are diagonal material matrices, which also include areas andedge lengths, Ce and Ch represent the discrete curl operators, Pe and Ph areprojection operators, and hn and en are the normal field unknowns scaled withthe respective edge lengths.

Proposition 4.5.1 The FVTD solver is stable on orthogonal grids.

Proof On orthogonal grids the two projection operators equal identity. If weeliminate hn from (4.11) and (4.12) we obtain the following second-order systemfor en:

∂2en

∂t2= −Mε ChMµ Ce en = −Men . (4.13)

The duality of the staggered grids implies that the curl operators satisfy,Ch = CT

e . Furthermore, the symmetric and positive definite material matricescan be split as

Mε = M1/2ε M1/2

ε , Mµ = M1/2µ M1/2

µ .

Therefore, we can factorize M as

M = M1/2ε

(M1/2

µ CeM1/2ε

)T (M1/2

µ CeM1/2ε

)M−1/2

ε

= M1/2ε

(Y T Y

)M−1/2ε .

(4.14)


Since, Y T Y and M1/2ε

(Y T Y

)M−1/2ε are spectrally equivalent we can conclude

that M is a positive semi-definite matrix. Thus, the semi-discrete FVTD solveris stable on orthogonal grids. ¥

On non-orthogonal grids it can be shown that the solver is stable if

Mµ = PhM−1µ (4.15)

Mε = PeM−1ε (4.16)

are symmetric and positive definite operators [69]. Unfortunately, on a generaltetrahedral grid it is very hard to construct operators, which fulfill these con-ditions. In 2D Schumann et al. [58] have constructed them by using a finiteelement type approach, but an extension to 3D of their proposed method doesnot seem possible. The projection operators defined in (4.5) and (4.6) do notsatisfy these properties.

On matrix form the first-order system (4.11–4.12) can be written as(

hn

en

)=

(0 AB 0

)(hn

en

), (4.17)

where

A = −Mµ Ce Pe , (4.18)B = Mε Ch Ph . (4.19)

Consider a grid consisting of 4 × 4 × 4 cubes, where each cube is split into fivetetrahedra in the same way as the cells in the transition layer, see Figure 6.3.Calculating the eigenvalues of the matrix in (4.17) numerically reveals that theyare all on the imaginary axis. In this case, the primary and dual grids aremutually orthogonal. Let us perturb the grid such that the orthogonality is lostby moving the midpoint along one of the Cartesian directions. The eigenvalues,scaled with the CFL number, are shown in the right part of Figure 4.2. Hence,the loss of orthogonality locally is sufficient to get eigenvalues with a nonzeroreal part.

For the fully-discrete case, let

zn =(E · nd|n H · np|n− 1

2 E · nd|n−1 H · np|n− 32 E · nd|n−2

)T

. (4.20)

After some straightforward rearrangements we are able to write (4.9) and (4.10)on matrix form as zn+1 = P(A,B)zn, where

P =

0BBBBBBBB@I + 625

576∆t2BA 23

24∆tB − 25

288∆t2BA 1

24∆tB 25

576∆t2BA

2524

∆tA I − 112

∆tA 0 124

∆tAI 0 0 0 0

0 I 0 0 0

0 0 I 0 0

1CCCCCCCCA . (4.21)


Analyzing the eigenvalues of the companion matrix P, for the perturbed 4×4×4grid, results in the eigenvalue spectrum shown in Figure 4.3, where all eigenvaluesare within the unit circle. The time step is chosen on the stability limit for theFVTD solver and is given by the largest eigenvalue of the semi-discrete problem.The eigenvalues to the left in Figure 4.3 are the first to move outside the unitcircle if the time step is increased. For a general unstructured grid an eigenvalueanalysis is not possible and hence the CFL condition is not known. A rule ofthumb, which works well in practice, is to decrease the time step given by thestability condition for stand-alone FDTD with a factor given by the quotientbetween the shortest primary edge and the shortest edge in the structured grid.Note that this type of stability condition is very similar to the one derived in2D, see (3.36). The only difference is that here ∆min is the shortest among theprimary edges, whereas in 2D it is the shortest among all unstructured edges.Building the dual grid using the circumcenters of the tetrahedra often createssome very short dual edges. However, in experiments we have seen that they donot affect the CFL condition.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Figure 4.3: Eigenvalues of the companion matrix P using ABS3 for the perturbed4× 4× 4 grid. The time step is chosen on the stability limit.

If ABS3 is replaced by the leap-frog scheme and the same time step is used,the largest eigenvalue is of the order 1.004, despite the fact that this time step iswell within the stability limit for leap-frog. Hence, not surprisingly the leap-frogscheme is unstable due to the fact that we have eigenvalues with a nonzero realpart.

However, using ABS3 does not guarantee long-term stability on general tetra-hedral grids in 3D. Eigenvalues to the semi-discrete problem might have a realpart that is too large for ABS3 to handle. One way of enhancing the numericalstability is to suppress the amplitude of the highest frequency components. Alow-pass filter is ideal for this purpose. The undesirable highest frequency com-ponents are eliminated, whereas the rest of the solution remains unchanged. We


use a spatial filter based on the Laplace operator. The filter is only applied tothe node values and therefore it does not act on edges, which are orthogonal totheir corresponding faces. For edges which are not orthogonal the larger the de-viation from orthogonality the more the filter affects the edge values. A filteredvalue Hr of Hr is given by

Hr = Hr +1

2N

N∑q=1

(Hq −Hr) , (4.22)

where the sum is taken over the nodes q, which share an edge with node r.The filtering for the electric node values is analogous. Note that this filteris not needed in 2D, where the use of ABS3 is sufficient to obtain long-termstability. However, the grids in 3D are in general much more complex for realisticapplications and the deviation from orthogonality could be large, which is thesource of the instability.

An analysis of this filter on uniform grids in 1D and 2D can be found in [19].In practice this filter improves stability considerably without deteriorating theaccuracy in the solutions [19]. However, it does not guarantee stability and latetime instabilities may still occur especially for highly resonant structures, seeSection 8.2.

4.6 Workload and memory requirements

The implementation in 3D follows the same strategy as in 2D, i.e. as much aspossible is computed initially and the update of the field variables is performedby matrix-vector multiplications. The matrices are stored in compress sparserow format, see Section 3.6.

To obtain a theoretical estimate of the memory requirements and numberof arithmetic operations required by the FVTD solver we have to make a fewassumptions. First of all we neglect boundary conditions. Note that in a realproblem only a few layers of unstructured cells are used to get a better geo-metrical representation of the object and hence to neglect boundaries should beconsidered a major simplification. We put σ equal to zero, ε and µ may vary inspace but are not dependent on frequency. The following is assumed about thegrid (based on the grids used in Chapter 8):

• There are n tetrahedra.

• There are four triangles in each tetrahedron and each triangle is shared bytwo tetrahedra ⇒ 2 n triangles.

• There are six edges in each tetrahedron and each edge belongs to fivetetrahedra ⇒ 6 n

5 edges and five dual edges per dual face.

4.6. WORKLOAD AND MEMORY REQUIREMENTS 39

• There are four nodes in each tetrahedron and each node is shared by 24tetrahedra ⇒ n

6 nodes.

• Each node belongs to fourteen edges ⇒ fourteen faces per dual cell.

• Half of the primary and dual edges are orthogonal to their respective faces.

The variables that have to be stored are H, E, H · np, E · nd, H · td andE · tp, where the last two are stored on three levels and the other four on onelevel.

Variable Floating point numbersH 3 · nE 3 · n

6H · np 2 nE · nd 6 n

5

H · td 3 · 2 n

E · tp 3 · 6 n5∑

16.3 n

Table 4.1: Memory requirements to store the field variables.

Variable Floating point numbers IntegersH 3 n · 4 = 12 n 12 n + 3 n = 15 nE n

2 · 14 = 7 n 7 n + n2 = 15 n

2H · np 2 n · 3 = 6 n 6 n + 2 n = 8 nE · nd 6 n

5 · 5 = 6 n 6 n + 6 n5 = 36 n

5

H · td 2 n · 4 = 8 n 8 n + 2 n = 10 n

E · tp 6 n5 · 4 = 24 n

524 n5 + 6 n

5 = 6 n

H 3 n · 5 = 15 n 15 n + 3 n = 18 n

E n2 · 15 = 15 n

215 n2 + n

2 = 8 n∑66.3 n 79.7 n

Table 4.2: Memory requirements to store the matrices.

The numbers in the Tables 4.1 and 4.2 are taken from (4.3–4.6) and (4.22),as well as from the grid assumptions and the way in which the node values arecalculated. Using the results in Tables 4.1 and 4.2 we are able to sum up thetotal memory requirements for the FVTD solver, which are approximately 82.6floating point numbers and 79.7 integers per tetrahedron. Using 64-bit precisionfor the floating point numbers and 32-bit precision for the integers this meansthat approximately 980 bytes per cell are needed. It is interesting to note thatthe cost in memory we have to pay for using ABS3 instead of leap-frog is onlyabout 50 bytes per cell, i.e. only about five percent of the total cost. The filter,


however, is responsible for a significant 30 percent of the total memory. Comparewith FDTD, whose memory requirements are six floating point numbers and sixintegers per cell, which results in 72 bytes per cell. Thus, the FVTD solver needsaround a factor of fourteen more memory per cell than FDTD.

Variable Arithmetic operationsH 3 n · 7 = 21nE n

2 · 27 = 27 n2

H · np 2 n · 6 = 12nE · nd 6 n

5 · 10 = 12 n

H · td 2 n · 12 = 24n

E · tp 6 n5 · 12 = 72 n

5

H 3 n · 10 = 30n

E n2 · 30 = 15 n∑

141.9 n

Table 4.3: Number of arithmetic operations per cell.

The number of arithmetic operations per cell used by the FVTD solver toperform one time step is given in Table 4.3. In addition to the equations usedto obtain the memory requirements we also use (4.9–4.10). Due to the fact thatthe update of the field variables are matrix-vector multiplications the number ofadditions and multiplications is either equal or there is one more multiplication.

Hence, from Table 4.3 we conclude that the FVTD solver performs approx-imately 142 arithmetic operations per cell for each time step. Also in terms ofarithmetic operations using ABS3 introduces only a small over-head of about tenpercent, whereas the filter adds about 30 percent extra arithmetic operations.FDTD makes 36 arithmetic operations per cell. Thus, FDTD is about a factorof four cheaper per cell compared to the FVTD solver.

Chapter 5

Finite Element method in3D

The Finite-Element Time-Domain (FETD) method is based on the followinginitial boundary value problem:

ε∂2E

∂t2+ σ

∂E

∂t+∇×

(1µ∇×E

)= −∂J

∂t, (5.1)

n× (∇×E) = 0 on ΓPMC , (5.2)

n×E = 0 on ΓPEC , (5.3)

n×E = n×Efdtd on ΓHyb , (5.4)

E = E0 at t = 0 , (5.5)

∂E

∂t= v0 at t = 0 . (5.6)


The weak formulation of the above initial boundary value problem can be statedas: Find E ∈ W , where W = H(curl, V ) =

w : w ∈ L2(V ),∇× w ∈ L2(V )

,

such that

∫

V

((ε∂2E

∂t2+ σ

∂E

∂t

)·w +

1µ

(∇×E) · (∇×w))

dV =

−∫

Γ

1µ

[n× (∇×E)] ·w dΓ−∫

V

∂J

∂t·w dV , (5.7)

41

42 CHAPTER 5. FINITE ELEMENT METHOD IN 3D

∀w ∈ W . The finite element process is to seek the best solution Eh of (5.7) ina finite-dimensional subspace Wh ∈ W .

As basis and test functions we use “edge” or “Whitney” elements [47]. Theseelements constitute the natural FEM analogue to the FDTD method in that theyyield the same method on a Cartesian grid if trapezoidal integration is used forthe elemental mass and stiffness matrices. They give a “physical” approximationin the sense that only tangential continuity across element edges is enforced, andnot normal continuity. This property allows edge elements to properly model nullspace vector fields for the ∇×∇×-operator consisting of electrostatic solutionsE = ∇φ. Therefore edge elements are free from spurious modes, which is not thecase for standard nodal elements. Another advantage is that it is straightforwardto enforce boundary conditions on material interfaces as well as on conductingsurfaces.

The basis function for a tetrahedral edge i joining node i1 and i2 is given by

ϕei = lei

(Φe

i1∇Φei2 − Φe

i2∇Φei1

), (5.8)

where Φei1

and Φei2

are the standard linear nodal basis functions for the twonodes and lei is the length of the edge. A sketch of the basis function for edge iis plotted in Figure 5.1.

n2

n4

n3

n1

Figure 5.1: The vector valued basis function ϕei for the edge connecting node 1

and 2. ϕei is illustrated for tetrahedron e. It is also defined in all other tetrahedra

that share this edge.

The electric field in each tetrahedra can be expanded as

Ee =6∑

i=1

Ei ϕei , (5.9)

where Ei denotes the tangential electric field along edge i. The stable cou-pling of the FDTD and FETD methods, presented in Section 6.3, requires alsohexahedral and pyramidal elements. Edge basis functions for hexahedral ele-ments are described in [37], whereas the ones for pyramidal elements are found


in [14, 29, 80]. The respective elemental matrices for the coupling of edge i andedge j are given by

Meij =

∫

V e

ε ϕei · ϕe

j dV , (5.10)

Keij =

∫

V e

σ ϕei · ϕe

j dV , (5.11)

Seij =

∫

V e

1µ

(∇× ϕei ) ·

(∇× ϕej

)dV . (5.12)

Assembling the elemental matrices yields the following system of ordinarydifferential equations:

Md2E

dt2+ K

dE

dt+ SE = fA , (5.13)

where the vector E contains the unknowns and

Mij =∑

e

Meij , Kij =

∑e

Keij , Sij =

∑e

Seij . (5.14)

The right-hand side fA contains the contributions from the boundary conditionsand source terms, i.e. we have fA = fD + fN + fJ . The general form of thecontributions from a time dependent Dirichlet boundary, ΓD, is

fDi = −

∑

j∈ΓD

(Mij

d2Ej

dt2+ Kij

dEj

dt+ SijEj

). (5.15)

The PEC boundary condition is a simple special case of this, where these termsequal zero. At a grid interface values calculated by FDTD enter the FETDdomain through (5.15). If n × (∇ × E) is a known function on a Neumannboundary, ΓN , then

fNi = −

∫

ΓN

1µ

[n× (∇×E)] · ϕi dΓ , (5.16)

has to be computed. At a PMC boundary n × (∇ × E) = 0. Finally, thecontributions from the source terms are given by

fJi = −

∫

V

∂J

∂t· ϕi dV . (5.17)



To discretize (5.13) we use the Newmark-Beta scheme. The time derivatives areapproximated by central differences according to

d2

dt2E|t=n∆t ≈ 1

∆t2(En+1 − 2En + En−1

), (5.18)

d

dtE|t=n∆t ≈ 1

∆t

(γEn+1 + (1− 2γ)En − (1− γ)En−1

), (5.19)

and the function is expressed as an average as

E|t=n∆t ≈ βEn+1 +(

12− 2β + γ

)En +

(12

+ β − γ

)En−1 . (5.20)

An unconditionally stable, second-order accurate scheme is obtained if and onlyif γ = 1

2 and β = 14 . Using these constants and substituting (5.18–5.20)

into (5.13) results in(

M +∆t

2K +

∆t2

4S

)En+1 = 2

(M − ∆t2

4S

)En

−(

M − ∆t

2K +

∆t2

4S

)En−1

+ ∆t2fA|t=n∆t .

(5.21)

The matrices M , K and S are symmetric, M is positive definite, and K andS are positive semi-definite (see Lemma 5.3.1). The properties of the matricesmake it possible to solve the system of equation (5.21) using a PreconditionedConjugate Gradient (PCG) method. As preconditioner we use an incompleteCholesky factorization. Convergence is typically reached after 10–20 iterations.


In this section we will prove that the FETD method is unconditionally stable.The proof is based on the symmetry and positive definiteness of the matrices,according to Lemma 5.3.1.

Lemma 5.3.1 The mass matrix M is symmetric and positive definite, the massmatrix K and the stiffness matrix S are symmetric and positive semi-definite.

Proof The symmetry of the respective matrices are easily realizable from (5.10–5.12). The components of the mass matrix are given by

Mij = (ϕi, ϕj) =∫

V

ε ϕi · ϕj dV . (5.22)


Let Eh =N∑

i=1

Ei ϕi then we have

ET ME =N∑

i,j=1

EiMijEj =N∑

i,j=1

Ei(ϕi, ϕj)Ej =

N∑

i=1

Eiϕi,

N∑

j=1

Ejϕj

=(Eh,Eh

) ≥ 0 ,

(5.23)

with equality only if Eh ≡ 0, that is only if E ≡ 0. Hence, M is positive definite,which is also the case for K if σ > 0. The stiffness matrix is similarly shown tobe positive semi-definite since E = ∇φ implies ∇×E = 0. ¥

Using Lemma 5.3.1 we are now able to prove Proposition 5.3.1:

Proposition 5.3.1 The FETD method is unconditionally stable.

Proof For simplicity we assume that σ = 0, which implies that K is zero andthat we have no sources such that fA = 0. Multiplying (5.21) from the left withM−1 gives

(I +

∆t2

4M−1S

)En+1 = 2

(I − ∆t2

4M−1S

)En

−(

I +∆t2

4M−1S

)En−1 .

(5.24)

Since M is symmetric and positive definite it can be factorized as M = CT C.Hence,

∆t2

4M−1S =

∆t2

4C−1C−T SC−1C = C−1PC , (5.25)

where

P =∆t2

4C−T SC−1 . (5.26)

The matrix S is symmetric and positive semi-definite and it is easily shown thatalso P has these properties. Therefore, P can be diagonalized by an orthogonalmatrix Q consisting of the eigenvectors of P such that QT PQ = Λ, where Λ holdsthe nonnegative eigenvalues of P . The same matrix diagonalizes also I + P andI − P . With QT CEn = zn we obtain

(I + Λ) zn+1 = 2 (I − Λ) zn − (I + Λ) zn−1 . (5.27)

We have now decoupled the equations and it is enough to treat the scalar equa-tion

(1 + λ) zn+1 = 2 (1− λ) zn − (1 + λ) zn−1 . (5.28)

The corresponding characteristic equation is given by

(1 + λ)µ2 − 2 (1− λ)µ + (1 + λ) = 0 , (5.29)


with solutions

µ =1− λ

1 + λ±

√(1− λ

1 + λ

)2

− 1 . (5.30)

By letting sin θ = 1−λ1+λ we obtain

µ = sin θ ±√

sin2θ − 1 = sin θ ±√−cos2θ = sin θ ± i cos θ . (5.31)

Hence, the magnitude of the two simple roots are one as expected for the losslesscase with σ = 0, which proves that using Newmark-Beta for time integrationresults in an unconditionally stable scheme. A similar analysis for σ > 0 showsthat the magnitude of the eigenvalues is strictly less than one in that case. ¥

Chapter 6

Hybridization

In this chapter we describe the coupling of the FVTD and FETD solvers withFDTD. The FDTD method is thoroughly described in the literature, see e.g.[63]. We simply note that it is an explicit, second-order accurate method usedon a staggered Cartesian grid. The coupling of the FDTD solver and FETDin 2D is described in [1]. The hybrid solver combining FDTD and FVTD isdenoted FD-FV, and the one combining FDTD and FETD is denoted FD-FE.

There are a number of reasons to use a hybrid solver. First of all the FDTDmethod is in general more efficient on simple geometries than unstructuredsolvers with the same order of accuracy, see e.g. Sections 3.6 and 4.6. Sec-ondly, unstructured grids can conform to complex objects, whereas Cartesiangrids have to use staircased approximations of the objects which often are notaccurate enough. By using a hybrid solver we can therefore take advantage ofthe efficiency of FDTD without suffering from the staircasing errors. Thirdly,accurate absorbing boundary conditions exist in FDTD, but are not as welldeveloped for the unstructured solvers.

6.1 Coupling of FDTD and FVTD in 2D

The unstructured grids have a semi-structured transition layer of split rectanglesas outermost cells. This gives a total collocation of hybrid components andtherefore interpolation between the grids is fully avoided. Coupling between theFDTD solver for structured grids and the FVTD solver for unstructured gridsis shown in Figures 6.1 and 6.2.

47

48 CHAPTER 6. HYBRIDIZATION

Hx Hy

A B ba

Hx,HyEzEz Hy~

x

y

Figure 6.1: Coupling between structured FDTD and unstructured FVTD. Thevariables along ΓA-Γa and ΓB-Γb are shared by the two solvers.

A bBPoints on Points ont

x

and a and

2

1

Γ Γ Γ

4

Γ

3

FVTD regionFDTD region

Figure 6.2: The timestepping mechanism for the FD-FV hybrid.

The timestepping algorithm for the hybrid solver is as follows:

1. The Hx and Hy components in the structured grid (to the left in Figure 6.1)are updated using the standard FDTD method.

2. The H and H ·nd components in the unstructured grid are updated by theFVTD solver using the Ez components along Γa as boundary condition.The H · tp components along Γb are sent to ΓB in the structured grid.

3. The FVTD solver updates the Ez components in the unstructured domain.

4. Finally, the time step is completed when FDTD updates the Ez compo-nents in the structured domain using the H · tp components along ΓB asboundary condition. The Ez components along ΓA are sent to Γa to beused in the next time step by the FVTD solver.

6.2. COUPLING OF FDTD AND FVTD IN 3D 49

6.2 Coupling of FDTD and FVTD in 3D

The hybridization technique presented here requires the unstructured grid tohave a transition layer consisting of semi-structured cells. This means that theoutermost unstructured cells are obtained by splitting a brick into five tetrahedra(the brick is of the same size as the FDTD cell). This outermost layer of cellsis called the transition layer since the unstructured grid and the structured gridoverlap here, see Figure 6.3. Hence, a number of components coincide in spaceand time for the solvers in the transition layer and it is these components thatare involved in the actual communication between the solvers.

Figure 6.3: The transition layer is based on coincidental structured and unstruc-tured cells. To the left an FDTD cell and to the right a cluster of five tetrahedraforming a brick is shown.

Based on Figure 6.3 the hybridization technique can be described as follows:

Left part of Figure 6.3, Right part of Figure 6.3,the structured cell the unstructured cells

The four Cartesian components to theleft (filled) are sent to the correspond-ing components in the unstructuredregion. The diagonal component (alsofilled) is calculated by interpolation ofthe four filled components and is alsosent to the unstructured region.

=⇒ The five filled components are re-ceived from the structured region andare treated by the unstructured solveras an inhomogeneous Dirichlet bound-ary condition.

⇓ ⇓(A full time step is taken) (A full time step is taken)

⇓ ⇓The remaining eight Cartesian com-ponents (hollow) are received from thecorresponding components in the un-structured region.

⇐= When the components are updated inthe unstructured region the remainingeight Cartesian components (hollow)are sent to the structured solver.

50 CHAPTER 6. HYBRIDIZATION

The remaining five diagonal components (ordinary arrows) in the right partof Figure 6.3 are not involved in the hybridization. Note that sending the eightCartesian components from the FVTD solver is equivalent in space to sendingthe four tangential H fields in the transition layer to FDTD.

It is worthwhile to point out that the primary and dual grids are mutuallyorthogonal in the transition layer. Without this orthogonality also node valueswould have to be sent from the structured region, to update the electric edgevalues in the transition layer according to (4.6). In that case a careful interpo-lation would have to be performed since there are no nodal values in the FDTDsolver.

6.3 Coupling of FDTD and FETD in 3D

It is well known that the FDTD method is stable as long as the CFL conditionis satisfied [52]. The FETD method is unconditionally stable, while the FVTDmethod is stable on orthogonal grids as long as the CFL condition is satisfied.Coupling of two stable methods does not necessarily result in a stable hybridmethod. The hybridization technique described in Section 6.2 is not stable. Thesource of the instabilities is the diagonal component that is not present in FDTDand therefore is calculated by interpolation. This breaks the symmetry of thehybrid method and makes it unstable.

To develop a symmetric interface we follow [57] and use a two-cell thick layerof hexahedral elements, such that cells neighboring the tetrahedral region aresubdivided into pyramidal and tetrahedral elements, see Figure 6.4. This givesa perfect geometrical coupling of the structured and unstructured grids and theneed for interpolation at the grid interface is fully avoided. Using trapezoidalintegration instead of exact integration on the hexahedral elements gives riseto an FETD method which is identical to FDTD. Hence, we can view the FD-FE method as an Explicit-Implicit algorithm, where the hexahedral elementsare treated explicitly, and the pyramidal and tetrahedral elements are treatedimplicitly. Therefore, the FD-FE method is stable as long as the time stepsatisfies the CFL condition for the FDTD method [34].

6.3. COUPLING OF FDTD AND FETD IN 3D 51

Pyramid

PyramidTetra

Tetra

Tet

ra

Hexa

Hexa

Hexa

Pyra

mid

FDTD

FETD

Dirichlet BC for

Dirichlet BC for

FETD E-field

FDTD E-field

FETD

FDTD

Figure 6.4: Hybrid grid interface to connect FETD and FDTD.

Based on Figure 6.4 the hybridization technique to couple the FDTD andFETD methods can be described as follows:

Left part of Figure 6.4, FETD Right part of Figure 6.4, FDTD

The filled components are receivedfrom the FDTD region and aretreated by the FETD method as an in-homogeneous Dirichlet boundary con-dition.

⇐= The filled Cartesian components aresent to the corresponding componentsin the FETD region.

⇓ ⇓(A full time step is taken) (A full time step is taken)

⇓ ⇓When the FETD region is updatedthe hollow Cartesian components aresent to the corresponding componentsin the FDTD region.

=⇒ The remaining Cartesian components(hollow) are received from the corre-sponding components in the FETDregion and are treated as an inhomo-geneous Dirichlet boundary condition.

The reason for having a different coupling of the FDTD and FVTD methodsis that the coupling described here does not solve the stability problem for theFD-FV method. This is due to the fact that the splitting of the hexahedralelements into pyramidal and tetrahedral elements results in a dual grid which isnot mutually orthogonal to the primary grid.

Chapter 7

Numerical experiments in 2D

In this chapter we validate the FD-FV solver in 2D for different circular cylindersand a PMC wall. We compare with results obtained by stand-alone FDTD andthe FD-FE hybrid solver [1].

7.1 Stability

A critical aspect of every numerical solver is whether or not it is stable. Both theFDTD and FVTD solvers are stable as long as they fulfill their CFL conditions.However, this does not guarantee stability for the hybrid solver since couplingof two stable solvers might result in an unstable solver.

The entire hybrid solver can be written as

un+1 = Aun + fn , (7.1)

where fn represent the source terms, which must be bounded. The vector ucontains the unknown values of both E and H fields and includes unknownsfrom several time levels from both the structured domain and the unstructureddomain. Furthermore u0 is given by initial values. The solver is stable if andonly if the absolute value of all eigenvalues of A are less than or equal to one.We examined the dominant eigenvalue of A by running the code for a very largenumber of time steps. This procedure is in numerical linear algebra known asthe power method.

We performed stability tests for several different grids. We ran the code forten million time steps. As Absorbing Boundary Condition (ABC) we used atwelve cell thick U-PML layer [26]. A plane wave was continuously fed into thecomputational domain using Huygens’ surfaces [63].

All grids tested were stable for the FD-FV hybrid as long as the time stepwas selected properly. The time step we used was a factor of

√2/4 smaller than

the stability limit for the FDTD solver. Using a time step twice as large proved

53

54 CHAPTER 7. NUMERICAL EXPERIMENTS IN 2D

to be unstable. This is not surprising since it violates the stability conditionin (3.36). All the grids used in these stability tests have a shortest edge lengthof slightly less than half the edge length in the structured grid.

7.2 Convergence

The order of convergence is at least two in both time and space for FDTDand FVTD. The hybridization technique presented in Section 6.1 is designedto preserve this property. We have validated this by doing calculations on fourdifferent circular cylinders: PMC, PEC, diamagnetic and dielectric.

We compare our hybrid results with FDTD solutions which use staircaseapproximations of the circular cylinder. In all cases, we have used a twelvecell thick U-PML layer and Huygens’ surfaces to create a plane wave, with aGaussian shape given by

f(t) = e−(t−t0)2/t2w , t0 = 20

c0, tw = t0

6 . (7.2)

(a) PEC/PMC (b) Material

Figure 7.1: The coarsest unstructured grids.

Figure 7.1 displays the unstructured grids for a circular cylinder with a radiusof two meters. These 8×8 meter grids were inserted in the center of a structuredFDTD grid with 40× 40 cells of the size ∆ = 1 m.

In all FDTD calculations the time step was chosen such that CFL = 1/√

2,while the time step for the FD-FV hybrid was a factor of two smaller for thePEC and PMC cylinders and a factor of four smaller for the material cylinders.

7.2. CONVERGENCE 55

The error was measured in the Ez components in the FDTD grid that werelocated closest to the unstructured grid. The error was defined as the differencebetween the computational solution and a reference solution.

The reference solutions were obtained by highly resolved FDTD calculations.We used Nx = Ny = 10, 240 for the PEC and PMC cylinders and Nx = Ny =5, 120 for the material cylinders. The errors in these reference solutions wereestimated by comparing them with solutions where the problem size was a factorof two smaller in each dimension. In all cases, the errors were less than 4 · 10−4.The Huygens’ surfaces were placed two meters from the outer boundary on allgrids and a refinement factor of two was used.

100 150 200 25010

−5

10−4

10−3

10−2

10−1

Mea

n(A

bs(E

rror

))

Time [ns]

PMC

Figure 7.2: Errors for three refinement levels for the PMC cylinder; solid lines:FDTD, dashed lines: FD-FE and dash-dotted lines: FD-FV.


The results from the convergence study are shown in Figures 7.2–7.5. Theplots display the mean value of the absolute error in the measure points.

100 150 200 25010

−5

10−4

10−3

10−2

10−1

Mea

n(A

bs(E

rror

))

Time [ns]

PEC

Figure 7.3: Errors for three refinement levels for the PEC cylinder; solid lines:FDTD, dashed lines: FD-FE and dash-dotted lines: FD-FV.

The peak of the Gaussian pulse passes the circular cylinder at t ≈ 125 ns.We note that in the beginning, the error is larger for the FD-FV hybrid. This iscaused by the shorter time step which gives a larger error in the FDTD solver [63].

For PEC and PMC, we note that around 110–115 ns, the error of the FDTDsolution is slightly smaller than the errors of the hybrid solutions. This is mostlikely due to reflections in the transition layer between the structured grid andthe unstructured grid. However, once the errors in the geometrical representationof the circular cylinder affects the solution, we clearly see that the hybrid solversare superior.

7.2. CONVERGENCE 57

100 150 200 25010

−5

10−4

10−3

10−2

10−1

Mea

n(A

bs(E

rror

))

Time [ns]

εr = 4

Figure 7.4: Errors for three refinement levels for the dielectric cylinder; solidlines: FDTD, dashed lines: FD-FE and dash-dotted lines: FD-FV.

We estimated the order of convergence by calculating the L1 norm of theerrors in Figures 7.2–7.5. The results are given in Table 7.1. The first value isobtained by comparing errors on the coarse and medium grids and the secondvalue by comparing errors on the medium and fine grids. The hybrid solversshow second-order convergence, while the FDTD solver does not.

FDTD FD-FE FD-FVPEC 2.23, 1.06 2.01, 1.98 1.99, 1.98PMC 1.33, 0.97 2.13, 2.08 2.01, 1.91εr = 4 1.50, 1.10 2.03, 2.00 2.03, 2.00µr = 4 1.54, 1.16 2.06, 2.01 2.04, 1.97

Table 7.1: Estimations of the order of convergence for the three solvers.


100 150 200 25010

−5

10−4

10−3

10−2

10−1

Mea

n(A

bs(E

rror

))

Time [ns]

µr = 4

Figure 7.5: Errors for three refinement levels for the diamagnetic cylinder; solidlines: FDTD, dashed lines: FD-FE and dash-dotted lines: FD-FV.

7.3 PMC wall

A classical paper demonstrating the errors caused by staircasing is the one byCangellaris and Wright [10]. They study waves propagating along PEC wallsin 2D and conclude that the TE modes suffer dispersion due to the staircaseapproximation of a PEC wall, while the TM modes do not.

One of their experiments involved a line source close to a PEC wall. Wemake a similar experiment here. Since we are working with the TM equations,we use the PMC boundary condition. The duality of the Maxwell equationsmakes this equivalent to the PEC boundary condition for the TE equations,which was shown by Cangellaris and Wright [10] to be the “bad” case.

The computational domain is a square, i.e. N = Nx = Ny and ∆ = ∆x = ∆y.

7.3. PMC WALL 59

The value of N was chosen such that effects from the ABC or from the edgesof the PMC wall did not reach the probing points during our probing window.We used ∆ = 0.1m and CFL = 0.75, which implied ∆t ≈ 0.176 ns. The FD-FVhybrid required a smaller time step (see (3.36)). We used CFL = 0.375.

The line source was applied as a current source, i.e.

Ez|n+1i,j = Ez|n+1

i,j +∆t

ε∆2e((n+1/2)∆t−t0)

2/t2w , (7.3)

where t0 = 10.56 ns and tw = t0/6 = 1.76 ns.In the same way as in [10] we made two different FDTD calculations: one

where the PMC wall aligned with the FDTD grid and one where the wall wasrotated 45 degrees compared to the grid axes. We will refer to the first case as“para” and the second as “dia”. They are illustrated in Figure 7.6.

In the “para” test case the line source and the probe point were separatedby a distance of d = 10.0m and placed at a height h = 0.65 m above the PMCwall.

The rotation of the Cartesian grid made it impossible in the “dia” test caseto have d = 10.0m. The best possible choice was d =

√2 · 71 ·∆ ≈ 10.04m. For

the same reason h = 4.25 · √2∆ ≈ 0.601 m was chosen.To obliterate the error introduced by the staircase approximation of the tilted

wall, we introduced an unstructured grid close to the wall, see Figure 7.6. Theline source and the probing point were located as in the “dia” case. The distanceh from the PMC wall to the source point and probe point was h = 4.5 · √2∆ ≈0.636m.

PMC w

alld h

Probe point

Source point

PMC wall

Source point Probe pointd

h

Figure 7.6: The horizontal wall and the tilted wall are shown to the left. Theunstructured grid used for the tilted wall is shown to the right.

The analytical solution and the errors of six different methods are shownin Figure 7.7. We clearly see that the hybrid solvers outperform the “dia”case. We also note that the errors of the two hybrid solvers are very similar.Furthermore, the hybrid solutions are actually slightly better than the “para”solution. This may be due to the fact that the error in the FDTD solver is


smaller for waves traveling diagonally than for waves traveling along a Cartesianaxis, see Chapter 5 in [63].

We also tested two contour path modeling schemes. The first scheme is theone described in Chapter 10.6 of [63]. It is labeled “Taf” in Figure 7.7. Thisscheme was found to be unstable, but the instability did not significantly affectthe solution in the probing point during our probing window. The instabilitywas generated at the wall. Finally, we also implemented the scheme of Dey andMittra [15]. This scheme was found to be stable for this test case.

0 20 40 60 80 100 120 140−30

−20

−10

0

10

20

30

40

50

60

t [ns]

Ez

Analytical solution

dia FE FV paraDey Taf

40 45 50−10

−5

0

5

10

Zoom

Figure 7.7: The analytical solution in the probing point and the errors of thesix solvers.

We note that these contour path modeling schemes produce much betterresults than the “dia” solution, but they are not quite as good as our hybridsolvers. Furthermore, even though the contour path modeling schemes wererather easy to implement for this very special 2D case, we feel that it would bea very complex and cumbersome task to make a general implementation in 3D.

Chapter 8

A comparison of the hybridsolvers

In this chapter we compare the two hybrid solvers in 3D in terms of accuracy,efficiency and stability. We present results from several scattering cases wherethe superiority of the hybrid solvers compared to stand-alone FDTD is clearlydemonstrated.

8.1 Grid reflections

Using a hybrid grid will inevitably lead to reflections at the grid interface dueto the fact that the methods posses different dispersion characteristics on thedifferent element types. For a reflecting target it is very important that thesenon-physical reflections are small compared to the physical reflections from thetarget, otherwise they could completely destroy the accuracy of the numericalsolution.

To study these reflections we let a unit amplitude Gaussian pulse impingenormally on a hybrid grid interface. The Gaussian pulse traveling in z-directionis defined by

Enx = e−(n∆t−z/c−t0)

2/t2w , (8.1)

where t0 = 120 ns, tw = t0/6 and c is the speed of light. The time step, ∆t, ischosen such that CFL = 0.99 for the FD-FE method and CFL = 0.33 for the FD-FV method, where CFL = 1 is the stability limit for FDTD. The unstructuredgrid consist of 8 × 8 × 8 cubes, where each cube is split into five tetrahedralelements, surrounded by a one or two layers thick interface region dependingon the chosen hybridization technique. This grid is inserted in the center of an24×24×24 FDTD grid with edge lengths ∆ = 1 m. The computational domainis truncated by twelve layers of U-PML [26] cells to ensure that the results are

61

62 CHAPTER 8. A COMPARISON OF THE HYBRID SOLVERS

not affected by reflections at the outer boundary.In Figure 8.1 we compare the instantaneous reflections in front of the grid

interface for the two hybrid solvers. For the FD-FE method we use the non-symmetric interface from Section 6.2, as well as the symmetric interface describedin Section 6.3. In [54] Riley reports that it is possible to reduce the reflections byusing trapezoidal integration on the pyramidal bases instead of exact integration,which is verified in Figure 8.1. The results clearly display that the symmetricinterface produces less reflections than the non-symmetric interface.

0.05 0.1 0.15 0.2 0.25 0.3−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−3

time [µs]

Ex [V

/m]

FD−FE (Trap. integr.)FD−FE (Exact integr.)FD−FE (Nonsymmetric) FD−FV

Figure 8.1: The instantaneous reflections at the grid interface for a normallyincident unit amplitude Gaussian pulse.

In Section 8.4.2 we study the reflections from the grid interface in a differentway by looking at the scattered field from a vacuum sphere.

8.2 Stability

To study the stability of the hybrid methods we use the same grid as in theprevious section, but this time the absorbing boundary condition is replaced byPEC walls. Hence, we get a lossless cavity, which is a very demanding case interms of stability for the numerical methods.

Using the same Gaussian pulse as in the previous section the FD-FV solverbecomes unstable after about 90,000 time steps. Note that this instability isentirely due to the non-symmetric hybridization, since the tetrahedral grid hasa mutually orthogonal dual grid in this case, which implies that the FVTDsolver is stable. The non-symmetric FD-FE solver becomes unstable alreadyafter less than 1000 steps. It is possible to stabilize both hybrid solvers by

8.3. CONVERGENCE IN VACUUM 63

using a dissipative time-integrator. However, the amount of dissipation needed isproblem dependent and for many problems the added dissipation has a significantimpact on the accuracy of the numerical solution. Thus, using the stable, non-dissipative FD-FE solver with the symmetric interface is much more appealingfrom an engineering point of view and as seen in Figure 8.2 it works as expectedin this case.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

−6

10−4

10−2

100

102

104

106

time [ms]

|Ex| [

V/m

]

FD−FEFD−FVFDTD

Figure 8.2: Electric field sampled at a point inside an infinitely resonant cavity.

8.3 Convergence in vacuum

We estimate the order of convergence for the hybrid solvers in vacuum, whereanalytical solutions are known. Hence, we only need two numerical solutions:one coarse grid solution and one fine grid solution. Since we consider the electriccomponents in the grids to be located at the center of each edge, and we wantto avoid spatial interpolation, the fine grid must have edges that are three timesshorter than the edges in the coarse grid.

In the structured grid we use cubical cells with edge lengths ∆ = 1 m for thecoarse grid and ∆ = 1

3 m for the fine grid. The unstructured grids have a size of4m × 4m × 3m and the tetrahedral cells are generated by splitting the cubesinto five tetrahedral elements. The grids are inserted in the center of an FDTDgrid with dimensions 14m × 14m × 13m.

The incident field traveling in z-direction is generated by Huygens’ surfaces,which are located two meters from the outer boundary. For FDTD and theFD-FE solver the CFL number equals

√3/2, and for the FD-FV solver it equals√

3/4. A twelve cells thick U-PML layer is used to terminate the computationaldomains to ensure that reflections from the outer boundary do not noticeably


affect our convergence estimates.We probe the solution in the Ez components in the FDTD grid that are

located closest to the transition layer. We calculate the l2-norm of the errors inall probing points, see Figure 8.3. These time dependent vectors enable us toestimate the order of convergence given in Table 8.1.

100 150 200 250 300 350 400 450 500

10−6

10−5

10−4

10−3

10−2

t [ns]

||E−E

a|| 2 [V/m

]

FD−FEFD−FVFDTD

Figure 8.3: Errors in l2-norm for two refinement levels and three solvers (thicklines: coarse grid, thin lines: fine grid).

FDTD FD-FE FD-FV1.97 1.97 1.06

Table 8.1: Estimates of the order of convergence in l2-norm for the two hybridsolvers and the FDTD solver.

Hence, the FD-FE hybrid solver is second-order accurate but the FD-FVsolver is not. Note that the FVTD solver is second-order accurate on the un-structured grid, which implies that it is the coupling that destroys the order ofaccuracy. Most likely this is due to the interpolation of diagonal componentsat the grid interface. The interpolation itself is second-order accurate but theinterpolated values are then used in what is basically difference approximations.If this happens only in one point it would not destroy the accuracy, but since ithappens on every diagonal on the outer surface of the transition region it doesaffect the order of accuracy globally. Note that the same thing happens if thenon-symmetric interface is used for the FD-FE solver [3].

By using the symmetric interface we believe that we would get second-orderaccuracy also for the FD-FV solver. However, as mentioned earlier, this sym-

8.4. SCATTERING RESULTS 65

metric interface does not solve the stability problems for the FD-FV solver andtherefore it has not been implemented.

8.4 Scattering results

In this section the hybrid solvers are compared to stand-alone FDTD in differ-ent scattering cases: a PEC sphere, a dielectric sphere and the generic aircraftRUND. For the PEC and dielectric spheres analytical solutions, Mie series, forthe RCS are known in frequency-domain. For the generic aircraft a solution ofthe Combined Field Integral Equation (CFIE) in frequency-domain using theFast Multipole Method [48] is used as a reference. The incident plane wavesin the form of Gaussian pulses are generated by Huygens’ surfaces. Since theestimation of RCS requires the scattered field intensity at infinity, the far-fieldsolutions are obtained by a near-to-far-field transformation [45]. The computa-tional domains are truncated by eight layers of U-PML cells. The grids usedfor stand-alone FDTD are generated such that all edges, which belong to a cellwhose centroid is inside the object are considered to be PEC or dielectric edges.The center of the sphere coincides with the center of an FDTD cell.

8.4.1 A PEC sphere

The sphere has a radius of three meters and an incident plane wave with horizon-tal polarization impinges the sphere. A thin layer of unstructured cells are usedin the near vicinity of the sphere to obtain a good geometrical representationof the sphere. A structured grid is used for the rest of the domain. The edgesin the unstructured region are of the same length as the ones in the structuredregion. To compare the memory requirements, execution times and accuracyfor the hybrid solvers and FDTD we use several different grids, a coarse gridwith edge lengths ∆=1m, an intermediate grid with ∆=0.5m and a fine gridwith ∆=0.25m. For stand-alone FDTD simulations we also use grids with edgelengths ∆=0.125m and ∆=0.0625m. The computational domains are resizedsuch that the distance from the sphere to the U-PML boundary is approximatelyten cells on all grids. The CFL number equals 0.99 for FDTD and FD-FE, and0.2475 for the FD-FV solver.

The unstructured solvers obviously require more memory and number ofarithmetic operations per cell. Therefore, we compare the error defined by thel2-norm of the difference between the numerical solution and the analytical Mieseries, as a function of memory requirements and execution times to determinehow much we gain by using the hybrid solvers. All simulations are run to thesame physical time and are performed using one processor of a 375 MHz Power3node on an IBM SP2. The memory requirements and execution times for thedifferent grids are collected in Table 8.2.


FDTD FD-FE FD-FV∆ Memory Time Memory Time Memory Time

1 m 11 8.5 15 30 16.5 440.5 m 15 28 24 135 31 1900.25 m 25 133 53 858 80 10010.125 m 52 688 - - - -0.0625 m 170 5940 - - - -

Table 8.2: Memory requirements in Mbytes and execution times in seconds forthe different solvers on the PEC sphere.

In Figure 8.4 we use the data in Table 8.2 to compare the accuracy at thefrequency 30 MHz for the different solvers as a function of memory requirementsand execution times. The FD-FE solver produces slightly better results than theFD-FV solver for the same cost in memory and execution time. However, bothhybrid solvers are much more efficient than stand-alone FDTD. The accuracyachieved by FDTD on the finest grid (maximum deviation of 0.6 dB) requiresroughly eight times more memory than the hybrid solvers. The execution timesfor FDTD are about a factor 30 longer than the FD-FV solver and a factor 40longer than the FD-FE solver. If better accuracy is wanted the comparison iseven more favorable for the hybrid solvers.

101

102

10−3

10−2

10−1

100

Memory [Mbyte]

||σ−

σ Mie

|| 2/||σ M

ie|| 2

FD−FEFD−FVFDTD

101

102

103

104

10−3

10−2

10−1

100

Execution time [s]

||σ−

σ Mie

|| 2/||σ M

ie|| 2

FD−FEFD−FVFDTD

Figure 8.4: Errors in bistatic RCS for a PEC sphere for the hybrid solvers andFDTD as a function of memory requirements (left) and execution times (right).


10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

Frequency [MHz]

σ [d

Bsm

]

Mie series FD−FE ∆=0.25mFD−FV ∆=0.25mFDTD ∆=0.125m

Figure 8.5: Monostatic RCS for a PEC sphere of radius three meters.

The monostatic RCS for the PEC sphere is shown in Figure 8.5. Both hybridsolvers show very good agreement with the Mie series for the whole frequencyinterval all the way to 100 MHz (ka = 2π), which corresponds to twelve pointsper wavelength. The maximum deviation from the Mie series is 0.76 dB forthe FD-FE solver and 0.56 dB for the FD-FV solver, which both occurs at100 MHz. The FDTD solution with twice the resolution of the hybrid solversshows deviations of this level already at the first dip at about 25 MHz.

8.4.2 A dielectric sphere

The sphere has a radius of three meters and a Gaussian pulse impinges thesphere. The relative permittivity, εr, of the sphere equals four. On the unstruc-tured grids tetrahedral elements are used throughout the inside of the sphere.For an electrically larger sphere tetrahedral elements should be used only at thesurface of the sphere, and structured cells should be used inside the sphere toreduce the memory requirements and execution time. However, in this case thesphere is small in terms of wavelengths and the extra computational cost forusing tetrahedral elements throughout the sphere is negligible.


0 20 40 60 80 100 120 140 160 180−4

−2

0

2

4

6

8

10

12

14

θ

σ [d

Bsm

]

Mie series FD−FE ∆=0.5m FD−FV ∆=0.5m FD−TD ∆=0.5m FD−TD ∆=0.125m

Figure 8.6: Bistatic RCS at 30 MHz for a sphere of radius three meters andεr =4.

In Figure 8.6 the bistatic RCS at 30 MHz is shown and the memory re-quirements and execution times needed by the respective solvers are collected inTable 8.3. The FDTD solver works much better in this case and a refinement ofa factor of four is enough to produce results of similar accuracy as the FD-FEsolver. Hence, although the hybrid solvers are more efficient than stand-aloneFDTD this case is a much closer race than it was for the PEC sphere.


0.5 m 14 25 27 156 40 2270.25 m 23 119 - - - -0.125 m 54 676 - - - -

Table 8.3: Memory requirements in Mbytes and execution times in seconds forthe different solvers on the dielectric sphere.

In Figure 8.7 the monostatic results from the hybrid solvers are compared tothe Mie series and to FDTD. The frequency 60 MHz implies a resolution of fivepoints per wavelength in the sphere and ten points per wavelength in vacuumwhen ∆=0.5m. This resolution is enough for the hybrid solver to obtain a goodcorrespondence with the Mie series for almost all frequencies in the interval,whereas FDTD needs about a factor of two finer resolution.


10 20 30 40 50 60−30

−25

−20

−15

−10

−5

0

5

10

15

Frequency [MHz]

σ [d

Bsm

]

Mie series FD−FE ∆=0.5mFD−FV ∆=0.5mFD−TD ∆=0.5m

Figure 8.7: Monostatic RCS for a dielectric sphere of radius three meters andεr =4.

To determine how much the hybrid solutions are affected by the non-physicalreflections at the grid interface we also run a simulation, where the sphere isfilled with vacuum. The results are illustrated in Figure 8.8 and comparing withFigure 8.7 we conclude that they do not affect any of the solvers significantly inthis case.

10 20 30 40 50 60−160

−140

−120

−100

−80

−60

−40

−20

Frequency [MHz]

σ [d

Bsm

]

FD−FE ∆=0.5mFD−FV ∆=0.5mFD−TD ∆=0.5m

Figure 8.8: Monostatic RCS for the vacuum sphere.


8.4.3 The generic aircraft RUND

The RCS from canonical objects like spheres are often used for validation sinceanalytical solutions are known and successive grid refinements are feasible. How-ever, from an industrial perspective spheres are not targets of practical interest.A more realistic case is scattering from the generic aircraft RUND. This aircraftis approximately 0.8m long, 0.8m between the wing tips and 0.25m high, seeFigure 8.9. The hybrid grid with 10 mm long edges in the FDTD region has

Figure 8.9: Part of the unstructured grid around the generic aircraft RUND.

roughly 150,000 tetrahedra and 500,000 FDTD cells. A refined hybrid grid with5mm long edges and roughly 460,000 tetrahedra and 3,300,000 FDTD cells isalso used. The edges in the unstructured region are approximately of the samelength as the ones in the structured region. The computational domains areresized such that the distance from the object to the absorbing boundary is ap-proximately ten cells on all grids. The CFL number is 0.99 for the FDTD andFD-FE solvers, and 0.2475 for the FD-FV solver.

The incident plane wave illuminates the aircraft head-on with vertical polar-ization. The time-domain solutions are compared to a solution of the CombinedField Integral Equation (CFIE) using the Fast Multipole Method (FMM) [48].The differences between the FMM solutions with edge lengths 10mm and 5mmare less than 0.2 dB for all angles at 1.5 GHz. The FMM solution with 5 mmlong edges is therefore considered to be a good reference solution. As shownin Figure 8.10 both hybrid solutions are in excellent agreement with the FMMsolution.


0 20 40 60 80 100 120 140 160 180−35

−30

−25

−20

−15

−10

−5

Azimuth

σ [d

B]

FD−FE ∆=10mmFD−FV ∆=10mmFDTD ∆=10mm FDTD ∆=2mm FMM ∆=5mm

Figure 8.10: Comparison of bistatic RCS for RUND at 1.5 GHz for verticalpolarization.

0 20 40 60 80 100 120 140 160 180−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Azimuth

σ [d

B]

FD−FE ∆=10mmFD−FE ∆=5mm FD−FV ∆=10mmFD−FV ∆=5mm FDTD ∆=4mm FDTD ∆=2mm

Figure 8.11: Differences in bistatic RCS between a FMM solution and the time-domain solutions on RUND at 1.5 GHz for vertical polarization.

The differences between the FMM solution and the time-domain solutionsare illustrated in Figure 8.11 and a careful inspection reveals that FDTD needs


roughly five times finer resolution to obtain results comparable in accuracy withthe two hybrid solvers. As seen in Table 8.4 the memory requirements to obtainthis accuracy using FDTD are about eight times larger than for the hybridsolvers. The execution time for FDTD is almost thirteen times longer than forthe FD-FV solver and almost a factor 23 longer than for the FD-FE solver.


10 mm 66 1288 186 7300 239 130005 mm - - 639 45500 840 1642044 mm 354 17350 - - - -2 mm 1733 166819 - - - -

Table 8.4: Memory requirements in Mbytes and execution times in seconds forthe different solvers on RUND.

For horizontal polarization it is significantly harder to predict the RCS. Thisis mainly due to the singularities at the front edge of the wings and the tailplane.These singularities have a much greater impact for horizontal polarization com-pared to vertical polarization. The nose of the aircraft is rotational symmetricand that singularity should therfore not affect one polarization more than theother. In Figure 8.12 we notice that the FD-FE solver needs a finer resolutionin this case in order to come close to the FMM solution around the monostaticangle. The results obtained by the hybrid solvers are very similar and thereforeonly the FD-FE solutions are shown. The FMM solutions using 10 mm and 5 mmedges, respectively, show a deviation of up to 1.5 dB at the dips, which impliesthat the FMM solution cannot be regarded as converged in this case. The un-certainty of the FMM solution makes it hard to draw any detailed conclusionsabout the accuracy of the hybrid solvers compared to FDTD.

8.5 Implicit versus explicit methods

In this section we discuss when to use an explicit method and when to use animplicit method by looking at the scalar wave equation given by

utt −4u = 0 , (8.2)

in some domain Ω. Assume for simplicity that the domain is discretized usingcubical elements. To analyze the discretization errors we make the plane waveansatz

u = eiωtf(x, y) , (8.3)

where ω is the wave number and the function f is a solution to Helmholtzequation.

8.5. IMPLICIT VERSUS EXPLICIT METHODS 73

0 20 40 60 80 100 120 140 160 180−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

Azimuth

σ [d

B]

FD−FE ∆=10mmFD−FE ∆=5mm FDTD ∆=10mm FMM ∆ = 5mm

Figure 8.12: Comparison of bistatic RCS for RUND at 1.5 GHz for horizontalpolarization.

The local numerical errors in cell j, with edge length ∆j , are for a second-order method proportional to

∣∣∆t2jutttt

∣∣ = ∆t2jω4 |fj | in time and (8.4)

∣∣∆2j (uxxxx + uxxxy + . . .)

∣∣ ≈ ∆2j

∣∣∣f (4)j

∣∣∣ in space . (8.5)

If we balance the local errors we get

∆t2jω4 |fj | = ∆2

j

∣∣∣f (4)j

∣∣∣ =⇒ ∆tj = ∆j1ω2

√√√√ |f (4)j ||fj | . (8.6)

The global time step is ∆t = minj∆tj . The time step for an explicit methodhas to satisfy ∆t ∝ minj∆j for stability, while an unconditionally stable implicitmethod has no restriction on the time step. Hence, we can draw the conclusion

that if 1ω2

√|f (4)

j |/|fj | is large an implicit method should be used, since an explicitmethod would be forced to use a very small time step. This occurs if only partsof a geometry needs to be finely resolved, e.g. close to singularities. In theseregions the spatial derivatives of the function are much larger than the function

itself, which implies that 1ω2

√|f (4)

j |/|fj | is large. However, if 1ω2

√|f (4)

j |/|fj | . 1an explicit method should be used since an explicit method is in general fasterper time step.


Another reason to use an implicit method is that the grid generator mightgenerate small cells where it is not needed. In this case the balance equation (8.6)is not valid. Instead we have

∆tj =1ω2

√tol

|fj | , (8.7)

where tol is the error tolerance. Hence, the small cells force the explicit methodto use an unnecessarily small time step based on the stability condition, whereasan implicit method can base the time step purely on accuracy requirementsfrom (8.7).

In our case the FD-FV solver is about twice as fast as the FD-FE solver pertime step, but that is more than compensated by the fact that the FD-FE solveris able to use the CFL condition given by the FDTD solver. Even for a smoothscatterer like a sphere a CFL number of 0.25 is generally needed by the FD-FVsolver, and for a more complex object including different space scales an evensmaller CFL number would have to be used.

The temporal and spatial errors in FDTD cancel to some extent and becausethis effect is largest at the CFL limit it is advantageous also from an accu-racy point of view for the hybrid solvers to use as large time steps as possible.However, using a local timestepping procedure in the FVTD domain would alsoenable us to use a CFL number close to unity in FDTD.

8.6 Conclusions

In this chapter two hybrid solvers for the Maxwell equations are compared interms of accuracy, efficiency and stability. A fair comparison of the solvers is pos-sible since they are implemented in the same code package. The hybrid solversare shown to be much more efficient than FDTD for objects where a Cartesiangrid is not able to capture the geometry properly. The artificial reflections fromthe hybrid grid interface are small, which makes it possible to use the hybridsolvers for low reflecting radar targets. In general the FD-FE solver producesslightly more accurate results and is more efficient than the FD-FV solver. Itmay come as a surprise that the implicit solver is more efficient than the explicitsolver. That is partly due to the fact that the iterative method used to solve thematrix-vector system arising in FETD converges fast and that the FVTD solveris heavily penalized by having to take shorter time steps. Furthermore, the up-date of the node values and the use of the nodal filter are costly, both in termsof memory requirements and number of arithmetic operations. Since the nodevalues also are responsible for the instabilities, alternative ways of calculatingthe edge projected fields on non-orthogonal grids are clearly needed.

We focus on solvers which are used by Swedish industry and therefore it isof fundamental importance to have a stable and robust solver. Currently this

8.6. CONCLUSIONS 75

implies that the stable FD-FE hybrid solver is our solver of choice for time-domain solution of the Maxwell equations.

Chapter 9

Frequency dispersivematerials

9.1 Introduction

The ability to model frequency dispersive materials is an important feature fora general purpose electromagnetic code. They are for instance used as radar ab-sorbing materials to minimize the RCS for an aircraft. Other examples includewaveguides filled with dispersive materials, antenna structures and microstripsystems based on dispersive substrates. One of the most common approachesto model these materials in FDTD is to use the Recursive Convolution (RC)method [44, 59]. This method is computer efficient and can be used for generallinear materials. Another approach is to use a Direct Integration (DI) method,where in addition to the Maxwell equations an auxiliary set of differential equa-tions are considered [38, 78]. A summary and analysis of both approaches arepresented in [79]. The main advantage of the RC method is that it allows a uni-fied treatment of a wide variety of dispersive materials, while the DI methodsrequire different formulations for different types of materials.

We will here generalize the second-order accurate RC method presentedin [59] to the unstructured FVTD and FETD solvers. The RC methods forour FVTD and FETD solvers are presented in the next section. A differentRC method for FETD has been proposed by Yeung [77]. The advantage of ourmethod is that the definiteness properties of the mass and stiffness matrices arekept. Furthermore, on hexahedral elements it reduces to the method by Schusteret al. [59], which we use for FDTD. This fact is important for the stability of theFD-FE solver when dispersive materials are present at the hybrid grid interface.

To enable better modeling of materials, which have a strong frequency de-pendence in their constitutive parameters a general material model is used [63].The susceptibility function in frequency-domain is in fractional form and consists

77

78 CHAPTER 9. FREQUENCY DISPERSIVE MATERIALS

of a sum of complex poles. The stability of the respective solvers is analyzed inSection 9.3. In Section 9.4 numerical results for scattering from spheres of com-plex permittivity obtained by the hybrid solvers are compared to stand-aloneFDTD and the respective Mie series. Finally, the last section summarizes thechapter.

9.2 Recursive convolution

For a linear isotropic material the polarization, P (r, ω), is proportional to theelectric field, E(r, ω),

P (r, ω) = ε0 χ(r, ω)E(r, ω) , (9.1)

where ε0 is the permittivity in vacuum and χ(r, ω) is the electric susceptibilityfunction [4]. The electric flux density vector, D(r, ω), is related to the electricfield as

D(r, ω) = ε0 ε∞E(r, ω) + P (r, ω) = ε0 (ε∞ + χ(r, ω)) E(r, ω) , (9.2)

where ε∞ is the limiting permittivity at infinite frequency. By applying theinverse Fourier transform to (9.2) we obtain the corresponding time-domainrelation

D(r, t) = ε0 ε∞E(r, t) + ε0

∫ t

0

χ (τ)E(r, t− τ) dτ . (9.3)

We consider frequency-domain susceptibility functions in fractional form

χ(ω) =Np∑p=1

Rp

jω − sp, (9.4)

where sp are the complex poles, Rp are the corresponding residues, j =√−1

and Np is the number of poles. The corresponding time-domain susceptibilityfunction can then be written as

χ(t) =Mp∑p=1

<eχp(t) , χp(t) = Γp espt u(t) , (9.5)

where u(t) is the unit step function. When all sp and Rp are real (such as Debyemedia) Mp = Np, Γp = Rp, whereas Mp = Np/2, Γp = 2Rp when there are Np/2complex-conjugate pole pairs (such as Lorentz media).

9.2. RECURSIVE CONVOLUTION 79

9.2.1 Finite Volume solver

When dispersive materials are present the FVTD solver is based on the followingdiscretization of the area integral formulations of Faraday’s and Ampere’s laws:

µ∂

∂t

∫

Api

H · npi dA = −

∑r

∫

Γpi,r

E · tpi,r dl , (9.6)

∂

∂t

∫

Adj

D · ndj dA =

∑

k

∫

Γdj,k

H · tdj,k dl − σ

∫

Adj

E · ndj dA . (9.7)

Note that the only difference compared to (4.3–4.4) is that we have replacedεE · nd

j with D · ndj in the left-hand side of (9.7).

Discretizing (9.3) in time and approximating Em as constant over the interval[(m− 1/2)∆t, (m + 1/2)∆t] results in

Dn = ε0 ε∞En + ε0En

∫ ∆t2

0

χ (τ) dτ

+ ε0

n−1∑m=0

En−m−1

∫ (m+ 32 )∆t

(m+ 12 )∆t

χ (τ) dτ ,

(9.8)

where Dn = D(n∆t). The explicit dependence on r has been omitted to simplifythe notation and it is assumed that E0 = 0. Evaluating the integrals in (9.7)and inserting (9.8) give on dual face j

D · ndj |n+1 −D · nd

j |n∆t

·Adj + σ

E · ndj |n+1 + E · nd

j |n2

·Adj =

∑

k

(2524

H · tdj,k|n+ 1

2 − 112

H · tdj,k|n−

12 +

124

H · tdj,k|n−

32

)∆d

j,k ,(9.9)

where ABS3 has been used for time discretization. To simplify the notation inthe update equation define,

χ0p =

∫ ∆t2

0

χp(τ) dτ = −Γp

sp

(1− esp

∆t2

), (9.10)

χm+ 1

2p =

∫ (m+ 32 )∆t

(m+ 12 )∆t

χp(τ) dτ = −Γp

sp

(1− esp∆t

)e(m+ 1

2 )sp∆t , (9.11)

∆χm+1p = χ

m+ 12

p − χm+ 3

2p = esp∆t∆χm

p , (9.12)

where χ0 =∑Mp

p=1 <eχ0

p

and accordingly for the other parameters. For the

expression in the right-hand side define,

H (H · tdj |n) =

∑

k

(2524

H · tdj,k|n+ 1

2 − 112

H · tdj,k|n−

12

+124

H · tdj,k|n−

32

)∆d

j,k .

(9.13)


Substituting (9.8) and (9.13) as well as the sum of the real parts of (9.10–9.12)into (9.9) yields after some rearrangements

E · ndj |n+1 =

1ε∞ + χ0 + σ∆t

2ε0

(ε∞ + χ0 − χ

12 − σ∆t

2ε0

)E · nd

j |n

+n−1∑m=0

∆χm+1 E · ndj |n−m−1 +

∆t

Adjε0

H(H · td

j |n)

.

(9.14)

Furthermore, define Ψdj |n =

∑Mp

p=1 <e

Ψdj |np

, where

Ψdj |np =

n−1∑m=0

∆χm+1p E · nd

j |n−m−1 . (9.15)

This implies

E · ndj |n+1 =

1ε∞ + χ0 + σ∆t

2ε0

(ε∞ + χ0 − χ

12 − σ∆t

2ε0

)E · nd

j |n

+ Ψdj |n +

∆t

Adjε0

H(H · td

j |n)

.

(9.16)

Luebbers et al. made the key observation that is is possible to update Ψdj |np

recursively as

Ψdj |np =

n−1∑m=0

∆χm+1p E · nd

j |n−m−1

= ∆χ1E · ndj |n−1 +

n−1∑m=1

∆χm+1E · ndj |n−m−1

= ∆χ1E · ndj |n−1 +

n−1∑m=1

∆χmE · ndj |n−m−1esp∆t

= ∆χ1E · ndj |n−1 +

n−2∑m=0

∆χm+1E · ndj |n−1−m−1esp∆t

= ∆χ1E · ndj |n−1 + esp∆t Ψd

j |n−1p .

(9.17)

Thus, the whole time-history of E ·ndj is not needed in order to update E ·nd

j

on the new time level. The extra memory requirements for each dispersive dualface are Mp complex values of Ψd

j |np .Our grid generator supplies information about the material in each tetra-

hedron and in each cube in the structured domain. But to update the electric

9.2. RECURSIVE CONVOLUTION 81

field we need the material on the dual faces. Hence, at material interfaces somekind of averaging has to be performed in order to get the material on the dualfaces. To preserve second-order accuracy this is accomplished by splitting thearea integral of D · nd

j over Adj into q sub-integrals, where q is the number of

tetrahedra sharing dual face j. Each sub-integral is taken over Adj,q, which is

the area of dual face j that is inside tetrahedron q. The contributions fromeach sub-integral sum up to the total integral. The same technique could beused to incorporate frequency dispersive materials into non-orthogonal FDTDformulations.

9.2.2 Finite Element solver

If we assume that J = 0 and replace εE with D the weak formulation of thesecond-order wave equation (5.7) can be stated as

∫

V

(D + σE

)·w +

1µ

(∇×E) · (∇×w) dV = 0 , (9.18)

where E denotes the time derivative of E. By taking two time derivatives of (9.3)we get

D(t) = ε0 ε∞ E(t) + ε0

Mp∑p=1

<e ¨Ip , Ip =

∫ t

0

χp(τ)E(t− τ) dτ . (9.19)

Expanding the electric field in edge basis functions and inserting (9.19) into (9.18)yield the following system of equations:

MεE(t) + KE(t) + SE(t) + L(t) = 0 , (9.20)

where E(t) is now the vector of expansion coefficients at time t. The respectivematrix components are given by

(Mε)ij =∫

V

ε0 ε∞ϕi · ϕjdV , (9.21)

Li = ε0

N∑

j=1

∫

V

Mp∑p=1

<e ¨Ipjϕi · ϕj dV , (9.22)

Kij =∫

V

σ ϕi · ϕj dV , (9.23)

Sij =∫

V

1µ

(∇× ϕi) · (∇× ϕj) dV . (9.24)


To discretize (9.20) we use the second-order accurate Newmark-Beta scheme

(Mε +

∆t

2K +

∆t2

4S

)En+1 = 2

(Mε − ∆t2

4S

)En

−(

Mε − ∆t

2K +

∆t2

4S

)En−1 −∆t2Ln . (9.25)

In [77] the second derivative of the convolution integral is computed analyticallyusing partial integration. This results in a contribution to the stiffness matrix,which for some materials makes it indefinite. To preserve the properties of themass and stiffness matrices we proceed differently and instead use a differenceapproximation. Thus, the ith component of the terms involving L is computedas

Lni = ε0

N∑

j=1

∫

V

Mp∑p=1

<e ¨Inpjϕi · ϕj dV , (9.26)

where¨Inpj = En

j χ0p +

n−1∑m=0

χm+1/2p En−m−1

j . (9.27)

The coefficients are defined by (9.10–9.11). Approximating En with a second-order difference yields

∆t2Ln = Md1

(En+1 − 2En + En−1

)+ Ψn , (9.28)

where Ψn =∑Mp

p=1 <eΨn

p and

Ψn

p =n−1∑m=0

Φmp

(En−m − 2En−m−1 + En−m−2

). (9.29)

The components in Md1 and Φmp are given by

(Md1)ij = ε0

∫

V

Mp∑p=1

<eχ0

p

ϕi · ϕjdV , (9.30)

and(Φm

p )ij = ε0

∫

V

χm+1/2p ϕi · ϕjdV . (9.31)

The recursive relation Φmp = esp∆t Φm−1

p now enables us to compute Ψn

p recur-sively for each pole as

Ψn

p = Φ0p

(En − 2En−1 + En−2

)+ esp∆t Ψ

n−1

p . (9.32)


Using (9.28), (9.25) can be rewritten as(

M +∆t

2K +

∆t2

4S

)En+1 = 2

(M − ∆t2

4S

)En

−(

M − ∆t

2K +

∆t2

4S

)En−1 −Ψn .

(9.33)

where M = Mε + Md1. Hence, the main computational difference in the FETDformulation for frequency dispersive materials is the computation of Ψn for whichone additional matrix-vector multiplication is needed.


In this section we analyze the stability of the solvers. For simplicity we assumethat σ = 0 and that we have a homogeneous dispersive material.

9.3.1 Finite Element solver

The assumption σ = 0 implies that K = 0. If we multiply (9.33) from the leftby M−1 we obtain with the use of (9.29) and (9.31)

(I +

∆t2

4M−1S

)En+1 = 2

(I − ∆t2

4M−1S

)En−

(I +

∆t2

4M−1S

)En−1

− I

ε∞ + χ0

n−1∑m=0

χm+1/2(En−m − 2En−m−1 + En−m−2

). (9.34)

An important property of the RC method described in Section 9.2.2 is that itdoes not destroy the matrix properties. Therefore we proceed in the same wayas in the proof of Proposition 5.3.1 and factorize M as M = CT C. Hence,

∆t2

4M−1S =

∆t2

4C−1C−T SC−1C = C−1PC , (9.35)

where

P =∆t2

4C−T SC−1 . (9.36)

The matrix S is symmetric and positive semi-definite and it is easily shown thatalso P has these properties. Therefore, P can be diagonalized by an orthogonalmatrix Q consisting of the eigenvectors of P such that QT PQ = Λ, where Λholds the nonnegative eigenvalues, λ, of P . The same matrix diagonalizes alsoI + P and I − P . With QT CEn = zn we obtain

(I + Λ) zn+1 = 2 (I − Λ) zn − (I + Λ) zn−1

− I

ε∞ + χ0

n−1∑m=0

χm+1/2(zn−m − 2zn−m−1 + zn−m−2

).

(9.37)


We have now decoupled the equations and it is enough to treat the scalar equa-tion

(1 + λ) zn+1 = 2 (1− λ) zn − (1 + λ) zn−1

− 1ε∞ + χ0

n−1∑m=0

χm+1/2(zn−m − 2zn−m−1 + zn−m−2

).

(9.38)

Using the fact the the sum is a geometric series we get

n−1∑m=0

χm+1/2(zn−m − 2zn−m−1 + zn−m−2

)=

Np∑p=1

χ1/2p

(zn − 2zn−1 + zn−2

) n−1∑m=0

z−memsp∆t =

Np∑p=1

χ1/2p

(zn − 2zn−1 + zn−2

) z − ensp∆tz−n

1− esp∆tz−1

=

Np∑p=1

χ1/2p

zn+1 − 2zn + zn−1 − ensp∆t

(z − 2 + z−1

)

z − esp∆t

.

(9.39)

Note that we sum over all Np poles to avoid having to take the real part ofthe sum for a Lorentz material. Since <esp < 0 and we are interested in thebehavior for large n we assume that ensp∆t = 0 for all poles.

For a Debye material with a single pole the characteristic equation is givenby

(1 + λ) zn+1 − 2 (1− λ) zn + (1 + λ) zn−1

+χ1/2

ε∞ + χ0

zn+1 − 2zn + zn−1

z − es∆t= 0 .

(9.40)

After some simple rearrangements (9.40) reads

(1 + λ) z3 − (2(1− λ) + v(1 + λ)− f) z2

+ (1 + λ + 2v(1− λ)− 2f) z − (v(1 + λ)− f) = 0 ,(9.41)

where

v = es∆t , f =χ1/2

ε∞ + χ0. (9.42)

For stability it is necessary that the roots of this third-order polynomial havemagnitude less than or equal to one. Numerical solution of (9.41), when v and fare given by the parameters for water in Section 9.4, reveals that the maximummagnitudes of the roots are less than or equal to one for all λ independent of∆t.


For higher-order materials we proceed in the same way and in general theorder of the characteristic equation is Np + 2. For the two Lorentz materials inSection 9.4 the roots of the corresponding characteristic equations are also lessthan or equal to one independent of ∆t.

An important property of the stable coupling of the FETD and FDTD solversis that they communicate over hexahedral elements, see Section 6.3. Usingtrapezoidal integration instead of exact integration on the hexahedral elementsgives rise to an FE solver which is identical to FDTD. It is of fundamentalimportance to keep this property also when frequency dispersive materials arepresent in the communication elements. It is easily verified that this is the case,since the RC method for the FETD solver and the one we use for FDTD areidentical on hexahedral elements if trapezoidal integration is used.

9.3.2 Finite Volume solver

A similar analysis can be performed for the FVTD solver if we assume that thegrid is orthogonal. We remove the dependence of the magnetic field in (9.14)and use the fact that the recursive sums are geometric series. This results incharacteristic equations given by a sixth-order polynomial for a single pole Debyematerial and a seventh-order polynomial for a Lorentz material with a single polepair. A numerical solution of these equations reveal that the FVTD solver isstable for the Debye and Lorentz materials given in Section 9.4 as long as theCFL condition is satisfied. Experimental results show that this still holds truefor inhomogeneous dispersive dielectric materials if the material averaging isperformed as described in Section 9.2.1.

9.4 Scattering results

In this section we demonstrate the accuracy of the hybrid solvers by computingthe RCS for homogeneous as well as layered spheres with different complex per-mittivities and comparing with analytic Mie series solutions. Unstructured cellsare used inside and in the near vicinity of the spheres to get a good geometricalrepresentation. Structured cells are used for the rest of the domains. The edgesin the unstructured region are approximately of the same length as the ones inthe structured region. In all cases a plane wave in the form of a Gaussian pulseimpinges on the spheres and the responses in the far-field are obtained usinga near-to-far-field transformation. The computational domains are truncatedby eight layers of U-PML cells. The CFL numbers equal 0.99 for FDTD andFD-FE, and 0.33 for FD-FV.


9.4.1 A Debye sphere

First-order dispersive materials are described by the Debye model, whose electricsusceptibility function is given by

χ(ω) =εs − ε∞1 + jωt0

, (9.43)

where εs is the permittivity at zero frequency and t0 is the relaxation time. Infractional form the residue and pole equal

R1 =εs − ε∞

t0, s1 = − 1

t0. (9.44)

A typical Debye material is water with the parameters in the Debye modelgiven by

ε∞ = 5.27 ,εs = 80.0 ,t0 = 1.0 · 10−11 ,σ = 0 .

(9.45)

In Figure 9.1 the real and imaginary parts of the relative permittivity as afunction of frequency are shown.

0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

80

Frequency [GHz]

Rel

ativ

e pe

rmitt

ivity

RealImag

Figure 9.1: Relative permittivity for water as a function of frequency.

In order to validate the hybrid solvers we compute the scattering from a watersphere and vacuum outside. The radius of the sphere is 420 µm, and the edgelengths are 140 µm. This resolution corresponds to ten points per wavelengthin the sphere at 50 GHz. Results for monostatic RCS are compared to theanalytical Mie series solution and to a stand-alone FDTD solution in Figure 9.2.


10 15 20 25 30 35 40 45 50−40

−35

−30

−25

−20

−15

−10

−5

Frequency [GHz]

σ [d

Bsm

]

Mie series FDTD ∆=140µmFD−FV ∆=140µmFD−FE ∆=140µm

Figure 9.2: Backscattering from a water sphere.

The hybrid results agree very well with the analytical solution with a mod-erate resolution of ten points per wavelength having in mind that the sphereis not particularly well resolved in that case. By increasing the resolution totwenty points per wavelength the numerical solutions for the two hybrid solversare hardly separable from the analytical one. FDTD requires approximately fourtimes finer resolution to obtain results comparable in accuracy with the hybridsolvers.

9.4.2 A Lorentz sphere

Lorentz materials are characterized by damped resonances and have susceptibil-ity functions with one or more pairs of complex conjugate poles. The susceptibil-ity function for a material described by a single pole pair is in frequency-domaingiven by

χ(ω) =(εs − ε∞)ω2

0

ω20 + j2ωδ − ω2

, (9.46)

where εs and ε∞ have the same meaning as for Debye materials, ω0 is theresonant frequency and δ is the damping coefficient. In fractional form theresidue and pole equal

Γ1 = −j(εs − ε∞) ω2

0√ω2

0 − δ2, s1 = −δ + j

√ω2

0 − δ2 . (9.47)


We perform the same numerical test here, i.e. scattering from a sphere ofLorentz material. The sphere is in this case characterized by a single pair ofcomplex conjugate poles with the parameters given by

ε∞ = 4.3 ,εs = 6.0 ,ω0 = 40 π · 109 ,δ = 0.1 ω0 ,σ = 0 .

(9.48)

The complex permittivity function is plotted in Figure 9.3, where we have aresonance around 20 GHz. The radius of the sphere is 900 µm, and the edge

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

Frequency [GHz]

Rel

ativ

e pe

rmitt

ivity

RealImag

Figure 9.3: Relative permittivity for the second-order dispersive material as afunction of frequency.

lengths are 300 µm. This resolution corresponds to ten points per wavelengthin the sphere at 50 GHz. The results for monostatic RCS are presented inFigure 9.4. We get very good agreement with the analytical solution for thewhole frequency spectrum. The FD-FE solver shows slightly better results thanthe FD-FV solver.

9.4.3 A General sphere

The susceptibility function for a Lorentz material consisting of two pairs ofcomplex conjugate poles is in frequency-domain given by

χ(ω) = G1(εs − ε∞) ω2

1

ω21 + j2ωδ1 − ω2

+ G2(εs − ε∞)ω2

2

ω22 + j2ωδ2 − ω2

, (9.49)


10 20 30 40 50−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

σ [d

Bsm

]

Mie series FDTD − ∆ = 300µmFD−FV − ∆ = 300µmFD−FE − ∆ = 300µm

Figure 9.4: Backscattering from a Lorentz sphere.

where ω1 and ω2 are the resonant frequencies and δ1 and δ2 are the dampingcoefficients. In fractional form the residues and poles equal

Γ1 = −jG1 (εs − ε∞)ω2

2√ω2

1 − δ21

, s1 = −δ1 + j√

ω21 − δ2

1 , (9.50)

Γ2 = −jG2 (εs − ε∞)ω2

2√ω2

2 − δ22

, s2 = −δ2 + j√

ω22 − δ2

2 . (9.51)

The parameters characterizing the sphere are given in Table 9.1. The complex

εs = 6.0 ε∞ = 4.3ω1 = 40 π · 109 ω2 = 100 π · 109

δ1 = 0.1 ω1 δ2 = 0.1 ω2

G1 = 0.4 G2 = 0.6

Table 9.1: Parameters for the fourth-order dispersive material.

permittivity function is illustrated in Figure 9.5. In this case we have two reso-nance frequencies one at 20 GHz and another one at 50 GHz.


0 10 20 30 40 50 60 70 800

2

4

6

8

Frequency [GHz]

Rel

ativ

e pe

rmitt

ivity

RealImag

Figure 9.5: Relative permittivity for the fourth-order dispersive material as afunction of frequency.

The radius of the sphere is 600 µm, and the edge lengths are 200 µm. Thisresolution corresponds to ten points per wavelength in the sphere at 80 GHz. Theresults for monostatic RCS are presented Figure 9.6. The hybrid solvers agreevery well with the Mie series and they are clearly much better than stand-aloneFDTD for the same resolution.

20 30 40 50 60 70 80−25

−20

−15

−10

−5

0

Frequency [GHz]

σ [d

Bsm

]

Mie series FDTD − ∆ = 200µmFD−FV − ∆ = 200µmFD−FE − ∆ = 200µm

Figure 9.6: Backscattering from a sphere of the fourth-order dispersive material.

9.5. CONCLUSIONS 91

9.4.4 A Layered sphere

Finally we show that we are able to handle layered materials, exemplified bya layered sphere. The core of the sphere has radius 0.7 mm and the mantelhas radius 2.1 mm. The core is filled with water and the mantel with theLorentz material used in Section 9.4.2. The edge lengths are chosen such thatthe resolution in water is ten points per wavelength at 50 GHz. To minimize thenumber of unstructured cells inside the sphere four layers of structured cells areused centered in the approximately ten cell layers thick mantel. This reducesthe number of unstructured cells from approximately 173,000 to 120,000. Thehybrid results for bistatic RCS in Figure 9.7 agree extremely well with the Mieseries and are much more accurate than the FDTD solution even though theFDTD solution shows good agreement for some angles.

0 30 60 90 120 150 180−10

−5

0

5

10

15

θ

σ [d

Bsm

]

Mie series FDTD ∆=140µmFD−FV ∆=140µmFD−FE ∆=140µm

Figure 9.7: Bistatic RCS at 50 GHz for a layered sphere with a water core anda mantel of Lorentz material.

9.5 Conclusions

We have presented a generalization of Luebbers et al. piecewise constant centeredRC method to treat frequency dispersive materials in two unstructured solvers,which both are interfaced to FDTD. The RC method for the FETD solver is animprovement compared to earlier methods, because it preserves the properties ofthe matrices as well as reducing to FDTD on hexahedral elements. Both theseissues are important for the stability of the FD-FE solver.


The numerical results for Debye and Lorentz as well as materials of a moregeneral type demonstrate excellent correspondence with the analytical Mie seriessolutions. The last experiment on the layered sphere shows very good results aswell as nicely demonstrating the versatility and strength of our hybrid solvers.Using stand-alone FDTD typically requires a grid refinement of a factor four toobtain results of similar accuracy as the two hybrid solvers. In general the FD-FE solver produces slightly more accurate results and is as efficient as the FD-FVsolver. The additional overhead for treating dispersive materials is roughly thesame for both solvers.

Chapter 10

Modeling thin wires inFETD

10.1 Introduction

The ability to model features that are small relative to the cell size is oftenimportant in electromagnetic simulations. In principle, an unstructured gridcould be used to resolve these small features. However, the increase in numberof unknowns can be prohibitive. Thus, the developement of accurate models thatcharacterize the physics of the feature without the need for a highly resolved gridis important.

Thin wires are often important parts of electromagnetic compatibility andantenna problems. A subcell model for thin wires in the FDTD method usingmodified telegraphers equations has been developed [33]. A generalization ofthis model to arbitrarily oriented wires has been proposed by Ledfelt [40]. Rileyhas developed a method that incorporates thin wires into the FETD methodby discretizing a second-order wave equation for the current [54]. By usingtwo-point, trapezoidal integration for the elemental matrices this results in anexplicit equation for the current. The method requires the wires to be alignedwith the edges in the unstructured grid.

In this chapter we present a new method, which is also based on a second-order equation for the current derived from modified telegraphers equations.A consistent discretization of this equation with linear nodal basis functionsand the use of a radial weighting function result in a symmetric spatial couplingbetween field and wire. When including thin wires in FDTD this symmetry is thekey to stability [40]. Here, we prove using the energy method that a symmetriccoupling is sufficient for stability of our semi-discrete field-wire system. For timediscretization we use the second-order accurate Newmark-Beta scheme, whichyields an implicit system of equations. A similar proof shows that the fully

93

94 CHAPTER 10. MODELING THIN WIRES IN FETD

discrete field-wire system is unconditionally stable.Furthermore, our method allows for arbitrarily located and oriented wires

with respect to the tetrahedral grid, which give considerable modeling flexibility.In fact, some grid generators do not even allow wires to be meshed simultaneouslywith the volume grid, which makes it hard to align wires to grid edges.

The outline of the rest of the chapter is as follows: In the next sectionwe present our new wire method and show how it is consistently incorporatedinto FETD. In Section 10.3 we take a closer look at the coupling between fieldand wire, in particular for bent wires. The iterative method used to solve theresulting system of equations is described in Section 10.4. The following sectionaddresses the important aspect of stability. In the results section the proposedmethod is applied to different configurations of dipole and loop antennas. Thefinal section concludes the chapter and discusses future extensions.

10.2 Thin wires in FETD

To derive the wire equation we study an infinitely long cylinder of radius arunning in the z-direction. To simplify the derivation we assume that σ = 0in the neighborhood of the wire. Following Holland et al. [33] we obtain incylindrical coordinates, see Figure 10.1,

∂Ez(r)∂t

+∂V inc

∂t= L

∂2I

∂t2− L

µε

∂2I

∂z2+ R

∂I

∂t, (10.1)

where I is the wire current, V inc is a voltage source per unit length, R is thewire resistance per unit length and L is the wire inductance per unit length(cf. (10.9)).

Figure 10.1: Three segments of a discretized wire described in cylindrical coor-dinates.

The current I along the wire can be expanded in basis functions as

I(z) =∑

j

IjΦj(z) , (10.2)

where Φj is the standard linear nodal basis function in 1D, see Figure 10.2,and Ij is the unknown current at wire node j. The current vanishes at an

10.2. THIN WIRES IN FETD 95

open termination, whereas the spatial derivative of the current vanishes whenthe wire terminates on a large perfect electric conductor. The first-order Murnonreflecting boundary condition is used to simulate a half-infinite wire.

The current density J is now expressed as

J(r, z) = I(z) g(r)z =∑

j

IjΦj(z) g(r)z , (10.3)

where r is the radial distance from the wire and g(r) is a weighting functionsatisfying ∫

r≥a

g(r) 2πr dr = 1 , (10.4)

and thus has dimension[

1m2

]. Furthermore, it is important that this function

decreases with r and equals zero for r ≥ r0, which gives a compact support. Thefunction used is given by

g(r) =

0 , r < a ,1+cos

π rr0

π(r2

0−a2)+ 2r20

π

−1−cos πa

r0−πa

r0sin πa

r0

, a ≤ r ≤ r0 ,

0 , r > r0 ,

(10.5)

where r0 is a grid dependent radial distance from the wire (see Section 10.3).

0 0.25 0.5 0.75 1 1.25 1.5 1.750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r

g(r)

Figure 10.2: The nodal basis function Φj (left) and the weighting function g(r)for r0 = 1.7 and a = 0.1 (right).

If we multiply the left-hand side of (10.1) by g(r)Φj(z) and integrate overall space we obtain

∂

∂t

∫

V

g(r)Φj(z)Ez(r) dV +∂

∂t

∫

V

g(r) Φj(z) V inc(z) dV

=∂

∂t

∫

V

∑

k

z · (Ek ϕk) g(r)Φj(z) dV +∂

∂t

∫

z

V inc(z)Φj(z) dz ,(10.6)


where we have expanded Ez in basis functions and used that∫

V

g(r)Φj(z) dV =∫

z

∫ 2π

0

∫

r≥a

g(r)Φj(z) r dr dθ dz =∫

z

Φj(z) dz , (10.7)

according to (10.4). Multiplying the right-hand side of (10.1) by g(r) Φj(z),integrating over all space and using (10.7) yield

∫

V

g(r)Φj(z)(

L∂2I

∂t2− L

µε

∂2I

∂z2+ R

∂I

∂t

)dV =

∫

z

Φj(z)(

L∂2I

∂t2− L

µε

∂2I

∂z2+ R

∂I

∂t

)dz . (10.8)

The inductance per unit length is approximated as

L =µ

2πlog

r0 + a

2a, (10.9)

where (r0 +a)/2 is an average distance from the wire to the surrounding electricfields used to drive the wire. Let the interpolation operator be defined by

(P E)j =∑

k

Ek

∫

V

z · ϕk g(r)Φj(z) dV , (10.10)

andV inc

j =∫

z

V inc(z)Φj(z) dz , (10.11)

where V inc is a voltage source. Therefore, inserting (10.2), (10.10–10.11) in (10.6)and (10.8) we finally arrive at

(P E)j + V incj =

∫

z

Φj(z)(

L∂2

∂t2− L

εµ

∂2

∂z2+ R

∂

∂t

) ∑

k

IkΦk(z) dz =

∑

k

(Ik

∫

z

L Φj(z)Φk(z) dz − Ik

∫

z

L

εµΦj(z)

d2Φk(z)dz2

dz

+ Ik

∫

z

R Φj(z)Φk(z) dz

).

(10.12)

The mass and stiffness matrices for the wire equation are thus given by

(Mw)jk =∫

z

L ΦjΦk dz , (10.13)

(Kw)jk =∫

z

R ΦjΦk dz , (10.14)

(Sw)jk = −∫

z

L

εµΦj

d2Φk

dz2dz =

∫

z

L

εµ

dΦj

dz

dΦk

dzdz . (10.15)

10.3. FIELD-WIRE COUPLING 97

Therefore, on matrix form (10.12) becomes

MwI + KwI + SwI = P E + Vinc

, (10.16)

where I is the vector of nodal current unknowns. By inserting (10.3) into (5.13)we obtain (cf. (10.10))

−∫

V

∂J

∂t· ϕk dV = −

∑

j

Ij

∫

V

z · ϕkΦj(z) g(r) dV = −(PT I

)k

. (10.17)

Hence, the field equations (5.13) can now be stated as

M E + K E + SE = −PT I . (10.18)

Thus, we have the following semi-discrete field-wire system to solve(

M 0

0 Mw

)(E

I

)+

(K PT

−P Kw

)(E

I

)+

(S 0

0 Sw

) (E

I

)=

(0

Vinc

). (10.19)

The field-wire system (10.19) is discretized in time by the second-order accurate,unconditionally stable Newmark-Beta scheme

(4M + 2∆tK + ∆t2S 2∆tPT

−2∆tP 4Mw + 2∆tKw + ∆t2Sw

)(En+1

In+1

)

=(

8M − 2∆t2S 00 8Mw − 2∆t2Sw

)(En

In

)

−(

4M − 2∆tK + ∆t2S −2∆tPT

2∆tP 4Mw − 2∆tKw + ∆t2Sw

)(En−1

In−1

)

+

(0

4∆t2 Vinc

)=

(bbw

).

(10.20)

10.3 Field-wire coupling

The coupling between field and wire is determined by the interpolation operatordefined in (10.10). In this section we take a closer look at the interpolationoperator P .

10.3.1 The interpolation operator

The operator in (10.10) is used to calculate the electric field along the wire ateach wire node. Due to the compact support of the nodal basis function Φj(z)and the weighting function g(r) only edge projected fields within a neighborhoodof the wire node contributes. To be more specific, each wire node is surrounded


by an interpolation cylinder of radius r0 and length equal to the sum of the twowire beams sharing the node. The first step is to determine which tetrahedralelements that are inside, or partly inside, the interpolation cylinder. The integralin (10.10) is calculated using a sixth-degree Gaussian quadrature formula fortetrahedral elements [39]. In each quadrature point, rq, the electric field isapproximated through

Eez(rq) =

6∑

i=1

Eei ϕe

i (rq) · z , (10.21)

where ϕei are the six basis functions in the tetrahedral element that the quadra-

ture point belongs to. Furthermore, g(r) is dependent on the radial distancefrom the wire to the quadrature point, whereas Φj(z) is dependent on the dis-tance along the wire to the quadrature point. Summing up the contributionsfrom all quadrature points in the tetrahedral elements within the cylinder yieldsthe electric field along the wire.

10.3.2 Bent wires

We would like to have a smooth interpolation cylinder that follows the wire.To accomplish this we introduce a curved cylinder at each node where the wirebends which connects the two cylinders of radius r0 surrounding the beams b1

and b2, see Figure 10.3. The projection of the curved cylinder in the plane withnormal n = z2 × z1 is a circular sector.

Figure 10.3: The circular sector in the wire plane belonging to wire node j.

The vectors tbi define the left and right boundaries of the sector and are givenby tbi = n× zbi. The sector angle ψ is obtained from cos ψ = tb1 · tb2. The pointrc is located at

rc = rj − r0

cos(ψ/2)tb1 + tb2

|tb1 + tb2|, (10.22)


where rj is the position of node j. The beams b1 and b2 cut the sector at

rb1 = rc + r0 tb1 , (10.23)

rb2 = rc + r0 tb2 . (10.24)

The strategy of bending the interpolation cylinder results in a slight shift of thelocation of the wire node. Its new position is given by

rj = rc + r0tb1 + tb2

|tb1 + tb2|. (10.25)

The length of the wire beams are adjusted for the shift in node locations.To determine the interpolation operator we need to be able to calculate

g(r)Φ(z) in an arbitrary quadrature point rq within the curved cylinder. There-fore, we project rq in the plane with normal n

rp = (rq − rc)− [(rq − rc) · n] n . (10.26)

The angle ϕ is then given by

cos ϕ = tb2 · rp

|rp| , (10.27)

and it determines the location of the quadrature point along the wire.The distance from the wire is determined in the plane orthogonal to the wire

defined by n, rc and rq, see Figure 10.4. Thus,

r =√|v|2 + (|rp| − r0)2 .

Figure 10.4: A slice of the interpolation cylinder orthogonal to the wire.

For the straight part of beam b1 we have in cylindrical coordinates∫ 2π

0

∫ zb1

zj−1

∫ r0

a

g(r)r drdzdθ =∫ zb1

zj−1

∫ r0

a

g(r)2πr dr

︸︷︷︸=1

dz

=∫ zb1

zj−1

dz = zb1 − zj−1 .

(10.28)


Hence, the volume integral equals the length of the straight part of b1. It isof fundamental importance that the volume integral over the curved cylinderequals the length of the bent part of b1. The volume element for the curvedcylinder is given by r (r0 + r cos θ) dr dψ dθ, where θ is defined in Figure 10.4.Thus, we have

∫ 2π

0

∫ ψ/2

0

∫ r0

a

g(r)r (r0 + r cos θ) drdψdθ =

∫ ψ/2

0

r0dψ

∫ r0

a

g(r)2πr dr

︸︷︷︸=1

=r0ψ

2.

(10.29)

Note that the approach described in this section is perfectly general. It isalways possible to define a curved interpolation cylinder which smoothly followsthe wire. This has also been found crucial in order to avoid undamped low fre-quency oscillations. However, to simplify the implementation we have assumedthat neighboring sectors do not overlap. Therefore, between two neighboringsectors there is always a straight part. A sector overlap could occur if the wirehas a sharp bend and is also dependent on the radius of the cylinder, r0, inrelation to the length of the wire beams. In practice, the overlap can be avoidedby using a coarser discretization of the wire close to sharp bends. For the wiresin Section 10.6 we use r0 = 1.7∆av, where ∆av is an average edge length in theunstructured grid.

For a grid with varying density, r0 should vary and depend on the edgelengths in a neighborhood of each wire beam. Some modifications of the pro-posed method are also needed if other geometry such as e.g. a PEC object fallswithin the chosen radius (but here the basic thin wire assumptions become ques-tionable). A general treatment of these issues requires further investigation.

10.4 Iterative method

We would like to solve (10.20) using a preconditioned conjugate gradient (PCG)method [28], but unfortunately the matrix in the left-hand side is neither sym-metric nor positive definite. However, by realizing that Aw = 4Mw + 2∆tKw +∆t2Sw is a tridiagonal matrix (or a tridiagonal, circulant matrix for a wire loop),we can solve the system of equations in (10.20) efficiently as follows:

1. The wire part of the system is given by

AwIn+1 = bw + 2∆tPEn+1 =⇒In+1 = (Aw)−1

(bw + 2∆tPEn+1

).

(10.30)

2. If we insert (10.30) into (10.20), then the field part of the system is sim-plified to

(A + H)En+1 = b− 2∆tPT (Aw)−1bw = b′, (10.31)


where A = 4M+2∆tK+∆t2S is positive definite, H = 4∆t2PT (Aw)−1P ispositive semi-definite and A+H is the Schur complement of Aw in (10.20).Since, A + H is a symmetric and positive definite matrix (10.31) can nowbe solved for En+1 by a PCG method.

3. Finally we insert En+1 in (10.30) and solve for In+1.

In the conjugate gradient (CG) method, vectors p are multiplied by A + H.Instead of explicitly computing (Aw)−1 in H the product (A+H)p is computedin two steps:

1. Awq = Pp. 2. (A + H)p = Ap + 4∆t2PT q.

Since Aw is tridiagonal it is an efficient procedure to calculate q in the first step.The rank of H is low and its elements are usually small in comparison to the

elements in A. Let λmin(B) and λmax(B) be the smallest and largest eigenvaluesof a symmetric matrix B. Since λmin(H) = 0, an eigenvalue λj of A+H satisfies[28]

0 < λmin(A) ≤ λj(A + H) ≤ λmax(A) + λmax(H). (10.32)

The convergence rate θ of the CG method depends on the condition numberκ = λmax/λmin of the system matrix [28] in the following way

θ ≤(√

κ− 1√κ + 1

)2

. (10.33)

We conclude from (10.32) and (10.33) that the convergence rate of CG withA + H compared to A is not very different since H is small.

The efficiency is improved by preconditioning A+H in the CG method withan incomplete Cholesky factorization [28] based on A. Since A+H is dominatedby A, this is a good preconditioner also for A + H. Furthermore, we cannot useA + H in the Cholesky factorization, because the wire matrix H is not knownexplicitly. The preconditioned matrix has a lower κ and a faster convergenceaccording to (10.33).

Hence, the conclusion is that the extra computational cost for including wiresis in general small.


In this section we prove that the semi-discrete field-wire system (10.19) is stableand also that the fully discrete field-wire system (10.20) is unconditionally sta-ble. The matrices M, Mw, K,Kw, S and Sw are all symmetric, M and Mw arepositive definite, whereas K, Kw, S and Sw are positive semi-definite. Sourceterms do not effect stability [30] and are therefore not included in the analysis.Let the total electromagnetic energy of the field-wire system be defined by

E(t) =12

(E

TME + ET SE + I

TMwI + IT SwI

). (10.34)


Then we have

Proposition 10.5.1 The semi-discrete field-wire system (10.19) is stable in thefollowing sense: The energy E(t) in (10.34) is non-increasing.

Proof If we let E = E and I = I we can write the system (10.19) on first-orderform as

M 0 0 00 Mw 0 00 0 S 00 0 0 Sw

˙E˙I

E

I

+

K PT S 0−P Kw 0 Sw

−S 0 0 00 −Sw 0 0

E

IEI

= 0 . (10.35)

Multiplying from the left by (E I E I) we obtain

E

IEI

T

M 0 0 00 Mw 0 00 0 S 00 0 0 Sw

˙E˙I

E

I

+

E

IEI

T


−S 0 0 00 −Sw 0 0

E

IEI

= 0 .

(10.36)

The second matrix can be split in a symmetric and a skew-symmetric part as


−S 0 0 00 −Sw 0 0

=

K 0 0 00 Kw 0 00 0 0 00 0 0 0

+

0 PT S 0−P 0 0 Sw

−S 0 0 00 −Sw 0 0

, (10.37)

where

E

IEI

T

K 0 0 00 Kw 0 00 0 0 00 0 0 0

E

IEI

= E

TKE

T+ I

TKwI

T ≥ 0 , (10.38)

and

E

IEI

T

0 PT S 0−P 0 0 Sw

−S 0 0 00 −Sw 0 0

E

IEI

= 0 . (10.39)


Therefore, in the original variables (10.36) simplifies to

dEdt

≤ −ETKE

T − ITKwI

T ≤ 0 . (10.40)

Thus, the total electromagnetic energy is non-increasing and the semi-discreteproblem is stable. ¥

For a conductive medium and/or a resistive wire we have a loss of electromagneticenergy, but if σ = 0 and R = 0 the total electromagnetic energy E is preserved.

The total electromagnetic energy at time step n + 1 is given by

En+1 =12

((En+1 −En

)T

∆tM

(En+1 −En

)

∆t

+

(En+1 + En

)T

2S

(En+1 + En

)

2

+

(In+1 − In

)T

∆tMw

(In+1 − In

)

∆t

+

(In+1 + In

)T

2Sw

(In+1 + In

)

2

).

(10.41)

Note that this energy is the discrete counterpart to the energy used in the semi-discrete case in (10.34). For the discrete energy we have

Proposition 10.5.2 The fully discrete field-wire system (10.20) is uncondition-ally stable in the following sense:

En+1 ≤ En, n = 0, 1, . . . .

Proof We will use a similar strategy as in the semi-discrete case to prove thatthe fully discretized system is stable. We can rewrite the system (10.20) as

M 0 0 0

0 Mw 0 0

0 0 S 0

0 0 0 Sw

En+1−2En+En−1

∆t2

In+1−2In+In−1

∆t2

En+1−En−1

2∆t

In+1−In−1

2∆t

+

K PT S 0

−P Kw 0 Sw

−S 0 0 0

0 −Sw 0 0

En+1−En−1

2∆t

In+1−In−1

2∆t

En+1+2En+En−1

4

In+1+2In+In−1

4

= 0 .

(10.42)


We proceed in exactly the same manner as in the semi-discrete case and multiplyfrom the left by the vector in the second part, which implies

En+1−En−1

2∆t

In+1−In−1

2∆t

En+1+2En+En−1

4

In+1+2In+In−1

4

T

M 0 0 0

0 Mw 0 0

0 0 S 0

0 0 0 Sw

En+1−2En+En−1

∆t2

In+1−2In+In−1

∆t2

En+1−En−1

2∆t

In+1−In−1

2∆t

≤

−(En+1 −En−1

)T

2∆tK

(En+1 −En−1

)

2∆t

−(In+1 − In−1

)T

2∆tKw

(In+1 − In−1

)

2∆t≤ 0 .

(10.43)

By multiplying through and simplifying the resulting expression it is easily shownthat (10.43) implies that

En+1 − En

∆t≤ 0 . (10.44)

Hence, En+1 ≤ En ≤ . . . ≤ E0, and our fully discrete system is therefore uncon-ditionally stable. ¥

As in the semi-discrete case the total electromagnetic energy of the system ispreserved if and only if σ = 0 and R = 0. Note that the key property forthe stability of the field-wire system is that we have a skew-symmetric couplingbetween field and wire.

10.6 Numerical results

10.6.1 Receiving dipole antenna

A thin wire of length 41 m is embedded in an unstructured tetrahedral grid withaverage edge lengths ∆ = 1.08 m. The wire radius is 10 mm and it runs in thex-direction discretized with 42 beams. The direction of the incoming plane waveis 26.6 degrees (arctan( 1

2 )) from broadside and a horizontal polarization is usedin order to excite the wire. The angle 26.6 degrees is chosen such that the firstten modes do not vanish, see Figure 10.5. Reciprocity implies that the sameangle excites the modes if the impinging plane wave contains the correspondingfrequency components. The incoming plane wave is a differentiated Gaussianpulse with the shape

f(t) = −2(t− t0)tw

e−(t−t0)2/t2w , t0 = 12.0× 10−8 , tw = 4.0× 10−8 . (10.45)

We use the time step ∆t = 0.5∆x/c0.

10.6. NUMERICAL RESULTS 105

k

E

Wire

(a) Problem setup.

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

m1234

(b) Radiation pattern.

Figure 10.5: A plane wave impinges the wire at an angle of 26.6 degrees withhorizontal polarization as indicated to the left. To the right the radiation pattern(linear) for the first four resonant modes.

To verify the consistency of the proposed wire method we run this case fora number of different locations and orientations of the wire with respect to thevolume grid. First the wire is moved transversally as in Figure 10.6a using0.25∆y (a25), 0.5∆y (a50) and 0.5∆y+0.5∆z (a55). Secondly, the wire is movedlongitudinally as in Figure 10.6b using 0.25∆x (b25) and 0.5∆x (b50). In thethird set of cases the wire is rotated in the xy-plane, Figure 10.6c, by 26.6degrees (c27) and 45 degrees (c45). Finally, the wire is rotated as in Figure 10.6dfrom the direction [1 0 0] to [1 0.5 0.5] (d24) and to [1 1 1] (d35).

In all these cases the incoming wave is designed such that the time when thewave reaches the midpoint of the wire is the same, and hence a direct comparisonof the currents can be performed. The current responses of the different casesare shown in Figures 10.7–10.8.

Assuming that the responses are described by a sum of damped oscillationswe can find the resonance frequencies using Prony’s method [63]. The firstthree resonance frequencies for the different cases are collected in Table 10.1. Itis obvious from the results in Figures 10.7–10.8 and Table 10.1 that the wirealgorithm is very consistent. The differences in Table 10.1 are of the order 0.1%.The results are a few percent below the theoretical values, but according toBalanis [5] the length of the dipole for first resonance is about 0.48λ for a thinwire. Adjusting the theoretical value for first resonance we get 3.51 MHz whichis exactly what we obtain for all cases.


55

00

50 25

E

k

(a) Transversal transla-tion.

k

E

Wire

0 2550

(b) Longitudal transla-tion.

27

45

4527

(c) Plane rotation.

3524

24

35

(d) 3D rotation.

Figure 10.6: Different locations and orientations of the dipole wire. Case a) isseen from the side, and in case b) the wire is moved in the length direction.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Cur

rent

[A]

Time step

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

Figure 10.7: The midpoint current for the different wire setups.


1600 1650 1700 1750 1800 1850 1900 1950 2000

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

Cur

rent

[A]

Time step

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

Figure 10.8: Detail of the midpoint current for the different wire setups.

Case f1 [MHz] f2 [MHz] f3 [MHz]a00 3.51 7.15 10.80a25 3.51 7.15 10.80a50 3.51 7.15 10.80a55 3.51 7.15 10.80b25 3.51 7.14 10.79b50 3.51 7.14 10.79c27 3.51 7.16 10.79c45 3.51 7.14 10.78d24 3.51 7.14 10.79d35 3.51 7.16 10.79

Theory 3.66 7.32 10.98

Table 10.1: The first three resonance frequencies for a 41 m long wire.

10.6.2 Transmitting dipole antenna

In this section we study the same dipole antenna as in the previous section butin transmitting mode instead of receiving mode. A transmitting dipole antennais characterized by the input impedances and input admittances.


The transmitting dipole antenna is excited at the midpoint of the antennawhere also the input impedance is registered. We excite the center currentnode using a voltage source. If we register the current at the midpoint we cancalculate the input impedance and input admittance for the dipole antenna. Theimpedance is calculated as the ratio between the input voltage and the currentin the frequency-domain and is given by

Z22(f) =V inc

22 (f)I22(f)

, (10.46)

and the admittance

Y22(f) =1

Z22(f). (10.47)

We can split the impedance and admittance into the real and imaginary parts.The real and imaginary parts of the impedance are resistance and reactance,respectively. For the admittance they are conductance and susceptance, respec-tively.

4 6 8 10 12 14 16 18 20 22 240

500

1000

1500

2000

2500

3000

3500

Frequency [MHz]

Rea

l(Z)

[Ω]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

MoM

Figure 10.9: The input resistance for the dipole. A Method of Moments solutionis used for comparison.


4 6 8 10 12 14 16 18 20 22 24−2000

−1500

−1000

−500

0

500

1000

1500

2000

Frequency [MHz]

Imag

(Z)

[Ω]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

MoM

Figure 10.10: The input reactance for the dipole. A Method of Moments solutionis used for comparison.

5 10 15 20 250

5

10

15

Frequency [MHz]

Rea

l(Y)

[mA

/V]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

MoM

Figure 10.11: The input conductance for the dipole. A Method of Momentssolution is used for comparison.


The impedance and admittance for the different wire cases (see Figure 10.6)are shown in Figures 10.9–10.11 where we have some very clear resonances. Theresults are compared to a Method of Moments solution obtained by NEC-3 andwe see a good agreement for most of the frequencies.

Case f [MHz] real(Z) [Ω]a00 3.54 71.2a25 3.54 71.2a50 3.54 71.2a55 3.54 71.2b25 3.53 71.0b50 3.53 71.0c27 3.53 71.0c45 3.53 71.0d24 3.53 71.0d35 3.53 71.0

Theory 3.66 73.0

Table 10.2: The frequency and resistance at half wavelength resonance for a41 m long transmitting wire. A solution obtained by NEC-3 gives 3.53 MHzand 72.2 Ω

The impedance for the first half wavelength resonance should equal 73 +j42.5 Ω [5]. The input impedance for the different wire cases are given in Ta-ble 10.2. We get very consistent results in all cases. They are a few percentless than the theoretical values, but we note that a solution obtained by NEC-3gives 3.53 MHz and 72.2 Ω, which is also slightly less than theoretical values.

10.6.3 Receiving loop antenna

In this section the thin wire model is used to simulate a circular loop antennain receiving mode. The loop diameter is 1m and the wire radius is 1.05 mm.It is discretized using 50 wire beams and located arbitrarily in an unstructuredtetrahedral grid with average edge lengths ∆ = 67.4 mm. The loop antennais illuminated by a horizontally polarized plane wave. The probed current isexpanded in a Prony’s series and the modes corresponding to nλ , n = 1, . . . , 4,are extracted. The results are compared to a Method of Moments solution inTable 10.3.

10.6.4 Shielded enclosure

In [42] measurements were performed on the shielding enclosure geometry shownin Figure 10.12. The inside of the geometry is discretized with an unstructuredgrid with average edge lengths 1.05 cm and roughly 85,000 unknowns. The


f1 [MHz] f2 [MHz] f3 [MHz] f4 [MHz]FETD 99.2 196.3 292.5 390.1MoM 98.3 195.2 292.4 390.0

Difference 0.9 1.1 0.1 0.1

Table 10.3: The resonance frequencies, fn, corresponding to the modes nλ for acircular loop antenna of diameter 1m.

source is located at the 50Ω load and has the form

V inc(t) = e−α2

1(f2−f1)2

t− α2f2−f1

2

cos[2π

f1 + f2

2

(t− α2

f2 − f1

)], (10.48)

where f1 = 0.7 GHz and f2 = 1.6 GHz are the starting and stopping frequenciesof the simulation, respectively, and α1 = 1.035 and α2 = 2.539. The wire hasradius 0.08 cm and is discretized with 28 beams, which are not aligned with thetetrahedral grid. It is attached to the walls of the enclosure at both ends. Thecoax feed line is modeled by the wire through the use of the 50Ω load. Thesource and the two loads are placed on the first and last of the wire beams,respectively.

! " # $ $ % & ' ( $) * % $ & $ % + , - . ' ( / ' 0 $ & ! % 1

Figure 10.12: Shielding enclosure geometry.

The time-history of the current at the 50Ω load, I0 (see Figure 10.13), andthe voltage across the 50Ω load, V0, are stored. A Fast Fourier Transform (FFT)is employed to obtain frequency-domain quantities and roughly 15,000 time stepsare needed for a good resolution. The real power delivered to the 50Ω load iscalculated as

P =12<e

(V0

V inc× I∗0

V inc

). (10.49)


In Figure 10.14 we compare the experimental results with our results obtainby the FETD method. We note a slight frequency shift of 1% for the broadfeed-probe (TEM) resonance at 1.08 GHz. The other resonances show less than0.5% deviations between the FETD and experimental results. Similar frequencyshifts have been observed in time-domain integral-equation solutions for thisproblem [46]. A color visualization showing the currents on the wire and thesurface currents on the walls of the enclosure is found in Section 12.6.

0 50 100 150 200 250 300−6

−4

−2

0

2

4

6

Time [ns]

Cur

rent

[mA

]

Figure 10.13: The current at the 50Ω load including a zoom of the long-timebehavior.

10.7 Conclusions

We have presented a subcell model for incorporating arbitrarily oriented thinwires in the FETD method. A symmetric coupling between field and wire makesit possible to prove that the resulting system is unconditionally stable. Theresults for the dipole antenna are very consistent for different wire orientationsand display good correspondence with a Method of Moments solution. Resultsof good accuracy have also been presented for a loop antenna and a shieldingenclosure.

A modification of the proposed wire method could be used to generalize thethin-slot model described in [54].

0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

Frequency [GHz]

Pow

er d

eliv

ered

to lo

ad [m

W]

Experimental dataFETD

Figure 10.14: Comparison between FETD and measurements for a shieldingenclosure.

Chapter 11

Modeling thin wires inFDTD

11.1 Introduction

The ability to model features that are small relative to the cell size is oftenimportant in electromagnetic simulations. Thus, the developement of accuratemodels that characterize the physics of the feature without the need for a highlyresolved grid is essential.

Thin wires are often important parts of electromagnetic compatibility andantenna problems. There are basically two approaches for constructing thinwire subcell models for the FDTD method. Holland et al. [33] separate the wirefrom the 3D volume and the current and voltage on the wire are described bymodified telegraphers equations. Umashankar et al. [65] include the effect of thewire directly into the stencil of the Yee scheme.

The algorithm proposed by Holland et al. is able to treat wires that arerunning parallel to the edges in the Cartesian grid, but for arbitrarily orientedwires both approaches have to rely on a staircased discretization, which oftenintroduces large errors. A generalization of the algorithm of Holland et al. toarbitrarily oriented wires has therefore been proposed by Ledfelt [40]. However,for bent wires this generalization suffers from problems with undamped lowfrequency oscillations.

Riley has developed a method that incorporates thin wires into the FETDmethod by discretizing a second-order wave equation for the current [54]. Themethod requires the wires to be aligned with the edges in the unstructured gridand may suffer from late time instability problems. In Chapter 10 we developedan unconditionally stable method for incorporating arbitrarily oriented wiresinto the FETD method.

In this chapter we present a similar method for arbitrarily oriented wires in

115

116 CHAPTER 11. MODELING THIN WIRES IN FDTD

FDTD. The outline of the rest of the chapter is as follows: In the next section weintroduce the equations and their discretization using the leap-frog scheme. Inthe following section we take a closer look at the coupling between field and wireand show how the algorithm proposed by Holland et al. can be generalized to takearbitrarily oriented wires into account. In Section 11.4 we prove that the semi-discrete field-wire system is stable. The following section discusses the possibilityof treating wires in the hybrid FD-FE solver. In the results section the proposedmethod is applied to different configurations of dipole and loop antennas. Thefinal section concludes the chapter and discusses future extensions.

11.2 Governing equations

The Maxwell equations for linear, isotropic and non-dispersive media are re-peated here for convenience

µ∂H

∂t+∇×E = 0 , (11.1)

ε∂E

∂t−∇×H = −σE − J , (11.2)

where E is the electric field, H is the magnetic field, J is the electric currentdensity, ε is the electric permittivity, µ is the magnetic permeability and σ is theelectric conductivity.

To derive the wire equation we study an infinitely long cylinder of radius arunning in the ξ-direction. Following Holland et al. [33] we obtain in cylindricalcoordinates, see Figure 11.1,

C∂V

∂t= −C

σ

εV − ∂I

∂ξ, (11.3)

L∂I

∂t= Eξ + V inc −R I − ∂V

∂ξ, (11.4)

where I is the wire current, V is the wire voltage, Eξ is the electric field along thewire, V inc is a voltage source per unit length, R is the wire resistance per unitlength, L is the wire inductance per unit length and C is the wire conductanceper unit length.

Figure 11.1: Three segments of a discretized wire described in cylindrical coor-dinates.


The current vanishes at an open termination, whereas the spatial derivativeof the current vanishes when the wire terminates on a large perfect electricconductor. The first-order Mur nonreflecting boundary condition is used tosimulate a half-infinite wire.

The field equations (11.1–11.2) are discretized by the well-known FDTDmethod [63]. The wire equations (11.3–11.4) are discretized by a leap-frogscheme on a staggered grid. The unknown voltages are located at the nodesof the wire, whereas the unknown currents are located at the midpoint of thewire segments, see Figure 11.1. If we assume that R = 0 and σ = 0, the dis-cretized wire equations are given by

V n+1j = V n

j − ∆t

C

In+ 1

2j − I

n+ 12

j−1

∆ξ, (11.5)

In+ 1

2j = I

n− 12

j − ∆t

L

[V n

j+1 − V nj

∆ξ− Einterp

ξj +V inc

j

∆ξ

], (11.6)

where ∆ξ is the length of the wire segments which are assumed equal to simplifythe description and V inc is a voltage beam source. The inductance per unitlength is approximated as

L =µ

2πlog

r0 + a

2a, (11.7)

where (r0 +a)/2 is an average distance from the wire to the surrounding electricfields used to drive the wire (cf. Section 11.3). Note that L is positive as long asr0 > a, which is necessary in order to have a well posed problem.

11.3 Field-wire coupling

The coupling between the field equations (11.1–11.2) and the wire equations (11.3–11.4) are through the source terms J and Eξ. Since we are focussing on wireswhich are not aligned with the edges in the grid we need to approximate theelectric field along the wire through interpolation of the surrounding field compo-nents. The current density, on the other hand, is approximated by distributingthe current from the wire to the surrounding field components.

The current I along the wire can be expanded in basis functions as

I(ξ) =∑

j

IjΦj(ξ) , (11.8)

where Φj is a constant basis function equal to one on segment j and zero on allother segments, and Ij is the unknown current at wire segment j. The currentdensity J is now expressed as

J(r, ξ) = I(ξ) g(r)ξ =∑

j

IjΦj(ξ) g(r)ξ , (11.9)


where r is a radial distance from the wire and g(r) is a weighting functionsatisfying ∫

r≥a

g(r) 2πr dr = 1 , (11.10)

and thus has dimension[

1m2

]. Furthermore, it is important that this function

decreases with r and equals zero for r ≥ r0, which gives a compact support. Thefunction used here is defined as

g(r) =

0 , r < a ,1+cos

π rr0

π(r2

0−a2)+ 2r20

π

−1−cos πa

r0−πa

r0sin πa

r0

, a ≤ r ≤ r0 ,

0 , r > r0 ,

(11.11)

where r0 = 1.7×max(∆x, ∆y, ∆z) is thus a grid dependent radial distance fromthe wire.

0 0.25 0.5 0.75 1 1.25 1.5 1.750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r

g(r)

Figure 11.2: The weighting function g(r) for r0 = 1.7 and a = 0.1 (right).

The electric field is expanded in basis functions as

E(r) =∑

k

EkNk(r) , (11.12)

where Nk are edge basis functions for hexahedral elements [37]. The sum istaken over all edges in the grid and Ek are the electric fields along the edges.Note that the FEM terminology is used in order to get a more compact notation.The approximation of the electric field inside a hexahedra according to (11.12) isnothing but a trilinear interpolation of the involved edge values. If we multiplyEξ with g(r)Φj(ξ) we can approximate the electric field along wire segment j as

Einterpξj =

∫V

Eξ(r) g(r)Φj(ξ) dV

∆ξ

=

∑k

Ek

∫V

ξ ·Nk(r) g(r)Φj(ξ) dV

∆ξ=

(P E)j

∆ξ.

(11.13)


The volume integral is taken over all space, but due to the compact supportof Nk(r), g(r) and Φj(ξ), only electric field components in a neighborhood ofsegment j contributes. The integral in (11.13) is calculated using a Gaussianquadrature formula.

In the same manner Jdistribk is approximated by

Jdistribk =

∫V

J(r) ·Nk(r) dV

∆x∆y∆z

=

∑j

Ij

∫V

ξ ·Nk(r)Φj(ξ) g(r) dV

∆x∆y∆z=

(PT I)k

∆x∆y∆z,

(11.14)

where the division by ∆x∆y∆z is explained by the fact that this is the volumeof the dual cell surrounding Jdistrib

k . Note that apart from a scale factor we havea skew-symmetric coupling between field and wire.

For a straight wire each wire segment has a corresponding interpolation cylin-der of radius r0, which determines the domain of support. For bent wires it hasbeen found crucial to slightly modify the domain of support in order to avoidundamped low frequency oscillations. We want to avoid that cylinders cor-responding to neighboring segments overlap and that we have regions outsidebends which are not covered by any cylinder. An overlap implies that too muchcurrent is distributed from the wire to this region, and a gap means that nocurrent is distributed to this region. In Figure 11.3 we show the projection ofthe cylinder on the plane with normal n = ξb2 × ξb1. The modified cylinder cor-responding to segment j−1 is bounded by planes with normals ξb1 and ξb1 + ξb2,respectively. The overlap below and the gap above the Vj node are indicated bydashed lines in Figure 11.3.

Figure 11.3: A slice of the interpolation cylinders at a wire bend. The dashedlines indicate the boundaries of the unmodified cylinders.

These modifications for bent wires are necessary in order to obtain accurate


results on the loop antenna in the results section and are also believed to be themajor reason why the approach described in [40] does not work well for bentwires.

Some modifications of our proposed method are needed if other geometrysuch as e.g. a PEC object falls within the chosen radius (but here the basic thinwire assumptions become questionable). A general treatment of these issuesrequires further investigation.


In this section we prove that the semi-discrete field-wire system is stable. LetH contain all the magnetic field unknowns, and accordingly for E, I, and V .Assume that R = 0, σ = 0, and ∆ = ∆x = ∆y = ∆z. We introduce the followingscaling of the variables:

h =√

µ H , e =√

εE , i =√

µ√

L√

∆ξ

∆3/2I , v =

√ε√

∆ξ√L∆3/2

V , (11.15)

where we have split L as L = µL. If we rename our variables and scaled oper-ators, and choose the old names we can write the field-wire system on matrixform as

H

E

I

V

=

0 A 0 0−AT 0 −PT 0

0 P 0 −BT0 0 B 0

HEIV

, (11.16)

where the operators take care of the spatial discretization and interpolation.Source terms do not effect stability [30] and are therefore not included in theanalysis. Let the total electromagnetic energy of the field-wire system be definedby

E(t) =12

(HT H + ET E + IT I + V T V

). (11.17)

Then we have

Proposition 11.4.1 The semi-discrete field-wire system (11.16) is stable in thefollowing sense: The energy E(t) in (11.17) is preserved.

Proof Multiplying from the left by (H E I V ) we obtain

HEIV

T

H

E

I

V

=

HEIV

T

0 A 0 0−AT 0 −PT 0

0 P 0 −BT0 0 B 0

HEIV

= 0 , (11.18)

11.5. THIN WIRES IN THE HYBRID SOLVERS 121

where the last equality follows from the fact that the matrix is skew-symmetric.Hence, we have

∂E∂t

= 0 . (11.19)

Thus, the total electromagnetic energy is preserved and the semi-discrete prob-lem is stable. ¥

For a conductive medium and/or a resistive wire we would have a loss of elec-tromagnetic energy. Note that the key property for the stability of the field-wiresystem is that we have a skew-symmetric coupling between field and wire.

To obtain stability for the fully discrete system we also have to fulfill theCFL condition. In general it is difficult to derive the CFL condition explicitlywhen wires are present. However, in practise for moderately thick wires we havenot seen any need to reduce the maximum time-step given by the CFL conditionfor FDTD without wires. A slight reduction of the time step is necessary forwires, where a is significantly larger than ∆, but it is doubtful whether suchwires satisfy the underlying thin wire assumptions.

11.5 Thin wires in the hybrid solvers

Since we are focussing on hybrid solvers we should therefore be able to handlewires which run both in the structured and unstructured domains. In Chap-ter 10 an unconditionally stable method for thin wires in FETD is developed.However, this method is based on a second-order wire equation for the current,where the unknown currents are located at the nodes of the wire and an implicittimestepping scheme is used. The different positions of the unknowns in spaceand time make it hard to treat a wire running through the transition layer. Abetter approach is to discretize the first-order system (11.3–11.4) in both do-mains by (11.5) and (11.6). For the part of the wire inside the FDTD domainthe coupling to the surrounding field is through (11.13) and (11.14). The onlymodification in the FETD domain is that the scaling of (11.14) with the volumeof the dual cell is removed. This is due to the fact that FETD is based on avolume integral formulation. For wire segments close to a grid interface spe-cial care must be taken since the interpolation cylinder may include hexahedral,tetrahedral as well pyramidal elements, see Section 6.3. Thus, to model wiresrunning through or close to the transition layer is only a bookkeeping problemand poses no fundamental difficulties.

A similar strategy could be used to model arbitrarily oriented thin wires inFVTD. However, a complication occurs when wires run through a domain wherethe grids are not mutually orthogonal. A straightforward coupling between thewires and the surrounding primary edges is not possible if the normal and edgevectors do not align. This complication requires further investigation.


11.6 Numerical results

11.6.1 Receiving dipole antenna

The first test case is the same as in Section 10.6.1, i.e. a thin wire of length 41 mwith radius 10 mm. The wire runs in the x-direction discretized with 41 beamsand is embedded in a Cartesian grid with edge lengths ∆ = 1 m.

To verify the consistency of the proposed wire method we run this case fora number of different locations and orientations of the wire with respect to thevolume grid, see Figure 10.6. The current responses of the different cases areshown in Figures 11.4–11.5 and the voltages in Figure 11.6.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time step

Cur

rent

[A]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

Figure 11.4: The midpoint current for the different wire setups.

Assuming that the responses are described by a sum of damped oscillationswe can find the resonance frequencies using Prony’s method [63]. Assume thatthe current response can be written as

I21(t) =p∑

l=1

Cl e(−γlt+j2πflt), (11.20)

where Abs(Cl) is the initial amplitude, Arg(Cl) is the phase, γ constitutes thedamping, and fl is the frequency of the oscillation. The results for the dominat-ing mode is collected in Table 11.1.


1600 1650 1700 1750 1800 1850 1900 1950 2000−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

Time step

Cur

rent

[A]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

Figure 11.5: Detail of the midpoint current for the different wire setups.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−8

−6

−4

−2

0

2

4

6

8

Time step

Vol

tage

[V]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

Figure 11.6: The midpoint voltage for the different wire setups.

It is obvious from the results in Figures 11.4–11.6 and Table 11.1 that the


Case f [MHz] γ [ 1µs ] ampl.[mA]

a00 3.52 1.21 27.12a25 3.52 1.21 27.12a50 3.52 1.21 27.12a55 3.52 1.21 27.12b25 3.51 1.21 27.11b50 3.51 1.21 27.11c27 3.51 1.21 27.07c45 3.51 1.20 27.03d24 3.51 1.20 27.04d35 3.51 1.20 27.01

Table 11.1: The dominant current mode for a 41 m long wire.

wire algorithm is very consistent. The differences in Table 11.1 are less than0.5%. The results are a few percent below the theoretical value (3.65 MHz),but according to Balanis [5] the length of the dipole for first resonance is about0.48λ for a thin wire. Adjusting the theoretical value for first resonance we get3.51 MHz, which is in close agreement with our values.

11.6.2 Transmitting dipole antenna

In this section we study the same dipole antenna as in the previous section butin transmitting mode instead of receiving mode. A transmitting dipole antennais characterized by the input impedances and input admittances. The dipoleantenna is excited at the midpoint using a voltage source with the shape of thedifferentiated Gaussian pulse given in (10.45). If we register the current at themidpoint we can calculate the input impedance and input admittance for thedipole antenna according to

Z21(f) =V inc

21 (f)I21(f)

, (11.21)

and

Y21(f) =1

Z21(f). (11.22)

The impedance and admittance are shown in Figures 11.7–11.9. The resultsare compared to a Method of Moments solution obtained by NEC-3 and we seea good agreement for most of the frequencies. The impedance for the first halfwavelength resonance should equal 73 + j42.5 Ω [5]. The input impedance forthe different wire setups are given in Table 11.2.


4 6 8 10 12 14 16 18 20 22 240

500

1000

1500

2000

2500

3000

3500

Frequency [MHz]

Rea

l(Z)

[Ω]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

MoM

Figure 11.7: The input resistance for the dipole. A Method of Moments solutionis used for comparison.

4 6 8 10 12 14 16 18 20 22 24−2000

−1500

−1000

−500

0

500

1000

1500

2000

Frequency [MHz]

Imag

(Z)

[Ω]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

MoM

Figure 11.8: The input reactance for the dipole. A Method of Moments solutionis used for comparison.


5 10 15 20 250

5

10

15

Frequency [MHz]

Rea

l(Y)

[mA

/V]

a00

a25

a50

a55

b25

b50

c27

c45

d24

d35

MoM

Figure 11.9: The input conductance for the dipole. A Method of Momentssolution is used for comparison.

We obtain very consistent results, they are a few percent less than the the-oretical values, but we note that a solution obtained by NEC-3 gives 3.53 MHzand 72.2 Ω, which is also slightly less than theoretical values.

Case f [MHz] real(Z) [Ω]a00 3.54 71.5a25 3.54 71.4a50 3.54 71.5a55 3.54 71.5b25 3.54 71.5b50 3.54 71.5c27 3.54 71.4c45 3.53 71.2d24 3.53 71.3d35 3.53 71.2

Theory 3.66 73.0

Table 11.2: The frequency and resistance at half wavelength resonance for a41 m long transmitting dipole. A solution obtained by NEC-3 gives 3.53 MHzand 72.2 Ω


11.6.3 Receiving loop antenna

In this section the thin wire model is used to simulate a circular loop antennain receiving mode. The loop diameter is 1m and the wire radius is 1.05 mm. Itis discretized using 50 wire segments and located in a Cartesian grid with edgelengths ∆ = 6.25 cm. The wire loop is moved in four different ways from a basicsetup (org), where the loop is centered in an xy-plane in the Cartesian grid.First the loop is moved laterally 0.25∆z (a), secondly it is moved transversally0.25∆x (b) and 0.25∆x+0.25∆y (c). Thirdly it is rotated around the y-axis by30 degrees (d30) and 45 degrees (d45). Finally, it is rotated around the x = y-direction by 30 degrees (e30) and 45 degrees (e45).

The loop antenna is illuminated by a horizontally polarized plane wave. Theincoming plane wave is a differentiated Gaussian pulse with tw = 1.25 × 10−9,i.e. the −3 dB bandwidth is 60 − 350 MHz. It is designed such that the timewhen the wave reaches segment 26 is the same, and hence a direct comparisonof the currents in this segment is possible. The time step equals ∆t = 0.5∆/c0.The current responses are shown in Figures 11.10–11.11.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time step

Cur

rent

[mA

]

org a b c d

30d

45e

30e

45

Figure 11.10: The current for the different wire setups.

The probed currents are expanded in a Prony’s series and the modes cor-responding to nλ , n = 1, . . . , 4, are extracted. The results are compared to aMethod of Moments solution and an FETD solution in Table 11.3. The differ-ences between the resonance frequencies are less than 0.1% for all setups. Theagreement with the FETD and MoM solutions are within 1%.


1800 1850 1900 1950 2000

−4

−2

0

2

4

6

x 10−4

Time step

Cur

rent

[mA

]

org a b c d

30d

45e

30e

45

Figure 11.11: Detail of the current for the different wire setups.

f1 [MHz] f2 [MHz] f3 [MHz] f4 [MHz]org 98.8 195.1 290.6 385.1a 98.8 195.1 290.6 385.1b 98.9 195.1 290.6 385.2c 98.9 195.1 290.6 385.1

d30 98.8 195.1 290.5 385.1d45 98.8 195.0 290.5 385.1e30 98.8 195.0 290.5 385.0e45 98.8 195.0 290.4 385.0

FETD 99.2 196.3 292.5 390.1MoM 98.3 195.2 292.4 390.0

Table 11.3: The resonance frequencies, fn, corresponding to the modes nλ for acircular loop antenna of diameter 1m.

11.6.4 Transmitting loop antenna

In this section we run the circular loop antenna in transmitting mode. One ofthe wire segments are excited with a voltage source with the same shape as thedifferentiated Gaussian pulse used for the receiving loop antenna. We registerthe current at this particular wire segment and calculate the input impedance


and input admittance according to (11.21) and (11.22). We compare with resultsobtained by the FETD solver developed in the previous chapter.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−1.5

−1

−0.5

0

0.5

1

1.5

Time [µs]

Cur

rent

[mA

]

org a b c d

30d

45e

30e

45

Figure 11.12: The current at the source segment of the loop.

100 150 200 250 300 350 4000

500

1000

1500

2000

2500

Frequency [MHz]

Rea

l(Z)

[Ω]

org a b c d

30d

45e

30e

45FETD

Figure 11.13: The input resistance for the loop.


The currents are shown in Figure 11.12, and the impedance and admittancein Figures 11.13–11.15.

100 150 200 250 300 350 400

−1000

−500

0

500

1000

Frequency [MHz]

Imag

(Z)

[Ω]

org a b c d

30d

45e

30e

45FETD

Figure 11.14: The input reactance for the loop.

100 150 200 250 300 350 4000

1

2

3

4

5

6

7

8

Frequency [MHz]

Rea

l(Y)

[mA

/V]

org a b c d

30d

45e

30e

45FETD

Figure 11.15: The input conductance for the loop.


The agreement with the FETD solution is within 1 − 2% in the frequencyrange 60−350 MHz, and the consistency of the different loop setups are excellent.The frequency and input impedance at the first full wavelength resonance arecollected in Table 11.4.

Case f [MHz] real(Z) [Ω]org 99.6 137.2a 99.6 137.0b 99.6 137.2c 99.6 137.2

d30 99.5 137.1d45 99.5 137.0e30 99.5 137.0e45 99.5 137.0

FETD 99.4 136.3

Table 11.4: The frequency and resistance at the first full wavelength resonancefor a transmitting circular loop of diameter 1m.

11.6.5 Shielded enclosure

In [42] measurements were performed on the shielding enclosure geometry shownin Figure 10.12. The inside of the geometry is discretized with a Cartesian gridwith edge lengths ∆ = 1.0 cm. The source is located at the 50Ω load and theform is given by (10.48). The wire has radius 0.08 cm and is discretized with 14beams. It is attached to the walls of the geometry at both ends. The coax feedline is modeled by the wire through the use of the 50Ω load. The source and thetwo loads are placed on the first and last of the wire beams, respectively.

The time-history of the current at the 50Ω load, I0, and the voltage acrossthe 50Ω load, V0, are stored. A Fast Fourier Transform (FFT) is employed toobtain frequency-domain quantities and roughly 15,000 time steps are neededfor a good resolution. The real power delivered to the 50Ω load is calculatedaccording to (10.49).

In Figure 11.16 we compare the experimental results with our results obtainby the FDTD method. Some deviations are observed around 900 MHz, but avery good agreement is observed for the rest of the interval. We note a slightfrequency shift of 1% for the broad feed-probe (TEM) resonance at 1.08 GHz.The other resonances show less than 0.5% deviations between the FDTD andexperimental results. Similar frequency shifts are also observed for the FETDsolution in Section 10.6.4 and in time-domain integral-equation solutions [46] forthis problem.


0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

Frequency [GHz]

Pow

er d

eliv

ered

to lo

ad [m

W]

Experimental dataFDTD

Figure 11.16: Comparison between FDTD and measurements for a shieldingenclosure.

11.7 Conclusions

We have presented an extension of the subcell model for thin wires in FDTDproposed by Holland et al. The new interpolation technique makes it possibleto accurately model wires not aligned with the edges in the Cartesian grid. Theresults for different orientations and locations of a dipole antenna show excellentconsistency and display good correspondence with a Method of Moments solu-tion. Results of good accuracy have also been presented for a loop antenna anda shielding enclosure.

A symmetric coupling between field and wire makes it possible to prove thatthe resulting semi-discrete field-wire system is stable. The fully discrete systemis stable under a CFL condition.

A modification of the proposed wire method could be used to generalize thethin-slot model described in [54]. Future work also includes to take bundledwires into account.

Chapter 12

Complex applications in 3D

In this chapter we apply the FD-FE solver to a number of different complexscattering cases. The results are compared to measurements, frequency-domainsolutions and highly resolved FDTD solutions. In Section 12.6 the shieldingenclosure geometry from Section 10.6.4 is visualized in a color plot. The chapteris concluded with a simulation of a lightning strike to a Saab 2000 aircraft.

12.1 Scattering from a metallic business card

A geometrically very simple but still challenging problem is to compute thescattering from a perfectely conducting business card shaped flat metallic plate.This problem is one of the EMCC benchmark problems [70] and the size of theplate is 2λ × 3.5λ, see Figure 12.1. It is discretized with a hybrid grid withaverage edge lengths ∆ = λ/20. A horizontally polarized plane wave impingesthe plate at elevation angle of ten degrees, which causes excitation of strongwaves along the edges of the metallic plate. These waves are important toresolve for an accurate prediction of the scattering. In Figure 12.2 we comparethe experimentally measured monostatic RCS with a number of computed values.Normal incidence to the 2λ edge is zero degrees. The agreement is good for thewhole azimuthal range well within the experimental error.

Business card

Figure 12.1: The metallic business card geometry.

133

134 CHAPTER 12. COMPLEX APPLICATIONS IN 3D

0 10 20 30 40 50 60 70 80 90−30

−25

−20

−15

−10

−5

0

5

Azimuth

σ [d

B λ

2 ]

Experimental results FD−FE ∆=λ/20

Figure 12.2: Comparison of monostatic RCS for a metallic business card.

12.2 Scattering from a metallic cone sphere

Another EMCC benchmark problem [71] is plane wave scattering from a per-fectly conducting cone sphere. The cone sphere consists of a 60.5 cm longcone with half angle 7 degrees, capped smoothly with a spherical cap of ra-dius 7.49 cm [71], see Figure 12.3. The major challenge in this problem is thesingularity at the very sharp apex. A grid with average edge lengths ∆ = 2 cmis used.

Figure 12.3: The surface grid for the metallic cone sphere.

12.2. SCATTERING FROM A METALLIC CONE SPHERE 135

0 20 40 60 80 100 120 140 160 180−40

−35

−30

−25

−20

−15

−10

−5

0

Azimuth

σ [d

B]

Zeus FD−FE

Figure 12.4: Comparison of monostatic RCS for the metallic cone sphere forvertical polarization at 869 MHz.

0 20 40 60 80 100 120 140 160 180−40

−35

−30

−25

−20

−15

−10

−5

0

Azimuth

σ [d

B]

Zeus FD−FE

Figure 12.5: Comparison of monostatic RCS for the metallic cone sphere forhorizontal polarization at 869 MHz.


We calculate the monostatic RCS for horizontal and vertical polarization at869 MHz as a function of angle of incidence. At this particular frequency theresolution is roughly 17 points per wavelength and the cone sphere is approxi-mately two wavelengths long. Azimuthal angle equal to zero degrees correspondsto normal incidence to the tip.

We compare with results [8, 67] from the integral equation solver Zeus devel-oped at Imperial College [9]. In [71] measurements are compared to a Methodof Moments (MoM) solver. However, the quality of the measurements are poorand they are therefore not included here. The monostatic RCS for horizon-tal and vertical polarization are presented in Figures 12.4–12.5. A very goodcorrespondence is observed for both polarizations.

12.3 Scattering from the generic aircraft RUND

Measurements of monostatic RCS for RUND (see Section 8.4.3) has been per-formed by the Swedish Defence Research Agency (FOI) in the frequency range6–16 GHz.

In Figures 12.6–12.7 we compare the results at 6 GHz for vertical and hor-izontal polarization obtained by the FD-FE solver with the measurements aswell as with results obtained by a Method of Moments (MoM) solution of theElectric Field Integral Equation (EFIE). The results agree remarkably well andare well within the experimental accuracy for most angles keeping in mind thatthe resolution is only ten points per wavelength. A significant deviation fromthe measurements are observed between 130 and 180 degrees for vertical polar-ization. However, since the results produced by the FD-FE solver and the MoMsolver are very similar there are reasons to believe that the numerical results aremore accurate than the measurements in this region.

Snapshots of the surface currents when the aircraft is illuminated head-onwith vertical and horizontal polarization are illustrated in Figures 12.8–12.9,respectively. We can clearly see in these figures that the impact of the wings aremore pronounced for horizontal polarization. The horizontally polarized wavecauses surface currents that travel along the wings. The vertically polarizedwave, however, is almost unaffected by the wings. This is due to the fact thatthe wings are thin and the direction of the incident electric field is orthogonalto the wing plane.

12.3. SCATTERING FROM THE GENERIC AIRCRAFT RUND 137

0 20 40 60 80 100 120 140 160 180−60

−50

−40

−30

−20

−10

0

10

Azimuth

σ [d

B]

Experimental results (FOI)FD−FE ∆=5mm MoM ∆=5mm

Figure 12.6: Comparison of monostatic RCS for RUND at 6 GHz for verticalpolarization.

0 20 40 60 80 100 120 140 160 180−60

−50

−40

−30

−20

−10

0

Azimuth

σ [d

B]

Experimental results (FOI)FD−FE ∆=5mm MoM ∆=5mm

Figure 12.7: Comparison of monostatic RCS for RUND at 6 GHz for horizontalpolarization.


Figure 12.8: Surface currents on RUND for vertical polarization when the mainpulse hits the tail of the aircraft.

Figure 12.9: Surface currents on RUND for horizontal polarization when themain pulse has reached the trailing edge of the wings.

12.4. SCATTERING FROM A MILITARY AIRCRAFT 139

12.4 Scattering from a military aircraft

A very challenging problem is scattering from a military aircraft. The aircraftis assumed to be a PEC shell including truncated air intakes. It is illuminatedhead-on with a horizontally polarized Gaussian shaped pulse. At 500 MHz theaircraft is roughly 20 wavelengths long and the air intakes form cavities roughlytwo wavelengths deep.

A grid with 5 cm long edges is used, although the grid generator generatessome much smaller edges. These smaller edges make it impossible to efficientlysolve this problem using the FD-FV solver, because a CFL number of less than0.01 would have to be used. There are about 900,000 tetrahedra, and more thanone million unknowns in the FETD region. About six million FDTD cells areneeded and the computational domain is truncated by eight layers of U-PMLcells. The simulation was run 7,000 time steps, which took about two daysusing one processor of a 375 MHz Power3 node on an IBM SP2. We also ranan FDTD simulation using three times finer resolution resulting in more thanhundred million FDTD cells. This simulation was run on sixteen processors onthe same IBM SP2 node and took about one day to complete.

The bistatic RCS at 500 MHz are compared to solutions of the ElectricField Integral Equation (EFIE) using the Fast Multipole Method (FMM) inFigure 12.10. We note an impressive correspondence of the time-domain solu-tions over a large dynamic range. The agreement with the FMM solution isalso very good considering the moderate resolution of only twelve points perwavelength.

0 20 40 60 80 100 120 140 160 180−40

−30

−20

−10

0

10

20

30

Azimuth

σ [d

B]

FD−FE ∆=5 cm FDTD ∆=1.67 cm FMM (EFIE) ∆=5 cm

Figure 12.10: Comparison of bistatic RCS for horizontal polarization at 500 MHzfor a military aircraft.


12.5 Scattering from the Saab Trainer aircraft

As this section we calculate the scattering from the conceptual Saab aircraft,Trainer, see Figure 12.11.

Figure 12.11: The Saab Trainer aricraft.

The aircraft is 11 meters long, 3.4 meters high and 8 meters between thewing tips. It is assumed to be a PEC shell including truncated air intakes. Itis illuminated head-on with horizontally as well as vertically polarized Gaussianshaped pulses. A hybrid grid with 4 cm long edges is used. There are about 1.8million unknowns in the FETD region and roughly nine million FDTD cells areneeded. The computational domain is truncated by eight layers of U-PML cells.The simulation was run 5,000 time steps, which took about 30 hours using oneprocessor of a 1.1 GHz Power4 node on an IBM SP2. We also ran an FDTDsimulation using ∆ = 1 cm resolution resulting in almost four hundred millionFDTD cells including the U-PML layers. This simulation was run 20,000 timesteps on sixteen processors on the same Power4 node and took about 56 hoursto complete. The memory requirements for the hybrid solver were 1.9 Gbytecompared to 23.4 Gbyte for FDTD. The bistatic RCS at 375 MHz and 500 MHzfor vertical as well as horizontal polarization are compared to solutions of theCFIE and EFIE using the Fast Multipole Method (FMM) in Figures 12.12–12.15.The agreement of the hybrid solutions and the highly resolved FDTD solutionsare very good in all four cases. However, the deviations from the FMM solutionsare quite large in some areas, in particular for vertical polarization at 500 MHz.The CFIE is known to have problems with sharp edges [56], which could explainthe differences in the FMM solutions.

Surface currents for horizontal polarization are displayed in Figure 12.16. Aview of the surface currents within the air ducts are shown in Figure 12.17.

12.5. SCATTERING FROM THE SAAB TRAINER AIRCRAFT 141

0 20 40 60 80 100 120 140 160 180−30

−20

−10

0

10

20

30

Azimuth

σ [d

B]

FD−FE ∆=4 cm FDTD ∆=1 cm FMM (CFIE) ∆=4 cm

Figure 12.12: Comparison of bistatic RCS for horizontal polarization at 375 MHzfor the Saab Trainer.

0 20 40 60 80 100 120 140 160 180

−30

−20

−10

0

10

20

30

Azimuth

σ [d

B]

FD−FE ∆=4 cm FDTD ∆=1 cm FMM (CFIE) ∆=4 cm

Figure 12.13: Comparison of bistatic RCS for horizontal polarization at 500 MHzfor the Saab Trainer.


0 20 40 60 80 100 120 140 160 180

−30

−20

−10

0

10

20

30

Azimuth

σ [d

B]

FD−FE ∆=4 cm FDTD ∆=1 cm FMM (CFIE) ∆=4 cmFMM (EFIE) ∆=4 cm

Figure 12.14: Comparison of bistatic RCS for vertical polarization at 375 MHzfor the Saab Trainer.

0 20 40 60 80 100 120 140 160 180

−30

−20

−10

0

10

20

30

Azimuth

σ [d

B]

FD−FE ∆=4 cm FDTD ∆=1 cm FMM (CFIE) ∆=4 cmFMM (EFIE) ∆=4 cm

Figure 12.15: Comparison of bistatic RCS for vertical polarization at 500 MHzfor the Saab Trainer.

12.5. SCATTERING FROM THE SAAB TRAINER AIRCRAFT 143

Figure 12.16: Surface currents on the Trainer after a horizontally polarizedGaussian pulse has hit the aircraft head-on. The main pulse has reach the tailof the aircraft.

Figure 12.17: Surface currents in the air ducts at the same physical time asabove.


12.6 Shielded enclosure

Figure 12.18: A visualization of the shielding enclosure geometry from Sec-tion 10.6.4. The enclosure is discretized by an unstructured grid and the thinwire subcell model presented in Chapter 10 is used. The currents are displayedon the wire and the surface currents are shown on the walls of the enclosure.The absolute value of the electric field is rendered volumetrically close to thewire.

12.7 Lightning strike to a Saab 2000 aircraft

Modern aircraft are vulnerable to effects of lightning due to the increasing use ofsensitive electronic equipments and composite materials. These effects are clas-sified into direct and indirect effects. The direct effects are the physical damagescaused at lightning attachment points and along the current flow paths. The in-direct effects are caused by the electromagnetic fields of lightning. Currents thatleak into the interior of the aircraft through windows and other openings couldinduce high currents on wires. These induced currents could damage electronicequipments if they are not designed to withstand these currents. The complexinner structure of aircraft as well as the fact that a lightning strike consistsof several current pulses make this problem very well suited for a volumetrictime-domain solver.

The aircraft in this case is the Saab 2000, which is roughly 27 meters long, 8meters high and 25 meters between the wing tips. Parts of the hybrid grid areshown in Figure 12.19. In this particular simulation we are interested in whathappens in the cockpit area and therefore an unstructured grid with averageedge lengths ∆ = 11 cm is used in the vicinity of the nose and a structured grid

12.7. LIGHTNING STRIKE TO A SAAB 2000 AIRCRAFT 145

with edge lengths ∆ = 10 cm is used for the rest of the domain. This results inapproximately five million FDTD cells and 125,000 unstructured unknowns.

Figure 12.19: Part of the hybrid grid for the Saab 2000 aircraft. An unstructuredgrid is used only in the vicinity of the nose and a structured grid is used for therest of the domain. Different colors are assigned to different parts of the object.Note that the outermost unstructured cells are part of the transition layer, cf.Figure 6.3.

A thin wire of radius 1 cm is attached to the nose of the aircraft and continuesto the outer boundary. A current source on the wire is used to simulate alightning strike. The exit point of the lightning current is modeled by a thinwire attached to the tip of the right wing and is also terminated at the outerboundary. Two wires are also placed in the interior of the aircraft to illustratethe effects of induced wire currents. They are attached in both ends to PECstructures (blue parts in Figure 12.21). To avoid that PEC structures fall withinthe interpolation radius (see Section 11.3), the radius is decreased to r0 = 0.25∆.

In Figure 12.20 the surface currents on the aircraft are displayed roughly70 ns after a lightning strike and the first pulse has reached the trailing edge ofthe wings. The surface currents on the wings at this time step are mainly due tothe electromagnetic fields surrounding the aircraft. These fields have propagated


in the air from the nose area to the wings. The magnitude of the magnetic fieldsare shown on a cutting plane through the wings and the lightning currents areshown on the wires.

Figure 12.20: The Saab 2000 aircraft is struck by lightning. The surface currentsare visualized on the fuselage. The magnitudes of the magnetic fields are shownon a cutting plane through the wings.

In Figure 12.21 the right half of the aircraft is removed to let use see intothe aircraft. The induced currents are shown on the two wires running throughthe beam structure in the floor.

Figure 12.21: A view of the interior of the aircraft. The currents are shown atthe same physical time as in the previous figure. Note the two wires runningthrough the beam structure in the floor.

Figure 12.22 offers an interesting view into the complex cockpit area. A lotof small details need to be resolved and a very fine structured grid would haveto be used to capture these details as good as the unstructured grid.

12.7. LIGHTNING STRIKE TO A SAAB 2000 AIRCRAFT 147

Figure 12.22: A close up of the cockpit area. The small details are nicely cap-tured by the unstructured grid. The two interior wires enter the cockpit throughholes under the pillar.

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