CHAPTER SEVEN Wavelets in Scattering and Radiation In this chapter we examine scattering from 2D grooves using standard Coiflets, scat- tering from 2D and 3D objects, scattering and radiation of curved wire antennas, and scatterers employing Coifman intervallic wavelets. We provide the error esti- mate and convergence rate of the single-point quadrature formula based on Coifman scalets. We also introduce the smooth local cosine (SLC), which is referred to as the Malvar wavelet [1], as an alternative to the intervallic wavelets in handling bounded intervals. 7.1 SCATTERING FROM A 2D GROOVE The scattering of electromagnetic waves from a two-dimensional groove in an infi- nite conducting plane has been studied using a hybrid technique of physical optics and the method of moments (PO-MoM) [2], where pulses and Haar wavelets were employed to solve the integral equation. In this section we apply the same formulation as in [2] but implement the Galerkin procedure with the Coifman wavelets. We first evaluate the physical optics (PO) cur- rent on an infinite conducting plane [3] and then apply the hybrid method, which solves for a local correction to the PO solution. In fact the unknown current is ex- pressed by a superposition of the known PO current induced on an infinite conducting plane by the incident plane wave plus the local correction current in the vicinity of the groove. Because of its local nature the correction current decays rapidly and is essentially negligible several wavelengths away from a groove. Because of the rapidly decaying nature of the unknown correction current, the Coiflets can be directly employed on a finite interval without any modification (peri- odizing or intervallic treatment). Hence all advantages of standard wavelets, includ- ing orthogonality, vanishing moments, MRA, single-point quadrature, and the like, are preserved. The localized correction current is numerically evaluated using the 299 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X 300 WAVELETS IN SCATTERING AND RADIATION x d h PEC z y H inc E inc φ inc b b κ ρ FIGURE 7.1 Geometry of the 2D groove in a conducting plane. MoM with the Galerkin technique [4]. The hybrid PO-MoM formulation is imple- mented with the Coiflets of order L = 4, which are compactly supported and possess the one-point quadrature rule with a convergence of O(h 5 ) in terms of the interval size h. This reduces the computational effort of filling the MoM impedance matrix entries from O(n 2 ) to O(n). As a result the Coiflet based method with twofold inte- gration is faster than the traditional pulse-collocation algorithm. The obtained system of linear equations is solved using the standard LU decomposition [5] and iterative Bi-CGSTAB [6] methods. For an impedance matrix of large size, the Bi-CGSTAB method performs faster than the standard LU decomposition approach, especially when sparse matrices are involved. 7.1.1 Method of Moments (MoM) Formulation In this section the Coifman wavelets are used on a finite interval without any modifi- cation. The scattering of the TM (z) and TE (z) time-harmonic electromagnetic plane waves by a groove in a conducting infinite plane is considered. The cross-sectional view of the 2D scattering problem is shown in Fig. 7.1. The angle of incidence φ inc is measured with respect to the y axis. The depth and width of the groove are h and d, respectively. For the TM (z) polarization of the incident plane wave, the induced current J s is z-directed and independent of z, that is, J s =ˆz · J z (x, y). For the TE (z) scattering case, the current J s is also z-independent and lies in the (x, y) plane. First, we consider the case of the TM (z) scattering. We split the geometry of our scattering problem into segments {l s }, s = 1, ,6, as shown in Fig. 7.2. The segments l 1 and l 5 are semi-infinite. We write J z in terms of four current distributions J PO , J PO L , J C ,and ˜ J C as J z = J PO − J PO L + J C + ˜ J C . (7.1.1) SCATTERING FROM A 2D GROOVE 301 PEC ~ J c ~ J c PO J PO J bb l 1 l l l 5 2 4 l 3 l6 d −J PO L FIGURE 7.2 Partition of the induced current J z . In (7.1.1) we partitioned the induced current J z into the following components: • J PO is the known physical optics current of the unperturbed problem (the cur- rent that would be induced on a perfectly infinite plane formed by  5 s=1 l s ). • J PO L is the portion of the physical optics current J PO residing on  4 s=2 l s . • J C is the unknown surface correction current on the groove region l 6 and its vicinity l 2 and l 4 . • ˜ J C is the unknown surface correction current, defined on l 1 and l 5 . The widths of the segments l 2 and l 4 are chosen sufficiently large to ensure that the induced current on the segments l 1 and l 5 is almost equal to the physical J PO optics current on an infinite plane. To find the induced current J z , we use the following boundary condition on the surface of the perfect conductor L s z (J z ) + E inc z = 0onl 1  l 2  l 6  l 4  l 5 , (7.1.2) where the operator L s z (·) denotes the scattered electric field component which is tangential to the surface of the groove scatterer and caused by the current J z .The electric field component E inc z is the tangential component of the incident electric field. From (7.1.1) and (7.1.2) we get the following: L s z (J PO − J PO L + J C + ˜ J C ) + E inc z = 0onl 1  l 2  l 6  l 4  l 5 . The operator L s z (·), which describes the scattered field, is a linear function of the induced current. Thus L s z (J PO ) − L s z (J PO L ) + L s z (J C ) + L s z ( ˜ J C ) =−E inc z on l 1  l 2  l 6  l 4  l 5 . (7.1.3) We should note here that the sum of the incident field and scattered field evaluated beneath the interface is equal to zero, according to the extinction theorem [7]. This means that L s z (J PO ) =−E inc z on l 1  l 2  l 6  l 4  l 5 . (7.1.4) 302 WAVELETS IN SCATTERING AND RADIATION Combining (7.1.3) and (7.1.4), we obtain L s z (J C + ˜ J C ) = L s z (J PO L ) on l 1  l 2  l 6  l 4  l 5 . (7.1.5) We can further simplify equation (7.1.5) by recalling that the induced current on l 1 and l 5 is essentially equal to the physical optics current J PO . This gives the following approximation: ˜ J C ≈ 0. (7.1.6) From (7.1.6) and (7.1.5) it follows immediately that L s z ( ˜ J C ) ≈ 0, and hence L s z (J C ) = L s z (J PO L ) on l 1  l 2  l 6  l 4  l 5 , (7.1.7) where the right-hand side is the known tangential electric field due to the current J PO L , while J C is the unknown correction current. The correction current J C is defined on l 2  l 6  l 4 , and therefore (7.1.7) can be rewritten in the following way: L s z (J C ) = L s z (J PO L ) on l 2  l 6  l 4 . (7.1.8) For the TM (z) scattering, the operator L s z (·) has the form L s z (J ( ␳  )) =− κη 4  l J ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  . Therefore we can rewrite (7.1.8) as  l 2 +l 6 +l 4 J C ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  =  l 2 +l 3 +l 4 J PO L ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  , (7.1.9) where ␳ ∈ l 2  l 6  l 4 , J PO L is the known physical optics current, and J C ( ␳  ) is the unknown local current. Equation (7.1.9) is sufficient for the determination of the local current J C .The unknown current J C is defined on the finite contour l 2  l 6  l 4 and is almost equal to the physical optics current J PO at the starting and end points of the integral path. The Coifman wavelets are defined on the real line. In order to apply the Coifman wavelets to the MoM on a finite interval, we change (7.1.9) into a slightly different form, such that the solution is almost equal to zero at the endpoints of the interval. This is due to the fact that the local current J C is approximately equal to the physical optics current J PO L at the endpoints of the interval l 2 and l 4 . We subtract the known current J PO ,defined on the intervals l 2 and l 4 , from the unknown current J L . Hence (7.1.9) becomes  l 2 +l 6 +l 4 J C ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  −  l 2 +l 4 J PO L ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  =  l 3 J PO L ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  . (7.1.10) SCATTERING FROM A 2D GROOVE 303 We define the new unknown current J p =  J C on l 6 J C − J PO L on l 2  l 4 . (7.1.11) Using the new definition, we rewrite (7.1.10) in a compact form:  l 2 +l 6 +l 4 J p ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  =  l 3 J PO L ( ␳  ) · H (2) 0 (κ| ␳ − ␳  |) dl  , ␳ ∈ l 2  l 6  l 4 . (7.1.12) The unknown current J p in (7.1.12) is solved by the MoM with Galerkin’s technique. First, we expand J p in terms of the basis functions {q i } N i=1 defined on l 2  l 6  l 4 as J p = N  n=1 a n q n . Then, we use the same basis as the testing functions to convert the integral equation (7.1.12) into a matrix equation [Z][I ]=[V ], (7.1.13) where Z m,n =  S m  S n q m (l)q n (l  )H (2) 0 (κ| ␳ − ␳  |) dl  dl, I n = a n , V m =  S m  l 3 q m (l)J PO L (l  )H (2) 0 (κ| ␳ − ␳  |) dl  dl. (7.1.14) In the previous equations, S m denotes the support of the basis function q m . By solving (7.1.13) numerically, we obtain the solution to the scattering problem of Fig. 7.1 with a finite number of unknowns. To calculate V m by using (7.1.14), we also need an expression for the physical optics current J PO . For the TM (z) scattering we find J PO L [3] J PO = 2 ˆn × H inc . The incident electric and magnetic field components are given by E inc =ˆz · η ·e jκ(x sin φ inc +y cos φ inc ) , H inc = (−ˆx · cos φ inc +ˆy ·sin φ inc ) · e jκ(x sin φ inc +y cos φ inc ) . Upon substituting (7.1.15) into (7.1.1), we obtain J PO L =ˆz · 2cosφ inc · e jκx sin φ inc . 304 WAVELETS IN SCATTERING AND RADIATION The same approach is employed to construct the integral equation for the TE (z) scattering. For the sake of simplicity, we will omit the detailed derivation of the TE (z) case and present only numerical results. 7.1.2 Coiflet-Based MoM The Coifman scalets of order L = 2N and resolution level j 0 are employed as the basis functions to expand the unknown surface current J p in (7.1.12) in the form J p (t  ) =  n a n ϕ j 0 ,n (t  ), where we have employed the parametric representation ␳ = ␳ (t) and ␳  = ␳  (t  ), and ϕ j 0 ,n (t  ) = 2 j 0 /2 ϕ(2 j 0 t  − n). Again, all equations are presented only for the TM (z) scattering, and the TE (z) case is treated in the same way. After testing the integral equation (7.1.12) with the same Coifman scalets {ϕ j 0 ,m (t)}, we arrive at the impedance matrix with the mnth entry Z m,n =  S m  S n H (2) 0 (κ| ␳ − ␳  |)ϕ j 0 ,m (t)ϕ j 0 ,n (t  ) dt  dt (7.1.15) and V m =  S m  l 3 ϕ j 0 ,m (t)J PO (t  )H (2) 0 (κ| ␳ − ␳  |) dt  dt, (7.1.16) where S n and S m are the support of the expansion and testing wavelets, respectively. The following one-point equation rule [8]:  S m  S n K (t, t  )ϕ j 0 ,m (t)ϕ j 0 ,n (t  ) dt  dt ≈ 1 2 j 0 K  m 2 j 0 , n 2 j 0  (7.1.17) is used to evaluate the matrix elements for which H (2) 0 (κ| ␳ − ␳  |) is free of singular- ity within the interval of integration. To be more specific, the one-point quadrature formula (7.1.17) is used to calculate elements of the impedance matrix for which |m − n |≥1. In addition to that, it is also used to construct the right-hand side vector (7.1.16). The error estimate of (7.1.17) can be found in Section 7.2.3. For all diagonal elements, the kernel of the integral (7.1.15) has a singularity at t = t  , where the diagonal elements are computed using standard Gauss–Legendre quadrature [5]. We used different number of Gaussian points with respect to t and t  in order to avoid the situation where t = t  . For the MoM with pulse basis, we used 4 and 6 Gaussian points for the integration with respect to t  and t. They are the minimum numbers of Gaussian points guaranteeing accurate and stable numerical results. For the Coiflet-based MoM, we split a support of each scalet into 5 small segments and used 4 and 6 points on each subinterval. In all numerical examples, the Coiflets are of order L = 2N = 4, this reflects a good trade off between accuracy and computation time. SCATTERING FROM A 2D GROOVE 305 It has also been noted that the accuracy of expression (7.1.17) depends on the res- olution level j 0 . The higher the resolution level is, the more accurate the results are. Here we mainly use the Coifman scalets with a resolution level j 0 = 5 to compute the MoM impedance matrix. We then perform the fast wavelet transform (FWT) of Section 4.8 to further sparsify the impedance matrix in standard form. 7.1.3 Bi-CGSTAB Algorithm For the solution of the linear algebraic system (7.1.13), one could use the standard LU decomposition in combination with backsubstitution, numerically available in many books. When the size of the impedance matrix Z becomes large, it is better to use the iterative method to speed up the numerical computation. In our numerical cal- culations we use the standard LU decomposition technique as well as the stabilized variant of the bi-conjugate gradient (Bi-CG) iterative solver, named Bi-CGSTAB [6]. It is very important to note that the Bi-CGSTAB method does not involve the transpose matrix Z T . The actual stopping criteria used in all numerical calculations is ||r i || L 2 < EPS ·||b − Ax 0 || L 2 with EPS = 10 −5 . It has been found from experiment that with this value of EPS we maintain accurate results in comparison with those of the LU decomposition. We have also employed the sparse version of the Bi-CGSTAB algorithm for the wavelet solution with a sparse standard matrix form. The row-indexed sparse stor- age technique has been implemented [5] to store a given sparse matrix in the com- puter memory. To be more specific, we have also used the special fast algorithm for production of the sparse matrix with a given vector at every iteration step of the Bi-CGSTAB. 7.1.4 Numerical Results We will first present the numerical results obtained from the TM (z) scattering with the following dimensions: b = 3.09375λ, h = 0.5λ,andd = 0.5λ. The number of unknowns for the pulse basis is 246. We used 256 Coifman scaling functions to expand the unknown current J p . The order of the Coiflets is L = 2N = 4 with the resolution level j 0 = 5. The obtained numerical results for different incident angles are presented in Fig. 7.3. We plotted the normalized correction current J c with respect to the length parameter (arclength) given in λ. The local current J L was obtained from (7.1.1) after we found the unknown current J p numerically. Numerical results for the case of TE (z) scattering are shown in Fig. 7.4. To demonstrate the advantage of the Coifman wavelets and Bi-CGSTAB algo- rithm, we present in Tables 7.1 and 7.2 the results of computation time. All numerical computations presented here were performed on a standard personal computer with 32-bit 400 MHz clock CPU from Advanced Micro Devices (AMD), 128 Mb RAM and Suse 6.3 Linux operational system. The public domain GNU g++ compiler was used to create executable codes. The following parameters were chosen to create the 306 WAVELETS IN SCATTERING AND RADIATION Length parameter 0 1 2 3 4 5 6 7 Normalized induced current Pulse basis Coiflets Length parameter 0 1 2 3 4 Normalized induced current Pulse basis Coiflets 0123 45 6 7 0123 45 6 7 FIGURE 7.3 Normalized induced current versus length λ, TM (z) case with: b = 3.09375λ, h = 0.5λ, d = 0.5λ, N p = 246, N c = 256. Left: φ inc = 0 ◦ ; right: φ inc = 60 ◦ . 0 0.5 1 1.5 2 2.5 3 Normalized induced current Length parameter Pulse basis Coiflets Length parameter Pulse basis Coiflets 0123 45 6 7 0123 45 6 7 0 0.5 1 1.5 2 2.5 3 Normalized induced current FIGURE 7.4 Normalized induced current versus length λ, TE (z) case with b = 3.09375λ, h = 0.5λ, d = 0.5λ, N p = 246, N c = 256, Left: φ inc = 0 ◦ ; right: φ inc = 60 ◦ . TABLE 7.1. Computation Time for the Pulse Basis, TM ( z ) Scattering LU Bi-CGSTAB Iteration, N p Time (s) time (sec) N it 1014 522.86 331.85 61 502 85.94 73.21 44 246 16.57 16.47 33 TABLE 7.2. Computation Time for the Coifman Wavelets, TM ( z ) Scattering LU Bi-CGSTAB Sparse Bi-CGSTAB Sparsity N c Time (s) Time (s) N it Time (s) N it (%) 1024 354.42 168.82 61 60.91 62 11.94 512 45.94 31.50 43 18.19 45 15.78 256 8.03 8.49 34 6.65 34 22.28 SCATTERING FROM A 2D GROOVE 307 data presented in Tables 7.1 and 7.2: b = 3.09375λ, h = 0.5λ, d = 0.5λ, φ inc = 60 ◦ , N p = 246, N c = 256. b = 6.34375λ, h = 1.0λ, d = 1.0λ, φ inc = 60 ◦ , N p = 502, N c = 512. b = 12.84375λ, h = 2.0λ, d = 2.0λ, φ inc = 60 ◦ , N p = 1014, N c = 1024. The numbers N p and N c denote the number of pulses and Coiflets in the MoM, N it is the number of iterations in the Bi-CGSTAB algorithm. We implemented the LU and Bi-CGSTAB methods to solve the system of linear equations. We also decompose the system matrix of the Coifman-based MoM into the standard matrix. The sparse version of the Bi-CGSTAB is used to solve the system of linear equations. Then the threshold level of 10 −4 · p is selected to sparsify the system matrix, where parameter p is the maximum entry in magnitude. The relative error of 10 −5 has been used as a stopping criterion for the Bi-CGSTAB. The sparsity of a matrix is defined as the percentage of the nonzero entries in the matrix. From Tables 7.1 and 7.2 it can be seen that the use of Coifman wavelet-based MoM in combination with the standard form matrix achieves a factor of approxi- mately 2.5to8.5 in the CPU time savings over the pulse-based MoM with the LU de- composition. This is due to the one-point quadrature formula, fast wavelet transform, and fast sparse matrix solver. Figure 7.6 illustrates the local current J L obtained from 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 nz=125204 FIGURE 7.5 Standard form matrix, TM (z) scattering. 308 WAVELETS IN SCATTERING AND RADIATION Length parameter 0 1 2 3 4 5 6 7 Normalized induced current Pulse basis Coiflets Length parameter Normalized induced current 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 0 1 2 3 4 5 Pulse basis Coiflets FIGURE 7.6 Normalized induced current versus length λ, TM (z) case. Left: b = 12.84375λ, h = 2.0λ, d = 2.0λ, φ inc = 60 ◦ , N p = 1014, N c = 1024; right: b = 6.34375λ, h = 1.0λ, d = 1.0λ, φ inc = 60 ◦ , N p = 502, N c = 512. the TM (z) scattering with the parameters in Tables 7.1 and 7.2. Figure 7.5 shows the standard form matrix with 1024 unknowns and five resolution levels. For all numerical results presented here, we made use of the Coiflets with reso- lution level j 0 = 5. This level has been chosen after a number of numerical trials indicating that this resolution level is the minimum at which there is good agreement between the pulse basis approach and wavelet technique. As the last numerical ex- ample we decrease the resolution level to j 0 = 4, thus obtaining fewer unknowns than in Fig. 7.3. Actually we used 123 pulse functions and 133 Coifman scalets to arrive at the results shown in Fig. 7.7. We can see that we still have good agreement Length parameter 0 1 2 3 4 5 6 7 Normalized induced current Pulse basis Coiflets Current Jp 01 23 45 67 FIGURE 7.7 Normalized induced current versus length λ, TM (z) case: b = 3.09375λ, h = 0.5λ, d = 0.5λ, φ inc = 0 ◦ , N p = 123, N c = 133. [...]... SCATTERING USING INTERVALLIC COIFLETS Periodic wavelets were applied to bounded intervals in Chapter 4 Nonetheless, the unknown functions must take on equal values at the endpoint of the bounded interval in order to apply periodic wavelets as the basis functions The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets The intervallic wavelets form an orthonormal basis and... to employ the intervallic wavelets, nor the periodic wavelets; instead, the standard wavelets are sufficient At the left edge, portions of the wavelets that are beyond the interval are circularly shifted to the right edge, and vice versa This procedure is similar to the circular convolution in the discrete Fourier transform In this example we employed the intervallic Coifman wavelets, although we could... concepts of intervallic wavelets were derived in Chapter 4 Here we will quickly review some major facts and then present the new material Starting from an orthogonal Coifman scalet with 3L nonzero coefficients (where L = 2N is the order of the Coifman wavelets) , we will assume that the scale is fine enough that the left- and right-edge bases are independent Since the Coifman wavelets have vanishing moment... INTERVALLIC COIFLETS 321 10 20 30 40 50 60 10 20 30 40 50 60 FIGURE 7.12 Magnitude of impedance matrix at level 6, generated by intervallic wavelets method x-axis The procedures described in the solution for Jz are then used to expand the current to Coifman intervallic wavelets Figure 7.12 shows the impedance matrix, which is produced by the intervallic Coifman scalet on level 6 In the figure the magnitudes... Jz that is produced by the vanishing moment properties of the Coifman wavelets We compare it with the current found by using the Gaussian quadrature for the calculation of matrix elements The magnitude of matrix elements, which are set to zero, does not exceed 0.1% of the largest element in the matrix In this example the scalets and wavelets are both chosen on level 6 with a total of 60 basis functions... moment property and reduce partially the double integration to single integration for the nonsingular part 314 WAVELETS IN SCATTERING AND RADIATION Using the Taylor expansion of the integral kernel, we can approximate the nonsingular coefficient matrix entries in (7.2.8), which contain complete wavelets and scalets For ease of reference, three basic cases are considered and relative errors are analyzed... transform In this example we employed the intervallic Coifman wavelets, although we could have used the standard wavelets This example is a typical onefold wavelet expansion It is mainly designed to demonstrate the fast construction of an impedance matrix for general problems in the confined interval 322 WAVELETS IN SCATTERING AND RADIATION 3 Current magnitude Coiflet solution Gaussian quadratures 2 1 0 0 0.2... through several examples of antennae and scatterers with complicated shapes The intervallic Coifman wavelets with L = 4 were employed The direct numerical integral algorithm has been implemented using Gaussian quadrature for the evaluation of the matrix elements The following examples are selected from [14] 332 WAVELETS IN SCATTERING AND RADIATION y (0,0.8 λ) − r a −′ r x o b (−1.6 λ,0) (1.6λ,0) (0,−0.8λ)... wavelets release the endpoints restrictions imposed on the periodic wavelets The intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis (MRA) of other usual unbounded wavelets The Coiflets possess a special property: their scalets have many vanishing moments As a result the zero entries of the matrices are identifiable directly, without using a truncation scheme of an... 7.9 Error distribution induced by Coifman zero moment approach on resolution levels 6 and 7 (Source: G Pan, M Toupikov, and B Gilbert, IEEE Trans Ant Propg., 47, 1189–1200, July 1999, c 1999 IEEE.) 316 WAVELETS IN SCATTERING AND RADIATION ing entries are plotted in Fig 7.9, where the solid lines are computed by Gaussian quadrature and the dashed/dashed-dotted lines are the error introduced by the zero . wavelets as the basis functions. The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets. The intervallic wavelets form an orthonormal basis and preserve the. Coifman wavelets) , we will assume that the scale is fine enough that the left- and right-edge bases are independent. Since the Coifman wavelets have vanishing moment properties in both scalets and wavelets, . evaluated using the 299 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X 300 WAVELETS IN SCATTERING AND RADIATION x d h PEC z y H inc E inc φ inc b b κ ρ FIGURE