A multi resolution study of the moment method solution to integral equations arising in electromagnetic problems

A multi-resolution study of the moment method solution to integral equations arising in electromagnetic problems by Wong Shih Nern (B.Eng (Hons) NUS) A Thesis Submitted For the Degree of Master of Engineering Department of Electrical and Computer Engineering National University of Singapore 2003 i Acknowledgemen The author is grateful to his supervisors, Prof M S Leong and A/Prof B L Ooi, for their constant guidance, support and encouragement throughout the course of this project In addition, the author would like to thank all the members working in the Radar and Signal Processing Laboratory, and the Microwave Research Laboratory for the pleasant working environment ii Tabl of Contents List of Figures v List of Tables vii List of Symbols viii Chapter Introduction 1.1 Fast solution methods 1.2 Wavelets 1.3 Motivation for a wavelet multi-resolution analysis of electromagnetic problems 1.4 Objectives 1.5 Scope of thesis 1.6 Related publications Chapter Introduction to wavelet theory 2.1 What is a multi-resolution analysis? 2.2 More about the Scaling function ϕ(x ) and Wavelet function ψ (x ) 2.2.1 Normalization 2.2.2 Orthogonality 10 2.3 The refinement equation 10 2.4 Mallat’s algorithm 10 2.5 FIR Filters 11 2.6 Periodic or Cyclic DWT 12 2.7 Regularity and vanishing moments 13 2.8 Some wavelet systems 14 2.8.1 Obtaining the derivatives of the scaling functions and wavelets 15 Chapter Analysis of the patch antenna using wavelet basis functions 18 3.1 Maxwell’s equations 18 iii 3.2 Cavity model 18 3.3 Contour integral formulation 19 3.3.1 ro not along the contour 23 3.3.2 ro along the contour 24 3.4 Solution using the method of moments 25 3.4.1 Integral equation 25 3.4.2 Triangular subsectional basis functions 26 3.4.3 Obtaining the wall impedance, Zw 29 3.4.4 Numerical example 34 3.5 Wavelet solution 35 3.5.1 Mapping to the interval [0, 1] 35 3.5.2 The wavelet basis 37 3.5.3 Testing with basis functions 38 3.5.4 The impedance matrix 39 3.5.5 Computing the integrals 40 3.5.6 Determining the value of Zw (Wavelets) 44 Chapter Numerical Results 47 4.1 Expansion in wavelet basis and comparison with known results 47 4.1.1 Sparsity graph 48 4.1.2 Effect of thresholding 48 4.1.3 Computed antenna parameters 48 Chapter Matrix compression/sparsification using the wavelet transform 54 5.1 The change of basis operation/similarity transform or W -matrix transform/waveletlike transform 54 5.2 Understanding the change of basis operation 56 Chapter Rectangular patch scattering 59 6.1 Theory 59 iv 6.2 Numerical solution of the problem 60 6.2.1 CN/LT basis functions 60 6.2.2 Testing functions 61 6.2.3 Formulation 62 6.2.4 Source functions 66 6.3 Radar Cross Section 66 6.4 Fast wavelet solution 68 6.5 Wavelet-like transform 69 6.5.1 Wavelet transform on augmented matrix 69 6.5.2 Wavelet transform on subblocks of matrix 71 Chapter Results and Discussion 73 7.1 Effect of threshold levels on sparsity and conditioning number using the modified W -transform method 73 7.2 Effect of thresholding/sparsification on the RCS 78 7.3 Advantages of the modified W -transform method 81 Chapter Conclusion 82 References 84 Appendix A Theory and Approximations 96 A.1 The Green’s function (or inverse operator) solution to the Helmholtz equation 96 A.2 Green’s theorem 97 A.3 Hankel functions of the Second kind 97 A.3.1 Approximation to H0(2) (x ) for small arguments 98 A.3.2 Approximation to H1(2) (x ) for small arguments 98 v Lis of Figures Figure 2.1 The Daubechies Wavelet system 14 Figure 2.2 The Coiflet system 15 Figure 2.3 Scaling functions and derivatives of the Coiflet-2 Wavelet system 16 Figure 2.4 Scaling functions and derivatives of the Coiflet-5 Wavelet system 17 Figure 3.1 Symbols used in model for microstrip patch 19 Figure 3.2 Limiting form of C2 when ro is excluded 22 Figure 3.3 Contours of integration when ro is an interior point 24 Figure 3.4 Contours of integration when ro is not an interior point 25 Figure 3.5 Partitioning of inner integral when evaluating the matrix elements for Galerkin’s method 28 Figure 3.6 Notation used for far-field radiation analysis 30 Figure 3.7 Description of patch used in [1] 35 Figure 3.8 Simulated input impedance (normalized to 50 ) for circular patch from Richards [1] computed using subsectional triangular basis functions (N = 25) Simulated frequencies are from 770 MHz to 840 MHz 36 Figure 3.9 Plot of surface electric field Ez (using equation (3.14) on page 23) for circular patch from Richards [1] using subsectional triangular basis functions (N = 25) 37 Figure 4.1 Discretization of circular patch from [1] at 800 MHz 47 Figure 4.2 Effect of thresholding on solution 49 Figure 4.3 Effect of thresholding on solution 50 Figure 4.4 Effect of thresholding on radiation pattern 51 Figure 4.5 Comparison of results with Ensemble (marked with *) 52 Figure 4.6 Comparison of results with Ensemble (marked with *) 53 Figure 4.7 Effect of thresholding on computed wall impedance, Zw 53 Figure 5.1 The wavelet-like transform matrix 55 Figure 5.2 The wavelet-like transform applied to a matrix (3-level decomposition) 57 vi Figure 6.1 Discretization of rectangular plate Patch is divided into Nx and Ny intervals in the x and y directions respectly a = A , Nx b= B Ny 60 Figure 6.2 Triangular rooftop basis functions 61 Figure 6.3 Computed surface current on square patch 67 Figure 7.1 Computed surface current on square patch after thresholding 74 Figure 7.2 Matrix sparsity and condition number for 2112 basis functions 76 Figure 7.3 Sparsity graph for a threshold of 5% using the modified W -transform (Nx = Ny = 33) 77 Figure 7.4 Computed backscattered RCS 78 Figure 7.5 Computed bistatic RCS 79 Figure 7.6 Computed surface current on square patch after thresholding 80 Figure A.1 Real and Imaginary parts of Hv(2) (x ) 98 Figure A.2 Real and Imaginary parts of H0(2) (x ) for small arguments 98 Figure A.3 Real and Imaginary parts of H1(2) (x ) for small arguments 99 vii Lis of Tables Table 6.1 Sparsity achieved, respectively, from matrices obtained from the wavelet transform on the augmented matrix, and on subblocks of the impedance matrix Table 6.2 70 Sparsity (of real and imaginary parts) achieved, respectively, from ma- trices obtained from the wavelet transform on the augmented matrix, and on subblocks of the impedance matrix 70 Table 6.3 Condition number of matrices from, respectively, matrices obtained from the wavelet transform on the augmented matrix, and on subblocks of the impedance matrix 71 Table 6.4 Sub-block size in each submatrix defined in equation (6.11) on page 63 72 Table 7.1 Sparsity achieved, respectively, from matrices obtained from the wavelet transform on the augmented matrix, and on subblocks of the impedance matrix Table 7.2 75 Condition number of matrices from, respectively, matrices obtained from the wavelet transform on the augmented matrix, and on subblocks of the impedance matrix 75 viii Lis of Symbols ro Observation point/Field point r′ Contour integration variable rs Source point/feed point/driving current Ez Electric field M Magnetic field Js Equivalent electric surface current Zw Wall impedance Ez r ′ Electric field on patch periphery Ms Equivalent magnetic current η Free space wave impedance h Substrate thickness Prad Radiated power Pwal l Power dissipation in wall impedance C1 and C2 Integration contours e j ωt Time dependence ϕ j ,k ξ Scaling function at a scale of j shifted by k ψ j ,k ξ The corresponding wavelet function ε Threshold in matrix sparsification Chapter  Introdu io Integral equations arise often in electromagnetic (EM) scattering problems A general procedure for finding a solution that is accurate enough for most practical purposes is the method of moments (MoM) [2, 3] We can regard the moment method as essentially a discretization scheme whereby a general operator equation is transformed into a matrix equation, so that it can be solved on a digital computer In numerical solutions, where the MoM is applied directly to integral equations arising in EM scattering problems, a complex dense (fully populated) matrix usually results The solution of this matrix equation often becomes unfeasible computationally, even for supercomputers, especially when the electrical size of the scatterer becomes large This is because the direct solution of an N × N complex dense matrix (from a discretization with N unknowns) using standard numerical techniques has a computational complexity of O N and a memory requirement of O N Here, O (•) denotes “the order of ” Even iterative solutions have a prohibitive complexity of O N Therefore, when the size of the scatterers or radiators is electrically large, the MoM becomes computationally too expensive (too much memory and CPU time) to be used in a numerical solution As a result, these traditional methods are limited to relatively small problems 1.1 Fast solution methods Numerous modifications to the moment method, as a solution method for electromagnetic 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The Helmholtz equation may be written in the form of a standard ordinary linear differential equation as follows: ∇ Ez (r ) + k2 Ez (r ) = j ωµ0 Jz (r ) ‫ތ‬Ez (r ) = φ (r ) , (A.1a) (A.1b) where ‫ + ∇ = ތ‬k2 is the linear Helmholtz operator, and φ (r ) = j ωµJz (r ) is the driving function The solution is then given by the following equation: Ez (r ) = ‫ 1−ތ‬φ (r ) , (A.2) where ‫ 1−ތ‬is the integral operator with the Green’s function as the kernel In this formulation, we start by writing: C G‫ތ‬Ez (r ) d l = [ .] C + C Ez (r ) ‫ ∗ތ‬Gd l , (A.3) where integration is carried out on r over an arbitrary contour C and ‫ + ∇ = ∗ތ‬k2 is the adjoint of ‫( ތ‬in this case, ‫ ތ‬is self-adjoint) Suppose we now impose the condition that ‫ ∗ތ‬G = δ ro − r (A.4) 97 Therefore, C G‫ތ‬Ez (r ) d l = [ .] C + Ez ro Ez ro = C G‫ތ‬Ez (r ) d l provided the boundary conditions for G are set such that the terms in the square brackets are zero The Green’s function for a dimensional region is then given by: j G r , ro = H0(2) k r − ro , (A.5) where H0(2) (x ) is the zeroth order Hankel function of the second kind The Green’s function is singular as r approaches ro and can be approximated as G r , ro → γ k r − ro ln 2π (A.6) “A Green’s function is the field due to a point source described by a delta function Once it is known, the field due to an arbitrary source can be calculated by a convolution integral involving the source distribution and the Green’s function.” A.2 Green’s theorem First form: V f ∇ 2g + ∇ f · ∇ g d v = V f ∇ 2g + ∇ f · ∇ g d v = S f ∂g ∂n d s, ∂g ∂n = n · grad g (A.7) = n · grad g (A.8) Second form A.3 S f ∂g ∂n d s, ∂g ∂n Hankel functions of the Second kind Hv(1) (x ) = Jv (x ) + jYv (x ), (A.9a) Hv(2) (x ) = Jv (x ) − jYv (x ) (A.9b) 98 1.5 ν=0 ν=1 ν=0 ν=1 0.8 Im(H(2)(x)) Re(H(2)(x)) 0.6 ν ν 0.4 0.2 0.5 0 −0.2 −0.4 −0.5 x 10 (a) Real part x 10 (b) Imaginary part Figure A.1: Real and Imaginary parts of Hv(2) (x ) Approximation to H0(2) (x ) for small arguments A.3.1 H0(2) (x ) ≈ − j π (ln x + γ − ln ) (A.10) 2.5 Exact Approximation Exact Approximation 0.9999 (2) Im(H0 (x)) 0.9997 Re(H(2)(x)) 0.9998 1.5 0.9996 0.9995 0.9994 0.005 0.01 0.015 0.02 0.025 x 0.03 0.035 0.04 0.045 0.05 0.05 (a) Real part 0.1 0.15 x 0.2 0.25 0.3 (b) Imaginary part Figure A.2: Real and Imaginary parts of H0(2) (x ) for small arguments A.3.2 Approximation to H1(2) (x ) for small arguments H1(2) (x ) ≈ 21 x − j − πx (A.11) 99 20 0.5 Exact Approximation 0.45 0.4 16 0.35 14 Im(H1 (x)) 0.3 (2) 12 (2) Re(H (x)) Exact Approximation 18 0.25 10 0.2 0.15 0.1 0.05 0.1 0.2 0.3 0.4 0.5 x 0.6 (a) Real part 0.7 0.8 0.9 0.05 0.1 0.15 0.2 0.25 x 0.3 0.35 0.4 (b) Imaginary part Figure A.3: Real and Imaginary parts of H1(2) (x ) for small arguments 0.45 0.5 ... formulations of electromagnetic problems, the application of the method of moments often leads to a dense matrix due to the nature of the integral kernel The reason is, in scattering problems, the. .. Location of observation point when evaluating inner integral integralright integralleft Figure 3.5: Partitioning of inner integral when evaluating the matrix elements for Galerkin’s method and... addition, the author would like to thank all the members working in the Radar and Signal Processing Laboratory, and the Microwave Research Laboratory for the pleasant working environment ii Tabl of