A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Dublin City University Accelerated integral equation techniques for solving EM wave propagation and scattering problems Dung Trinh, M. Conor Brennan School of Electronic Engineering Dublin City University April 2014 Dissertation Committee: Dr. Conor Brennan, supervisor Prof. Claude Oestges Prof.
Liam Barry Dr. Noel Murphy i To my family ii Declaration I hereby certify that this material, which is now submit for assessment on the programme of study leading to the award of Doctor of Philosophy is entirely my own work, that I have exercised reasonable care to ensure that the work is original, and does not to the best of my knowledge breach any law of copyright, and has not been taken from the work of others save and to the extent that such work has been cited and acknowledged within the text of my work. Signed: _______________ ID No. Date: April 8th, 2014 iii Acknowledgments First of all, I would like to express my sincere gratitude to my supervisor, Dr.
This dissertation would not have been possible without his continuous and patient guidance from the beginning of my experience as a master student in the Dublin City University through my preparation and completion of this dissertation. I also would like to thank my other committee members: Prof. Claude Oestges, Prof. Liam Barry and Dr.
Noel Murphy for their valuable comments on the dissertation. In the RF propagation modeling and simulation group, I would like to acknowledge all former and current members for sharing their knowledge: Marie Mullen, Patrick Bradley and Vinh Pham. Finally, I would like to express my deepest gratitude to all my family and my girlfriend for their support, encouragement throughout my life. iv Contents Declaration iii Acknowledgments iv 1 Introduction 1 1.1 EM wave propagation in rural and urban areas .2 EM wave scattering from random rough surfaces.
8 2 Integral Equation Formulations 10 2.1 Maxwell’s equations and the scattering problem .2 Surface Integral Equations for homogeneous scatterers .1 Surface equivalence principle .2 Surface Integral Equations for homogeneous scatterers .3 Method of Moments .4 Wave scattering from infinite cylinders .1 TM-wave scattering from homogeneous dielectric cylinders [1, 2] .2 TE-wave scattering from homogeneous dielectric cylinders .5 3D wave scattering problem formulation .1 Scattered field in the far zone. 43 3 Improved Tabulated Interaction Method for Electromagnetic Wave Scattering From Lossy Irregular Terrain Profiles 46 3.2 Wave scattering from 1D dielectric surfaces .3 The Improved Tabulated Interaction Method .1 Basis function definition .2 Derivation of ITIM linear system .4 Derivation of underpinning approximations .2 Interaction between groups .3 Two-level Improved Tabulated Interaction Method (TL-ITIM) .5 Calculation of pathloss and computational complexity .1 Rural terrain profile .2 Mountainous terrain profile .7 Efficient numerical method for computing ITIM basis functions .1 Complexity Analysis of the New FFT Based Method .3 Investigation of convergence versus problem size .4 Convergence comparison with Krylov methods. 81 4 Fullwave Computation of Path Loss in Urban Areas 82 4.2 Description of the algorithm for extracting vertical plane profiles from 3D city map .3 The Generalized Forward Backward Method (GFBM) .1 Accuracy of the forward scattering assumption .2 Comparison with slope diffraction method and measurement data. 94 5 Improved Forward Backward Method with Spectral Acceleration for Scattering From Randomly Rough Lossy Surfaces 95 5.1 Forward Backward Method .2 Improved Forward Backward Method .3 Reduction of computational complexity of improvement step .4 Spectral Acceleration of matrix-vector products .5 Scattered wave, Normalised Bistatic Scattering Coefficient, Emissiv- ity and Brightness temperature .6 Absorptivity, Reflectivity and Energy Conservation Check .1 Gaussian Correlation Function .2 Exponential Correlation Function .3 Emissivity and energy conservation .4 Comparison against measurement data.
117 vi Contents 6 Block Forward Backward Method with Spectral Acceleration for Scattering from Two Dimensional Dielectric Random Rough Surfaces 120 6.2 Block Forward Backward Method with Spectral Acceleration .1 Wave scattering by dielectric surfaces .2 Tapered incident wave .3 Block Forward Backward Method .1 A brief review of Forward Backward Method for 2D Ran- dom Rough Surface Scattering .2 Block Forward Backward Method for 2D Random Rough Surface Scattering .4 Spectral Acceleration (SA) for 2D lossy surface .5 Normalized Bistatic Scattering Coefficient, Emissivity and Bright- ness Temperature .6 Absorptivity, Reflectivity and Energy Conservation Check .1 Comparison against 2D model and measurement data .2 Convergence of the BFBM-SA .3 Emissivity, Reflectivity and Energy Conservation. 148 7 Conclusions 150 Appendix A 154 Appendix B 157 Appendix C 158 Appendix D 175 Publications 177 Bibliography 178 vii Abstract This dissertation focuses on the development of the robust, efficient and accurate numerical methods of EM wave propagation and scattering from urban, rural areas and random rough surfaces. There are four main contributions of this dissertation. - The Improved Tabulated Interaction Method (ITIM) is proposed to compute EM wave propagation over lossy terrain profiles using a coupled surface integral equation formu- lation.
The ITIM uses a common set of basis functions in conjunction with a simple matching technique to compress the original system to a reduced system containing con- siderably smaller number of unknowns and therefore provide a very efficient and accurate method. - Initial efforts in using the full-wave method to compute EM wave propagation over urban areas. The un-accelerated full-wave method has a massive computational burden. In order to reduce the computational complexity, Generalized Forward Backward Method (GFBM) is applied (note that the conventional Forward Backward Method diverges in this scenario).
- The Improved Forward Backward Method with Spectral Acceleration (FBM-SA) is pro- posed to solve the problem of 2D wave scattering from random lossy rough surfaces. - An efficient and accurate iterative method is proposed for computing the 3D wave scat- tering from 2D dielectric random rough surfaces. The proposed method referred to as the Block Forward Backward Method improves the convergence of the 3D FBM, makes it converge for the case of 2D dielectric surfaces. In addition the Spectral Acceleration is also modified and combined with the BFBM to reduce the computational complexity of the proposed method.
viii List of Figures 1.1 Illustration of full 3D ray tracing method.2 Illustration of horizontal and vertical ray tracing method.3 Illustration of Soil Moisture Active Passive (SMAP) mission, scheduled to launch.by NASA in 2014 [3] .1 Classification of integral equation formulations used in this dissertation .2 The scattering Problem .5 Equivalent exterior problem associated with the homogeneous object in Figure 2.6 Equivalent interior problem associated with the homogeneous object in Figure 2.7 An infinite cylinder illuminated by an incident wave (a) Infinite cylinder (b) Cross section of the infinite cylinder .8 Discretisation of the cylinder contour (a) A cylinder illuminated by an in- cident wave (b) Cylinder contour is divided into cells .9 Evaluation of the diagonal elements of impedance matrix .10 Example of one-dimensional randomly rough surface .11 Example of two dimensional dielectric rough surface profile illuminated by an incident wave .12 Near and far field geometry .1 Wave impinging upon a dielectric surface .2 A terrain profile (Hjorring - Denmark) is considered to consists of connected identical linear segments.3 K + 1 direction vectors êk are defined on a reference group and are used to (k) (k) define the set of common basis functions φ0 and φ1 .4 Far Field Approximation of Incidence Field. Circular dots represent centre of Q pulse basis domains while square dot represent centre of group. x̂ is unit vector tangent to surface of group.5 Incident field on group can be expressed in terms of two plane waves with amplitudes based on linear interpolation. 59 ix List of Figures 3.6 Far Field Approximation of interaction between two groups i and j.7 Near neighbour group j is sub-divided into H sub-groups with H discreti- sations each.
Each scatters its own plane wave to group i.8 Pathloss generated by proposed method, precise solution and measured data over Hadsund terain profile. Length of profile: 8km. (a) Hadsund terrain profile, (b) Pathloss at 144M Hz with T M z Polarization.9 Pathloss generated by proposed method, precise solution and measured data over Jerslev terain profile. Length of profile: 5.
(a) Jerslev terrain profile, (b) Pathloss at 435M Hz with T M z Polarization.10 Pathloss generated by proposed method and precise solution over moun- tainous terain profile. Length of profile: 6km. (a) Wicklow terrain profile, (b) Pathloss at 300M Hz with T M z Polarization .11 Pathloss generated by proposed method and precise solution over moun- tainous terain profile. Length of profile: 6km.
(a) Wicklow terrain profile, (b) Pathloss at 300M Hz with T E z Polarization .12 Pathloss generated by proposed method and precise solution over Wicklow terain profile. Length of profile: 6km. Operating frequency: 300M Hz. (a) Pathloss generated by TL-TIM with block size of 25m, (b) Pathloss generated by standard TIM with block size of 25m, (c) Pathloss generated by standard TIM with block size of 12.13 Spectral Radius of matrix N M .14 Comparison of convergence rate between proposed method, GMRES-FFT and block-diagonal preconditioned GMRES-FFT .1 A example of transmitter, receiver and the associated intersection points.2 A example of vertical plane profile extraction from the intersection points shown in Figure 4.4 Pathloss generated by proposed method, precise solution over a sample pro- file (a) Sample profile extracted from Munich city (b) Pathloss at 945MHz with T M z Polarization.5 Map of Munich City with 3 Metro routes .6 (a) Partial map of Munich city and Metro 200 (b) Comparison between mea- surements, GFBM with forward scattering assumption and Slope Diffrac- tion Method.7 (a) Partial map of Munich city and Metro 201 (b) Comparison between mea- surements, GFBM with forward scattering assumption and Slope Diffrac- tion Method.
92 x List of Figures 4.8 (a) Partial map of Munich City and Metro 202 (b) Comparison between measurements, GFBM with forward scattering assumption and Slope Diffrac- tion Method.1 Bistatic scattering coefficient of a flat surface. root mean squared height hrms = 0.0λ and correlation length: lc = 0. (a) Flat surface (b) Bistatic scattering coefficients of the surface .2 Bistatic scattering coefficient of a rough surface. root mean squared height hrms = 0.1λ and correlation length: lc = 0.
(a) Rough surface (b) Bistatic scattering coefficients of the surface .3 Bistatic scattering coefficient of a flat surface. root mean squared height hrms = 0.5λ and correlation length: lc = 1. (a) Rough surface (b) Bistatic scattering coefficients of the surface .4 One dimensional dielectric rough surface profile z = f (x) illuminated by an incident wave .5 (a) Eigenvalues of iterative matrix M for random rough surface (b) Eigen- (2) values associated with dominant value of βn , that is the dominant error coefficients after two iterations.6 Strong and weak regions in the forward and backward scattering direction (a) Forward Scattering (b) Backward Scattering .7 Comparison of residual error norm of proposed method (IFBM-SA) and FBM-SA .8 Comparison of Run Time between IFBM-SA, reference method and FBM- SA .9 Comparison of averaged TE and TM NBSCs of proposed method and Direct Matrix Inversion (DMI) over 100 realisations. Relative dielectric constant: εr = 20+4i.
Autocorrelation function is Gaussian with Gaussian spectrum.1 Two dimensional dielectric rough surface profile z = f (x, y) illuminated by an incident wave .2 A 8λ × 8λ surface illuminated by a tapered plane wave with g = Lx/2 = Ly/2.3 A 8λ × 8λ surface illuminated by a tapered plane wave with g = Lx/3 = Ly/3.4 A 8λ × 8λ surface illuminated by a tapered plane wave with g = Lx/6 = Ly/6.7 Strong and weak regions in the FS direction .8 Case 1: Field point is NOT the first point of the block.9 Case 2: Field point is the first point of the block. 136 xi List of Figures 6.10 Comparison of the convergence rate of the proposed method (BFBM-SA) and GMRES.11 Comparison of run time to achieve the desired residual relative error ver- sus number of unknowns between BFBM-SA and GMRES-SA for the 2D dielectric problem.
